Class Book. F8 a Copyright N°_Jia_ CDEXRIGHT DEPOSfT. CAMS ELEMENTARY AND ADVANCED BY FRANKLIN DeRONDE FURMAN, M.E. Professor of Mechanism and Machine Design at Stevens Institute of Technology Member of American Society of Mechanical Engineers ELEMENTARY CAMS FIRST EDITION (THIRD IMPRESSION) TOTAL ISSUE FOUR THOUSAND CAMS-ELEMENTARY AND ADVANCED FIRST EDITION NEW YORK JOHN WILEY & SONS, Inc. London: CHAPMAN & HALL, Limited 1921 3®° (p Copyright, 1916, 1921 BY FRANKLIN DeRONDE FURMAN FEB -3 !32i w. / PRESS OF BRAUNWORTH & CO. BOOK MANUFACTURERS BROOKLYN. N. Y. ©CU608208 i » ( 'Wo I PREFACE TO THE ENLARGED EDITION The first five sections of this book were published about three years ago under the title of "Elementary Cams." The chief features of this earlier book were that it pointed out a classification, an arrange- ment, and a general method of solution of the well-known cams in such manner as has been generally developed in other specialized branches in technical engineering work; and also it gave a series of cam factors for base curves in common use, which enabled designers to compute proper cam sizes for specific running conditions, offering numerous examples in the use of these factors in the several kinds of cam problems. The factors, with the exception of the one for the 30° pressure angle for the Crank Curve were new, so far as the author is aware. The " Elementary Cams " will continue to be sold as a separate volume. A further development of the subject is given in the present work which is under the title of "Cams." The chief original features of this advanced work include the development or use, or both, of the logarithmic, cube, circular, tangential and involute base curves, the establishing of cam factors for such of these curves as have general factors, and the demonstration that the logarithmic base curve gives the smallest possible cam for given data. The new material now introduced into the book includes, further, comparisons of the characteristic results obtained from all base curves, in which the relative size of each cam, and the relative velocity and acceleration produced by each, is shown graphically in one combined group of illustrations, thus enabling the designer to glance over the entire field of theoretical cam design and quickly select the type that is best adapted for the work in hand. From these diagrams one may observe, for example, which form of cam is best adapted for gravity, spring or positive return, which is best for slow or fast veloci- ties at various points in the stroke, and which ones are apt to develop "hard spots" in running. The involute curve is found to have its chief and characteristic theoretical advantage when it is used with an offset follower. The nature of the contact between cylindrical, conical and hyperboloidal roller pins, when used in connection with grooved cylindrical cams, has been investigated and pointed out. The subject of pure rolling contact between various forms of oscillat- ing cam arm surfaces, and of the nature and amount of sliding action iii iv PREFACE TO THE ENLARGED EDITION of such surfaces has been developed so that the effects of wear due to rubbing may be confidently considered when such types of cams are under design. While the whole purpose of this work has been to present the subject matter in graphical form and in the simplest possible manner so as to make it available to the greatest number, much mathematical investigation has been necessary and in this I have been greatly aided by my colleague Professor L. A. Hazeltine, M. E., head of the department of Electrical Engineering at Stevens, to whom I express my deep appreciation. The details of these investigations are not necessary here and are not set down, but their results are. These results are given in various formulas that are used in the solution of a number of the problems. These final formulas avoid the use of calculus and are mostly in such form as to be readily used by designers generally. In closing, the author desires to introduce a personal thought that has grown up, and which is inseparable, with this book. Some years ago, before any special study was given by the writer to the sub- ject of cams, it appeared that the whole subject of mechanism was so thoroughly covered by various text books and technical papers that the time in engineering development had arrived when there was but little for an instructor to look forward to in the way of produc- tion of extended original work on any given topic. To say the least such a thought was not at all encouraging, and so it is a pleasure now to the author, and it is hoped that it will be an inspiration particularly to the younger readers, to record that the study of this subject of cams has brought forth a great wealth of new and practical material which had not previously been brought to light and set down in the literature of the subject. Now that this work is done, the vastness of the " unknown," even in this present era of great accomplish- ments, is realized as it never was before, and it only remains to suggest that not only this topic of cams but many other topics in the science of engineering may offer opportunities for much further development and perfection on the part of those who have the desire for such work and the time to pursue it. F. DeR. Furman. Hoboken, N. J., April, 1920. CONTENTS PAGES Section I. — Definitions and Classification 1-19 Cams Follower Surfaces Radial or Disk Cams Side or Cylindrical Cams Conical and Spherical Cams Names of Cams — Periphery, Plate, Heart, Frog, Mushroom, Face, Wiper, Rolling, Yoke, Cylindrical, End, Double End, Box, Internal, Offset, Positive Drive, Single Acting, Double Acting, Step, Adjustable, Clamp, Strap, Dog, Carrier, Double Mounted, Multiple Mounted, Oscillating Definitions of Terms Used in the Solution of Cam Problems — Cam Chart, Cam Chart Diagram, Time Chart, Base Curve, Base Line, Pitch Line, Pitch Circle, Pitch Surface, Working Surface, Pitch Point, Pressure Angle Formula for Size of Cam for a Given Maximum Pressure Angle Table of Cam Factors for All Base Curves for Maximum Pressure Angles from 20° to 60° Section II. — Method of Construction of Base Curves in Common Use 20-24 Straight Line Base Straight-Line Combination Curve Crank Curve Parabola Elliptical Curve Section III. — Cam Problems and Exercise Problems 25-74 Problem 1, Empirical Design Problem 2, Technical Design. Advantages of Technical Design Problem 3, Single-Step Radial Cam, Pressure Angle Equal on Both Strokes Omission of Cam Chart Problem 4, Single-Step Radial Cam, Pressure Angles Unequal on Both Strokes Pressure Angle Increases as Pitch Size of Cam Decreases Change of Pressure Angles in Passing from Cam Chart to Cam Cam Con- sidered as Bent Chart. Base Line Angles Before and After Bending Limiting Size of Follower Roller Radius of Curvature of Non- Circular Arcs Problem 5, Double-Step Radial Cam Determination of Maximum Pressure Angle for a Multiple-Step Cam Problem 6, Cam with Offset Roller Follower Problem 7, Cam with Flat Surface Follower Limited Use of Cams with Flat Surface Followers Problem 8, Cam with Swinging Follower Arm, Roller Contact — Extremities of Swinging Arc on Radial Line Problem 9, Cam with Swinging Follower Arm, Roller Contact — Swinging Arc, Con- tinued, Passes Through Center of Cam Effect of Location of Swinging Follower Arm Relatively to the Cam VI CONTENTS PAGES Problem 10, Face Cam with Swinging Follower Problem 11, Cam with Swinging Follower Arm, Sliding Surface Contact Data Limited for Followers with Sliding Surface Contact Problem 12, Toe and Wiper Cam Modifications of the Toe and Wiper Cam Problem 13, Single Disk Yoke Cam Limited Application of Single Disk Yoke Cam Problem 14, Double Disk Yoke Cam Problem 15, Cylindrical Cam with Follower that Moves in a Straight Line Refinements in Cylindrical Cam Design Prob- lem 16, Cylindrical Cam with Swinging Follower Chart Method for Laying Out a Cylindrical Cam with a Swinging Follower Arm Exercise Problems, 3a to 16a Section IV. — Timing and Interference of Cams 75- 78 Problem 17, Cam Timing and Interference Location of Key- ways Exercise Problem 17a Section V. — Cams for Reproducing Given Curves or Figures . . 79- 87 Problem 18, Cam Mechanism for Drawing an Ellipse Prob- lem 18a, Exercise Problem for Drawing Figure 8 Problem 19, Cam for Reproducing Handwriting Using Script Letters Ste Method of Subdividing Circles into Any Desired Number of Equal Parts Section VI. — Advanced Group of Base Curves 88-137 Complete List and Comparison of Base Curves, Their Appli- cations and Characteristic Motions Velocity and Acceleration Diagrams Showing Characteristic Action of Various Cams All- logarithmic Curve Gives Smallest Possible Cam for a Given Pres- sure Angle Problem 20, All-logarithmic Cam General Analysis Detail Construction of Logarithmic Curve and Cam by Analytical and Graphical Methods Problem 21, Logarithmic-combination Cam with Parabolic Easing-off Arcs Problem 22, Cam with Straight-Line Base Straight-Line Combination Curve Crank Curve Effect of Crank Curve Following Its Tangent Closely Parabola Gravity Curve Curve of Squares Perfect Cam Action Comparison of Parabola and Crank Curves Problem 23, Tangential Cam, Case 1 Graphical and Ana- lytical Methods Characteristic Retardation Problem 24, Circular Base Curve Cam, Case 1 Elliptical Base Curve Effect of Varying Axes of Ellipses Elliptical Base Curve Equivalent to Nearly All Other Base Curves Cube Curve Symmetrically and Unsymmetrically Applied Problem 25, Cube Curve Cam, Case 1 CONTENTS Vll PAGES Cams Specially Designed for Low-Starting Velocities Problem 26, Circular Base Curve Cam, Case 2 Problem 27, Cube Curve Cam, Case 2 Problem 28, Tangential Cam, Case 2 Section VII. — Cam Characteristics 138-156 Methods of Determining Velocities and Accelerations Time- Distance, Time-Velocity and Time-Acceleration Diagrams Degree of Precision Obtained by Graphical Methods Comparison of Relative Velocities and Forces Produced by Cams of Different Base Curves Cam Follower Returned by Springs Rela- tive Strength of Spring Required for Cams of Different Base Curves Special Adaptation of Cube Curve Cam for Follower Returned by a Spring Na'ure of Pressure between Cam Surface and Spring-Returned Follower Accuracy in Cam Construction Regulation of Noise High-Speed Cams Balancing of Cams Pressure Angle Factors, Nature of Application and Method of Determination for All Base Curves Varied Forms of Funda- mental Base Curves Chart Showing Values of Intermediate Pressure Angles from 20° to 60° for all Cams Section VIII. — Miscellaneous Cam Actions and Constructions . 157-229 Variable Angular Velocity in Driving Cam Shaft Problem 29, Oscillating Cam Having Variable Angular Velocity, Toe and Wiper Type Problem 30, Wiper Cam Operating Curved-Toe Follower Problem 31, Sliding Action between Cam and Flat Follower Sur- faces Rate of Sliding Measured Velocity of the Folbwer Measured Problem 32, Sliding Action between Cam and Curved-Toe Fol- lower Problem 33, Sliding Action where Driving Cam Ha^ Variable Angular Velocity Elimination of All Sliding Action between Cam and Flat or Curved Surface Follower The Princip e of Pure Rolling Action between Cam Surfaces Well- Known Curves that Lend Themselves Readily to Pure Rolling Cam Action Problem 31, Pure Rolling with Flat-Surface Follower Use of Logarithmic Curve for Pure Rolling Action Charac- teristic Properties of the Logarithmic Curve Problem 35, Pure Rolling with Logarithmic Curved Cam Arm Angular Motion of Each Arm Tangency of Logarithmic Cam Surfaces Regulation of Pressure Angle when Logarithmic Rolling Cams are Used Derived, or Computed Curves for Rolling Cam Arms Vlll CONTENTS PAGES Problem 36, The Use of a Derived Curve for Rolling Cam Arms 174 Rolling Cam Arms Useful for Starting Shafts Gradually Regulation of Pressure Angle with Derived Rolling Cams Elliptical Arcs for Pure Rolling Cam Arms Problem 37, Elliptical Rolling Cam Arms, Angles of Action Equal Determination of Major and Minor Axes of Ellipses Construc- tion of Ellipse Pressure Angle in Rolling Elliptical Cam Arms Problem 38, Elliptical Rolling Cam Arms, Angles of Action Unequal Pure Rolling Parabolic Cam Surfaces for a Reciprocating Motion Problem 39, Rolling Parabolas Construction of Parabola Pure Rolling Hyperbolic Cam Arms where Centers are Close Together Problem 40, Rolling Hyperbolas Construction of Hyperbola Detail Drawing of Cylindrical Cams The True Maximum Pressure Angle in Cylindrical Cams Drawing of Groove Out- lines, Approximate and More Exact Methods Forms of Follower Pins for Cylindrical Grooved Cams Line of Contact between Pin and Groove Surface, at Rest and Moving The Cylindrical Follower Pin The Conical Follower Pin The Hyperboloidal Follower Pin Plates for Cylindrical Cams Adjustable Cylindrical Cams for Automatic Work Double-Screw Cylindrical Cams Periods of Rest of More than One Revolution in Cylindrical Cams Slow-advance and Quick-Return Secured by Double-Screw Cam Straight-Sliding Plate Cams Involute Cams Construction of Involute Curve Pressure Angle with Involute Cam Involute Cam Specially Adapted for Flat-Surface Follower Problem 41, Involute Cam with Radial Follower Oscillating Positive-Drive-Single-Disk Cam Cam Shaft Acting as Guide Positive Drive with Cam Shaft as Guide Positive- Drive Double-Disk Radial Cam with Swinging Follower Rotary- Sliding Yoke Cams Giving Intermittent Harmonic Motion, and Reciprocating Motion Rotary Sliding Yoke Cam, General Case Cam Surface on Reciprocating Follower Rod Problem 42, Definite Motion where Cam Surface is on Follower Rod Problem 43, Cam Surface on Swinging Follower Arm Effect of Swinging Transmitter Arm between Ordinary Radial Cam and Follower Angular Velocity Curve for a Swinging Follower Arm Velocity Curve for a Follower Rod with Com- parison of Results Obtained by Using Transmitter Arms with Sliding and Roller Action Diagram of Pressure Angles Meas- urement of Rubbing Velocities in Cams Having Sliding Action Boundary of Follower Surface Subjected to Wear in Sliding Cams 221 CONTENTS IX PAjGES Cam Action Different on Forward and Return Strokes with Sliding 222 Cams Problem 44, Small Cams with Small Pressure Angles Secured by Using Variable Drive Variable Drive by Whit worth Motion Swash Plate Cam Uniformly Rotating Cam Giving Inter- mittent Rotary Motion The Eccentric a Special Type of Cam An Example of a Time-Chart Diagram for Eleven Cams on One Shaft of an Automatic Machine 229 ; ELEMENTARY GAMS SECTION I.— DEFINITIONS AND CLASSIFICATION Definitions 1. Cams are rotating or oscillating pieces of mechanism having specially formed surfaces against which a follower slides or rolls and thus receives a reciprocating or intermittent motion such as cannot be generally obtained by gear wheels or link motions. Various forms of cams are illustrated at C in Figs. 1 to 10. The follower in each case is shown at F, all having roller contact except the ones shown in Figs. 7 and 8. The former has a V edge and the latter a plane surface in contact with the cam and both have sliding action. 2. Follower edges or rollers may have motion in a straight line as from D to G, Fig. 7, or in a curved path depending on suit- ably constructed guides or on swinging arms. The total range of travel of the follower may be accomplished by one continuous motion, or by several separate motions with intervals of rest. Each motion may be either constant or variable in velocity, and the time used by the motion may be greater or less, all according to the work the machine has to do and to the will of the designer. Classification 3. Cams may be most simply, and at the same time most com- pletely, classified according to the motion of the follower with re- spect to the axis of the cam, as: (a) Radial or disk cams, in which the radial distance from the cam axis to the acting surface varies constantly during part or all of the cam cycle, according to the data. The follower edge or roller moves in all cases in a radial, or an approximately radial, direction with respect to the cam. Various forms of radial cams are illus- trated in Figs. 1, 2, 7, 8, and 9. (b) Side or cylindrical cams, in which the follower edge or roller moves parallel to the axis of the cam, or approximately in 1 ELEMENTAEY CAMS / F s " this direction. Several types of side cams are shown in Figs. 3, 4, and 10. Nearly all the cams referred to in the above figures illustrating the two general classes of radial and side cams respectively have special or local trade names which will be pointed out in a succeed- ing paragraph. (c) Conical and (d) Spherical cams, in which the follower edge or roller moves in an inclined direction having both radial and longitudinal components with respect to the axis of the cam as illustrated in Figs. 5 and 6. 4. Names of cams. Cams, in popular usage, have come to be known by a wide range of names, the same cam often being designated by a number of different names accord- ing to geographical loca- tion and personal prefer- ence and surroundings of the cam builder or user. This is an unfortunate con- dition, and in the general classification in the preced- ing paragraph an endeavor is made to establish a fun- damental basis for clarifying and simplifying the nomenclature of cams as much as possible. In a treatise of this kind, however, it is essential that, at least, the more common of the ordinary working terms be recognized and defined, and that the cams under their popular names be properly placed in the fundamental classification given in the preceding paragraph. The following specially named cams fall under the classifica- tion of radial cams: (e) Periphery cams, in which the acting surface is the periphery of the cam, as illustrated in Figs. 1, 7, and 9. While these are ex- amples of true periphery cams, it must be recorded that the cylin- drical grooved cam, shown in Fig. 3, is also known to some extent as a periphery cam, due no doubt to the fact that in designing this ' r 1 \ — 1 .._. i i / c End Front Fig. 1. -Radial Cam and Follower, Roller Contact DEFINITIONS AND CLASSIFICATION 3 cam the original layout for the contour of the groove is first made on a flat piece of paper, which is then wrapped on to the surface or "periphery" of the cylinder. Since the contour line of the groove which lies on the periphery is merely a guiding line for cutting the groove, and since the side surface of the groove is the working sur- face, it is, to say the least, a misnomer to designate such a cam as a periphery cam. (/) Plate cams, in which the working surface includes the full 360°, and forms either the periphery of the cam, or the sides of a E>'D FlJOXT Fig. 2. — Face Cam and Follower groove cut into the face of the cam plate, as illustrated in Figs. 1 and 2 respectively. Figs. 7 and 9 also show plate cams. (g) Heart cams, in which the general form is that which the name implies. See Fig. 7. In this type of cam there are two distinct symmetrical lobes, often so laid out as to give uniform velocity to the driver. In this case each lobe would be bounded by an Archimedean spiral with the ends eased off. (h) Frog cam, in which the general form includes several lobes more or less irregular, as illustrated, for example at C in Fig. 9. (i) Mushroom cam, in which the periphery of a radial or disk cam works against a flat surface, usually a circular disk at right angles to the cam disk, instead of against a roller, see Fig. 44. 0') Face cam, also called a Groove, but more properly a Plate Groove cam, to distinguish it from the Cylindrical Groove cam, in which a groove is cut into the flat face of the cam disk. In 4 ELEMENTARY CAMS this form of cam shown in Fig. 2 the roller has two opposite lines of contact, one against each side of the groove, when the roller has a snug fit. The plate or disk in which the groove is cut is generally circular; but it may be cast to conform with the contour of the groove, or it may be built with radial arms supporting the irregular grooved rim. In the latter case it lacks resemblance to the face Top End Front Fig. 3. — Cylindrical Cam and Swinging: Follower cam, but nevertheless it must, because of the nature of its action, be classed with it. The face cam, as ordinarily considered and as illustrated in Fig. 2, is better adapted for higher speeds because of its more nearly balanced form of construction. Against this, how- ever, must be considered one of two disadvantages, either the high rubbing velocity of the roller against one side of the groove when the roller is a snug fit, or lost motion and noise as the working line of contact changes from one side of the groove to the other when the roller has a loose fit. The most important advantage of the face cam, that of giving positive drive, will be considered in para- graph 9. The term groove cam might be applied, with advantage in clearness of meaning, to such face cams as are cut or cast on non-circular plates. DEFINITIONS AND CLASSIFICATION (k) Wiper cam, which has an oscillating motion, and is con- structed usually with a long; curved arm in order that it may "wipe" or rub along- the plane surface of a long projecting "toe," or follower. The wiper cam is used generally to give motion to a follower which moves straight up and down as shown from F to F' in Fig. 8. This, however, is not essential and the follower may also have a swinging Fig. 4. — End Cam and Follower motion. The disadvantage of sliding friction, which is inseparable from the wiper cam, is balanced to some extent by the fact that the very sliding permits, within certain range, of the assignment of specified intermediate velocities between the starting and stopping points which cannot be obtained with similar forms of cams which have pure rolling action. (/) Rolling cam, which greatly resembles the wiper cam in general appearance, but which is totally different in principle, for the curves of the cam and follower surfaces are specially formed so as to give pure rolling action between them. The rolling cam is specially well adapted to cases where both driver and follower have an oscillating motion and where the velocities between the starting and stopping points are not important and are not specified. 6 ELEMENTARY CAMS (m) Yoke cam, a form of radial cam in which all diametral lines drawn across the face and through the center of rotation of the cam are equal in length. This form of cam permits the use of two opposite follower rollers whose centers remain a fixed distance apart, to roll simultaneously on opposite sides of the cam, and thus give positive motion to the follower. For illustration, see Fig. 9. Fig. 5. — Conical Cam and Recipro- cating Follower Fig. 6. -Spherical Cam and Swinging Follower Yoke cams may be, and frequently are, made of two disks fixed side by side, the peripheries being complementary to each other and the two rollers of the yoke rolling on their respective cam surfaces, as shown in Fig. 56. The advantage of yoke cams is that they give positive motion with pure rolling of the follower roller, there being contact on only one side of the roller in contradistinction to the double contact of th^ roller which exists in face and groove cams. 5. The following specially named cams fall under the general classification of side cams. These include cams that have been made from blank cylindrical bodies by using a rotary end cutter with its axis at right angles to the axis of the cylinder and by moving the axis of the rotary cutter parallel to the axis of the cylinder while the cylinder rotates. A groove of desired depth is thus left in the cylinder, Fig. 3, or the end of a cylindrical shell is thus milled to a desired form, Fig. 4. A side cam may also be formed by screwing a number of formed DEFINITIONS AND CLASSIFICATION clamps on to a blank cyl- inder, the sides of the clamps thus acting as the working surface as illus- trated in Fig. 11. All types of side cams may properly be considered as derived from blank cylindrical forms, and, therefore, the name "cyl- indrical cam" could be regarded as synonymous with side cam; but gen- eral custom has limited the use of the term cyl- indrical cam to the "bar-" rel" or "drum" type mentioned below: (n) Cylindrical cam, also called Barrel cam, Drum cam, or Cylindrical Groove cam, in which the fig. 7 groove, cut around the cylinder, affords bearing surface to the two opposite sides of the follower roller, thus giving positive motion, as illustrated in Fig. 3. (o) End cam, in which the working surface has been cut at the end of a cylindrical shell, thus re- quiring outside effort such as a spring or weight to hold the follower roller against the cam surface during the return of the follower. An end cam is shown in Fig. 4. (p) Double end cam, in which a projecting twisted thread has been left on a cylindrical body, against both sides of which separate rollers on a follower arm may operate, and thus secure positive motion. Instead of cutting down a cylinder to leave a projecting twisted thread, fig. s.— toe and wiper cam it may be cast integral with a -Heart Cam and Follower, Sliding Contact 8 ELEMENTARY CAMS warped plate, as illustrated in Fig. 10, but this in no way changes its characteristic action. There are a number of names in common use for cams, that cover both radial and side cams. Most prominent in this connection are those mentioned in paragraphs 6 to 14. 6. Box cam, which designates a cam in which the follower roller is encased between two walls as in the face cam, Fig. 2, or the cylin- drical cam, Fig. 3. Literally, box cams would also include yoke cams, in which the yoke would be the "box." Box cams, because of their form of construction, give a positive drive in all cases. 7. Internal cam, in which there is only one working surface, and this is outside of the pitch surface. The internal cam cor- responds to the internal gear wheel in toothed gearing. It may also be considered as a face cam with the inside surface of the groove removed, thus requiring that the follower roller should always be in pressure contact on the outside surface of the groove by means of a spring or weight, etc. Under some conditions of structural arrange- ments of the cam machine, the internal cam may be used to advan- tage where it will give a positive motion to a follower on the opposite stroke to that of the periphery cam; and it will also sometimes Fig. 9. — Yoke Cam permit of a larger roller than the periphery cam, as explained in paragraphs 56 and 62. 8. Offset cam, in which the line of action of the follower, when extended, does not pass through the center of the cam, see Fig. 43. 9. Positive-drive cam is one in which the cam itself drives the follower on the return as well as the forward motion. Most DEFINITIONS AND CLASSIFICATION 9 cams drive only on the forward motion of the follower and depend upon gravity or the action of a spring to drive the follower in its return motion; such cams are illustrated in Figs. 1, 4, 5, 6, 7, and 8. Cams having positive drive, and therefore independent of gravity or springs, are illustrated in Figs. 2, 3, 9, and 10. It will be noted that positive-drive cams include the face, yoke, cylindrical, and double-end types of cams; also that the box cam, although it in- cludes some of these, should also be considered as a group name of the positive-drive type. 10. Single-acting and double-acting cams comprise all forms of cams, the single-acting ones giving motion only in one direction and depending on a spring or gravity to return the follower. Double- acting cams have the follower under direct control all the time and are the same as positive-drive cams described in the preceding paragraph. 11. Step cams. Cams which give continuous motion to the llkiTH^ Fhont Fig. 10. — Double-End Cam follower from one end of the stroke to the other are called single- step cams. When the follower's motion in either of its two general directions is made up of two entirely separate movements it is called a double-step cam with reference to that stroke. If three or more separate movements are given to the follower while it moves in one general direction it is generally referred to as a multiple step cam, or as a triple-step, quadruple-step cam, etc. Since a cam may be, for example, a double-step cam on the out or working stroke, and Front ,End Fig. 11. — Barrel Cam Fron,t End Fig. 12. — Adjustable Plate Cam Front Fig. 13. — Dog Cam DEFINITIONS AND CLASSIFICATION 11 a single-step cam on the return stroke, such a cam may be referred to as a two-one step cam, always giving the number referring to the working stroke first. 12. Adjustable cam, also called clamp cam, strap cam, dog cam, and carrier cam, in which specially formed pieces are directly bolted or clamped to any of the regular geometrical surfaces, usually to either the plane or cylindrical surfaces. In Fig. 12 the clamps are shown at C and D fastened to a disk. The cam, considered as a whole, belongs to the radial class. In Fig. 13 the clamps are shown at C and D, also fastened to a disk, but in this case the clamps, or dogs, as they are usually called when used in this way, are so formed as to give a sidewise motion to the follower, and therefore this cam belongs to the side cam class. In Fig. 11 clamps are shown at C, D, E, and F fastened to a cylinder, and they are shaped to give the same action as a regularly formed end-cam in the side-cam class. The type of cam illustrated in Fig. 11 is also known as an adjustable cylindrical or "barrel" or "drum" cam and is very widely used for regulating the feeding of the stock, and in operating the turret in automatic machines for the manufacture of screws, bolts, ferrules, and small pieces generally that are made up in quantities. 13. Double-mounted or multiple-mounted cams are some- times resorted to where several movements can be concentrated into small space. This consists merely in placing two or more of any of the cam surfaces described in the preceding paragraphs on one solid casting or cam body. For example, a face cam, a cylin- drical, and an end cam may all be cut on one piece. 14. Oscillating cams, in which the cam itself turns through a fraction of a turn instead of through the entire 360°. While any type of cam may be designed to oscillate instead of rotate, it is usually the toe-and-wiper and rolling forms of the radial type of cam that are known as oscillating cams. With oscillating cams the follower may move forth and back on a straight line, or it may oscillate also. 15. Cams falling in the conical class have no special name other than the one here used. The spherical cams are sometimes termed globe cams. Cams in conical and spherical classes are particularly useful in changing direction of motion in close quarters and in directions other than at right angles. In both Figs. 5 and 6, end action of the cam is shown, but it is apparent that with thicker walls on both the cone and the sphere, grooves could be cut in them, thus giving positive driving cams in both cases. 12 ELEMENTARY CAMS 16. Summing up the general and special names for cams we have in tabular form: p. Periphery / Plate g Heart h Frog i Mushroom j Face or Plate Grooved k Toe and Wiper I Rolling s m Yoke or Duplex n Cylindrical, Grooved, Barrel, or Drum o End p Double End Cams Box Internal Offset Positive Drive Single Acting Double Acting Step Adjustable or Strap Dog or Carrier Multiple Mounted . Oscillating a Radial or Disk Side, or. Cylindrical c Conical d Spherical or Globe Definitions of Terms Used in the Solution of Cam Problems 17. Cam chart. Illustrated in Fig. 14. The chart is a rectangle the height of which is equal to the total motion of the follower in one direction, and the length equal to the circumference of the pitch circle of the cam. The chart length represents 360° and is sub- o J> H * X % 1 c S 4 ^Se \ Pitch Line cLUMT ?• V 10 20 30 40 50'60 70 '80^90° 180" 270" 360- ~ F Fig. 14. — Cam Chart divided into equal parts marking the 5°, 10 c points, or the }/§, 34 • • • points, or any other convenient subdivision, according to the requirements of the problem. On the cam chart are drawn the base curve and the pitch line. The former becomes the pitch surface of the cam and the latter the pitch circle. 18. Cam chart diagram. Illustrated in Fig. 15. The cam chart diagram is a rectangle, the height of which represents the total motion of the follower in one direction. The length of the diagram represents the circumference of the pitch circle of the cam. DEFINITIONS AND CLASSIFICATION 13 In the cam chart diagram the scales for drawing the height and the length of the rectangle are totally independent of each other and independent also of the scale of the cam drawing. In drawing the diagram no scale need be used at all, and the entire chart diagram with its base curve and pitch line may be drawn entirely freehand with suitable subdivisions marked off entirely "by eye" according to the requirements of the problem. The base curve may be drawn roughly as a curve or it may be made up of a series of straight lines. The cam chart diagram frequently serves all the purposes of the cam chart. It saves time, and permits of chart drawings being It0° 270° 3 60 ■Represents tenft/i of circumference of pitch circ/e ofcam^ Fig. 15. — Cam Chart Diagram made on small available sheets of paper, w T hereas the more precise cam chart often requires large sheets of paper which are usually impracticable and unnecessary in many circumstances. 19. Time charts. Illustrated in Figs. 16 and 17. Time charts are the same as cam charts or cam chart diagrams, and are con- structed in the same way as described in the two preceding para- graphs. The term "time chart," however, is most appropriately applied to problems where two or more cams are used in the same machine and where their functions are dependent on each other. ,1 f fi N^ """* ^- ____ j'lT 4 T Yvy --!> c £' c U. 0° 90° 180° 270° 36(r Fig. 10. — Time Chart Diagram, Base Curves Superposed The time chart permits of allowances being made for avoiding possible interference of the several moving parts, and for the desired timing of relative motions for each part. The time chart contains two or more base curves according to the number of cams used. When the base curves are superposed as in Fig. 16, the time chart consists of a single rectangle whose height is equal to the greatest 14 ELEMENTAKY CAMS follower motion. The superposing of curves and lines often leads to confusion and error, and it is better, in general, that the time chart should consist of a series of charts or rectangles all of the same length and one directly under the other as in Fig. 17. Where there are many base curves it is desirable to separate the rectangles \ 1 $n v 1 1 i*"*^ S^\ 0° 90° 180° 270° 360 Fig. 17. — Time Chart Diagram, Base Curves Separated by a small space to avoid any possibility of confusion due to different base curves running together. In many cases the term "time chart diagram," or " timing diagram," will be more appropriate than "time chart" in just the same way as the cam chart diagram is more ap- propriate than the cam chart. 20. Base curve. Illustrated in Fig. 14. A base curve is made up of a series of smooth continuous curves, or a combination of curves and straight lines, which represent the motion of the follower, and which run in a wave-like form across the entire length of the cam chart or diagram. The base curve of the cam chart becomes the pitch surface of the cam. 21. Base line. Illustrated in Fig. 15. A base line is made up of a series of inclined straight lines, or a series of inclined and hori- zontal lines, in consecutive order, which zigzag across the entire length of the chart. The base line when used on the cam chart indicates the exact motion of the follower, but when used on a cam chart diagram it is merely a time-saving substitute for the drawing of the base curve. The base line of the cam chart diagram represents the pitch surface of the cam. 22. Names of base curves or base lines in common use, see Figs. 18 and 19: 1. Straight line 4. Parabola. 2. Straight-line combination 5. Elliptical curve. 3. Crank curve. DEFINITIONS AND CLASSIFICATION 15 23. Pitch line. Illustrated in Fig. 14. A pitch line is a horizontal line drawn on the cam chart or diagram, and it becomes the pitch circle of the cam. The position, or elevation, of the pitch line on the chart varies according to the base curve which is specified, and according to the data of the problem. For cams which give a Fig. 18. — Comparison of Base Curves in Common Use Showing Varying Degrees of Maximum Slope When Drawn in Same Chart Length continuous motion to the follower during its entire stroke, or throw, the pitch line will pass through the point on the base curve which has the greatest slope, starting from the bottom of the chart. This does not apply to all possible base curves, but it does apply to all 1 Straight Line 2 Straight Line Combination 3 Crank Curve 4 Parabola 5 Elliptical Curve Fig. 19. — Comparison of Base Curves in Common Use Showing Uniform Maximum Slope of 30° When Drawn in Charts of Varying Length those mentioned in the preceding paragraph, a minor exception being made of the crank curve which will be referred to in para- graph 34. When the cam causes the follower to move through its total stroke in two or more separate steps the position of the pitch line on the chart must be specially found as will be explained in problem 5. 16 ELEMENTARY CAMS 24. Pitch circle. Illustrated in Fig. 20. A pitch circle is drawn with the center of rotation of the cam as a center, and its circumfer- ence is equal to the cam chart length. Its characteristic is that it passes through that point A, Fig. 20, of the pitch surface of the cam where the cam has its greatest side pressure against the follower. This applies to all cams in which the center of the follower roller moves in a straight radial line. For other motions of the follower roller, and for flat-faced followers, the pitch circle must be specially considered, as will be explained in some of the problems covering these types. 25. Pitch surface. Illustrated in Fig. 20. The pitch surface of a cam is the theoretical boundary of. the cam that is first laid down in constructing the cam. When the follower has a V-shaped edge, as at D in Fig. 7, the pitch surface coincides with the working surface of the cam. When the follower has roller contact, as in Fig. 20, the pitch surface passes through the axis of the roller and the working or actual surface of the cam is parallel to the pitch surface and a distance from it equal to the radius of the roller. 26. Working surface. Illustrated in Fig. 20. The working surface of the cam is the surface with which the follower is in actual contact. It limits the working size and weight of cam. For exact compliance with a given set of cam data, the cam has only one theoretical size which is bounded by the pitch surface, but the working size may be anything within wide limits which depend on the radius of the follower roller and the necessary diameter of the cam shaft. The working surface is found by taking a compass set to the radius of the roller and striking a series of arcs whose centers are on the pitch surface. Such a series of arcs is shown in Fig. 20 with their centers at B, A, etc. The curve which is an envelope to these arcs is the working surface. 27. Pitch point of follower. Illustrated in Fig. 20. The pitch point of the follower is that point fixed on the follower rod or arm which is always in theoretical contact with the pitch surface of the cam. If the follower has a sharp V-edge the pitch point is the edge itself. If the follower has a roller end, the pitch point is the axis of the roller. The pitch point is constantly changing its position from C to D as the follower moves up and down. 28. Pressure angle. Illustrated in Fig. 20. The pressure angle is the angle whose vertex is at the pitch point of the follower in its successive positions and whose sides are the direction DEFINITIONS AND CLASSIFICATION 17 of motion of the pitch point and the normal to the pitch surface. Pressure angles exist when the surface of the cam presses sidewise against the follower; they cause bending in the follower arm and side pressure in the follower guide and in the bearings. The pres- Maximum Pressure Angle^-\ Normalto .Pitch Surface Radial Line Fig. 20. — Showing Names of Surfaces, Lines, and Points of a Cam sure angle is constantly varying in all cams as the follower moves up and down, except where a logarithmic spiral is used. In assign- ing cam problems the maximum permissible pressure angle is usually given. In Fig. 20 the pressure angle is zero at C, it will be equal to a when B reaches J, and will be a maximum when A reaches K. 29. Formula for size of cam for a given maximum pressure angle. The radius of the pitch circle of the cam may be found directly by the formula: 360 . . . . . 1 hX- V XfX 2 = 57.3 hf or, ^ X T X ' X A = .159 hf (1) (2) 18 ELEMENTARY CAMS in which, r = radius of pitch circle of cam. h = distance traveled by follower. / = factor for a given maximum pressure angle. b = angle, in degrees, turned by cam while follower moves distance h. e = angle, in fraction of revolution, turned by cam while follower moves distance h. 30. Cam factors for maximum pressure angle. The factors, or value of /, for various maximum pressure angles for cams using the several base curves in common use are: Table of Cam Factors Maximum Pressure Angle and Values OF / Name of Base Curve 20° 30° 40° 50° 60° Straight line 2.75 3.10 4.32 5.50 6.25 1.73 2.27 2.72 3.46 3.95 1.19 1.92 1.87 2.38 2.75 .84 1.77 1.32 1.68 1.95 .58 Straight-line combination* . . . Crank curve 1.73 .91 Parabola 1.15 Elliptical curvef 1.35 These factors, for 30°, are illustrated in Fig. 19 where each of the base curves is given such a length, in terms of the height, that they will all have the "same maximum slope. The values given in this table are also shown, graphically, in Fig. 21, thus enabling one to find the proper cam factor for any intermediate pressure angle between 20° and 60°. * For case where easing off radius equals follower's motion. t For case where ratio of horizontal to vertical axes of ellipse is 7 to 4. DEFINITIONS AND CLASSIFICATION 19 60 B L DQ F 50 10 30 20 \ \ TVT \ v\\ \ \A\ \ 1, \ A- \ \ \ JV\ V S \ A tA s \*v hr X" c k <* u k%> $ \<3, ^ \< &_P§ /c *, ^■•^V-4 \ s < '#' N fV% ^s A. r? Ss "*->-* /■: M 1 2 .4 3JS Cam Factors Fig. 21. — Chart Showing Relation Between Pressure Angles a.nd Cam Factors for the Ordinary Base Curves SECTION II.— METHOD OF CONSTRUCTION OF BASE CURVES IN COMMON USE 31. Detail construction of base curves. The method of constructing the several base curves for a rise of one unit of the follower will be explained in the succeeding paragraphs. The curves will be constructed to give a pressure angle of 30° by selecting factors from the 30° column in the table in the preceding paragraph. Should the base curve for any other pressure angle be desired the factor should be taken from the corresponding column. 32. Straight-line base. Fig. 22. Lay off A B equal to the follower motion, which will be taken as 1 unit in these illustra- tions. Multiply this by the factor 1.73 from paragraph 30, and lay off the distance A R equal to it. Complete the parallelogram and draw the diagonal. This will be the straight line base and the Fig. 22. — Straight Base Line Fig. 23. — Straight-Line Combination Curve angle R AC will be 30°. A R will be the pitch line. These base lines and curves are laid off from right to left so that they may be used in a natural manner later on in laying out the cam so that it will turn in a right-handed or clockwise direction. The straight-line base gives abrupt starting and stopping velocities at the beginning and end of the stroke and causes actual shock in the follower arm. The velocity of the follower during the stroke is constant. The acceleration at starting and retardation at stopping is infinite and is zero during the stroke. 33. Straight-line combination curve. Fig. 23. Construct the rectangle with a height of 1 unit and a length of 2.27 units. With B and R as centers draw the arcs A E and C N, and draw a straight line E N tangent to them. The angle FEN will then equal 30° and the line A C will be a base curve made up of arcs and a 20 CONSTRUCTION OF BASE CURVES IN COMMON I SE 21 straight line combined to form a smooth curve. DF will be the pitch line. The straight-line combination curve, being rounded off at the ends, does not give actual shock to the follower at starting and stop- ping, but it does give a more sudden action than any of the base curves which follow, and the maximum acceleration and retardation values are comparatively larger. 34. Crank Curve. Fig. 24. Construct the rectangle. Draw the semicircle R G C and divide it into any number of equal parts. Six parts are best for practice work for this curve, but in general in practical work the greater the number of divisions the more accurate will be the curve and the smoother the action of the cam. 5-^< ^C Bk u r ^ gI F ^\U3 I H D \ gj£ K a\ k— ' Fig. 24. — Crank Curv; The six equal divisions of the semicircle are readily obtained by taking G as a center and F C as a radius and striking arcs at 1 and 5, then with R and C as centers mark the points 2 and 4 respectively. Divide the length of the chart into six equal parts, as at H, I, E, etc. From these points drop vertical lines, and from the corresponding divisions on the semicircle draw horizontal lines, giving intersecting points, as at K, on the desired crank curve. The tangent to the curve at E will then make an angle of 30° with the line E F. The pitch line will be D F. When the crank curve is transferred from the chart to the cam it gives an angle which is a fraction of a degree greater than 30° at the point E on the cam in practical cases. This is not enough greater to warrant the special computations and drawing that would be necessary to be exact. Therefore the method of laying out the crank curve and the pitch line, as given above, will be adhered to in this elementary consideration of cam work, because of its simplicity. The crank curve gives a slightly irregular increasing velocity to the follower from the beginning to the middle of its stroke; then a decreasing velocity in reverse order to the end of the stroke. The 22 ELEMENTARY CAMS acceleration diminishes to zero at the middle of the stroke and then increases to the end. The maximum acceleration and retardation values are much less than for the straight-line combination curve, and are only a little greater than for the parabola. 35. Parabola. Fig. 25. Construct the rectangle. Draw the straight line R S in any direction and lay off on it sixteen equal divisions to any scale. From the sixteenth division draw a line to F, the middle point of the chart; draw other lines parallel to this through the points 9, 4, and 1, thus dividing the distance R F into four unequal parts which are to each other, in order, as 1, 3, 5, and 7. From these division points draw horizontal lines, and from H, I, and J drop vertical lines. The intersecting points, as at K, Fig. 25. — Parabola will be on the desired parabola. The points H, I, and J divide the distance D E into four equal parts. The parabola gives a uniformly increasing velocity from the beginning to the middle of the stroke; then a uniformly decreasing velocity to the end. The acceleration of the follower is constant during the first half of the stroke and the retardation is constant during the last half. The acceleration and retardation values are equal and are less than the maximum value of any of the other base curves. This means that the direct effort required to turn a positive- acting parabola cam is less than for any other type of positive cam. 36. To better understand the smooth action given by the cam using this curve, consider, 1st, D H as a time unit during which the follower rises one space unit ; 2d, H I as an equal time unit during which the follower rises three space units; 3d, 7 J as the time unit during which the follower rises five space units, etc. Inasmuch as the follower travels two units further in each succeeding time unit, it gains a velocity of two units in each time unit, and this is uniform acceleration. The distance from F to C would be divided the same as from F to R and points on the part of the curve from E to C similarly CONSTRUCTION OF BASE CURVES IN COMMON USE 23 located. This curve will be identical with E A, but in reverse order, and will give uniform retardation. The tangenl to the curve A C at the point E will make an angle of 30° with E F, and D F will be the pitch line. Eight construction points were taken in developing the curve A C. Eight points will be sufficient for beginners for practice work Fig. 20. — Elliptical Curve and later six points may be used. When using six points only nine equal divisions should be laid out on the line R S, the remaining construction being the same as described above, except that D E should be divided into three parts instead of four. In practical work many more construction points should be used for accuracy and smooth cam action. 37. Elliptical curve. Fig. 26. Draw rectangle A B C R. Draw semi-ellipse making F G equal to -r F C. To draw the ellipse, take a strip of paper with a straight edge and mark fine lines at P, T, and S, Fig. 26a, making P T = C F and P S = G F. Move the strip of paper so that S will always be on the line R C, and T on the line F G; P will then describe the path of the ellipse. Having the semi- ellipse, divide the part R G, Fig. 26, into four equal arcs as at 1, 2, 3. This is quickest done by setting the small dividers to a small space of any value and stepping off the distance from R to G. Suppose that there are 18.8 steps. Set down this number and divide it into four parts, giving 4.7, 9.4, and 14.1. Then again step off the arc from R to G with the same setting of the dividers, marking the points that are at 4.7, 9.4, and 14.1 steps. The compass setting- being small, the fractional part of it can be estimated with all prac- tical precision. Divide D E into four equal parts as at H, I, J. Draw vertical lines from these points and horizontal lines from the Fig. 2Ga. — Showing Method of Draw- ing Semi-Ei.lip.se 24 ELEMENTARY CAMS corresponding points at 1, 2, and 3. The intersections, as at K, will give a series of points on the elliptical base curve. The curve E C is similar to A E but in reverse order. The tangent to the curve at E makes an angle of 30° with E F, and D F is the pitch line. The elliptical base curve gives slower starting and stopping velocities to the follower than any of the other curves, but the velocity is higher at the center of the stroke. The acceleration is variable and increases to the middle of the stroke, where its maximum value is greater than that of the crank curve but less than that of the straight- line combination curve. The retardation values decrease in reverse order to the end of. the stroke. SECTION III.— CAM PROBLEMS AND EXERCISE PROBLEMS 38. Problem 1. Empirical design. Required a radial cam that will operate a V-edge follower: (a) Up 3 units while the cam turns 90°. (b) Down 2 " " " " " 60°. (c) Dwell " " " " 120°. (d) Down 1 unit " " " " 90°. 39. Applying the simplest process for laying out cams, it is only necessary, in starting, to assume a minimum radius C D, Fig. 27, for Fig. 27. — Empirical Design of Cam for Data in Problem 1, V-Edge Follower the cam, and then lay off the given or total distance of 3 units as at D B. The assigned angle of 90° is next laid off as at D C Di and the point Di marked so as to be 3 units further out than D. Any desired curve is then drawn through the points I) and l) x and part of the cam layout is completed. The same operations are repeated for obtaining the points D 2 and D 3 and the entire cam is finished. If the follower had roller contact instead of V-edge contact, a 25 26 ELEMENTARY CAMS minimum radius C D, Fig. 28, would be assumed as in the previous case, and D would be taken as the center of the roller. The closed curve D, D 4 , Z>i . . . would be obtained as before and another closed curve E, Ei . . . would be drawn parallel to it at a distance equal Fig. 28 — Empirical Design of Cam for Data in Problem 1, Roller Follower to the assumed radius of the roller. The latter closed curve would be the actual outline of the cam. The closed curve E E\ . . . would be known as the working sur- face and the curve 2) Di ... as the pitch surface of the cam. In Fig. 27 the pitch and working surfaces coincide because the follower has a V-edge. 40. Cams are sometimes designed with no more labor than that entailed in the previous preliminary problem. And it may be added that where one has had a sufficient experience good practical results may be obtained by following only this simple method. The method of cam construction described above, however, does not enable the cam builder or designer to hold in control the velocity or acceleration of the follower rod D G as it moves up its 3 units; nor does it enable him to know the variable and maximum side pres- sures which exist between the follower rod and the bearing or guide CAM PROBLEMS AND EXERCISE PROBLEMS 27 F, Fig. 27, as the rod moves up. In order that these things may be known, this preliminary problem will now be redrawn with ad- ditional specifications. 41. Problem 2. Technical desicn. Required a radial cam that will operate a roller follower: (a) Up 3 units while the cam turns 90°. (b) Down 2 " " " " " 60°. (c) Dwell " " " " 120°. (d) Down 1 unit " " " " 90°. (e) The follower, in all its motions, shall move with uniform acceleration and uniform retardation. (f) The maximum side pressure of the cam against the follower rod shall be 40°. Items (a), (b), (c), and (d) are the same as in Problem 1. 42. Inasmuch as this problem is given at this place simply to show that velocity and acceleration and side pressure can always be controlled with very little additional labor beyond that necessary for the simple layout shown in Fig. 28, the full explanations of the •formula and figures used will not be given here. They will be taken up in their proper order in subsequent paragraphs. For this problem the only necessary computation is : 57.3 hf 57.3 3 X 2.38 b "'•" 90 C H, Fig. 29. The reference letters, h, f, and b are defined in paragraph 29. Lay off C H in Fig. 29, and then lay off the- follower motion of 3 units equally distributed on each side of H, as at H B and H D. Divide D H into nine equal parts and take the first, fourth, and ninth parts; do like- wise with B H. Divide the 90° angle B C Dx into six equal parts by radial lines as shown, and swing each of the six di- vision points between D and B = 4.55 = Radius of pitch circle = around until they meet succes- FlG - 29 — technical design of cam for data in Problem 2, Drawn to Same Scale s - as Fia. 28 sively the six radial line 28 ELEMENTARY CAMS A curve through the intersecting points will be the pitch surface of the cam, as shown by the dash-and-dot curve D Hi D\. . . . The working surface will be E Ei . . . which is found as described in paragraph 26. The pitch surface Z>i Z) 2 is obtained in the same way as D Di was found. The curve D 2 Ds is an arc of a circle, and the curve D 3 D is found in the same manner as D D\. 43. Advantages of the TECHNICAL DESIGN. With the cam constructed as above the follower will start to move with the same characteristic motion as has a falling body starting Fig. 29.— (Duplicate) Technical Design of f rom rest anc J t } ie follower will Cam for Data in Problem 2, Drawn to . same Scale as fig. 28 be stopped with the same gen- tle motion in reverse order. It- will be definitely known also that the greatest side pressure of the cam against the follower is at an angle of 40° as specified, and that this pressure will occur when Hi of the pitch surface of the cam is at H, or when the roller is in contact with the working surface at H 2 . Where the cam form is assumed as in Fig. 28, nothing is known positively of the starting and stopping velocities of the follower. Further, as may be found by trial, the maximum angle of pressure of the cam against the rod runs up to 47° in Fig. 28, as shown at D 4 . The minimum radius of the cam in Fig. 28 was taken equal to that in Fig. 29 for comparison. 44. The two previous problems have been given as brief exercises without going into all the detail necessary to a full understanding, in order to give an idea of the method of producing cams on a scientific basis. In the problems which will follow, the several steps in building cams of various types will be explained. In many of the problems the same data will be used so that comparisons of different forms of cams which produce the same results may be made. 45. Problem 3. Single-step radial cam, pressure angle equal on both strokes. Required a single-step radial cam in which the center of the follower roller moves in a radial line. The maximum pressure angle to be 30°, and the follower to move: CAM PROBLEMS AND EXERCISE PROBLEMS 29 (a) Up 3 units in 90° with uniform acceleration and retardation. (b) Down 3 units in 90° with uniform acceleration and retardation. (c) At rest for 180° 46. The first step in the solution is to determine the total length of the cam chart for a parabola chart curve and for a 30° maximum pressure angle. From the table, paragraph 30, the factor for this case is found to be 3.46. Since the travel of the follower is 3 units in 34 revolution, the total length of chart will be 3 X 3.46 X 4 = 41.52, which, therefore, is the length of the chart A A' in Fig. 30. This length represents the 360° of the cam. Lay off A W equal to 90°, according to item (a) in the data. Construct the parabolic curve A E C. Completing the entire chart, the base curve is found tobe AC M N A'. The next step is to find the radius of the pitch circle. The circumference of this circle is equal to the length of the pitch . 41.52 line D D f . Its radius is, therefore, equal to ~ — = Z IT 6.61, and this value is laid off at D, Fig. 31, and the pitch circle D F Q W drawn. The quadrant D F is divided into the same number of parts as D F in Fig. 30. The vertical construction lines H Hi, II i, J J i . . . in Fig. 30 now become the radial lines correspondingly lettered in Fig. 31, and the pitch surface is drawn through the points A Hi 7i J\. . . . The positions of maximum pres- sure are shown at E and Q; at all other points it will be less. The working surface B G R P is found by assuming a radius A B for the roller, and by striking a series of arcs as shown at H 2 , L, J 2 . . • with the points H h I h Ji ... as cen- ters, and then drawing the working curve tangent to these arcs. With the same specifications for the up and down motions of the follower, as given by items (a) and (b) in the data, this type of cam will be symmetrical about the line Y C. 30 ELEMENTARY CAMS Fig. 31. — Problem 3, Cam Laid Out from Cam Chart Fig. 32. — Problem 3, Cam Laid Out Independently of Cam Chart CAM PROBLEMS AND EXERCISE PROBLEMS 31 47. Omission of cam chart. When the relation between pics- sure angle, chart base and pitch lines, and cam pitch and surf; ice lines is understood and fixed in mind, the actual drawing of the chart for the graphical construction of simple cams and particularly of single-step cams may be omitted with full confidence when the elementary base curves are used. For example, the problem in the previous paragraph is shown completely worked out in Fig. 32 without any reference whatever to the chart of Fig. 30. The radius D of the pitch circle, Fig. 32, is obtained directly from the formula, h f r = 57.3 -j- given in paragraph 29. Substituting the data as given 3 X 3.46 in the previous paragraph, r = 57.3 ^ — = 6.61 and is laid off at D 0. The assigned motion of the follower is laid off symmetri- cally on both sides of the pitch point D, as at A V, and the distances A D and V D are divided into the desired number of unequal parts, as at 1, 4, 9, 16. The quadrant D F is divided into the same number of equal parts as at H, I, J . . . and indefinite radial construction lines drawn through the points. Circular construction arcs are next drawn through the points 1, 4, 9 . . . until they intersect the radial lines, thus obtaining points H h I h J x . . . on the cam pitch surface. In general, a neater construction is obtained by omitting the full length of the construction arcs, as from V to C . . . and simply drawing short portions of the arc at the intersecting radial lines as shown in the lower left-hand quadrant between C and M . 48. Exercise problem 3a. Required a single-step radial cam in which the center of the follower roller moves in a radial line. The maximum pressure angle to be 40°, and the follower to move: (a) Out 6 units in 135° on the crank curve. (b) In 6 " " 135° " " " (c) At rest for 90°. 49. Problem 4. Single-step radial cam, pressure angles unequal on the two strokes. Required a single-step radial cam in which the center of the follower moves in a radial line. The maximum pressure angle not to exceed 30° on the outstroke nor 50° on the return stroke, and the follower to move: (a) Out 2 units in /i 6 revolution on the crank curve. (b) In 2 " " % 6 " " " (c) At rest for y 2 revolution. 32 ELEMENTARY CAMS 50. The diameter of pitch circle of the cam that will be necessary to fulfil the requirements on the outstroke will be: 2 X 2.72 X 16 d a = — » * - — = 5.54 units, or from formula paragraph 29. r = .159 2X2 - 7 5 2X16 = 2.77, and the diameter of pitch circle required for the instroke will be , 2 X 1.32 X 16 ... . b = 3 14 X 3 — = units. Inasmuch as there can be only one pitch circle for a cam, the largest one resulting from the several specifications must be used. In this problem then the diameter S D of the pitch circle in Fig. 33 Fig. 33. — Problem 4, Maximum Pressure Angle Different on the Two Strokes equals 5.54 units. The follower's motion of two units is laid out at A V and the pitch surface A E C M N constructed. The working surface of the cam B KG, etc., is then drawn. Since a larger diameter of pitch circle had to be used for the return stroke than the require- CAM PROBLEMS AND EXERCISE PROBLEMS 33 merits called for, it follows that the pressure angle will not reach 50° on that stroke, and it may be of some interest to determine what the maximum pressure angle on the return stroke will be. Sub- 7 f stituting the diameter used, 5.54, in the formula d = — and solving for/, / is found to be equal to 1.63. From the chart in Fig. 21 it is shown that a factor of 1.63 for the crank curve corresponds to a maximum pressure angle of nearly 44°, and this angle may be drawn in its proper position at Q in Fig. 33. 51. Exercise problem 4a. Required a single-step radial cam in which the center of the follower roller moves in a radial line. The maximum pressure not to exceed 30° on the up stroke nor 40° on the down stroke, and the follower to move: (a) Up 3 units in 135° on the parabola curve. (b) At rest for 45°. (c) Down 3 units in 90° on the parabola curve. (d) At rest for 90°. 52. Pressure angle increases as pitch size of cam decreases. This is illustrated in Fig. 34, where the large pitch cam represented by D, D 2 . . . gives exactly the same motion to a follower as the small pitch cam d, d 2 . . . . It will be noted that the pressure angle for the large cam, at the start, is H D G, while for the small cam it is increased to h d g. Likewise the maximum pressure angle for the large cam, when the follower is near the end of its stroke, is &i, while for the small cam the maximum pressure angle is b, which is larger than b u From these observations it may be said, in general, that the larger the pitch surface of the cam the smaller will be the pressure angle. The size of the roller has no effect whatever on the pressure angle. Two cams of the same pitch size may be of totally different actual sizes for the same work, one cam having a large roller and the other a small roller. Therefore it is important to remember that, in general, the pressure angle may be regulated by changing the size of the pitch surface only and not the working surface. 53. Change of pressure angle in passing from chart to cam. The circumference of the pitch circle of the cam, it will be recalled, is equal to the length of the pitch line on the chart. It will also be remembered that the pitch line may be at various heights on the chart, paragraph 23. It is now important to consider: 1st. That the pressure angle at the pitch circle on the cam must be the same as the pressure angle at the pitch line on the chart. 34 ELEMENTARY CAMS 2d. That the pressure angle at any point on the pitch surface of the cam outside of the pitch circle will be less than the pressure angle of the corresponding point on the base curve of the cam chart. 3d. That the pressure angle at any point on the pitch surface d^ — — — _ x> 5 Fig. 34. — Showing Relation Between Pressure Angle and Size of Pitch Cam of the cam inside of the pitch circle will be greater than the pressure angle of the corresponding point on the base curve of the cam chart. These statements, which are theoretically true for nearly all cases, and practically so for all other cases where the usual base curves are employed, are demonstrated in the following paragraph. 54. Cam considered as a bent chart. Consider that the cam itself is the cam chart bent in its own plane so that the pitch line CAM PROBLEMS AND EXERCISE PROBLEMS 35 becomes the pitch circle. Then the line D D' ', Fig. 30, becomes the circle DFOW, Fig. 31; the line V V f is stretched to become the circle V C S Y, and the straight line A MA' is compressed to become the circle A MA. This means, in a general way, that the rectangle D V V f D', Fig. 30, is so distorted that if an original diagonal had been drawn from D to V' it would have an increased length and a decreasing slant after the bending had taken place. With a decreasing slant of the pitch surface the pressure angle will decrease. Likewise, a diagonal drawn from D r to A in the original rectangular chart would be decreased in length and would have an increasing slant, and the pressure angle would be increasing toward A. This is illustrated in detail in Figs. 35 and 30. 55. Base line angles, before and after bending. The pres- sure angle of 30° at E in Fig. 35 is reduced to 23° in Fig. 30, and the Fia. 35. — Section of Cam Chart Be- Fig. 36. — Section of Cam Chart After fore Bending Bending, BC Constant in Both Figubbs 30° at D are increased to 41°. Fig. 35 represents a cam chart with a straight base line D E, and Fig. 30 is a corresponding cam sector with D E as the pitch surface. If B C, Fig. 35, is taken as the pitch line, B C, Fig. 30, will be part of the pitch circle. The uniform pressure angle of 30° from A to E, Fig. 35, will grow smaller beyond A in Fig. 30 for the reason that the radial components of the tan- gential triangles remain constant, as illustrated at L M, while the tangential components grow longer as illustrated from A N to E L, which are respectively equal to the arcs A Y and E L\. Con- sequently, the angles grow smaller from the angle N A P to L E M. Similarly it may be shown that they grow larger from NAP to QDR. 50. Limiting size of follower roller. The radius of the follower roller may be equal to, but in general should be less than 36 ELEMENTARY CAMS the shortest radius of curvature of the pitch surface, when measured on the working-surface side. If the radius of the roller is not so taken, the follower, when put in service, will not have the motion for which it was designed. 57. Case 1. Radius of roller equal to radius of curvature of pitch cam. In Fig. 37, A B E F A is the pitch surface of a cam. Fig. 37. — Limiting Size of Follower Roller CAM PROBLEMS AND EXERCISE PROBLEMS 37 G A is the radius of curvature at A and A G is the radius of the roller. In this case both radii are equal and the working surface has a sharp edge at G. 58. Case II. Radius of roller greater than radius of curvature of pitch cam. From B to C, Fig. 37, the radius of curvature of the pitch surface is H B, which is less than the roller radius. In this case the working surface will be undercut at / in generating the cam, and if the cam is built the center of the roller will mark the path BiJiCi instead of Bi J Cj, and the follower will fail to move the desired distance by the amount J J±. 59. Special application of case II. Effect of an angle in the pitch surface outline. This is illustrated at R F Q in Fig. 37, and is a special application of Case II, in w-hich the radius of cur- vature of the cam's pitch surface is reduced to zero. Undercutting is here illustrated by considering that a cutter, represented by the dash circular arc, is moving with its center on the pitch surface arc E F. It then cuts the working surface M S. As the center of the cutter is moved from F toward A, the part W S of the working surface which was previously formed is now cut away, leaving the sharp edge W on which the follower roller will turn when the cam is placed in operation. The center of the follower roller will then move in the path R T Q instead of R F Q, and the follower will fall short of the desired motion by the amount T F. 60. Case III. Radius of roller less than radius of curva- ture of pitch cam. From D to E, Fig. 37, the radius of curvature of the pitch surface is K D, which is greater than the roller radius. In this case, which is the practical one, although close to the limit, a smooth curved working surface is provided for the roller from Lto M. 61. Radius of roller not affected by radius of curvature on non-working side. From Ci to D, Fig. 37, the radius of curva- ture of the pitch surface is less than the radius of the roller, but this short radius is not on the working side of the pitch surface, and therefore the roller will roll on the surface I L while its center travels on the pitch curve Ci D. 62. Rollers for positive-drive cams. When the largest roller for a positive or double-acting cam is being determined the radius of curvature on both sides the pitch-surface curve must be con- sidered and the smallest radius used. For example, in Fig. 37, if A J E T A were the pitch surface for a double-acting cam, N C would be the maximum roller radius, whereas H J would be 38 ELEMENTARY CAMS the maximum radius if it were for an external single-acting cam. 63. Radius of curvature of non-circular arcs. In illustrat- ing the above cases the pitch surface was assumed as being made up of straight lines and arcs of circles in order to show more effec- tively and more simply the limits of action in each instance. Where the pitch surface contains curves of constantly varying curvature, and they generally do in practice, the shortest radius of curvature of the pitch surface may be found with all necessary accuracy by trial with the compass, using finally that radius whose circular arc agrees for a small distance with the irregularly curved arc. For example, in Fig. 38, let G H D J B be a portion of a pitch surface Fig. 38. — Limiting Size of Follower Roller Working on Non-Circular Cam Curves made up of non-circular arcs. The shortest radius of curvature on both sides is found, by trial, to be F H. The center F is marked and the osculatory arc X H Z drawn in. Then H F is the largest possible radius of roller for a double-acting cam, and with this roller the working surfaces will be V F T W and Vi Fi T x W\. CAM PROBLEMS AND EXERCISE PROBLEMS 30 If a larger roller is used, with a radius I) I\, for example, the working; surfaces of the groove will be S E and l\ A'i A',, and the new pitch surface, after cutting the cam, will be GC DLB, if the roller is kept always in contact with the inner surface of the groove. If it is kept always in contact with the outer surface of the groove, the original pitch surface will be changed to G C H DiJ B. In either case the original desired follower motion is not obtained if the roller is too large, and if a positive-drive cam is run with the larger roller the follower's motion will be indeterminate, the center of the roller having any possible position between C D L and C H J L. 64. Problem 5. Double-step radial cam. Required a double- step radial cam in which the center of the follower roller moves in a radial line. The maximum pressure angle to be 30°, and the follower to move: (a) Up 4 units in }/$ revolution on the crank curve. (b) At rest for 34 revolution. (c) Up 4 units in y% revolution on the parabola curve. (d) Down 2 units in % revolution on the elliptical curve. (e) At rest for Y% revolution. (f) Down 6 units in \i revolution on the parabola curve. 65. In Problem 3 there are only two motion assignments, (a) and (b), in the data, and they were the same except for direction. Con- sequently only one computation was necessary. When two or more dissimilar assignments are made in the data, as in the present problem, it is advisable to make a computation for the length of the chart diagram for each motion specification, as follows: (a) 4 X 2.72 X 8 = 87.04, which is the length of chart and of the pitch circle circumference = 13.86 pitch circle radius. (c) 4 X 3.46 X 8 = 110.72, which is the length of chart and pitch circle circumference = 17.62 pitch circle radius. (d) 2 X 3.95 X 8 = 63.20, which is the length of chart and pitch circle circumference = 10.06 pitch circle radius. (f) 6 X 3.46 X 4 = 83.04, which is the length of chart and pitch circle circumference = 13.22 pitch circle radius. Inasmuch as there is a different length of chart and a different pitch line for each item in the data one can not tell which pitch line to take without some preliminary computation. For this purpose 40 ELEMENTARY CAMS a chart diagram is well adapted, as follows: Construct a rectangle, Fig. 39, with a height A T equal to the total motion of the follower in one direction, 8 units in this case. Make the length A A' of rectangle any convenient value entirely independent of any of the values computed above and label this according to the longest chart length as computed above. Lay off straight lines to represent the component parts of the base curve as assigned in the data and label them as shown at A C, C B, B H, etc. Draw the several pitch lines as at F D, J I, etc. 66. For general procedure, consider the pitch line which passes through the point calling for the longest chart length. This will be the pitch line J I passing through G, Fig. 39, which calls for a chart length of 110.72 and a pitch radius of 17.62. If G is to be at T H * u D ~u I .$*y n^ -Lj , yr* j vl i\t U M, f L\ B 1 S f C 4 Q si *X f F > ■ the curve B L being used to lift the arm, and the curve L E to lower the arm, the swinging velocities of the arm being the same in both directions. 106. Data limited for followers with sliding surface con- tact. The data for this type of cam construction are extremely limited when the swinging velocity of the arm is assigned. The limitations are that the working surface of the cam must be drawn tangent to every construction line in succession, and that it must be convex externally at all points. In most arbitrary assignments of data the construction line through Cg, for example, would intersect the line through d before it cut the line through Cg. In this case it would be impossible to draw a smooth working curve tangent, successively, to the lines through d, C%, and Cg. This is illustrated more clearly in Fig. 51 and will be more evident after the limiting- case is described. The limiting case for flat surface followers with sliding contact occurs where three or more of the construction lines meet in a point, as at N in Fig. 50. In this case the working surface of the cam Fig. 50. — Limiting Case for Straight Edge Fig. 51. — Impossible Case for Straight Follower with Sliding Contact Edge Follower with Sliding Contact 60 ELEMENTARY CAMS would have a sharp edge. In this type of cam it is necessary to use more construction lines than in other types, because it is pos- sible to have the construction lines so far apart that such a case as is shown in Fig. 51 might not evidence itself at all. For example, if the distance C 9 C7 were the unit space for construction lines, in- stead of C 9 C 8 , the smooth convex curve F N L could be drawn tangent to lines through C 9 , C 7 . . . without the error showing itself. 107. If it is required of this cam only that it shall swing a follower arm through a given angle in a given time, without regard to the Fig. 50. — (Duplicate.) Limiting Case for Straight Edge Follower with Sliding Contact Fig. 51. — (Duplicate.) Impossible Case for Straight Edge Follower with Sliding Contact intermediate velocities of the arm, it may be as widely used as any other type of cam. In this case only the innermost and outermost positions of the arm would be drawn, as at C A, C& V h and C i2 E, Fig. 49, and a smooth convex curve drawn tangent to these lines. Such construction, however, might give an irregular or jerky motion to the follower. Whether it did or not could be readily determined by laying off a number of equal divisions, as at C h C 2 . . . C i2 ; drawing lines, such as C% Ji, tangent to the assumed smooth convex working surface; and revolving C 3 Ji back to C J. After doing this with other construction lines a series of points, such as H, I, J . . . would be determined and the spaces between them would represent the distances traveled by A on the follower arm during successive equal intervals of time. 108. Exercise problem 11a. Required a radial periphery cam for a swinging follower arm, sliding surface contact. Arm to be 10 units long to the point which is used to measure the angular velocity, and this point to move through an arc which is measured by a chord of 4 units. The arm is to : CAM PROBLEMS AND EXERCISE PROBLEMS Gl (a) Swing full out with uniform acceleration and retardation while the cam turns % revolution. (b) Swing in with the same angular motion in Y% revolution. (c) Remain stationary for J^ revolution of the cam. 109. Toe and w t iper cams. In this form of cam construction the cam or " wiper" OC, Fig. 52, oscillates or swings back and forth through an angle of 120° or less, instead of rotating con- tinuously the full 360° as it does in all cams thus far considered. The follower or "toe" A W is usually a narrow flat strip resting on the curved periphery of the cam, and moving straight up and down. There is sliding action between the wiper and the toe. 110. Problem 12. Toe and wiper cam. Required a wiper cam to operate a flat toe follower which shall move: (a) Up 4 units with uniform acceleration all the way while the cam turns counterclockwise 45° with uniform angular velocity. (b) Down 4 units with uniform retardation all the way while the cam turns clockwise 45° with uniform angular velocity. 111. The detail of construction for this class of problem is iden- tical with that described for the mushroom cam in Problem 7, it being observed that the two cams differ only in that the mush- room cam turns through the full 360° instead of 45° as in this problem, and the mushroom follower is circular instead of rectangular. Neither of these differences nor the offset of the mushroom follower affect the similarity of construction for the two types of cams. There- fore, only a brief review of the general method of construction for the present problem will be given here. 112. Inasmuch as the line of pressure between cam and follower is always parallel to the direction of motion of the follower in prob- lems such as this, there is no pressure angle in the ordinary sense. If a computation for size of cam is made in the usual way, the radius of the pitch circle will figure to be unnecessarily large, due princi- pally to the fact that only a 45° degree turn of the cam is allowed for the upward motion of the follower. A radius O A, Fig. 52, which allows for radius of shaft, thickness of hub, etc., is assumed, and the follower motion of 4 units is laid off at A V. This distance is divided into four unequal parts at H, I . . . which are to each other as 1, 3, 5, and 7, thus giving uniform acceleration all the way up. The angle A B of the cam is laid off 45° and is divided into four equal time parts. The follower or toe surface A W is then moved up the distance A H and revolved through the angle A 1 to the position H x H 2 which is marked. Simi- 62 ELEMENTARY CAMS larly A W is next moved to I I 3 and revolved to h 7 2 . The smooth- est possible convex curve is then drawn to the lines Hi H 2 , I\ h . . . and this curve becomes the working surface of the wiper. The necessary working length for the wiper is found to be A V 2 , and, adding a small arbitrary distance, V 2 C, the total length is taken Fig. 52. — Problem 12, Toe and Wiper Cam as A C. The total length of the toe A W will be equal to V\ C. The long dash lines in Fig. 52 indicate the highest position of the toe and wiper, and the short dash-line curve marks the locus of contact between the wiper and toe. This curve is obtained by making, for example, J J 3 equal to Ji J 2 - 113. Modifications of the toe and wiper cam. The toe and wiper cam constructions are commonly used. In the present ele- mentary problems the cam or wiper is assumed to oscillate with uniform angular velocity, whereas in practice it usually has a variable angular velocity due to the fact that it is operated through a rod which is connected at the driving end to a crank pin or eccentric CAM PROBLEMS AND EXERCISE PROBLEMS 03 whose diameter of action corresponds to the swing of the wiper cam. The follower toe may be built with a curved instead of a straight line, by a slight modification in detail which consists in draw- ing the curved toe line in place of the straight lines, Hi 11*, l x I 2 . . . as shown in Fig. 52. These points, together with a consideration of the amount of slip between the surfaces in this type of cam and a discussion of the necessary modification to secure pure rolling in cams of this general appearance, are subjects for more advanced work than is covered by the present elementary problems. 114. Exercise problem 12a. Required a wiper cam to operate a flat toe follower which shall move: (a) Up 3 units with uniform velocity while the cam turns G0° in a counterclockwise direction with uniform angular velocity. (b) Down 3 units with uniform velocity while the cam turns G0° in a clockwise direction with uniform angular velocity. 115. Yoke cams. Yoke cams are simple radial periphery cams in which two points of the follower, instead of one, are in contact with the working surface. The contact points are usually diametri- cally opposite to each other. Roller contact is generally used and the centers of the rollers are a fixed distance apart. The yoke cam gives positive motion in both directions, and does not depend on a spring or on gravity to return the follower as do all other cams thus far considered, excepting the face cam. 116. Problem 13. Single-disk yoke cam. Required a single radial cam to operate a yoke follower with a maximum pressure angle of 30° : (a) Out 4 units in 45° turn of the cam, on crank curve. (b) In 4 " " 90 c (c) Out 4 " " 45 c 117. With a single radial cam for a yoke follower, data may be assigned only within the first 180°. The reason for this will appear presently. Compute the radius of pitch circle as in ordinary radial cam problems. It is found to be 13.86 units and is laid off at D, Fig. 53. The pitch surface, A D1V1A1 V 2 , is found in the usual way. Then the diametral distance, AV 2 , will be the fixed distance between the centers of the rollers, and if this distance is laid off on diametral lines, as from h, Ki . . ., the points W, X ... on the complementary pitch surface will be located. A size of roller A B is next assumed and the working surface B B 2 is constructed. The maximum radius of the working surface is finally located, as at B 2 . A small amount jO u £0 tt 64 ELEMENTARY CAMS is added to this for clearance and the total laid off at Z, thus giving the width of yoke necessary for an enclosed cam. 118. Limited application of single-disk yoke cam. In yoke cams constructed from a single disk the data are limited in two ways : First, data can be assigned for the first 180° only, because the pitch surface for the second 180° must be complementary to the pitch surface in the first 180°. Secondly, the complementary pitch surface cannot approach any nearer to the center of rotation of the cam than does the pitch surface TT-W4 1 - — np ±: rrh — " T H" ? Fig. 53.— Problem 13, Single-Disk Yoke Cam in the first 180°, otherwise the follower will have a greater motion than that which was assigned to it. To illustrate this second case, assume that item (c) had been changed in the data for Problem 13 so as to specify that the follower should remain at rest while the cam turns 45°. Then the pitch surface of the cam for the first 180° would have been AViAiC, Fig. 54, instead of AVi A x V 2 . The diametral distance A C would then have been the distance between roller centers, and would have been also the distance used in determining the complementary pitch surface C E x A s A which, it will be noted, approaches closer to O than does A Vi C. When E x of the complementary surface CAM PROBLEMS AND EXERCISE PROBLEMS 65 reaches the center line D, the center A of the roller will be at E and the roller will have traveled the distance A E in addition to the travel A V which was assigned. Furthermore, the pressure angle will be very high when F crosses the line D 2 . With the data which gives the pitch surface AV\C, the yoke follower will move just twice the assigned distance. This double motion will not be continuous, as the follower will be at rest for a definite period represented by A X C. Even if the data were such that A i should fall at C there would be a momentary pe- riod of rest for the follower at the middle of its stroke. Summing up, the desired travel, pressure angle, and follower velocity will be ob- tained in single-disk yoke cams, only when the data are such as to have the fol- lower at the extreme oppo- site ends of its stroke at the zero and 180° phases. In other cases increased travel, increased pressure angle, and irregular follower velocities will have to be considered. All of the limitations of the single-disk yoke cam may be avoided by using the double disk cam as illustrated in Problem 14. 119. Exercise problem 13a. Required a single-disk radial cam to operate a yoke follower with a maximum pressure angle of 30°: (a) In 6 units in 60° turn of the cam on parabola curve. (b) Out 6 " " 45° " " " " " (c) At rest for 30° " " " " (d) In 6 units in 45° " " " " " 120. Problem 14. Double-disk yoke cam. Required a double- disk cam to operate a yoke follower with a maximum pressure angle of 30°: (a) To the right 6 units in 150° turn of the cam, on the crank curve. (b) To the left 6 units in 90° turn of the cam, on the crank curve. (c) To remain stationary for 120° turn of the cam. Fig. 54.- i illustrating limited application of Single-Disk Yoke Cam 66 ELEMENTARY CAMS 121. The detail of construction for the primary disk is the same as in previous problems involving radial cams. In this problem, then, the radius of the pitch circle is 6 X 2.72 X 4 X o"T7 X ~k = 10.4 units and this is laid off at D, Fig. 55. The forward driving pitch surface, A Hi Vi 7i A , is constructed in the regular way as indicated by the construction lines. 122. The diameter D C of the pitch circle is next taken as a constant and its length is laid off on diametral lines from successive Fig. 55. — Problem 14, Double-Disk Yoke Cams, Detail Construction points on the primary pitch surface, thus giving the secondary or return pitch surface. For example, the point P on the secondary surface is found by making A P = DC; the point M by making Hi M = D C. . . . This second cam disk has a pressure angle of 30° at D A , the same as the primary disk has at D 2 . Had any diam- etral length other than D C been taken in this problem as a constant for constructing the second cam, the pressure angle at D 4 would have been greater or less than the assigned 30°. It does "not follow that the diameter of the pitch circle should be used as a constant for generating the complementary cam. The determining factor, in selecting a constant diametral length is that the maximum pres- sure angle on the second cam should not exceed the assigned value. CAM PROBLEMS AND EXERCISE PROBLEMS 67 123. To avoid intricate line work, only the detail drawing for the construction of the pitch surfaces for this problem is shown in Fig. 55. The pitch surfaces are then redrawn in Fig. 56 and the working surfaces and the yoke constructed. The working surface of the primary or forward-driving cam is shown at B E F G B, Fig. 56, and is constructed in the same way as Fig. 56. — Problem 14, Double-Disk Yoke Cams Showing Strap Yoke and Rollers in previous problems by drawing it as an envelope to successive roller positions. The working surface of the return cam is shown at S Q R S. A special caution to be observed at this point is that the working surface of the second cam cannot be obtained directly from the working surface of the first cam by using the diametral constant; the second cam pitch surface must be obtained first. 124. The form of yoke in yoke cams may vary, as illustrated for example by the box type which encloses the cam, Fig. 53, and by the strap type, Fig. 56. In the latter illustration the strap W X has a longitudinal slot T U permitting it to move back and forth astride the shaft without interference. The guide arms of the 68 ELEMENTARY CAMS yoke are shown at Y and Z. In all yoke constructions it is desirable to have all the forces acting in as nearly a straight line, or in a plane, as possible.' In Fig. 53 this is obtained, as may be noted in the top view where the longitudinal center lines of cam disk, cam roller, yoke and yoke guides are all in the same plane. In Fig. 56 the yoke guides, Y r and Z r , are placed in a line lying between the cam disks, B f and S f , so as to have the forces balanced to a greater degree than they would be if the guides were in line with the strap W f X' '. 125. Exercise problem 14a. Required a double-disk cam to operate a yoke follower with a maximum pressure angle of 30°, as follows : (a) To the right 4 units in 90° on the parabola base. (b) Dwell for 30°. (c) To the right 4 " " 105° " " (d) " " left 8 " " 135° " " 126 Problem 15. Cylindrical cam with follow t er that moves in a straight line. Required a cylindrical cam to operate a reciprocating follower rod : (a) To the right 4 units in 120° on the crank curve. (b) " " left 4 " " 120° " " (c) " dwell 120°. The maximum surface pressure angle to be 30°. 127. The size of cylinder is found by a computation similar to that for radial cams, and in this problem the radius of the cylinder is, 4 X 2.72 X 3 X t^tt X 2 = 5.2 units. This distance is laid off at O f A r in Fig. 57, and the circle drawn. The distance A V, the travel of the follower, is laid off equal to 4 units and subdivided, according to the crank circle, at H, I . . . The radius of the follower pin is assumed as at A S and this distance is laid off at S C, thus locating the edge of the cylinder. Make V D equal to A C. The circle representing the cylinder is next divided into three 120° divisions at A f , M f , and Q', as specified. A r M r is divided into six equal parts by the points H f , V . . . which are projected over to meet the vertical construction lines through H, I . . .at H 2 , li. . . . The latter points mark a curve on the surface of the cylinder. This curve is a guide for the center of the tool which cuts the groove. The finding of this curve and the construction of the follower pin and rod constitute the remaining essential work on this problem. If it is desired to show the groove CAM PROBLEMS AND EXERCISE PROBLEMS 69 itself, the directions in paragraph 134 will give an approximate method. The follower pin is attached to a follower rod X which is guided by the bearings Y and Z. The assigned pressure angle of 30° is shown in its true size at D J G; J D being parallel to the direction of motion of the follower rod, and J G being a normal to the cutting-tool curve M N J P. . . . In general, the pressure angle will not show in its true size, and if it is then desired to illus- trate it, the cylinder may, in effect, be revolved until the correct point of the cutting-tool curve is projected on the horizontal center line. The exact point E where the cutting-tool curve comes tangent to the bottom line of the cylinder may be found by locating E\ relatively to Ki and Li, the same as E' is located relatively to K r and Z/, and projecting E x down to E. A small clearance is allowed between the end B f of the pin and the inner surface of the groove, which is represented by the dash circle passing through F r . 128. Refinements in cylindrical cam design. It will be noted that the " maximum surface pressure angle" was given in the data for this problem instead of the term " maximum pressure angle" that has been used thus far. The reason for this is that the pressure angle varies along the length of the pin and is always greatest at the outer end, that is, at the point B in Fig. 57. This is not important in most practical cases. Further, the term "pitch cylinder" is not mentioned in the simple form of practical construc- E 'K' M£' L' Fig. 57. — Problem 15, Cylindrical Cam with Follower Sliding in a Straight Line 70 ELEMENTARY CAMS tion here used. Since the pitch cylinder should pass through the point where maximum pressure angle exists, the pitch cylinder in cams of this type would be one having a radius O f B' '. The pitch surface of the cylindrical cam would be a warped surface, known as the right helicoid, and the intersection of this surface with the surface of the cylinder is the curve R A E P R and is the guide curve for the cutting tool in milling out the groove for the pin. The sides of the groove are the working surfaces of the cam; they are indicated in the sectioned part of the front view of Fig. 57. More exact methods for drawing the sides of the groove in a cylindrical cam, together with a more exact method for determining the maximum pressure angle, involve a knowledge of projections and an intricacy in drawing that make such work a proper subject for advanced study, and it will therefore be omitted in this elemen- tary treatment, as it is totally unnecessary in most practical work. 129. Exercise problem 15a. Required a cylindrical cam to operate a sliding follower rod, with a maximum pressure angle of 40°: (a) 6 units to the right in 90° on the parabola base. (b) 6 " " " left " 270° " " 130. Problem 16. Cylindrical cam w t ith swinging follower. Required a cylindrical cam to operate a swinging follower arm: (a) To the left 40° in 120° turn of the cam on the crank curve. (b) " " right 40° " 120° " " " " " " (c) Dwell for 120° turn of the cam. The length of the follower arm is to be 9 units and the approximate maximum pressure angle is to be 30°. 131. The diameter for the cylindrical surface is found in the same manner as the diameter of the pitch circle in radial periphery cams. The data in this problem do not give directly the travel of the follower and so this value must be found first. The chord of a 40° arc hav- ing 9 units radius will be 9 X 2 X sin 20° = 9 X 2 X .342 = 6.2 units. \D L_I ^< | G If a trigonometrical table is not at Fig. 58. — Determining Fol- , , ,, , , . . lower travel in swinging hand the arc may be drawn out as m dSSe F ter C ° MPUTING PlTCH Fi S- 58 where half the S iven an ^ le is laid out at AY J by the simple expedient of subdividing a 30° arc by means of a dividers. The half chord A D is drawn and measured. It is equal to 3.1 units, thus mak- ing the chord of the whole arc of travel equal to 6.2 units. CAM PROBLEMS AND EXERCISE PROBLEMS 71 This value is used in obtaining the diameter of the surface of the cyl- inder as follows: G.2 X 2.72 X 3 X 3.14 16.12 units. The circle A' Q r M f , Fig. 59, is drawn with a radius of 8.06 units. 132. The 120° angles assigned in the data are next laid out but not from the center line R r as in previous problems. In mechan- isms of all kinds where there is a Swinging follower, it is a rule, unless otherwise specified, that the swinging pin should be the same dis- Fig. 59. — Problem 16, Cylindrical Cam with Swinging Follower Arm tance above a center line at the middle of its swing as it is below at the two extremities of its swing. In this case, then, the point G, Fig. 58, will be marked midway between J and D and the distance G J laid off at G J in Fig. 59. Y will be the center of swing of the follower arm and the arc of swing of the follower pin will be A J V. J will be as much above the center line as A and V are below. The practical advantage of this detail in the layout is that it gives a maximum bearing length between the follower pin and the side of the groove. 133. The arc A J V, Fig. 59, is next divided at the points marked H, I . . . according to the crank curve assignment, and vertical construction lines are drawn through these points. The point A is now projected to A' and the radial line, A' 0, is drawn. This becomes the base line from which to lay off the three 72 ELEMENTARY CAMS assigned timing angles of 120°, as shown at A' M', M' Q', and Q f A'. The arc A' M' is next divided into the desired number of equal construction parts, as at i7 3 , 7 3 , J3. . . . When Hz reaches A', the pin A will have swung not only over to H } but it will have moved up the distance A' H' measured on the surface of the cylinder. Therefore, when H z reaches A', it is the line through H$ (i7 3 H$ = A' H') on the groove center line that will be in contact with the pin center line. For this reason H b , instead of i7 3 , is projected over to meet the construction line at H 2 . This latter point is on the guide curve for the cutting tool on the surface of the cylinder. Other points are found in the same way. Time may be saved by marking the points A' H' I' J' on the straight edge of a piece of paper and transferring these marks at one time so as to obtain the points 7 5 , J 5 . . . P 5 . . . . 134. If it is required to show the surface bounding lines of the side of the groove it may be done quickly, although approximately, by laying off on a horizontal line, as at 7 2 , the points 7 4 and 7 6 at distances equal to the radius of the pin. These will represent points on the curve. If it is required to show the bottom lines of the groove it may be done by projecting from 7 7 and finding, for example, the point 7 8 in the same way as 7 4 was found. 135. Exercise problem 16a. Required a cylindrical cam to operate a swinging follower arm: (a) To the right 6 units (measured on chord of follower pin arc) while cam turns 150°. (b) Dwell while cam turns 120°. (c) To the left 6 units while cam turns 90°. The follower arm to be 8 units long and its rate of swinging to be controlled by the crank curve with a maximum approximate pres- sure angle of 40°. 136. Chart method for laying out a cylindrical cam with a swinging follower arm. This method is illustrated in Figs. 60 and 61. The data in this problem will be taken the same as in Problem 16, namely, that a follower arm of 9 units length shall: Swing through an angle of 40° to the left while the cam turns 120°; through the same angle to the right while the cam turns 120°, on the crank curve in both directions; remain stationary while the cam turns 120°. The maximum pressure angle is to be approxi- mately 30°. 137. To find the length of the chart, the chord that measures the arc of swing of the follower pin is first determined to be 6.2 CAM PROBLEMS AND EXERCISE PROBLEMS 73 K:/ M< y* K Qi units as explained in paragraph L31. The length of chart is 6.2 X 2.72 X3 = 50.G units, and this is laid off at J J h Fig. 60. The length of the follower arm is then laid off at J Y, and the follower-pin arc A V drawn. This arc is subdivided at H, I . . . according to the crank curve. The distance Y F 6 is then laid off to represent 120° and its length will be equal to one-third the length of the chart. As many construction points as were used from A to V are then laid off between Y and Y G . With these as centers and YA as a radius draw a series of arcs to which the points H, I . . . are pro- jected, thus giving the base curve through the points H h 1 1. . . . Tangent to the series of arcs on the chart draw straight lines and mark the intercepts H4H2, h h- • . . 138. Upon completing the chart, the surface of the cam is drawn as in Fig. 61, with a diam- eter E' T' = |^j = 16.12. The width C N of the cylinder may be taken equal to the chord A V of the arc of swing of the follower pin, plus twice the diameter of the pin. 139. The simplest general plan for trans- ferring the cam chart to the surface of the cam is to consider the chart lines to be on a strip of paper, and that this paper is simply wound around the cylindrical surface of the cam, starting the point G of the chart at G on the center line of the cam. G on the chart is midway between J and D. Then the points H 2 , li . . . of the base curve in Fig. 60 will fall at H 2 , 1 2, in Fig. 61, giving the surface guide curve for the center of the cutting tool. 140. The detail necessary to actually lo- cate the points H 2 , 1 2 in Fig. 61 is accomplished by projecting J to J' and laying off the as- 74 ELEMENTARY CAMS signed 120° divisions, and also the subdivisions from this latter point. The 120° divisions are shown at M', Q', J'; the equal subdivisions at H 3 1 3 . . . . Prom these latter points, lines are pro- jected to the front view and the lengths H± H 2 , 1 4 1 2 are transferred Fig. 61.— Cylindrical Cam with Swinging Follower Drawn from Chart from Fig. 60. To find the point of tangency at E, make i£ 4 E\ of Fig. 60 equal to K 3 E' of Fig. 61, then draw E l E in Fig. 60 and lay off this distance from the center line G Y in Fig. 61, thus giving the point E. To find the point of tangency at M, lay off at M ' M s a distance equal to the chart distance from M 2 to M in Fig. 60 and project M z of Fig. 61 to M. SECTION IV.— TIMING AND INTERFERENCE OF CAMS a 141. In machines where two or more cams are emploj'ed it is generally necessary to lay down a preliminary diagram showing the relative times of starting and stopping of the several cams, in order to be assured that the various operations will take place in proper sequence and at proper intervals. The same preliminary diagram is also used to avoid interference and to make clearance allowances for follower rods whose paths cross each other. 142. Problem 17. Cam timing and interference. Required two cams that will operate the follower rods A and E, Fig. 62, lying in the same plane, so that: (a) Rod A shall move 16 units to D, dwell for 30°, return 8 units to B and again dwell 30°, all to take place in 180° turn of the cam. The cam to produce the same motions in the second 180° but in reverse order. (b) Rod E shall cross path of rod A and move 4 units be- yond it and back again during the time that rod A is moving from D to B to D. All motions to be on the crank curve with maximum pres- sure angles of 40°. 143. Before taking up the solution of this problem in de- tail it should be noted: 1st, that fig. G2.— problem 17, preliminary layout any convenient type of cam may £££ FOR PaOBLEM IN CAM lNTER " be used in problems of this kind ; 2d, that usually only general motions of followers or objects are given in the preliminary data, as above, and that the cam designer must supply data and restate the problem in terms of angles for each of the movements after studying the preliminary data with the aid of a timing diagram. 144. The first step leading to a restatement of the problem is to determine the number of degrees in which rod A may move the 75 76 ELEMENTARY CAMS 16 units, and also the number of degrees in which it may move the 8 units in order that the pressure angle will be 40° in both cases. Since there are two 30° dwells in the first 180° there will be 120° left for the two motions of which the first -j n motion will require ^r °f 120° or 80°, and the second, 40°. The length of chart for cam A may now be computed as 16 X 1.87 X t^t = 134.6 and laid off as at A A h Fig. 63. The height of the chart AD is 16 units. The chart is next divided into degrees of any con- venient unit, 0, 10°, 20° . . . being used in this case. For the present the base line may be made up of a series of straight lines as at A D h D, D 2 , D 2 Bl . . . 145. The amount of clearance between the moving arms must now be decided upon. Let it be the designer's judgment that the end of the follower rod E should lie at rest 1 unit to the left of rod A as shown in Fig. 64, and that rod E should not begin to move until the rod A is one unit out of the way. Then A will be at O , Fig. 64, moving down, when E starts, assuming the rod E to be 3 units wide and that it is so placed that its top edge is one unit below D. The point C is then 5 units from the top of the stroke and if this distance is laid off in Fig. 63, as shown, the line C Ci is obtained cutting the crank curve, which should now be drawn at C. C is at the 133° point and this, then, is the time when the follower E should start to move. 146. The total motion for rod E is 4 + 5 + 1 = 10 units, assuming width of rod A to be 5 units. The time during which this motion can take place, outward, is 180° - 133° = 47° as represented at E x E 2 , Fig. 63. If the crank curve E\ F is now drawn it will be intersected by the one-unit clearance line GiGatG which rep- resents, in this case, a rotation of approximately 11° of the cam that drives rod E. The total clearance for the two rods which cross each Fig. 63.— Problem 17, Timing Diagram for Avoiding Interfer- ence of Cams TIMING AND INTERFERENCE OF CAMS 77 other's paths is now found to be 3° for cam follower A and 11° for cam follower E, or 14° of the machine cycle. These clearances are in- dicated in Fig. 63. If it is the judgmenl of the designer thai errors in cutting keyways and in assembling, and thai the wear of the parts will fall within these limits, the cams may now be drawn. 147. The cam chart for cam E was made the same length as rarv i;j„L_. J. i u T 1 Fig. 64. — Problem 17, Design for Definite Timing and Non-Interference of Cams Operating in Same Plane the chart for cam A in order to make clearance allowance. The true length of this chart, for a 40° pressure angle, would be: 360 10 X 1.87 X 47 143.2 units, instead of 134.6 as now drawn. If an exact clearance allowance in degrees were required, it would be necessary to redraw the crank 47 curve Ei F, making the distance EiE 2 equal to ^x of 143.2. It is now 47 o«7j of 134.6. With a new and exact drawing the crank curve E\ G 78 ELEMENTARY CAMS would not rise quite so rapidly and the intercept at G would show a small fraction over the 11° taken above. In some problems where the lengths of the true charts differ considerably it may be necessary to redraw this part of the base curve to be sufficiently accurate in obtaining the clearance in degrees. 148. The radius of the pitch circle for the cam operating rod A will be „ q = 21.4 units as drawn at H I, Fig. 64. The pitch surface of the cam and the working surface are drawn in the same way as the ordinary radial cams in previous problems. The length of the rod A± A may be assumed. The radius for the pitch circle for the cam operating rod E will be 143.2 n ' = 22.8 and this is laid off at M S. The location of M and the 6.28 length of the rod N E will either enter into the layout of the frame- work of the machine in a practical problem, or will be determined by the framework if pre- «- ? viously laid out. In the D ci present case it will only !_ I |__ be necessary, in deter- - 4 >] ! \t 4 — J 1 1 4 — >| I mining the length of rod — I — t- N E and the position i ii i B C of M, to make certain i Ii i i j i 1. that the shafts M and H Fig. 65.-Probi.em 17a, Diagram Showing Appli- are Sufficiently far apart cation of data to keep the cams from striking when turning. 149. Location of keyw t ays. It is important to locate the keyway exactly by giving its position in degrees so as not to destroy the clearance values already made. Since the working surfaces of cams frequently approach close to the hub or shaft it is a good plan to place the keyway at the center of the longest lobe of the cam, as illustrated in both cams. 150. Exercise problem 17a. Assume a stack of blocks at A, Fig. 65. Required that the bottom block shall be delivered with one stroke at C, the next block at D, being moved first to B and then to Z>, the next block at C, the next at D, etc. Let the sizes of the blocks and the distances they must be moved be as shown in Fig. 65. Lay out cam mechanism to secure this result, keeping the maximum pressure angle at 30°. SECTION V.— CAMS FOR REPRODUCING GIVEN CURVES OR FIGURES 151. Problem 18. Cam mechanism for drawing an ellipse. Required a cam mechanism that will reproduce the ellipse A C B D in Fig. 66, the marking point to move slowly at the extremities A and B of the major axis and rapidly at C and Z>, the rate of increase and decrease of velocity being uniform. 152. Divide A C into three parts which are to each other as 1, 3, and 5; C B into three parts which are as 5, 3, and 1 ... in order that the marking point shall move through increasing spaces in equal times in moving from A to C. . . . For greater accuracy A C would be divided into a greater number of parts. 153. In devising the mechanism assume that the marking point shall be at the end of a rod which shall be controlled by two com- ponent motions that are horizontal and vertical, or nearly so. This suggests the rod A E F, with marking point at A, with horizontal motion supplied from a bent rocker attached at F and with vertical motion supplied from a reversing straight arm rocker L K J, attached through a link E J at the point E. The lengths of the links and of the arms of the rockers, and the positions of the fixed centers of the rockers will have to be assumed, the lengths of the arms and links being such that none of them will have to swing through more than 60°. With more than 60° swing the angle between an arm and a link is liable to become too acute for smooth running. Where rocker arms are connected to links the ends of the rocker should, in general, swing equal distances above and below the center line of the link's motion, as for example, the points F and 6 on the arc of swing of F should be as much above the line A M as the point 3 is below. Also the arc 3 J 9 should swing equally on each side of J T in order to secure best average pressure angles for the mechanism. 154. Let each of the rocker arms be assumed to be controlled by single-acting radial cams. The center of roller H will be required to swing on an arc 6 H which, continued, passes through M. This gives small pressure angles while A is traveling to B, especially when A is at C and is moving fastest. It gives large pressure angle, how- ever, while A is traveling from B to D to A. If A is assumed to do heavy work along AC B and to run light along B D A this is the 79 80 ELEMENTARY CAMS better arrangement. If A did the same work on both strokes it would be better to place the rocker arm G H so that H and 6 rested on a radial line. The center of roller L will be assumed to travel on an arc whose extremities are on a radial line, or nearly so. With A F as a radius and A, 1, 2 . . . as centers, strike short arcs intersecting F 6 at F, 1 , 2 . . . numbering the arcs as soon as drawn to avoid confusion later on. Lay off points on H 6 corre- sponding to those on F 6. 155. Inasmuch as the point H does not move in accordance with the law of any of the base curves no precise computation can be made for the size of the pitch circle for any given pressure angle and it may be omitted. Instead, a minimum radius M H of the pitch surface may be assumed. If it is desired to control the pres- sure angles it may be done by first constructing the pitch surface, H V W, and then measuring the angles at the construction points. Some of these are shown in Fig. 66, at H, 3, 6, and 8, and are 20°, — 12°, 48°, and 57°, respectively. If these angles should prove un- satisfactory a larger pitch circle, or a differently proportioned rocker, may be used. Or, an approximate computation for radius of pitch circle by the method which is explained to advantage in connection with the next problem, paragraphs 164 and 165, may be used. 156. To construct the second cam, take the distance A E as a radius and A, 1, 2 . . . as centers and mark the points E, 1,2. . . . Again, with the latter points as centers and E J as a radius, mark the points J, 1, 2 . . . and transfer these to L, 1, 2. . . . With the latter points marked, the pitch surface of the second cam, P Q R, is constructed in the same way as was the first cam. The angle between the keyways, marked 393/2° in Fig. 66, must be carefully measured and shown on the drawing. 157. Exercise problem 18a. Required a cam mechanism that will draw the numeral 8, the marking point moving with uniform velocity. 158. Problem 19. Cams for reproduction of handwriting. Required a cam mechanism to reproduce the script letters S t e. 159. The first step in the solution of this problem is to write the letters carefully, for if the machine is properly designed it will re- produce the copy exactly as written. The copy is written at A in Fig. 67. 160. The next step is to decide on the kind of mechanism and the type of cams to be used, for the problem may be solved by a number of different combinations. The mechanism for this problem -H^rL^ Fig. GG. — Problem 18, Cam for Drawing an Ellipse Fig. 67. — Problem 19, Cams for Reproducing Script Letters, etc. CAMS FOR REPRODUCING GIVEN CURVES OR FIGURES 83 :t> r will consist of two radial single-acting cams mounted on one shaft, and a swinging rocker arm mounted on a pivot which is moved forth and back on a radial line as shown in Fig. 67. This mechanical combination is selected for this problem because it involves methods of construc- tion not used in any of the preceding problems. 161. The actual work of construction is started by marking off a series of dots along the lines of S the entire copy, as shown at A, and marked from a zero to 64. Inasmuch as there is some latitude fi in the spacing, and consequently in the number s of these dots, as will be explained presently, it is o advisable to use a total number of dots whose o least factors are 2 and 2, 2 and 3, or '2 and 5. | This is not essential but it will facilitate the work > later on. 162. The matter of placing the dots is per- il haps the most important item of the entire prob- 2 lem, for on this depends the size of the roller and smooth action. In fact, with some methods of spac- ing, no roller can be used at all and a sharp V- edge sliding follower will have to be used if true reproduction is desired. The basic considerations in selecting the points are: First, that a point should be located at the extreme right and extreme left of each right and left throw, as at - 7, 7 - 16, 16 -20 . . . in Fig. 67, A, and at the top and bottom of each swing, as at - 8, 8 - 13, 18 - 18 . . .; and, Secondly, that the marking point should start slowly and come to rest gradually on each stroke, considering both of the component directions of its motion at the same time. On account of this it is impossible to secure ideal conditions at all times and com- promises must frequently be made. For example, the component motions of the marking point D are: First, a horizontal one due to Cam No. 1 ; and secondly, a vertical curvilinear one due to Cam No. 2 and the rocker arm H G D. The intermediate points 0-7 on the upper swing of the letter S are so selected as to give increasing and 84 ELEMENTARY CAMS decreasing spaces in the horizontal projections on D E, and the same points, together with point 8, are selected at the same time so as to give increasing and decreasing spaces when projected onto the arc D F. Each space between a pair of adjacent numbers represents the same time unit. On this basis the entire spacing of the copy is done. 163. With each of the points in the group at A, Fig. 67, as centers, and with a radius, D G, mark very carefully the corresponding points on G L in group B. To avoid confusion it is essential here to adopt some method of identifying points so marked for later reference. A satisfactory method is shown at B, all the motions to the right being indicated below, and the motions to the left, above G L. 164. The sizes of the cams are to be next computed. To do this select the largest horizontal space in section A. This is found between 56 and 57 and is equal to .46 of the unit of length that happened to be selected in this problem. Assuming that the marking point moves with uniform velocity over this distance, and that a pressure angle of 40° is suitable in this instance where no heavy work is done, the factor of 1.19 is taken from the table in paragraph 30. Since there are 64 time units the length of circumference of pitch circle for Cam No. 1 will be .46 X 1.19 X 64 = 35.03, and the radius 5.58. 165. Before calculating the size of Cam No. 2 the length of the rocker arm G H must be decided upon and this will be taken in this problem at 5 units, the same as the arm G D. Then the total swing of the follower point H will be H K, equal to D F, and the greatest swing in any one direction in any one time unit will be during the periods 10-11 and 61-62, shown at A, Fig. 67, both equal to .48 units. Making the same computation as for Cam No. 1, .48 X 1.19 X 64 _ 3.14 X 2 - 5 '* Z equals the pitch radius of Cam No. 2. 166. The position of the cam shaft relatively to the pivot arm G depends on what is desired for the position of the arc H K with reference to the cam center. If it is desired that the points H and K shall be on a radial line from the center of the cam, which gives best practical average results for both in and out strokes, proceed as follows : Draw chord D F at A in Fig. 67 ; bisect it at J and measure distance G J which is 4.93 units. Then the distance CAMS FOR REPRODUCING GIVEN CURVES OR FIGURES 85 GO will be the hypothenuse of a right angle triangle of which one side is 4.93 and the other 5.82. This may be separately drawn and the length of the hypothenuse found graphically or it may be figured as follows: GO = V 5.82 2 + 4.93 2 = 7.03. 167. The pitch circles for both cams may be taken in problems of this kind to pass through the midpoint of the total travel. Then M is the radius of the pitch circle of Cam No. 1 and N P the total range of travel of the roller center; and Ji is the radius of the pitch circle of Cam No. 2 and H K the total range of travel of the roller center relatively to G. 168. To find points on the pitch surface of the cams proceed in the usual way for Cam No. 1, by dividing the circle whose radius is N into as many equal parts as there were dots on construction points at A. Draw radial lines, and on these lay off the distances secured from B in Fig. 67 ; for example, the distance 3 N\ is laid out equal to G 3. The point Ni, and other points secured in similar manner, will lie on the pitch surface of Cam No. 1. 169. The construction of the pitch surface for Cam No. 2 is different from that of Cam No. 1, and is different also from anything done in the preceding problems. In this case the resultant motion of the arm G H is made up of rectilinear translation and rotation and both components must be considered in laying out the pitch surface, for example, as follows : With G H as a radius and point 4 of B as a center draw an arc intersecting the horizontal line through H at 4- Then when G is moved to 4 by Cam No. 1, H would be at 4 if the rectilinear component motion due to cam No. 1 were the only one acting. During the period represented by G 4, however, Cam No. 2 must move the rocker arm through an arc Q 4, shown at A, and this arc must now be laid off at 4 R> The point R is then revolved to its proper position at T as follows: Divide the circle OG into sixty-four equal parts. This is readily done in this problem because G is taken on the same radial line with N and the radial divisions already made on the circle having N f or a radius need only be extended. Lay off the distance G 4 at 4 S. With S as a center and G H as a radius draw the arc 4 T. Then T will be a point on the pitch surface of Cam No. 2. 170. Having determined the pitch surfaces of the two cams the largest possible roller for each is found by searching for the shortest radius of curvature on the working side of each pitch surface. For 86 ELEMENTARY CAMS Cam No. 1 the size of the largest roller that can be used is that of the circle whose center is at U; and for Cam No. 2 it is that of the circle whose center is at V. In order to avoid sharp edges on the cams, rollers slightly smaller than these circles will be used. 171. For assembling the cams the angles between them and the angles for the keyways should be carefully measured and placed on the drawing as shown in Fig. 67. A front view showing the elevations of the cams, lever arm, slide, and plate is given in Fig. 68. 172. Method of subdividing circles into any desired num- ber of equal parts. The matter of subdividing the circle having Fig. 69. — Method of Subdividing Circles into Any Desired Number of Equal Arcs radius N, Fig. 67, into sixty-four equal parts was a simple matter of subdivisions. If it is required to divide the circle into eighty-seven equal parts the work is just as simple if a proper start is made as follows : Let it be required that the circle B D, Fig. 69, be divided into eighty-seven equal parts. Find the number next lower than eighty-seven whose least factors are 2X2, 2X3, or 2X5. Such a number is 80. Assume that the circle is 6 inches in diameter; then the circumference is 18.84 inches and 7 / 8 7 of this is 1.516 inches, which is laid off to scale on the tangent at B F. With a pair of small dividers, set to any convenient small measuring unit, step off divisions CAMS FOR REPRODUCING GIVEN CURVES OR FIGURES 87 from F to the next step beyond B. Assume that there are 11 steps from F to G, then go forward 11 steps on the arc to K. Divide the large part of the circle K D B into eighty parts by the process of sub- division with the dividers as indicated by the divided angles 80, 40, 20, 4, 2, and 1, in Fig. 09. Then B II is % of K D B, or % 7 of the entire circle, and the length B II will go exactly seven times into the arc B K. In this work nothing is said of the use of a protractor for laying off a large number of small subdivisions on a circle, al- though it may be used. The process of subdivision, however, always using the small dividers, gives automatically remarkably accurate results. SECTION VI.— ADVANCED GROUP OF BASE CURVES FOR CAMS 173. The previous sections of the book have dealt with the simpler base curves which are in common use, and with their ele- mentary application to various types of cams. In the present sec- tion the simpler forms of base curves are further considered, other forms are treated, and new ones are proposed; all are brought together for comparisons. 174. Complete list of base curves. The base curves which have been used in the previous sections are: Straight Line, Figs. 22 and 78. Straight-Line Combination, Figs. 23 and 82. Crank Curve, Figs. 24 and 86. Parabola, Figs. 25 and 90. Elliptical Curve, Figs. 26 and 102. Other base curves which will be considered in following para- graphs are: All-Logarithmic Curve, Fig. 70. Logarithmic-Combination Curve, Fig. 74. Tangential Curve, Case 1, Fig. 94. Circular Curve, Case 1, Fig. 98. Cube Curve, Case 1, Fig. 106. Circular Curve, Case 2, Fig. 110. Cube Curve, Case 2, Fig. 114. Tangential Curve, Case 2, Fig. 118. 175. Comparison of base curves, their applications, and their characteristic motions. Figs. 70 to 121 illustrate: (1) The forms of each of the base curves, Column 1. (2) The form and true relative size of cam, all having the same data, Column 2. (3) The velocity diagram for each cam, Column 3. (4) The acceleration diagram for each cam, Column, 4. 176. The data for all of the cams and diagrams illustrated in Figs. 70 to 121 are as follows: (a) The follower to rise 1 unit in 60° turn of the cam, (b) " " " fall 1 unit in 60° " " " " (c) The follower to remain at rest for 240° turn of the cam, (d) " maximum pressure angle to be 30°. ADVANCED GROUP OF BASE CURVES FOR CAMS 89 177. All of the cam charts illustrated in Column 1, Figs. 70 to 121, include only the first item in the above data and they show, therefore, only one-sixth of their full length. In Column 2 the entire cam is shown in each case, and it is drawn to one-third of the scale used for "the chart in Column 1. 178. Velocity and acceleration diagrams showing char- acteristic ACTION OF CAMS HAVING DIFFERENT FORMS OF BASE curves. All of the diagrams in Column 3, Figs. 70 to 121, show the velocity given to the follower by the cam at every instant during the follower stroke. In each case the length of the diagram A C represents the time required by the cam to turn through 60°, the cam shaft being assumed to be turning with uniform angular velocity. The numbered scale on each diagram shows the relative velocity given by each cam at any phase of the stroke. 179. All of the diagrams given in Column 4 show the acceler- ation given to the follower by the different cams. These diagrams have a special interest when it is remembered that force = mass X acceleration, and if the mass is the same in all cases the ordinates of the diagrams represent the forces necessary to move the follower at any instant. A diagram with a distinctively long ordinate indi- cates that the cam will "run hard " at the phase where the long ordinate occurs. The scale numbers shown on the diagrams are based on the uniform acceleration given by the parabola cam as shown in Fig. 93. 180. The characteristic actions of different cams built from the various base curves will be considered, in order, in the fol- lowing paragraphs. 181. The all-logarithmic curve, Fig. 70, gives the smallest possible cam for a given pressure angle. It differs from all other cam curves in that it gives the maximum pressure angle all the time that the follower is moving, whereas the others give a maximum pressure angle for an instant only. One of the disadvantages of the all- logarithmic cam is that it causes the follower to attain nearly its full velocity instantaneously, and causes it to come to rest in a similar manner, thus giving a shock at the beginning and end of the stroke. This gives excessively large acceleration and retardation at the ends of the stroke and causes the cam to " pound " or " run hard " at these phases of its action. Another disadvantage is that a roller cannot be used with it because the pitch surface has a sharp edge, or angle, on the working side as shown at C, Fig. 71. The rea- 90 CAMS COMPARISONS OF CAMS FOR DIFFERENT < Column 1 Cam. Charts and Base Curves tor One-sixth of Cam Column 2 Relative Sizes of Cam/ B V A C 30 X Pitch Line 'X ^rifT H I\ J\^ E X r] * / — >- /Eig. 9i. Tangential Base Curve, Case 1 ~FlQ. Qnfit^W^ ADVANCED GROUP OF BASE CURVES FOR CAMS 91 BASE CURVES, ALL HAVING SAME DATA Column 3 Velocity Diagrams Column 4 Acceleration Diagrams L Fig- 73? Y eM 2 -1 D ■ — Fta.Tf. ll r-l U Z rr E"-2~ ■ ^ P c Fig F 81. 8 J 92 CAMS COMPARISONS OF CAMS FOR DIFFERENT Column 1 Cam Charts and Base curves for One-sixth of Cam r~ r x Column 2 Relative Sizes of Cams B V ^uvn B c Pitch Line «--&»*£- r Fy H \I K\Js* A\ R, A _J*^Tk -^/"jv - 3 V^H Fig. 98. — Circular Curve, Case 1 Fig. 99, FG:FC::7:i Fig. 102. — Elliptical Curve Fig. 103. -^j- D Pitch Line s H. 27V 84; — Base Curve^fe^ E r 4 1 A ,. w „ co I ,v .r v « — 4.20- > Fig. 106. — Cube Curve, Case 1 Fig. 114.- Cube Curve, Case 2 VLH I. p Jf - IT c Pitch Line M^'fT Ft // 1/ 0.0/ """ > Fig* 118.— Tangential Base Curve, Case 2 ADVANCED GROUP OF BASE CURVES FOR CAMS 93 BASE CURVES, ALL HAVING SAME DATA— Continued Column 3 Column 4 Velocity Diagrams Acceleration Diagrams Fig. 108 94 CAMS son why a roller cannot be used under these conditions is explained in paragraph 59, page 37. The construction of the all-logarithmic cam is explained in the following paragraphs. 182. Problem 20. Required an all-logarithmic cam causing: (a) The follower to rise 1 unit in 60° turn of the cam, (b) " " " fall 1 " " 60° " " " " (c) ' ' " " remain stationary for 240° turn of the cam, (d) A uniform pressure angle of 30°. 183. A brief general analysis for the method of procedure in solving an all-logarithmic cam problem is : (1) To construct a logarithmic spiral having a constant normal angle of 30°. The spiral is shown at B H, Fig. 122, and the constant B ] dr i\ Y&fr~ y^ \ Pitch Line B v Fig. 70. — (Enlarged) All-Logarithmic Base Curve Fig. 71— (Enlarged) All- Logarith- mic Cam angle is noted at JDK, where D K is a radial line and D J a line normal to the curve. (2) To lay out the assigned working angle during which the follower motion takes place, on a piece of tracing cloth or tracing paper, as at b in Fig. 123. (3) To mark on each leg of the angle a scale to measure the fol- lower's motion, as at 0' M and 0' N in Fig. 123. (4) To lay the tracing cloth represented by Fig. 123 over the loga- rithmic spiral with the apex O r of the angle always at the pole O of the spiral, and to rotate the tracing cloth until the two legs of the angle cut the spiral at such points that the difference in length of the two legs is equal to the assigned follower motion. This is illus- trated in Fig. 122 where the shaded area represents the tracing-cloth with the assigned angle of 60° shown at b, while C minus O A equals the assigned follower motion of 1 unit. ADVANCED GROUP OF BASE CURVES FOR CAMS 95 (5) To mark the included part of the logarithmic spiral A C and use it as the surface of the cam as shown at A C in Fig. 124. 184. The detail construction necessary to lay out the all- logarithmic cam for Problem 20 is as follows : Construct a logarithmic spiral with a constant normal angle of 30°. This may be done mathematically by laying off computed values which method will be taken up first, or it may be done graphically as will be explained later. A 1 Fig. 122. — Logarithmic Curve Giving Constant Pressure Angle of 30 Degrees Fig. 123. — Assigned Working Angle, to be Drawn on Tracing Cloth In the mathematical method the first step is to solve the following equation : r 10 °- 4343 rio 6tan ( 90 °- a ) where a is the assigned press are angle and b is a unit angle taken aD any value which may be conveniently used later in starting the drawing of the spiral. The values of r and r' are shown at D and H respectively in Fig. 122. The angles a and b are also shown. A convenient angle to assume for b, in general, is G0° and it is so taken in this problem. Then — equals the number whose logarithm is 96 CAMS 0.4343 X ^~ X 60° X tan (90° - 30°). Solving, the value of the logarithm is 0.2623 and the number corresponding to this is 1.83. Therefore, / -=1.83 r and O H , Fig. 122, is made 1.83 times D, the included angle being 60° in accordance with the above assumption for b. The length O D may be taken any length in starting the construction of the spiral. The two points of the spiral may now be laid down as at D and H with O as the pole. 185. Intermediate points on the logarithmic spiral as at G may be found by bisecting the angle D O H and making G a mean proportional between O D and O H. Then OD : OG : : OG : OH If O D is taken as 3 units, then G = V 3 X (1.83 X 3) = 4.06. To find points on the spiral at closer intervals bisect angle DOG and find the mean proportion 7 which is equal to V 3 X 4.06 = 3.52. To find other points outside of a given angle, such as at 5, lay off the angle D 5 equal to angle D 7 and make 5a fourth propor- tional to 7 and D as follows : 05 : OD : : OD : 07 Then 05 = ~ = 2.58. o.oZ If points are desired still closer together, or if it is desired to extend the spiral in either direction, it may be done by the above- described processes, or, it may be done graphically as described in paragraph 187. 186. The next detail step in the solution of Problem 20 is to draw an angle M O ' N, Fig. 123, on tracing cloth, equal to the assigned angle of 60° as given at (a) in the data, and lay off a scale on each leg of the angle as shown. Then lay Fig. 123 over Fig. 122, ' always at 0, and rotate the tracing cloth until the spiral B H inter- cepts the lines ' M and ' N at such points that ' C ' minus O' A' equals the assigned follower motion which is 1 unit as stated at (a) in Problem 20. This occurs when Fig. 123 is at the position ADVANCED GROUP OF BASE CURVES FOR CAMS 97 shown by the section lines in Fig. 122 where A equals 1.18 and C equals 2.18. The intercepted part A C of the logarithmic spiral // Fig. 122. — (Duplicate) Logarithmic Curve Giving Constant Pressure Angle of 30 Degrees Fig. 123. — (Duplicate) Assigned Working Angle, to be Drawn on Tracing Cloth becomes a portion of the cam pitch surface as shown at A C in Fig. 124 and its distance from the center of rotation of the cam is the same as the distance from the spiral arc to the pole of the spiral. Fig. 124. — Problem 20. All-Logarithmic Cam for Assigned Data Other portions of the cam surface are found in a similar manner. As shown at A, E, and C, in Fig. 124, the pressure angle is 30° at all points. 98 CAMS 187. Intermediate points on the logarithmic spiral may be found graphically, instead of by computation as given in paragraph 185, as follows: From any point of a straight line, Fig. 125, lay off D and H in opposite directions, D and H being the values ob- tained by computation in paragraph 184 and shown in Fig. 122. At 0, Fig. 125, erect a perpendicular line. Find the midpoint 0\ on the line D H, and with this as a center for the compass draw the semicircle D G H. Then G will be a mean proportional between D and H and may be laid out as the ordinate of the loga- rithmic spiral, as at G, Fig. 122, where G bisects the angle DO H. D s o o, o, H Fig. 125. — Graphical Method for Finding Intermediate Points ox Logarithmic Curve To find a fourth proportional graphically proceed as follows: Lay off the two known values, D and H, which are shown in Fig. 122, at right angles to each other as shown at D' and H in Fig. 125. Find the point O2 on OH that is equidistant from D f and H, and with this as a center draw the semicircle H D' 3, giving the length 3 as the fourth proportional. This latter distance is laid off at 3 in Fig. 122 where the angle D 3 is equal to angle DOH. 188. A GRAPHICAL METHOD FOR CONSTRUCTING A LOGARITHMIC SPIRAL WHICH HAS A GIVEN CONSTANT NORMAL ANCLE is illustrated in Fig. 126. This method, referred to in paragraph 184, is based on the following theoretical property of the logarithmic spiral, namely, that all pairs of radiants having a common difference embrace equal lengths of arcs en the spiral. ADVANCED GROUP OF BASE CURVES FOR CAMS 99 189. The principle stated in the previous paragraph may be graphically applied only approximately, bul with all accessary pre- cision, by first drawing the lines .1/ /' and P N, Fig. L26, making the desired angle with each other. This angle will be 30° if a spiral having a constant normal angle of 30° is required, 40° if a constant pressure angle of 40° is required, etc. From a point 0, where the vertical intercept D is equal to about the estimated short radius of the cam, draw a series of equidistant vertical lines as at B, C, E, etc. With B F as a radius and as a center draw the short arc 1 ; with D F as a radius and D as a center draw arc 2. The intersection of arcs 1 and 2 will give the point H on the spiral. Again, with C G as a radius and Fig. 126. — Graphical Method ion Constructing a Logarithmic Curve Having a Given Constant Normal Angle as a center draw arc 3; and with F G (equal D F) as a radius and H as a center draw arc ■ 4. The intersection of arcs 3 and 4 will give a second point L on the logarithmic spiral. It will now be noted that the two pairs of radiants H 0- DO and LO-H have a common difference, and that the logarithmic arcs D H and H L are equal (approximately), which accords with the general principle laid down in the preceding paragraph. 190. To be exact, in the matter of the graphical construction of the logarithmic spiral, it must be noted that it is the chords from D to H and from H to L that are equal according to this method of con- struction and not the arcs as they should be theoretically; but where the vertical construction lines are taken close together and where the distance D F is, therefore, small, the error in the curve is negligible. 100 CAMS In the present case the ultimate distance R when drawn with aver- age care to a scale several times that shown in Fig. 126, varied from the computed value by less than .01 inch. The part of the curve from D to Q will depart from theoretical values faster than the part from D to R, due to the sharper curvature of D Q, but the effect of this may be overcome, if desired, by making the vertical con- struction lines to the right of D closer than those to the left oiOD. 191. The all-logarithmic cam may be constructed by a purely graphical method, and without any mathematical com- putation whatever. In Problem 20, for example, it would only be necessary to follow the directions in paragraphs 188 and 189, making the angle a of Fig. 126 equal to 30° which is the assigned pressure angle in the problem. This would give the proper logarithmic curve identical with the one in Fig. 122. From this point on, the direc- tions given in paragraph 186 apply. If a pressure angle of any other size were desired, say 45°, the angle M P N Fig. 126, would be made 45°. 192. Exercise problem 20a. Required an all-logarithmic cam which will cause a follower to : (a) Rise two units in 45° turn of the cam. (b) Remain stationary for 135° " " " " (c) Fall two units in 45° « << << « (d) Remain stationary for 135° " " " " (e) The constant pressure angle to be 30°. 193. A logarithmic-combination cam may be used to overcome the disadvantages (paragraph 181) of the all-logarithmic cam and at the same time to sacrifice very little in the matter of increased size. This is accomplished by substituting rounded surfaces for the angular surfaces formed by the all-logarith- mic curve. When the rounded sur- face thus substituted is derived from parabolic base arcs the best results are obtained. A cam in which this has been done is shown in Fig. 75, where the curves A Y and Z C are arcs of a parabola base and the center portion Y Z is an Fig 75. — (Enlarged) Logarithmic- Combination Cam ADVANCED GROUP OF BASE CURVES FOR CAMS 101 arc of a logarithmic curve. To illustrate an actual case, a prob- lem having the same general data as Problem 20 will be discussed in the following paragraphs. 194. Problem 21. Required a logarithmic-combination cam causing the follower to : (a) Rise 1 unit in 60° turn of the cam. (b) Fall 1 " " " " " " " (c) Remain stationary for 240° turn of the cam. (d) The maximum pressure angle to be 30°, and the easing-off base curves to be parabolic arcs. 195. A brief general analysis of the method of procedure in solving problems of this kind is: (1) To draw a general logarithmic curve on rectangular coordi- nates, the longest and shortest ordinates of which will correspond to the estimated longest and shortest radii of the cam, or the longest and shortest radii of a series of cams if a series should happen to be under design. (2) To compute the length of rectangular cam chart, as directed in paragraph 198, and to draw the rectangle on tracing cloth or tracing paper. (3) To construct parabolic arcs within the rectangular cam chart as directed in paragraph 199. (4) To place the cam chart as now drawn on tne tracing cloth, over the logarithmic curve, so that the logarithmic curve will be tangent to the two parabolic arcs while the bottom line of the chart is parallel to the abscissa of the logarithmic curve. The distance between the bottom of the chart and the abscissa will be the shortest radius of the cam. 196. The first step in the detail of the solution of Problem 21 is to construct a logarithmic curve on rectangular coordinates as in Fig. 127. This curve is a perfectly general one and if it is drawn with a wide enough range of ordinates will do for all possible log- arithmic-combination cams, independently of all specific data. To construct the logarithmic curve draw a horizontal abscissa line 0', Fig. 127, and erect a series of ordinates one unit apart as on both sides of r, making their length a geometrical progression. To do this, make the first ordinate drawn, say L, equal to 1 unit and all succeeding ordinates such as ri, r-2 longer than the preceding ordinate by using any common multiplier throughout; also, all preceding 102 CAMS ordinates such as r', r", if tney are necessary, shorter by the inverse of the same ratio. For example, if L equals 1 and if the common multiplier is taken as 1.25 (it may be any convenient number), then n = 1X1.25 = 1.25, r 2 = 1.25X1.25 = 1.5625. r 3 = 1.5C25 X 1.25 = 1.953, etc.; also r' = 1 X _L = .8, r" = .8 X r ~ = .04 etc. 1.2o 1.2o The lengths should be accurately computed up to the length of the maximum radius of the largest cam that is likely to be used and the curve L G carefully drawn. Gy 1 \^ L^ " r" r' r ] y^ r 2 r 3 r a •r 7 0" C ) C\ B Fig. 127 -General Logakithmic Curve Showing Subtance:;t, V— ?ul in Solving a Wide Range of Logarithmic Cam PROBi.r.:.:^ 197. The length of the sub-tangent, s, in Fig. 127, is next found by the formula, s = .434 log. m , where m is the common multiplier used in laying out the logarithmic curve L G. Since the value of m is 1.25 434 in this problem, s = ^— = 4 .48. This value of the sub-tangent may also be found graphically by drawing tangents to the logarithmic curve, by eye, at several points and taking an average of the sub- tangents thus found. This average value will probably be close enough for most practical work. The tangent line at A is shown at A C, Fig. 127. The length of the sub-tangent, B C, will be the same for each tangent line if it is accurately drawn. ADVANCED GROUP OF BASE CURVES FOR CAMS 103 198. A special form of rectangular diagram, Fig. 128, depending on the data is now constructed, its length being: b ir s tan a ~180 ' where I = length of diagram, b = assigned angle of action, s = length of sub-tangent of the logarithmic curve as found in the preceding paragraph, a = assigned pressure angle. Fig. 128. — Rectangular Chart Used in Design of Logarithmic-Combination Cam Taking the figures from the data for this problem, and the value of s as found and substituting in the above formula, 1 = 60 X 3.14 X 4.48 X .577 180 2.71. The height of the diagram is the continuous motion of the fol- lower in one direction and is 1 unit in this problem as indicated at R C, Fig. 128. Draw the rectangle, as shown at A R C at or near the top of a piece of tracing cloth or tracing paper, leaving a length under it equal at least to what the short radius of the cam is estimated to be. 199. Parabolic easing-off arcs for logarithmic-combination cam. The length of the rectangular diagram is now divided into at least 8 equal parts which are sufficient for practice problems, but in practical applications at least 16 divisions should be taken. A diagram divided into 8 parts is shown in Fig. 128. Construct a para- bola with vertex at A and passing through the midpoint of the dia- gram as at P. This is done as explained in detail in paragraph 35 104 CAMS and, briefly as follows: Divide A B into a series of equal parts, the total number of parts being equal to the square of the number of construction spaces between A and J. In this problem there are four construction spaces and so A B is divided into 16 equal parts and the 1st, 4th and 9th division points are projected horizontally to M, N and which are points on the parabola. Construct the similar parabolic arc C J in the same way. Lay the rectangular diagram constructed as above on tracing cloth over Fig. 127 and manipulate it, always with the line A R, Fig. 128, parallel with the line 0', Fig. 127, until the logarithmic curve L G, showing through the tracing cloth, is tangent to the two- parabolic arcs. This occurs, in this problem, when A R is 1.55 units above 0' , and 1.55, therefore, is the shortest radius of the pitch surface of the cam. For precision work later on, mark the points Y and Z, Fig. 128, where the logarithmic arc comes tangent to the parabolic arcs, B P V S c 16 o a w V - a * K. 9 II 4 1 ' M N i? " / »1 A L HXl J ' K R o Fig. 128. — (Duplicate) Rectangular Chart Used in Design of Loga- rithmic-Combination Cam Fig. 129. — Problem 21. Logarithmic- Combination Cam with Parabolic Arcs at Ends 200. The cam may now be constructed, drawing first the circle, Fig. 129, having a radius Q A of 1.55 units. Lay out the angle A Q C equal to the assigned 60° and divide it into equal spaces by as many radial lines as there are ordinates in Fig. 128. Transfer the ordinates L M, H F, etc., from Fig. 128 to Fig. 129 and draw the pitch surface of the cam through the points A, M, F, etc. The working surface would be a parallel curve distant from the pitch surface by the radius of the follower roller. With this cam there would be uniform accel- eration of the follower from A to Y where the pressure angle reaches ADVANCED GROUP OF BASE CURVES FOR CAMS 105 30°. This angle remains constant until Z comes into action, when the follower is uniformly retarded to zero at C. 201. If it is desired to know the pitch circle of the cam it may be found by noting, in Fig. 128, where the logarithmic arc comes tan- gent to the starting parabolic arc. This is at Y and in this problem it is .06 unit from the bottom of the diagram. This distance is laid off at A S in Fig. 129 to obtain the pitch circle S T. If it is desired further, to obtain the cam chart which is necessary to draw the veloc- ity and acceleration diagrams, it may be found as represented in Fig. 74 where the length D F r is equal to the length of the arc S T in Fig. 129 when both are drawn to the same scale. D F r is the pitch line of the chart, and A R is .06 unit below it, this value being taken from Fig, 128, The length of the ordinates, LM,H F } etc., in Fig. 74 v K -30^ K ^ Z \ F^ A i/ Pitch Line F' { \ D A L H I J R Fig. 74. — (Enlarged) Logarithmic-Combination Base Curve are equal to those in Fig. 128 when both figures are drawn to the same scale. It will be noted that no factor is given in connection with the cam chart for the logarithmic cam as it is for other cams. There is no constant factor; it varies with each problem. 202. The rates of acceleration and retardation that will be given by the cam at the ends of the stroke are arbitrarily determined in Fig. 128 by causing the parabolic arcs to pass through P and J. With the parabolic arcs so taken good average results will be ob- tained, as compared with other small cams. If different accelera- tions and retardations are desired for the follower the point P may be located further up, or further down, and the cam will be either smaller or larger. 203. Exercise problem 21a. Required a logarithmic-com- bination cam with parabolic easing-ofT arcs which will cause a fol- lower : (a) To rise 3 units in 90° turn of the cam. (b) To remain stationary for 180° turn of the cam. 106 CAMS (c) To fall 3 units in 90° turn of the cam. (d) The maximum pressure angle to be 35°. 204. The characteristics of a cam having a straight base line have already been considered in the early part of this book, in paragraph 32. A sharp or V-edge sliding follower is the only kind that can be used with the straight base line for true results; a roller cannot be used for reasons explained in paragraph 59. The form of the pitch surface of the cam that is derived from the straight base line is the Archimedean spiral. The straight base line gives the smallest simple cam for a given maximum pressure angle. Its method Fig. 78. — (Enlarged) Straight Base Line Fig. 79. — (Enlarged) Problem 22. Straight Base Line Cam of construction is illustrated in Figs. 78 and 79 for a problem of the following data: 205. Problem 22. Required a cam with a straight-line base in which the follower : (a) Rises 1 unit in 60° turn of the cam. (b) Falls 1 " " 60° " " " " (c) Remains stationary for 240° turn of the cam. (d) The maximum pressure angle to be 30°. 206. In accordance with formula (1), paragraph 29, the radius 1 X 1 73 of the pitch circle will be 57.3 — — - — = 1.65 which is drawn at D 60 in Fig. 79. The given angle of 60° for the rise is laid off at D C and divided into any convenient number of construction parts, six being shown by the radial extension lines in the Figure. The first line is $ of F C, the second f of F C, etc. Inasmuch as no roller ADVANCED GROUP OF BASE CURVES FOR CAMS 107 can be used with this cam the pitch and working surfaces coincide, and a V-edge follower must be used for true results. The max- imum pressure angle occurs at the start and grows smaller towards the end of the stroke; in this problem it diminishes to 16° as indicated in the Figure. 207. Exercise problem 22a. Required a cam with a straight- line base in which the follower: (a) Rises 3 units in 120° turn of the cam. (b) Falls 3 " " 120° " " " " (c) Remains stationary for 120° turn of the cam. (d) The maximum pressure angle to be 30 a . 208. The straight-line combination base curve, Fig. 82, gives increasing velocity and acceleration at the beginning of the c yS\l .i Pitch Line F \ i -f E ' ^'n X R ~T' Fig. 82 (Enlarged) Straight-Line Combination Base Curve stroke, uniform velocity and zero acceleration during a large middle portion of the stroke, and decreasing velocity and retardation at the end. The length of the period for uniform velocity and the amounts of acceleration and retardation depend entirely on the length of the easing-off radius. This may be taken at w any value. The acceleration diagram in Fig. 85 is based on a radius equal to the follower motion as shown at B A, Fig. 82. The shorter this radius is taken, the nearer the straight-line combination curve ap- proaches the cam having a straight base line, Fig. 78, and the action at the beginning and at the end of the stroke becomes more violent. The longer the easing off radius is taken, the nearer the combination curve approaches the circular base ii Fig. 85. — (Duplicate) Accel- eration Diagram for Straight - Line - Combina- tion Cam 108 CAMS Fig. 89. — (Duplicate) Ac- celeration Diagram for Crank Curve Cam curve of Fig. 98 and the smoother the action will be, but in this case the cam will be relatively large. The combination curve cannot be laid out directly on the cam itself; the chart must be constructed first and the ordinates transferred to the cam drawing. The con- struction of a cam from the combination curve is illustrated in Problem 10, page 55. 209. The crank curve base, Fig. 86, described in paragraph 34, gives increasing variable velocity during the first half of the stroke and decreasing variable velocity during the last half. The acceleration and re- tardation are also variable, being greatest at the ends as may be noted by an in- spection of Fig. 89. The suddenness of the starting action compares with that of a body starting to fall under the action of gravity, approximately as 1.23 is to 1.00. 210. The crank curve is sometimes called the harmonic curve due to the fact that it gives to the follower a motion similar to that described by the foot of a perpendicular let fall on the diameter of a crank circle from a crank pin moving with uniform velocity in that circle; or, in other words, a motion similar to that of a crosshead which is operated from a uniformly rotating crank with a T-headed or " infinite " connecting rod. It will also be observed that the crank curve is a projection of a helix onto a plane surface parallel to the axis of the helix, and is, further, a sine curve, or sinusoid, in which the length or pitch is not necessarily equal to the circumference of the construction circle. 211. Effect of crank curve following its tangent line closely. The crank curve has the marked characteristic, under ordinary conditions, of following its tan- gent so closely, as, for example, on each side of E, Fig. 86, that when the crank curve chart is bent to form the cam, as explained in paragraphs 54 and 55, a j^g maximum pressure angle slightly greater than 30° is produced in the cam. In the case illustrated in Fig. 87 the pressure angle would still be 30° at E but it would be 30° 27' just to the left of E towards A . If it were desired to keep the maximum pressure angle exactly 30° instead of 30° 27', it could be done by moving all the points from A to C, . — (Duplicate) Crank Chart Curve ADVANCED GROUP OF BASE CURVES FOR CAMS 109 Fig. 87, outward radially by the amount d given in the following formula : .5h d = V 1 + jr, cot 2 a Fig. 87. — (Enlarged) Crank Curve Cam where d = distance the points on the pitch surface, as obtained in the ordinary way, would have to be moved out radially to obtain exact size of crank curve cam for a given max- imum pressure angle. h = total rise of follower. b = angle turned by cam during the follower's total rise, in radians. If b is taken in degrees the number 180 must be used in place of tt. a = pressure angle in degrees. The maximum pressure angle of 30° would then occur where the enlarged pitch surface crosses the pitch circle which would be slightly to the left of E, Fig. 87. The cam would be .09 larger in maximum radius, or 3.19 units from to C instead of 3.10 as shown and as used in practice. 212. Another way of obtaining exact results with the crank curve would be to compute the length of the chart from the following formula : Z = .5 6 >w 7T 2 1 + 7^ COt 2 110 CAMS For the case in hand I would equal 2.77, which it will be noted is .05 larger than the practical value used in Fig. 86. With this length of chart the crank curve base line would not reach a 30° angle in the chart but the cam pitch surface would, at a point just inside of the pitch circle. 213. Parabola. This chart curve, Fig. 90, already discussed in paragraphs 35 and 36, gives uniformly increasing velocity to the B G ■4-\. c — 7 Pitch \ yr lime J F y< -jj==a. H I \.J^ E X R " ■ A s^N Q Fig. 90. — (Enlarged) Parabola Base Curve follower up to mid stroke when the velocity is twice that produced by the straight base line as illustrated in Figs. 92 and 80, respectively. The follower has uniformly decreasing velocity during the second half of its motion. Both the acceleration and the retardation are uniform throughout the entire stroke as shown by the horizontal lines BD and FH in Fig. 93. i» Fig. 80.' FiQ. 92. Fig. 80. — (Duplicate) Velocity Diagram for Straight Base Line Cam Fig. 92. — (Duplicate) Velocity Diagram for Parabola Cam Fig. 93. — (Duplicate) Acceleration Diagram for Parabola Cam 214. Perfect cam action. The parabola is the only base curve that gives a theoretically perfect motion so far as inherent smooth- ness of action is concerned. It gives to the follower the same gentle motion on starting as a falling body has when starting from rest, and it brings the follower to rest at the end of its stroke with the same gentle action reversed. For this reason the curve is sometimes called the " Gravity Curve." The curve for the parabola cam is also referred to by some as the curve of squares from the fact that one set of ordinates of the curve vary as the square of the time, as may be noted from the fact that the construction numbers 1, 4, 9, and 16 in Fig. 90 are the squares of 1, 2, 3, and 4, respectively. In Fig. 106 ADVANCED GROUP OF BASE CURVES FOR CAMS 111 which will be described later, a curve is used in which the ordinates of the curve vary as the cube of the time. 215. The parabola base curve will also operate a follower with the least amount of effort of any of the base curves, due to the fact that the acceleration is constant. Since the mass is also constant in cases under comparison, the force required to move the follower will be constant and may be represented by 1.0 as shown in Fig. 93 in comparison with a maximum of 1.8 for the logarithmic-combina- tion cam, Fig. 77; 2.0 for the straight-line combination curve, Fig. 85; 1.2 for the crank curve, Fig. 89; 1.6 for the tangential curve, Case I, Fig. 97; 1.5 for the circular curve, Case I, Fig. 101; and 1.7 for the elliptical curve, Fig. 105. These figures are for symmetrical chart curves. Among the unsymmetrical chart curves shown in Figs. 110, 114, etc., much larger direct forces even may be required to operate the cam as illustrated by the relative maximum values of 2.9 for the circular curve, Case II, Fig. 113; 4.8 for the cube curve, Fig. 117; and 6.4 for the tangential cam, Case II, Fig. 121. 216. Comparison of parabolic and crank base curves. While the parabola base curve combines the two highest theoretical con- siderations, namely smoothest possible motion and least power for operation, it has not become so widely used as the crank curve. This may be due to the experience of builders of cams who have found that the crank curve permits of a smaller cam for a given pressure angle than does the parabola; or for the same size cams the pressure angle is the smaller for the crank curve and, therefore, does not " stick " or "run hard " so much as the parabola cam of equal size. Figures on which the above state- ments are based may be seen in Fig. 87 where it is shown that a maximum radius of 3.1 inches is required for a lift of 1 inch in 60° with a maximum pressure angle of 30° when the crank curve is used; while in Fig. 91 a parabola cam is shown to require a maxi- mum radius of 3.8 inches for the same data. The crank curve has obtained some undue comparative credit over the "parabola" curve FlG - 91.— (Enlarged) Parabola on account of the fact that the " parabola" was constructed with spaces in some other ratio than 1, 3, 5, etc. While, for example, a true parabola may be constructed with spaces of 1, 2, 3, instead of 1, 3, 5, as used in paragraph 35, 112 CAMS the parabolic curve of the cam surface in the former case will not be tangent to the circular part of the cam surface, or, in other words, the base curve E A in Fig. 90 will not be tangent to the horizontal base line of the chart at A but will intersect it at that point. A " parabola" cam, therefore, with ordinates that are in B G ( -* ^— 5 -*- 7 r;tch A^° >^ Line f F ~ H H I \, 7^ E . X R ' ' A /^N Q Fig. 90. — (Duplicate) Parabola Base Curve any other ratio than 1, 4, 9, etc., will naturally show " bright spots " and rapid wear at the beginning and end of the parabolic surface, and this has actually been erroneously charged against the true practical parabola cam. 217. A further comparison of the parabola and crank base curves shows that their velocity and acceleration lines, Figs. 88, 89, 92 and 93, do not differ in their maximum values to such an extent, as to RP Fig. 88. — (Duplicate) Velocity Diagram for Crank Curve Cam Fig. 89. — (Duplicate) Acceleration Diagram for Crank Curve Cam ~t-'z --^ — R Tig. 93. Fig. 92. — (Duplicate) Velocity Diagram for Parabola Cam Fig. 93. — (Duplicate) Acceleration Diagram for Parabola Cam make a noticeable difference in the action in many cam applications, particularly where the smoothest motion is not essential nor where there is a surplus of driving power. Furthermore, the drawing of the crank curve has appeared to some builders as a much easier and better-understood procedure and this has accounted some for the use ADVANCED GROUP OF BASE CURVES FOR CAMS 113 of the crank curve. It may be observed, however, that the parabola is really no more difficult to draw than the crank curve, and when it is fully understood it is quite certain that the parabola cam will come into a more general use in all cases except where space is extremely limited, or where special considerations of the follower motion as to spring or gravity action or as to low striking or seating velocity, etc., become especially desirable. The subjects of spring action and low striking velocities will be treated in paragraph 273, et seq. 218. Tangential base curve. This base curve differs from the others in that it cannot be readily used to construct the cam. The cam itself is drawn first by using straight lines as the side boundaries of the cam lobe, the straight lines being rounded off at the ends by arcs of circles or other smooth curves as shown in Fig. 95. At the inner ends, the straight lines are tangent to a circle which has the center of rotation of the cam as its center. The base curve for this cam is useful only where it is desired to find graphically the velocity and acceleration diagrams, and when it is so used, it must be derived from the cam drawing as explained in paragraph 225. The tangential cam is perhaps the easiest of all cams to draw when one is not par- ticular about the maximum pressure angle, but it is apt to give the highest velocities and the greatest accelerations of all the cams when it is laid out "by eye " by an inexperienced person. To keep the tangential cam under control when being designed, requires either a preliminary graphical construction, or a series of computations by means of formulas which will give results that may be laid out directly. 219. Problem 23. Tangential cam, case I. Required a tan- gential cam in which the follower : (a) Rises 1 unit in 60° turn of the cam. (b) Falls 1 " " 60° "' " " " (c) Remains at rest for 240° turn of the cam. (d) The maximum pressure angle to be 30° and the end of the lobe to be rounded off by a circular arc. Find: The shortest radius of pitch surface of cam, the length of the straight-line portion of the cam lobe, the radius of the rounding off curve at the end, and the largest size roller that may be used. 220. The graphical method of construction for the tangen- tial cam is as follows: In a preliminary and separate drawing, con- struct an angle AO E, Fig. 130, equal to the given pressure angle; 114 CAMS draw a line A E at right angles to A at any distance out, and con- tinue A E until it intersects E; draw an angle A C equal to the assigned angle of action; drop a vertical line from E to C; draw the arc E C with L as a center ; draw the arc C G with as a center, and measure the distances G A and A 0. Then G A : h : : A : s, where h is the assigned motion of the follower and s is the correct radius at which to draw the line A E in the direct drawing of the cam. Fig. 130. -Tangential Cam, Preliminary Sketch in Graphical Method of Con- struction for Definitely Assigned Data In the present illustration G A, Fig. 130, is 1.33 units and A is 4 units. Therefore, in the direct drawing of the cam, Fig. 95, h X A = 1 X 4 33 GA 1 3.00, an.d this value is laid off at A Fig. 95. The pitch surface of the cam A E C is then drawn by repeating the operations in precisely the same order as in the preliminary drawing described above. The maximum pressure angle will be 30° at E where the circular easing-off arc is tangent to the straight line. The maximum radius of the roller would be E L, but as this would leave a sharp edge on the working surface of the cam, a value of z /± E L is taken as the radius, thus giving W N P as the working surface of the cam. ADVANCED GROUP OF BASE CURVES FOR CAMS 115 221. Analytical method of construction of the tangential cam. A direct drawing of the tangential cam may U 4 made from value obtained from a series of formulas having the following nota- tion, in which all linear dimensions are in inches and all angular Cir^ e Fig. 95. — (Enlarged) Problem 23. Tangential Base Curve Cam, Case 1 dimensions in degrees unless otherwise specified. All symbols are illustrated in Fig. 131 which is for a general case: h = total motion of follower. x = fraction of follower's motion while rolling on the straight sur- face of the cam, or, fraction of stroke during which acceleration takes place. a = maximum pressure angle. b = time allotted by the data to the follower motion, measured in angular motion of the cam in degrees. s = radius of pitch surface to which the straight pitch line is drawn tangent. t = length of straight edge of cam on both pitch and working surface. p = radius of pitch circle. d = largest radius of pitch surface of cam. c = angle turned through by the cam when the full motion of the follower is reached, c will equal b when the straight part of the cam is not assigned in the data. 116 CAMS e = radius of circular arc for rounding-off outer corner of pitch cam. r = radius of roller. w = radius of working surface to which the straight working line of the cam is drawn tangent. Fig. 131. — Tangential Cam, Showing Terms Used in the Direct Construction bt the Analytical Method 222. When the length of the straight part of the cam is not assigned in the data, c and b will be equal. When the length of the straight part is assigned c will figure out differently from b; if it comes less the problem is possible with the assigned data; if more, the length of the straight part must be reduced. The general formulas are : s = x h sec a . • (1) t = s tan a (2) sin a cctc = ifi~S ' r < e (3) (5) (7) s + h (4) e = d sin c w = s—r . . . (6) • • . (8) ADVANCED GROUP OF BASE CURVES FOR CAMS 117 223. With the data of the present problem, equation (5) must be solved first, for it is the only one in which all the terms but one are known. This formula is solved for t. With t known, formula (2) may be solved for s, then formula (1) for x, and so on in order with equations (3), (4), (6), (7), and (8). These formulas give the following values in the present problem: t =1.73 5 =3 x = .46 V = 3.46 d = 4 e = 2 r = 1.5 w = 1.5 224. With the above values, the cam in Fig. 95 is laid out in the following manner: Lay off given angle of 60° at DOC, draw circle Fig. -(Duplicate) Problem 23. Tangential Base Curve Cam, Case having radius A equal to s, draw straight part of cam A E equal to t, draw circular arc E C with center on C and with radius of L C equal to e, call r = .75e and make A W equal to it. Then W N P is the working surface of the cam where A W is the radius of the roller. The length W N of the straight part of the working surface is the same as the length of the straight part of the pitch surface, and the circular arc N P of the working surface has the same center as the arc E C of the pitch surface. The values d and w are automatically included in the process of the above described layout. 118 CAMS 225. If it is desired to construct the cam chart, Fig. 94, for the tangential cam in order to find the velocity and acceleration diagrams, the pitch circle of the cam, Fig. 95, should be drawn with the radius equal to E as computed above, and radial intercepts should be placed at regular distances as shown at H, I, etc., in Fig. 95. Then draw part of the cam chart with length equal to pitch arc D F, when both are to same scale, and with height equal to h. Draw Fig. 94.-(Duplicate) Tangential Pitch line DF Oil the chart at Base Curve, Case i a distance above A R equal to D A on the cam when both are to the same scale. In general the pitch line on the chart will not be half way up, although it appears so in this problem. Take the lengths of the radial lines at H, I, etc., which are shown on the cam in Fig. 95 and lay them off at equally spaced distances on the chart, Fig. 94, and draw the chart base curve A E C through the extremities of these lines. 226. The tangential cam for this case has a characteristic retardation curve in that it is convex downward as shown from F to H in Fig. 97, while the retardation curves for all other cams that have intermediate maximum ordinates are either straight or con- cave. This characteristic may be an advantage in some cam appli- cations and will be referred to in paragraph 273 et seq. on the use of springs for returning the follower. The pressure angle factors for this curve, for the data given in this problem, are: 5.28 for 20°, 3.62 for 30°, 2.82 for 40°, 2.36 for 50°, and 2.09 for 60°. These factors are used for the ordinates of curve No. 9 in Fig. 132 which shows that the tangential cam, for the data of Problem 23, has the advantage of smaller size over the parabola, circular, elliptical and cube cams when the lower range of pressure angles are used, but that it begins rapidly to lose this advantage at angles of about 36°. 227. Further characteristics of this tangential cam that may be used to advantage in assigning data, are that if the angle turned through by the cam is twice the pressure angle, the maximum retardation for the circular easing-off arc of the cam will occur at the end of the stroke as shown at C H, Fig. 97 ; and that the retarda- tion at the point on the cam where the arc joins the straight line will be, .866 C H as shown at E F, Fig. 97. If the angle turned through by the cam during the motion of the follower is greater than twice ADVANCED GROUP OF BASE CURVES FOR CAMS 119 D r* 77 1 A E 2 6' ***-«. 'fe — \ C r Fig. 97.— (Duplicate) Accel- E RATION Diagram for Tan- GENTIAL Cam the pressure angle the retardation value will still be a maximum at the end but will be less than .866 of this value at the point where retardation begins, that is, E F will be still shorter in comparison with C H than it is shown in Fig. 97. This condition has the practical value in that it allows a lighter-weight, or smaller spring to return the follower where a spring is used. If the angle turned through by the cam during the motion of the follower is less than twice the pressure angle the re- tardation at E F will be greater than .866 C H, and if it is much less the retardation value will be a maximum at the point where the easing-ofT arc joins the straight line, that is, E F will be greater than C H. 228. Exercise problem 23a. Tangential cam, case I. Re- quired a tangential cam in which the follower : (a) Rises lJ/2 units in 50° turn of the cam. (b) Falls 13^ units "50° " " " " (c) Remains at rest for 260° turn of the cam. (d) The maximum pressure angle to be 30°, and the end of cam lobe eased off by a circular arc. 229. Circular base curve, case I. This curve, Fig. 98, is made up simply of two equal circular arcs as shown at A E and E C. Fig. 98. — (Enlarged) Circular Base Curve, Case 1 It is the limiting case of the straight-line combination curve in which the two easing-ofT arcs are so large as to meet and eliminate the intermediate straight line entirely. The circular base curve 120 CAMS gives variable velocity and acceleration to the follower the first half of the follower stroke, and also variable velocity and retardation during the last half, as shown in Figs. 100 and 101. It will be noted rpD -1 A— J&-K C ik ^~. ^ _ II ^^F E^^^ — 2— P Fig. 101 Fig. 100. — (Duplicate) Velocity Diagram for Circular Base Curve Cam Fig. 101. — (Duplicate) Acceleration Diagram for Circular Base Curve Cam that the circular curve, and the elliptical curve shown in Fig. 102, give nearly the same sized cams and that the velocity and acceleration diagrams for each are quite similar. With the circular base curve, the radial distances on the cam at D, H, I, J, Fig. 99, cannot be found Fig. 99. — (Enlarged) Problem 24. Circular Base Curve Cam, Case 1 directly except by means of the chart or by computation. For graphical construction it is necessary to draw the chart, Fig. 98, first and it is then a simple matter to transfer the ordinates at H, I, J, to Fig. 99. The length of the chart for a maximum pressure angle of 30° is 3.73 times the motion of the follower. 230. The length of radius for the equal arcs in the circular base curve is 3.73 times the follower motion for a 30° maximum pressure angle. To find the length of radius for any other maximum pressure angle, use the formula: h r ~ 2(1 - cos a)' ADVANCED GROUP OF BASE CURVES FOR CAMS where r = the desired radius, a = the desired maximum pressure angle, and h = the given follower motion. Table for Circular Base Curve 121 For Maximum Pressure Alible of Radius of Arc is 20° 30° 40° 50° 60° 8.29/i 3.73 h 2.14 A 1.40 h 1.00 h 231. Problem 24. Required a circular base curve cam that will cause the follower to: turn of cam. a it it (a) Rise 1 unit in G0° (b) Fall 1 " " 60° (c) Remain stationary for 240 (d) With a maximum pressure angle of. 30 O t I it It o 232. The general description of the circular base curve given in the two preceding paragraphs will doubtless give all the necessary information for the solution of this problem so that only a brief order of procedure will be given here. The total length of chart is 1 X 3.73 X ^ = 22.38. One-sixth of this length is shown in Fig. 98. The radius of the cir- cular arc A E, which is the same as E C, is 1 2Q - cos 30°) 2(1 - .866) = 3.73. Draw eight equally spaced ordinates as at //, /, J, etc., Fig. 98. The radius of the pitch circle of the cam is, 22.38 2 X 3.14 = 3.56, 122 CAMS as drawn at D in Fig. 99. Divide the assigned arc of action D F, which is 60°, into eight equal parts as at H, I, J, etc. On the radial lines at each of these points lay off the corresponding ordinates from H, I, J, etc., in the chart, Fig. 98, thus obtaining the pitch surface A EC, Fig. 99. 233. In some cases it may happen, when the circular base curve is assigned, that the length and height only of the rectangular chart enclosing the circular curve will be known and it may be desired to compute the radius and the pressure angle for the circular arc that must be used. For example, in Fig. 98, assume that A R and R C are the only known values and it is desired to find the proper radius of the arc EC and the pressure angle that will exist at E. The radius may be readily computed by simple geometry, for, the two s s Fig. 98. — (Duplicate) Circular Base Curve, Case 1 triangles C F E and C T S will be similar in all cases and, therefore, SC : EC : : TC : FC. Since E F and F C are equal to one- half of A R and R C, respectively, their values are known and EC = V EF 2 + FC 2 . The length of T C is one-half of E C. The radius of the circular arc will be SC EC X TC FC ' 234. In order to obtain the pressure angle, for the case given in the preceding paragraph, simple trigonometry is required, and in using the trigonometry, the length of the radius may also be obtained even more readily than by geometry. The method is as follows: ADVANCED GROUP OF BASE CURVES FOR CAMS 123 In Fig. 98 the angles C S T and EST are each equal to one-half the angle C S E which is the pressure angle and is designated by a in the following formulas. The triangles C E F and C ST are similar in all cases. Therefore, a may be found by the following formula : 1 CF i ™2 a = YF With a known, the radius of the arc E C may also be found as follows: E F ES = =-?- = C S. sin a 235. Exercise problem 24a. Required a circular base curve cam which will cause the follower to : (a) Move out 3 units in 90° turn of the cam. (b) Remain stationary for 195° " " " " (c) Move in 3 units in 75° " " " " (d) With a maximum pressure angle of 40°. 236. Elliptical base curve. The elliptical base curve gives variable velocity and variable acceleration to the follower. By using different ratios for the horizontal and vertical axes of the ellipse on which the curve is based, the velocity of the follower may be made to increase rapidly or slowly at the start, and the cam may be made small or large and still not exceed a given maximum pressure angle. 237. Elliptical base curve, ratio 7 to 4. As stated in the pre- ceding paragraph the elliptical cam may be based on ellipses having various proportions between their major and minor axesr When the proportions are as 7 : 4, as in Fig. 102 where F G = 7 and F C = 4, the length of the chart will be 3.95 times the travel of the follower for a maximum pressure angle of 30°. The cam will be larger, but the velocity of the follower will be less at starting and stopping and greater at midstroke than for any of the cams described thus far. If a still lower starting and stopping velocity is desired with an elliptical cam, it may be obtained by making the ratio of horizontal to vertical axes on the chart as 8 : 4, 9 : 4, or greater, instead of 7 : 4 as here used. The drawbacks to increasing the ratios above 7 : 4 are increased size of cam and high velocity at midstroke for a given pressure angle. 124 CAMS 238. Elliptical base curve, ratio 2 to 4. The cam produced from the elliptical base curve is shown, in the preceding paragraph, to give a certain characteristic action to the follower when the ratio FG:.FC:.:.1A Fig. 102. — (Enlarged) Elliptical Base Curve of the horizontal axis to the vertical axis is 7 to 4. When the ratio is 2 to 4, a totally different characteristic follower action is obtained as may be determined by a process of construction similar to that shown in Figs. 102 and 103. The cam itself, with a ratio of 2 to 4, will be much smaller for a given pressure angle, as may be seen by comparing the abscissae of curves 5 and 11 in Fig. 132. Where it is desired to use a very small cam for a given pressure angle, the 2 : 4 ellip- tical curve will have an advantage over the ordinary straight-line com- bination curve above 27° as may be noted from an inspection of curves 5 and 6, Fig. 132; but it is at a disadvantage compared with the log- arithmic-combination cam at all pressures angles as is shown by a comparison of curves 2 and 5. 239. Elliptical base curve may be made equivalent to nearly all other base curves. Since the elliptical base curve may be constructed with any ratio of horizontal to vertical axes, it has a range of usefulness over the entire field covered by all the other base curves except the logarithmic curve. When the horizontal axis of the ellipse is zero, the elliptical base curve coincides exactly with the straight-line base. As the horizontal axis increases in length, the vertical axis remaining constant, the elliptical base curve crosses the straight-line combination curve. When the horizontal axis of the ellipse equals the vertical axis, the elliptical base curve is identical with the crank curve. As the horizontal axis continues to increase, the elliptical curve approximates very closely indeed to the parabola Fig. 103. — (Enlarged) Elliptical Base Curve Cam ADVANCED GROUP OF BASE CURVES FOR CAMS 125 when the ratio of horizontal to vertical axes is as 11 to 8. A further general characteristic of the elliptical curve is that the starting and stopping velocities grow smaller, and also the accelerations or start- ing and stopping forces grow smaller as the horizontal axis of the ellipse grows larger. 240. Cube base curve, symmetrically applied. The cube base curve, Fig. 106, is similar in method of construction to the Fig. 100. — (Enlarged) Cube Base Curve, Case 1 parabola base curve, the only difference being that the cubes of the numbers 1, 2, 3, etc., instead of the squares, are used as ordinates of the curve. The cube curve gives extremely low and slowly increasing motion to the follower at the start as may be noted by an inspection of the velocity curve A E, Fig. 108, which shows the distinguishing characteristic that the velocity curve is tangent to the base line. The cube curve is the only one that gives uniformly increasing accel- eration to the follower, starting from zero, as indicated by the straight Eia, 108. Fig. 108. — (Duplicate) Velocity Diagram for Cube Cam Fig. 109. — (Duplicate) Acceleration Diagram for Cube Cam inclined line A D in Fig. 109. The disadvantage of the cube curve, however, is that it gives an extremely large cam for a given maximum pressure angle, if it is used in the same way that the preceding curves are used, that is, if it is made up of two similar arcs placed in reverse order. If the cube curve were so drawn it would be made up of two arcs similar to A E, Fig. 106, and the pressure angle factor would be 5.20 as compared, for example, with 3.46 for the parabola, and the maximum radius of the cam would be 5.47 against 3.80 for the para- bola. Because of the similarity of method of construction of the 126 CAMS cube curve and the parabola, and because the large size of the sym- metrical cube cam renders it impractical for most cases, its drawing will be omitted, and instead, a modified and more practical con- struction of the cube cam will be illustrated and explained in the following paragraphs. 241. Cube base curve unsymmetrically applied for best advantage. This modified cube curve will be referred to as cube curve, case I. Its features are that it retains the very low starting values of the regulation or symmetrical cube cam, and at the same time keeps down the size of the cam by using che regulation cube curve for the first half of the follower's motion and then using a short arc of another cube curve for the retardation in such a way that the maximum acceleration and retardation values shall be equal. In order to use this base curve several formulas are necessary and they, together with their notation, are given in the following par- agraph. 242. Notation and formulas for cube curve cam, case I : h = distance moved by the follower. a = pressure angle. I = length of part of cam chart corresponding to follower's motion. x = length of cam chart during which acceleration takes place. Xi, 0:2 . . . = arbitrary lengths of cam chart taken for purposes of constructing chart base curve. 2/i, 2/2 • • . — length of ordinates of cam chart corresponding to the values of x\, X2 . . . . r = radius of pitch circle of cam. b = angle turned through by cam in degrees during follower's mo- tion. The general formulas are: I = 2.427 h cot a ... (1) x = .618 I (2) -■(f y = — 7=^ — from zero to x (3) 2V5-4 »(i)-(i)'- (Vif - ,) V = h — — — 7= from x to I (4) 180 1 ,.. ADVANCED GROUP OF BASE CURVES FOR CAMS 127 243. Problem 25. Cube curve cam, case I. Required a cube curve cam with unsymmetrical cube curve arcs in which the follower shall : (a) Rise 1 unit in 60° turn of the cam. (b) Fall 1 " " 60° " " " " (c) Remain stationary for 240° turn of the cam, and (d) The maximum pressure angle shall be 30°. Substituting the values given in the data in the formulas in the preceding paragraph, I = 4.20, x = 2.60 and r = 4.0. With these values, the rectangle A B C R, Fig. 106, for the cam chart may be 2T D Pitch Line pv, «~^ S I P.nsn pil-VO 1- *" K^ E X T RV < H 7 rn l N J 60 > ^ 2.60 " Fig. 106. — (Duplicate) Cube Base Curve, Case 1 drawn, A R being made equal tol,AX equal to x, and R C equal to h. The curve A E may be drawn graphically by dividing A X into four equal parts, A D into four unequal parts, as shown in Fig. 106, and projecting the division points until they meet, as at K. A D, which is one-half of A B, is divided into the four unequal parts as follows : Draw a straight line A G in any convenient direction about as shown ; make its length 64 units according to any convenient scale; with the scale still in place mark the 1st, 8th and 27th division points on A G and from each of these points draw lines parallel to G D until they intersect the side A D of the rectangle; from the latter points draw horizontal lines until they intersect their corresponding ordinal cs, as at K. Or, the values of these ordinates, as at J K, may be com- puted by formula (3) of the preceding paragraph by substituting the following values for x : x\ = }4 X , %2 = %x, x% = %x. The computed values of yi, ?/2, 2/3, are .008, .063, .211, respectively, and these are laid off at H, I, and J in Fig. 106. 244. The portion of the cube curve from E to C, Fig. 106, is found by taking a series of any number of equally spaced ordinates, four being used in this problem and one of them marked at T S. The values of these ordinates are computed from formula (4) of para- graph 242, and arc as follows: y± = .50, y 5 = .71, y% = .87 (shown 128 CAMS at aS T), and 7/7 = .95. The corresponding values of x±, X5. . . which were substituted for x in equation (4) in obtaining these values were x± = x, X5 = x + J<£ (I — x), xq = x + Y 2 {1 — x), etc. 245. The pitch circle of the cam is drawn with D, Fig. 107, as a radius and is equal to r = 4.00, obtained from equation 5. The values as found for the cam chart may be now transferred to the cor- Fig. 107. — (Enlarged) Problem 25. Cube Base Cam, Case 1 respondingly placed radial lines from A to R, or the values as com- puted from formulas (3) and (4) may be laid off directly on these radial lines without drawing the cam chart at all. 246. The characteristic velocities, accelerations and retardations produced by this case of the cube curve cam are shown in Figs. 108 and 109, respectively. From the latter it may be seen that the Fig. 109. Fig. 108. Fig. 108. — (Duplicate) Velocity Diagram for Cube Cam Fig. 109. — (Duplicate) Acceleration Diagram for Cube Cam acceleration and retardation lines, A D and F H, respectively, are straight inclined lines, characteristic of the cube curve, as pointed out in paragraph 240. When the retardation line F H is extended, as shown by the long-dash line, Fig. 109, it passes through the zero point of the diagram. A cam with this characteristic may have particular advantages in some instances, one of which will be referred to later in the discussion of the relative strength of springs necessary \o return the follower. ADVANCED GROUP OF BASE CURVES FOR CAMS 129 247. Exercise problem, 25a. cube curve, cam, case I. Re- quired a cube curve cam in which the follower: (a) Moves up 1 unit in 50° turn of the cam. (b) Moves down 1 " " 50° " " " " (c) Remains stationary for 260° turn of the cam, and in which (d) The maximum pressure angle shall be 30°. 248. Cams specially designed for low-starting velocttii;<. In cams where the change in velocity of the follower during the latter part of its travel may take place rapidly the early motion of the fol- lower may be made both very low and very gradual. These condi- Fig. 114. — (Duplicate) Cube Base Curve, Case 2 tions as to velocity may be obtained by giving more than half the stroke to the acceleration of the follower, instead of one-half as has been the case in all preceding problems. In Figs. 110 and 114, are illustrated special cases of the circular and cube base curves in which the follower is permitted to accelerate during % of its stroke, while its retardation takes place in the last quarter of the stroke. In these Fig. Fig. 112. FiG. 116. 112. — (Duplicate) Velocity Diagram for Circular Base Curve Cam, Fig. 11G. — (Duplicate) Velocity Diagram for Cube Cam, Case 2 Case 2 two cases the velocities at midstroke are approximately 1.2 and 1.0, respectively, as may be noted from the dash line construction in Figs. 112 and 116, respectively, against 2.2 and 1.7 as shown for similar basic curves in Figs. 100 and 108. 249. Problem 26. Circular base curve cam, case II. Re- quired a cam with a circular base curve in w r hich the follower shall: (a) Rise 1 unit in 60° turn of the cam. (b) Fall 1 " " 60° " " " " 130 CAMS (c) Remain stationary for 240° turn of the cam. (d) Accelerate for % of its stroke, and in which (e) The maximum pressure angle shall be 30°. 250. For a graphical method of construction of case II of the circular base curve cam, draw the cam chart as in Fig. 110 making its length I = h X -^ X 36Q , where I = total length of chart. h = height of chart. / = pressure angle factor. b = angle during which follower motion takes place. In this problem I 1 X 3.73 X 360 60 22.38. 251. One-sixth of the chart is shown in Fig. 110 at A R. Lay off the height A B equal to one unit and mark the point D so that Fig. 110. — (Enlarged) Circular Base Curve, Case 2 Fig. 111. — (Enlarged) Problem 26. Circular Base Curve Cam, Case 2 AD — t, where t equals fraction of stroke assigned for acceleration. Draw D F. Mark the point X on A R so that A X = t X A R. Draw X E. Through E draw an inclined line making an angle with X E equal to the assigned pressure angle. Where this inclined line meets the lines A B and C R will be the centers for the circular arcs making up the base curve. These center points will be at Y and at S respectively. Draw the circular arcs A E and E C. Divide D E into a convenient number of equal parts, as at H, I . . . and draw ordinates to the circular arc A E. Do the same with E F. ADVANCED GROUP OF BASE CURVES FOR CAMS 131 Construct the pitch circle of the cam with a radius, I 22.38 OD = 2 X 3.14 G.28 3.5G, as shown in Fig. 111. Lay off D F equal to the assigned motion angle, which is 60° in this problem. The arc D F will be equal in length to the line D F in the chart when both are drawn to the same scale. Make D E on the arc equal to D E on the chart and divide the arc D E into the same number of equal parts as the line D E. Draw radial lines at the division points H, I, J, . . . and transfer the ordinates from the chart to these radial lines, thus obtaining the pitch surface of the cam from A to E. Do likewise to obtain the arc E C of the cam. 252. The circular base curve, case II, gives a smaller cam than does case I, although both have the same pressure angle factor and the same chart length. The maximum radius of the cam for Fig. 99. — (Duplicate) Ciecular Base Curve Cam, Case 1 case II is 3.81 against 4.06 for case I as shown in Figs. Ill and 99 respectively. The reduction in size in case II is due to the fact that the pitch line D F on the cam chart is higher up in the present case, and, consequently, that more of the pitch surface falls inside of the pitch circle than in Fig. 99. The pitch circle is the same size in both cases. 253. Computation for the lengths of the radii for the arcs A E and E C in the cam chart in Fig. 110 may be made by the fol- lowing formulas if desired, instead of finding them graphically as explained in paragraph 251. A Y = hi cos a and C S h(l - t) 1 — cos a' 132 CAMS where a equals the assigned pressure angle, h equals follower motion, and t equals fraction of stroke assigned to acceleration. 254. Exercise problem 26a. Circular base curve cam, case II. Required a cam with a circular base curve in which the follower shall: (a) Rise 2 units in 75° turn of the cam. (b) Fall 2 *■' il 75° " " " " (c) Remain stationary for 210° turn of the cam. (d) Accelerate for .7 of its stroke, and in which (e) The maximum pressure angle shall be 30°. 255. The use of the cube curve for obtaining extremely low starting velocities is illustrated in Fig. 115. The cam is built up Fig. 114.— (Enlarged) Cube Base Curve, Case 2 D\ H i _ D ~r2 -1 ^^-"""' E a Q u N ^ M v. R — 1 —2 y^4 " p F G L-5 Fig. 115. — (Enlarged) Problem 27. Cube Base Curve Cam, Case 2 Fig. 117. — (Duplicate) Acceleration Diagram for Cube Cam, Case 2 from a specially long arc of the cube base curve and it has a short circular base arc for easing off at the end. The chart and the base curve for this cam are shown in Fig. 114. The low-starting velocities are due to the fact that the follower has % of its stroke to reach max- imum velocity. This gives only \i stroke for retardation which attains a very high value near the end of the stroke ranging from 4.8 to 3.2, as shown in Fig. 117. This, of course, becomes the acceleration ADVANCED GROUP OF BASE CURVES FOR CAMS 133 value at the beginning of the return stroke. Herein lies the disad- vantage of this cam. It is useful only where extremely slow starting velocity is required a1 one end of the stroke and where a rapid change of velocity at the other end of the stroke is immaterial. It would require a powerful spring to keep the follower roller in contact with the cam at high speeds, and if it were used on a positive drive cam would cause rapid wear at the beginning of the return stroke. 256. Problem 27. Cube curve, case II. Required a cube curve cam with a circular arc for easing-off radius in which the fol- lower : (a) Rises 1 unit in 60° turn of the cam. (b) Falls 1 " " 60° " " " " (c) Remains stationary for 240° turn of the cam. (d) Accelerates during J^ of the stroke. (e) The maximum pressure angle to be 30°. 257. In solving the above problem the length A X, Fig. 114, of that part of the chart which is given over to the cube curve is first found by the formula, 3th . Xi = where tan a, t = the fractional part of the follower's motion devoted to accelera- tion. h = the total motion of the follower. a = the pressure angle. .ri = the length of chart under the cube curve. xo = the length of chart under the circular easing-off arc. Substituting the values given in problem 27, 3 X .75 X 1 Q on xi = === = 3.90. .57/ 258. The length X R of chart, Fig. 114, necessary for the easing- off circular arc may be computed by the formula, _ h(l - t) ^25 X2 ~ tan Y 2 a " .268 ' J6 ' Or, the length XR may be found directly by drawing NEK so that it is tangent to the cube curve at E. The angle KEF will 134 CAMS then be equal to the pressure angle. Make K C equal to E K. The point C will then be at the end of the chart. The center for the arc E C will then be on the line C R extended, and at a point S which must also be on the perpendicular to N E K. 259. To find points on the cube base curve A E, Fig. 114, divide D E into any convenient number of equal parts, six being used in the illustration. Draw vertical lines through each of the division points as at H, I, ... Draw a line A G inclining upward from A in any convenient direction and make the distance A G equal to the cube of the number of construction parts. Six parts having been chosen in this problem, A G will be equal to the cube of 6, or 216 units in length laid off to any convenient scale. At the same time lay off the division points 1, 8, 27, etc., which are the cubes of 1, 2, 3. etc. Draw Fig. 114. — (Duplicate) Cube Base Curve, Case 2 the line G D, and then draw lines parallel to it through the points 1, 8, 27, etc., until they intersect A D. Project these intersecting points horizontally until they meet the corresponding verticals from H,I, . . . , thus giving points on the cube base curve A E. 260. The radius for the pitch circle of the cam will be, IX 360 = 4.83 X 360 2 X 7T X b ~~~ 6.28 X 60 4.62, where I = length of chart used for rise of follower and, b = angle during which the follower is moving. With the above value of r the circle through D is drawn in Fig. 115. The arc D E F will be equal in length to the line D E F in Fig. 114 when drawn to the same scale, and it should be similarly divided and the radial lines at #, /, . . . made equal to the similarly lettered ordinates in the chart. The curve A E C thus obtained will be the pitch surface of the cam. ADVANCED GROUP OF BASE CURVES FOR CAMS 135 261. Exercise problem 27a. Cube curve, case II. Require' a cube curve cam with a circular easing-off arc in which the follow* (a) Rises 3 units in 90° turn of the cam. (b) Falls 3 " " 90° " " " " (c) Remains stationary 180° " " " " (d) Accelerates during .70 of its stroke. (e) The maximum pressure angle to be 30°. Fig. 115. — (Duplicate) Cube Base Curve Cam, Case 2 262. Tangential cam, case II. The tangential cam, as stated in paragraph 218, is made up of straight-line sides with a circular arc for rounding off the end of the lobe. When the length of the straight surface of the cam is not specified, or when the portion of the stroke during which the follower accelerates is not given in the data, the tangential type of cam works out to good advantage. But when either of the above items is included in the data for the tangential cam it may conflict with the proper cam angle which should be allowed for the follower motion, as illustrated in the following prob- lem, which contains the same data as the two previous problems. The possible difficulty met with in using the tangential cam arises from high accelerations that may be produced. 263. Problem 28. Tangential cam, case II. Required a tan- gential cam with a circular easing-off arc in w r hich the follower : (a) Rises 1 unit during 60° turn of the cam. (b) Falls 1 " " 60° " " " " (c) Remains stationary for 240 ° " " " " (d) Accelerates during J4 of its stroke. (e) The maximum pressure angle to be 30°. 136 CAMS 264. The cam may be constructed directly by substituting values given in the data in the general formu- las given in paragraph 222, and then laying out the results as in Fig. 119. In the present problem A, Fig. 119 equals s as found in paragraph a* 222, A E = t, D = p, C = d, angle D C = b, angle D K = c, and L E = e. The radius r of the roller and the minimum radius w of the working surface are not shown in the illustration but may be readily added if called for. The radius of the roller, however, cannot be greater than E L. The numerical results found by substituting the values given in the data in the series of formulas referred to above are as follows: Fig. 119.— (Enlarged) Pkoblem 28 Tangential Base Curve Cam Case 2 S = 4.84 t = 2.79 5.58 d = 5.84 c = 39^- e = 1.47. 265. If it is desired to construct the cam chart for the purpose of determining the velocity and acceleration diagrams later, it may readily be done: (1) By making the length of chart A R, Fig. 118, equal to the length of the arc D F on the cam drawing, X h. . C D Pitch Line X s^f~\ 4 E X G Y A r ' ' n.-> 2.94 ^ < d.Ht Fig. 118. — (Enlarged) Tangential Base Curve, Case 2 (2) by laying off the pitch line D F on the chart and subdividing the same as the arc D F on the cam is subdivided, (3) by transferring the radial lines at H, I, . . . from the cam to the chart and drawing them as vertical lines, thus obtaining points for the base curve A E K C. ADVANCED GKOUP OF BASE CURVES FOR CAMS 137 266. It will be noted that an attempt to construct a tangential cam in cases such as the one here represented may result in extremely large retardation or acceleration values, as shown in Fig. 121, the practical result of which will be a "hard-turning " spot at a point on Fig. 121. — (Duplicate) Acceleration Diagram for Tangential Cam, Case 2 the cam corresponding to E, Fig. 119, and continuing, in lessening degree, to K. 267. Exercise problem 28a. Tangential cam, case II. He- quired a tangential cam with a circular easing-ofT arc in which the follower : (a) Rises 2 units during 75° turn of the cam. (b) Falls 2 " " 75° " " " " (c) Remains stationary for 210° " " " " (d) Accelerates during .70 of its stroke. (e) The maximum pressure angle to be 30 °, SECTION VII.— CAM CHARACTERISTICS. 268. Method of determining velocities and accelerations. The velocity and acceleration values in the diagrams shown in Figs. 72 to 121 may be found by graphical methods which are simple and quite accurate enough for most practical purposes if precision in drawing is followed. The graphical method applies to all forms of cams and starts with the cam chart. Its application, however, is illustrated only in connection with the circular cam chart in Fig. 98, it being unnecessary to add similar lines to all the other chart drawings, as the constructions would be the same in every case. 269. The use of time-distance and time-velocity diagrams. The chart curve A E C, Fig. 98, for our present purpose, may be \ B D \ C J > Pitch Line J§5 > V H I K V A R \ -LI T ' i ^-^^N \ \ J- if / s «o \ \ \ . 1 s s Fig. 98. — (Duplicate) Circulak Base Curve, Case 1 termed time-distance curve in which the abscissa A R represents time, and the ordinates parallel to A B represent distances traveled by the follower at corresponding times. If, then, the time-distance curve were a straight inclined line, the velocity of the follower would be constant. We may consider, for the instant, that the time-distance curve is straight at E and draw a straight line, E P, tangent at that point. If this were the time-distance line and if it were continued for a time period represented by E D, the follower would have moved the distance P D in the time represented by E D. If E D is consid- 138 CAM CHARACTERISTICS 139 ered as a unit of time, then P D becomes a measure of velocity and its length is laid off in Fig. 100, at XE which is at the center of the time- velocity diagram. The length A C of the velocity diagram may be any convenient value for the purpose of comparison. The distance D E, Fig. 98, or one-half the length of the cam chart, was selected as a time unit because it is a convenient length and because the length of one-half of each cam chart represents the same amount of time in each of the chart drawings. This is because the data are the same in all the cams represented in Figs. 71 to 119. To find other points on the time-velocity diagram, divide the time-distance curve by a number of equally spaced ordinates as shown at J, I, H, Fig. 98. The tangent to the curve at K, on the ordinate J V, is K M, and the time unit K L is equal to D E. Then, from the same reason- ing as given above for the point E, L M becomes a measure of the velocity of the follower at K, and it is laid off at M L in Fig. 100. Similar constructions are repeated at the other points and the time-velocity diagram completed. 270. The time acceleration diagrams are found graphically from the time-velocity diagrams by similar constructions. In Fig. 100 a tangent E S is drawn to the time-velocity curve at E and if the ~ M Fig. 100. — (Enlarged) Velocity Diagram for Circular Base Curve Cam velocity of the follower is continued along this line for a time repre- sented by E Q it will lose a velocity of Q S in the time E Q. Such loss in velocity is retardation and consequently the distance S Q is laid off at E D at the center of the time-acceleration diagram in Fig. 101. The line E S in Fig. 100 was drawn to the left, and consequently downward to make the drawing more compact. In this way retarda- tion instead of acceleration was found logically. Had the tangent line E S been drawn to the right, and consequently upward, the value Q S would have been found just the same and would have been called acceleration. The length of the acceleration diagram, A C, in Fig. 101 may be taken any value; also, the time unit E Q in Fig. 100 may be taken any value entirely independent of the time unit used in Fig. 98, so long as the same length of line is taken in all the 140 CAMS velocity diagrams as the time unit, in making comparisons. If a definite speed is assigned to the cam then all the lines in the time- distance, time-velocity and time-acceleration diagrams will have a definite value in feet and in seconds, and by closely following these values, the diagrams may be scaled so as to interpret them in the ordinary units of feet and seconds, even if arbitrary time lines have Fig. 101. — (Enlarged) Acceleration Diagram for Circular Base Curve Cam been used in constructing the diagrams. For example, suppose that the cam in Fig. 99 is turning at 120 revolutions per minute. Then it will require V12 second to turn through the 60° angle DOC, and D E in Fig. 98 will represent V24 second. D P measures .5625 inch or .0469 foot. Therefore the velocity of the follower at E will be ten |Ctt Fig. 99. — (Duplicate) Circular Base Curve Cam, Case 1 .0469 foot per V24 second, or 1.125 feet per second. The scale on X E, Fig. 100, would then be graded so that a mark at 1.125 would fall at E. In Fig. 100, A C represents V12 second, and Q E, 1 / 4 s second. Since X E represents 1.125 feet per second in this example, Q S repre- CAM CHARACTERISTICS 141 sents .750 foot per second to the same scale. Therefore the accelera- tion is .750 foot per second per l ,. s second or 36.00 feel per second per second. The scale on ED, Fig. 101, would then be graded so that a mark at 36.00 would fall at D. Another set of construction lines for obtaining an ordinate in the acceleration diagram is shown at L T V, Fig. 100, where L T is the same length as E Q, and V T is equal to the acceleration and is laid off at V T in the acceleration diagram, Fig. 101. 271. Degree of precision obtained by graphical method. In Fig. 98 the tangent lines may be drawn with precision because the curve A E is an arc of a circle, but in the other curves the center of curvature for each of the construction points is not known and the tangent must, therefore, be drawn by eye. Even here considerable precision may be obtained if, in so drawing the tangent, it is remem- bered that the tangent at L, Fig. 100, for example, will be practically the same distance from U as it is from E when it passes each of these points, provided U and E are on ordinates equally spaced, and pro- vided also that the curve A E has a fairly uniform rate of curvature on both sides of L. If the radius of curvature to the right of L should grow noticeably shorter than the radius of curvature to the left of L, the tangent at L would pass a little closer to U than to E. If, in addition to using such judgment as here indicated in the drawing of tangents to irregular curves, a sufficient number of points are taken closely together, and if the newly derived curve is drawn smoothly through the average positions of plotted points, a remarkable degree of accuracy may be obtained by the graphical method of obtaining velocity and acceleration diagrams. 272. Comparison of relative velocities and forces produced by cams having different base curves. This comparison, which may be made by studying the several velocity and acceleration diagrams in Figs. 72 to 121, is also shown in the accompanying table where the maximum velocities of the follower are shown in Column 2, and the maximum acceleration and retardation values in Columns 3 and 4. Since force equals acceleration multiplied by mass, the direct effort required to move the follower is proportional to the acceleration, and, therefore, the relative direct force needed to operate the follower for various cams is also shown in Columns 3 and 4. The retardation values in Column 4 represent the relative pressures exerted by the follower against the cam surface in slowing up where a positive drive cam is considered. They also represent the relative 142 CAMS sizes of counterweights where a gravity return is used. In the cam with the straight-line base there would be violent shock at the start and the cam would " stick " and require considerable direct power, but after that it would be necessary only to overcome friction. The parabola, it will be noted from the table and from Fig. 93, requires the least direct effort, considering the entire cycle of the follower. This effort is represented by unity for purpose of comparison. The circular base curve cam, Case II, Fig. 113, requires a trifle less effort than the parabola cam while on acceleration on the forward stroke, but 2.86 times the effort of the parabola while the follower is on acceleration during the return stroke where a double-acting cam is used. For a single-acting cam the values given in Column 4 show the relative forces necessary to sufficiently accelerate the follower on the return stroke so as to keep it in contact with the cam. TABLE SHOWING RELATIVE MAXIMUM VELOCITY, ACCELERA- TION AND POWER FOR EACH TYPE OF CAM Form of Cam Col. 1 All-logarithmic Logarithmic combination Straight line Straight-line combination curve (r = h) Crank curve Parabola Tangential curve, Case I Circular curve, Case I Elliptical curve Cube curve, Case I Circular curve, case II Cube curve, Case II Tangential curve, Case II Relative Amounts of Direct Force Needed to Operate Cam During 273. Cam follower returned by springs. Although the cam built from the parabola chart pitch curve gives the smoothest motion and requires the least direct power to operate it so far as the cam and follower only are concerned, there may be other considerations in the CAM CHARACTERISTICS 143 design that make or appear to make some other form of chart pitch curve more desirable. For example, when a follower is returned by a positive drive parabola cam, or when it is returned by gravity, the parabola cam gives the best action because the pull on the follower is constant all the time, but when the follower is returned by a spring, the spring reacts on the cam with a uniformly increasing pressure during the out stroke as represented by the straight inclined clash- line & P in Fig. 93, and with a reverse uniformly decreasing pressure during the instroke. 274. If the spring pressure acting on the cam is zero when the follower is at rest in its lowest position, the spring compression line would be represented by the straight line A N, Fig. 93, starting at A and inclined so as to touch the retardation line as at F. Inasmuch as there should jp always be some compression in the spring, s ^'^•■y even when the follower is at rest, a margin ^r of compression will be taken as illustrated at A S. The practical spring compression Fig. 93.— (Duplicate) Accel- line will, therefore, be S P parallel to A N. pakabola Cam**' As the follower moves out, its acceleration during the first part of the stroke produces increasing pressure between the cam surface and the spring-actuated follower as represented by the increasing length of the ordinates from S B to R D. At mid- stroke the follower begins to slow up. In the case shown in Fig. 93, the slope of the spring pressure line was taken so as to have the same spring pressure (R F = S A) on the cam at midstroke as it has at the beginning. The line S P could have been given a steeper slope if a larger margin of pressure than R F had been desired at midstroke. This would have required a heavier spring. From midstroke to the end there is again an increasing margin of pressure, the maximum being represented by the difference between the ordinates P H and R F. The full strength of the spring which would have to be used would be represented by the ordinate P C. 275. Relative strength of spring required for crank, tan- gential, CUBE AND PARABOLA BASE CURVE CAMS. Although the parabola cam, with its perfect action as described in paragraph 214, permits of the use of a light spring when a single spring is used to return the follower, the crank curve, tangential curve and cube curve cams may each be designed to operate with somewhat lighter springs. Spring compression lines for each of the three last-men- 144 CAMS tioned cams are shown at S P in Figs. 89, 97, and 109, and the max- imum compression required of a single spring in each case is 1.75, 2.35, and 2.30 as compared with 2.40 for the parabola cam as shown in Fig. 93. The return spring pressure between the follower roller and the cam surface, when the crank base curve is used, is more nearly uniform throughout the entire stroke than it is with any other D : 2 1 A. E S L -^ ^--^ F — 1 Fig. 109. ~R Q 5^2^*^- -2 RP Fig. 89. — (Duplicate) Acceleration Diagram for Crank Curve Cam Fig. 97. — (Duplicate) Acceleration Diagram for Tangential Cam Fig. 109. — (Duplicate) Acceleration Diagram for Cube Cam type of cam, as may be noted from the maximum and the average ordinates between the acceleration-retardation curve and the spring pressure line, S P, in the several diagrams. 276. Cube curve cam specially adapted for a follower returned by a spring. The cube curve cam possesses one characteristic over the others in that the pressure between the cam and the follower is absolutely uniform during the latter part of the up-stroke and the first part of the down-stroke when the follower is returned by a rpring, as shown by the parallel lines F H and R P, Fig. 109. This gives an advantage of smooth running and uniform wear when the spring is under its greatest compression. 277. The pressure between the spring-actuated follower and the cam is variable throughout the stroke in all cams except during part of the stroke with the cube curve cam. And it may readily happen that the acceleration called for by the cam is so great that the spring will not be strong enough to keep the fol- lower roller against the cam surface as may be specially noted at or near the beginning of the return stroke. This is illustrated in (Duplicate) Accel- Fig. 113 where the spring pressure against eration diagram for Cir- | ne follower which would be necessary to cular Base Curve Cam, . ,11 n tti Case2 hold it to the cam is represented by t L, whereas, if a spring of the same strength as for the cube curve cam, Fig. 109, were used the pressure at the CAM CHARACTERISTICS 145 phase E, Fig. 113, would be only R E. This means that the cam will "run away" from the follower, because the spring is not strong enough during the part of the stroke represented by TF R to press the follower against the rapidly receding cam surface. 278. In order to keep the follower roller against the cam surface where cams with large retardation values are used, as in Figs. 77, 85, 113, 117, and 121, a comparatively heavy spring is required which will be unnecessarily strong during a very large part of the stroke, or else two springs will be required, the second one to come into action when needed. Both cases are illustrated in Fig. 113. A single heavy spring that will exert a pressure represented by W V :nr r 2 1 Fig. 77? r-1 z Fig. 85.br Fig. 77. — (Duplicate) Acceleration Diagram for Logarithmic-Combination Cam Fig. 85. — (Duplicate) Acceleration Diagram for Straight-Line-Combination Cam Fig. 117. — (Duplicate) Acceleration Diagram for Cube Cam, Case 2 Fig. 121. — (Duplicate) Acceleration Diagram for Tangential Cam, Case 2 will keep the follower roller against the cam surface at all times, the minimum pressure between the two occurring at F G. Or, a single and much lighter spring exerting a pressure represented by S P, Fig. 113, may be used, and then a second and shorter spring with an initial compression represented by M E may be so placed as to come into action at E so that the combined pressure of the two springs on the follower is M E plus R E equal F E. This means that the combined pressure of the two springs will be just sufficient to keep the follower roller against the cam at phase E, and that the total pressure of the two springs at the end of the stroke will be represented by C V, thus giving an excess pressure represented by H V at the end of the stroke. 146 CAMS The base curves that are best suited for spring-return followers will be seen by comparing the acceleration diagrams in Figs. 73 to 121 to be the crank curve, parabola, tangential curve, Case I, and the cube curve, Case I. The logarithmic combination and straight-line com- bination curves come next in order. 279. Accuracy in cam construction. It need scarcely be pointed out that the pitch surfaces of cams should be constructed with considerable accuracy and the working surfaces carefully fin- 15 r Fig. 111. — (Duplicate) Circular Base Curve Cam, Case 2 ished, if definite results are required, for, it may be seen by com- paring the pitch surfaces of several of the cams illustrated in Figs. 71 to 119 that a relatively small difference in form may make a large difference in the velocity, acceleration, and force or pressure, under which the follower operates. For example, the circular curve cam, Fig. 103. Fig. 107.- -(Duplicate) Elliptical Base Curve Cam -(Duplicate) Cube Base Curve Cam, Case 1 Case II, Fig. Ill, and the cube curve cam, Case II, Fig. 115, are apparently quite similar in form, though varying in sizes, yet the maximum accelerations which they impose on the follower on the return stroke are quite different, being 2.9 and 4.8 respectively, as shown in Figs. 113 and 117. Also the cube curve cam, Case I, Fig. 107 and the elliptical cam, Fig. 103, are much alike, yet their velocity CAM CHARACTERISTICS 147 lines and their acceleration lines, Figs. 109 and 105, arc different in every way and if a spring were used to return the follower, the one for the elliptical cam would have to be enough heavier to carry 1.7 more compression at the end of the stroke than the one for the cube cam, assuming an initial pressure of A S, in each one. The value 1.7 is found by comparing the lengths C P in Figs. 109 and 105. FlG. 105T Fig. 105. — (Duplicate) Acceleration Diagram for Elliptical Base Curve Cam Fig. 109. — (Duplicate) Acceleration Diagram for Cube Cam 280. Regulation of noise. If a cam follower, as for example a cam-operated disk valve, comes to rest on a seat at one end of its stroke, it is evident that it would be desirable for the follower to have the least possible velocity for at least a short distance before it reaches the seat, in order to provide against unnecessary striking velocity. Noise will be in some proportion to the velocity of the follower at the instant of seating. With this in mind, an examination of the velocity diagrams in Figs. 72 to 120 will show that the cube base curve, Case I,' Fig. 106, gives by far the best results, for, the vertical ordinates of the velocity curve in Fig. 108 are very much smaller as the follower approaches A than they are in any other diagram, excepting Case II of the cube curve, Fig. 116, but in this instance the advantage is more than offset by the high retardation values at the end of the stroke as shown in Fig. 117. The circular curve, Case II, comes next in the matter of giving small velocity to the follower, Fig. 112, but it does not possess the advantage of the cube curve when a spring is used to return the follower. The crank curve cam is least adapted of all the cams where quiet seating of a follower is desired, as may be observed by noting that the velocity curve, Fig. 88, for this cam is convex upwards, whereas the others are straight or convex downwards and thus have smaller initial vertical ordinates and, therefore, lower velocity. The full practical ad- vantage of cams which give low-sealing velocities and consequently a more quiet follower action, is offset to a considerable extent where the follower operates a valve which must admit a comparatively large volume of gas or fluid quickly. 148 CAMS 281. High speed cams. Cams intended for use on high-speed machines should give the smoothest possible motion to the follower, that is, should be free from sudden variations of velocity during the stroke and from shock due to sudden starting and stopping. A study of the velocity diagrams, Figs. 72 to 120, shows that the all- logarithmic and the straight-line base curves, Figs. 72 and 80, give extreme velocity right at the start in all cases; and that the logarith- mic-combination and straight-line combination cams will also give relatively high velocities at the start, Figs. 76 and 84. Therefore none of these cams would, in general, be suitable for high-speed work. Among the other cams some have an advantage at one end of the follower stroke where the rate of change in velocity is low, but they lose it at the other end where it is high as, for example, the cube cam Case II, as shown in Fig. 117; or they lose their advantage at the center or some intermediate point as in the elliptical cam, Fig. 105. 282. The cams specified in the preceding paragraph give rela- tively large sudden change of velocity to the follower either at one end of the stroke or the other, or at intermediate positions; and of the remaining cams, the parabola cam is the only one that gives absolutely uniform rate of change of velocity to the follower. The crank curve, the circular curve, Case I, and the tangential curve, Case I, give relatively good results, all being at a slight disadvantage compared with the parabola due to variations in acceleration of the follower. . This disadvantage, however, is small, and these three cams, together with the parabola cam, should give best results where there is high speed, provided they are accurately designed and made. 283. Balancing of cams. In addition to the forms of the curves here discussed for the pitch surfaces of cams that are to run at high speed, it is necessary to design the cam and so place the weight that the cam will be as nearly balanced as possible. This matter of bal- ancing is one of the greatest drawbacks to the use of the cam in high- speed work, for the very nature of a cam implies irregularity in form and hence difficulty in balancing. The face cam cut on a full cir- cular disk as illustrated in Fig. 2 comes nearest to a natural balance of any of the forms of radial cams. The trouble due to lack of natural balance in ordinary radial cams may easily be so decided as to render them quite impracticable in many cases where high speed and large stroke are required, unless elaborate balancing problems are solved in connection with the cam design. Small radial cams with small strokes have been made to run at exceedingly high speeds. CAM CHARACTERISTICS 149 The cylindrical cam, because of its natural balanced form with respect to the axis of rotation, is we'll adapted to high speeds. 281. Pressure angle factors for 20°, 30°, 40°, 50°, and (i()° for various FORMS of CAMS. Most of the base curves for cams are of such nature that it is only necessary to multiply the follower motion by a given factor and then multiply the product by 3(30 and divide by the number of degrees the cam rotates during the follower motion, to obtain the circumference of the pitch circle and the proper size of the cam for a given pressure angle. The logarithmic and tan- gential base curves are of such a nature that no one factor can be used for all data that include a common pressure angle. When these base curves are used the length of chart, if desired, must be com- puted by separate formulas for each problem. The logarithmic and tangential base curves are most easily applied by constructing the cam pitch surface directly from calculated values in each problem without the use of any chart whatever. 285. The factors for pressure angles for all base curves, excepting the logarithmic and tangential, are given in the accompanying Table of Factors for 20°, 30°, 40°, 50° and 60°. These factors are also laid off graphically in Fig. 132, thus enabling one to use inter- mediate values if desired. For partial comparison of the curves which have no general factor with those which have, the special fac- tor in each case for the single comparative problem which has been used throughout in designing the cams in Figs. 70 to 121 is given in the following paragraphs, and these factors are plotted to give the dash lines in the accompanying chart for factors. 286. Varied forms of fundamental base curves. Several of the base curves are, or may be, used in practical work with variations II \ ^V J Pitch Line \ t F \ , E _A ( 2.S V It Fig. 82. — (Duplicate) Straight-Line Combination Base Curve in details of construction, as, for example, in the straight-line com- bination curve, Fig. 82, the easing-off arc A E has a radius A B equal to the total rise of the follower, whereas it would be equally correct 150 CAMS c < O W Uh <1 K « H N C hJ l> + 1> i-H O ^H O 1-1 rH^^^^^O^C CO oo GO 00 CM 00 ^ CM r+l 03 N OO co CM OS ^ CM oo tH • CM ^H O _|_ lO H CM CO CM © CM I CM CO CO ^ O lO CO co co © © CO ,' CO ,' t- + 00 + o _J © oo ^ © CO CM M ^cm" o o ?MOOO(N(NNHO £©CMCO_l_CMCOlcO oq co co ^t 1 lo CM X m a 3 13 -^ faC -g o S 3 a i= S3 o .2 .2 "8 3 05 O bC faf 'c3 < uj X' Oj & .2 ^ " > -£ be M W x O C3 C3 "^3 'o S C3 ^03 PLh H § s o c3 "* > .2 § 03 o o T3 +3 fl r^ 03 03 2 >> H CO CO HH HH 0) L_( CO 03 CD r^ co w 03 - U « oT % > o 05 3 Si ° o CU o oT J8 > 6 s o a o u o r: 03 T3 3 c vC 03 o 0) fat; u X? c — 3 o3 u H i-i CN CO T)i lO CO 1> 00 OS CAM CHARACTERISTICS 151 <0 Go >T5 Of '-t ^ ^ r r 1 / * c> *** s 1 "9i / 1 / / #■ dv .5 / 57 V V // c7 / 6y / ^ v / fep/ &Mnl\h / / J / c j/ <3j / ■%7l V / ' 1 / / / IJ 7/ / 1 j/ ' / r / A^ // / j^ J ,f / 1 // {/ / // / / f / . / / */ / v/ ^^ ■A J* 1*2 .©* * / s7 $y .j- '^jtf s& 77\ y\ , ^ t — - "" l^' >f "<^ .4 t; ^**\ S V ^ y /&* •S ,>•/ £' 'Y _^ • ' ^ <-'' .v&#> -*"* c '0° V / • <»3 " Si . Is H or d w § « W r » - w « > w K 2 o a CO fn W 3 oiSuy eanssoJd 152 CAMS in principle to make this radius Y^AB. In this latter case the cam would be smaller for a given pressure angle, but the shock on starting and stopping would be greater. This case is not illustrated in Figs. 70 to 121 but is included under item 4 in the Table of Pressure Angle Factors, and also in the Chart of Pressure Curves, Fig. 132. Like- wise the factors for the elliptical base curve having a ratio of 2 to 4 instead of 7 to 4, are given in item 5 in the Table and also in the Chart, Fig. 132. The factors for a cube base curve made up of sym- metrical cube curves are also given in item 13 in the Table where it may be noted that this base curve gives an extremely large cam where small pressure angles are desired. 287. Methods of determining the cam factors. The methods of computing the cam factors for various base curves are briefly described in the following paragraphs. The letter h in the following formulas represents the motion of the follower, and the letter a the maximum pressure angle. 288. All-logarithmic and logarithmic-combination curves. These base curves do not have a constant factor for each pressure Fig. 78. — (Duplicate) Straight Base Line angle. The radius for the pitch circle in each problem is found by computation and graphics as described in paragraph 182 et seq. The factors for the data used in the charts shown in Figs. 70 and 74 are: For all-logarithmic cam: 20°, 2.28; 30°, 1.28; 40°, .76; 50°, .42; 60°, .21. For logarithmic-combination cam: 20°, 2.76; 30°, 1.69; 40°, 1.04; 50°, .62; 60°, .34. ?89. Straight-line base. Fig. 78. AR = FC cot a = hX cot a = IX 1.73 = 1.73. CAM CHARACTERISTICS 153 290. Straight-line combination base curve, Fig. 82. AR = 2AN + 2N X = 2h tan ( | j -f kota = 2 X 1 X .268 + 1 X 1.73 = 2.27. 291. Crank curve, Fig. 86. This curve may be regarded as the projection of a helix and, therefore, D Q equals the length of the Fig. 86. — (Duplicate) Crank Base Curve quadrant R which in turn is equal to^irh. The line E Q is tangent to the base curve at E. AR = 2DE = 2DQXcota = 1.57 h cot a = 1.57 X 1 X 1.73 = 2.72. 292. Parabola, Fig. 90. In a parabola, the subtangent D Q is equal to twice the projected length of the curve A E, and, therefore, DQ = h AR = 2 DE = 2 DQ cot a = 2 h cot a = 2X 1 X 1.73 = 3.46. c Fig. 90. — (.Duplicate) Parabola Base Curve 293. Tangential curve, case I, Fig. 94. This curve has no common factor for a given pressure angle and the radius of its pitch Fig. 94. — (Duplicate) Tangential Base Curve, Case 2 surface must be computed directly by formulas given in paragraph 222 without the intervention of a cam chart. For purposes of com- parison with other curves the following factors are given ; they apply 154 CAMS only for the data that have been used in the cams illustrated in Figs. 71 to 119. 20°, 5.28; 30°, 3.62; 40°, 2.82; 50° 2.36; 60°, 2.09. These values are shown in the dash line curve, No. 9, in Fig. 132. 294. Circular curve, case I, Fig. 98. The chord E C is per- pendicular to the line S T which bisects the angle C B E. This angle f Y \ Fig. 98. — (Duplicate) Circular Base Curve, Case 1 is equal to the pressure angle. The line E F is perpendicular to C S. Therefore angle C E F equals one-half of the pressure angle. Then E F = F C cot Y 2 a and A R = 2 E F = h cot Y 2 a = 1 X 3.73 = 3.73. 295. Elliptical curve, Fig. 102. The length of the cam chart for the elliptical curve for a pressure angle of say 30° may be most Tti Tve C L y Fig. 102. — (Duplicate) Elliptical Base Curve readily found by constructing several arbitrary elliptical charts, say four, each with a pressure angle factor, or length, of 2, 3, 4, and 5 re- spectively and each having a common height equal to the rise of the follower. Having constructed the elliptical curve in each of the charts, draw tangents in each case as at E, Fig. 102, and measure the angle E N X which will be the pressure angle corresponding to the factor or length assumed. Then, on any coordinate paper plot a curve with the pressure-angle factors as ordinates and the corresponding measured angles as abscissas. This curve will cross the ordinate CAM CHARACTERISTICS 155 which passes through the assigned pressure angle, in this case 30°, and the length of ordinate will give the desired cam factor. 296. Cube curve, case I, Fig. 106. The pressure angle factors for this case in which two unsymmetrical cube curve arcs are used Fig. 106. — (Duplicate) Cube Base Curve, Case 1 are specially computed by the formulas given in paragraph 242. The value of I in formula (1) when h = 1, will give the factor for whatever pressure angle is assigned to a. For a pressure angle of 30° I = 2.427 h cot a = 2.427 X 1 X 1.73 = 4.20. 297. Circular base curve, case II, Fig. 110. The complete factors for this curve are the same as for the circular base curve, Fig. 110. — (Duplicate) Circular Base Curve, Case 2 Case I, and are found in the same general way. In Case I the two arcs making up the base curve are equal; in the present case, they are unequal, and the formula deduced in paragraph 294 must be used for each arc. In this case, the first circular arc is required to lift the follower during % of its stroke, and, therefore, the distance A X in Fig. 110 will be, A X = .75 h cot Y 2 a = .75 X 1 X 3.73 = 2.80. The second circular arc is used for the balance of the stroke and, therefore, the distance, X R = .25 h cot Y 2 a = .25 X 1 X 3.73 = .93. 298. Cube curve, case II, Fig. 114. In this case the cube curve is used for % of the stroke and a circular arc for the remainder of the 156 CAMS stroke. The formula x = is used to compute the part ilof Tan a the cam chart length. The value of h is the follower's total motion Fig. 114. — (Duplicate) Cube Base Curve, Case 2 and that of / is the fractional part of the follower's motion during which acceleration takes place. Then AX = 3X 'J!L X 1 = 3.90. .577 The length X R is found in the same manner as in the preceding paragraph and is the same value, namely .93. 299. Tangential curve, case II, Fig. 118. This curve, like Case I of the tangential cam has no cam chart, unless it is specially Fig. 118. — (Duplicate) Tangential Base Curve, Case 2 desired to lay it out after the cam is drawn by making special com- putations based on the pitch circle as described in paragraph 225. For purposes of comparison the data used in this cam, as drawn in Fig. 119, are the same as for all other cams in Figs. 71 to 119, and for the data so used the pressure angle factors are: 20°, 13.02; 30°, 5.86; 40°, 3.36; 50°, 2.20; 60°, 1.57. These values are shown in the dash line curve, 16, in Fig. 132. SECTION VIII.— MISCELLANEOUS CAM ACTIONS AND CONSTRUCTIONS 300. Variable angular velocity in the driving cam shaft. The subject of variable angularity velocity in the drive shaft of a cam applies to all types of cams, but it is rarely met with except in oscil- lating cams. The reason for this is that in machinery, in general, the shafts that make a full turn do so with practically uniangular velocity except in slow-advance and quick-return motions and in some special cases, and, therefore, the shaft that operates a cam, in general, is considered to have uniform angular velocity. But with the oscillating cam the motion must come through a crank and connecting rod, or eccentric and beam, or some other device, from a shaft which, in general, turns with uniform angular velocity, and which gives to the oscillating cam a variable angular velocity as illustrated in Fig. 133 where the unequal arcs B\ G\, G\K\, K\L\ represent the distances traversed by the cam pin B\ while the main- shaft crank pin turns through the equal arcs B G,G K and K L. The method of building a cam which has variable angular velocity will be illustrated in the following problem. 301. Problem 29. Oscillating cam having variable angular velocity, toe and wiper type. Required an oscillating wiper cam, operated by a crank and connecting rod from a n ain shaft to raise and lower a straight-toe follower through a distance of one unit while the crank shaft turns through 120°. Assume the following dimensions : Main crank radius, C B, 4 units, Fig. 133; connecting rod length, B B\, 20 units; cam-arm radius, B\ 0, 5 units; shortest cam surface radius, A, 2 units. Find the distance the follower will move during each of three equal periods of time on the up-stroke. 302. The first step in the solution of the problem is to lay out the main crank center as at C in Fig. 133; then the crank-pin circle with a radius C B of 4 units, and next the connecting rod length of 20 units on the centerline as at E J. Lay off the assigned 120° of crank- shaft motion symmetrically about the main centerline as at BCD and with B and D as centers and the length of the connecting rod 157 158 CAMS a < K Q & „ o CO O fad w S >. H § § « a Ah 3 I M 00 CO MISCELLANEOUS CAM ACTIONS AND CONSTRUCTIONS 159 as a radius draw two arcs intersecting on the horizontal centerline, thus locating B\. With C as a center and the connecting-rod plus the crank as a radius, draw the arc passing through «/; with C as a center and the connecting rod minus the crank as a radius, draw the arc passing through J] . 303. To find the center of the cam shaft, Fig. 133, take Bi as a center and the assigned cam-arm radius of 5 units, and draw an arc, on which the point will be found later. On this arc find a point, by trial and error with the compass, which is the center of an arc which passes through B\ and which intersects the two arcs through J and Ji at the same elevation, as, for example, at L\ and l\. The center point so found is the point 0. The arc L\B\Fi will then be the arc of swing for the center of the cam-arm pin, and the angularity of action between the connecting rod and the cam arm at the two extreme ends of the cam-arm swing will be approximately the same. Draw a vertical line through and mark the assigned distance A which is the shortest radius of the cam surface. The horizontal line through A will be the lowest position of the flat-surface follower toe. The distance A V is equal to the assigned motion for the follower. 304. Having completed the general layout of the assigned data, the cam surface A V2 is found as follows: Draw the arc Bo Lo with a radius equal to Bi, and make the length B2 L2 equal to B\ L\. Revolve V about until it meets the radial line drawn from L2 to 0, thus determining the point V\. At this latter point draw a line Vi V2 perpendicular to V\. With the aid of any smooth-edged curved ruler draw a curved line tangent to A W at A and also tan- gent to Vi V2 at the point where it happens to come. Such a curved line is shown at A V2 in Fig. 133. Any other curved line tangent to the straight lines A W and V\ V2 would have done the work in the same time but would have given slightly different intermediate velocities to the follower as will be explained in a later paragraph. The actual working length A W of the follower toe is readily obtained by revolving the point of tangency V2 about until it meets the horizontal line through V at Vs. Projecting V3 down to A W and adding a short distance W W\ to prevent a sharp-edge action, the practical length A W\ is obtained. If the toe shaft is offset a distance A Y the total length of follower toe will be Y W\. 305. To find the distances moved by the follower toe during each of three equal periods while on the upstroke, divide B L, Fig. 133, into three equal parts as at G and K. With these points as centers 160 CAMS and with the connecting rod length as a radius construct short arcs intersecting B\ L\ as at Gi and K\ . Lay off the arcs Bi G\ and B\ K\ at B2 G2 and B2 K2 and draw the radial lines G2 and K2. Per- pendicular to these radial lines draw other straight lines, tangent to the curved cam surface A V2, thus obtaining the lines Hi H2 and 1 1 1 2. Revolving H\ and I\ back to the vertical line, the points H and / will be obtained and the distances moved by the follower during the three equal time periods on the upstroke will be A H, H I and I V respectively. 306. The path of contact between the cam wiper and the toe is shown by the curved dash line A Vs, Fig. 133. Points on this curve, such as at Is, are obtained by revolving the point of tangency 1 2 around until it meets the horizontal line through /. 307. Other considerations relating to variable angular velocity drive, brought out in this problem (Problem No. 29) are that the follower toe takes a longer time for the down-stroke as shown by the length of arc L D as compared with L B, Fig. 133. This could be rectified and both times made the same, if desired, by placing the center of the cam so that the points B\ and L\ would be on the horizontal line through C. This would only be possible with certain limited combinations of lengths of crank arms and rods, and in any event the intermediate velocities of the follower would be different on the up- and down-strokes. If it were desired to know the distances moved by the follower during three equal periods on the down- stroke the equally spaced points M and N, Fig. 133, would be obtained and used in exactly the same way as explained for G and K in para- graph 305. The point F is the outward dead center position of the driving crank pin and is found by continuing the straight line through F\ and C to F. When the driving crank pin is at F, the cam surface is in the position shown by the dash line A3 W5 and A\ is at A. While B is moving from D to F, A\ is moving to A and the follower toe is at rest, being supported by the cylindrical surface A A\ rub- bing against it, or it may be supported by a resting block indicated at £7. It is sometimes thought that this toe-and-wiper cam is prac- tically free from rubbing action especially where the length of the toe surface equals approximately that of the wiper, but it will be seen from the velocity diagram shown just above the cam and described in paragraphs 317 and 318, that there may be considerable rubbing. MISCELLANEOUS CAM ACTIONS AND CONSTRUCTIONS 1G1 o 5 * 2 .2 3 a 00 Q ° / k ^ •I ■ / I ' ' * ' I ' ' ' M ~i CO l-< O >— < •oag Jod •} j 162 CAMS There must be some sliding in all flat-toe followers where the acting surface is perpendicular to the right-line motion of the follower, as it is in Fig. 133. 308. Exercise problem 29a. Oscillating cam having vari- able angular velocity. Required an oscillating wiper cam, oper- ated by a crank and connecting-rod from a main shaft, to raise and lower a straight-toe follower through a distance of three units while the crank shaft turns through 150°. Find, also, the distances that will be traversed by the follower toe during equal intervals of time on the up-stroke. Assume the following dimensions: Main crank radius, 5 units; connecting-rod length, 30 units; cam-arm radius, 7 units; shortest cam surface radius, 4 units. 309. TOE-AND-WIPER CAM WHERE TOE IS CURVED. In the toe- and-wiper cam explained in the paragraphs immediately preceding, a flat surface toe Y W, Fig. 133 was used. A curved toe such as is shown at A W, Fig. 134 may be used as illustrated in the following problem. 310. Problem 30. Required a wiper cam to operate a curved-toe follower which shall move: (a) Up 4 units on the elliptical base curve where the ratio of axes is 2 to 4, while the cam turns 45° in a counter-clockwise direc- tion with uniform angular velocity. (b) Down 4 units on the same base curve, while the cam turns 45° in a clockwise direction with uniform angular velocity. 311. While the follower toe may have the form of any smooth curve which is convex to the cam wiper, an arc of a circle will be assumed because of the ease in drawing. The general principles of construction are the same for this problem as in Problem 12. The shortest radius A of the wiper cam, Fig. 134 is assumed. The form of the curved toe is the circular arc A W, with its center at Ai. It is convenient in such a problem as this to work with the center points of the follower arc, and, therefore, the 4 units of travel are laid off first at A\ Vi instead of A V. The semi-ellipse in which h Ui: IiVi : : 2 : 4 is drawn and the perimeter divided into equal parts at J', Ui, H' . Only four construction points are used in this problem in order to secure as much simplicity as possible in the illustration. In practice, more construction points should be used. The four construction centers at Hi, h, Ji, Vi, are revolved to their corre- sponding positions relatively to the cam at H2, 1 2, J 2, and V2 and MISCELLANEOUS CAM ACTIONS AND CONSTRUCTIONS 163 B _ Fig. 137. Diagram or Sl'dino Velocixt a* v< ./, H t Fig. I'M. — Pkublem 30. Oscillating Cam with Ctjkved-Toe Follower 164 CAMS the toe-arcs drawn as shown at #3, h, Js, and V3. The wiper cam curve A C is then drawn tangent to these arcs and the tangent points revolved back to their actual positions at Hi, h, J a, and F4, thus obtaining the locus of contact between the wiper and toe. This locus is shown by the dashline curve A H4 F 4 . The necessary length V Y\ of the follower arc is also obtained by projecting the extreme point Y on the locus to Y\ and adding an arbitrary distance such as Y\ W\ to avoid wear at the tip end. 312. If an irregular curve had been used for the form of the toe instead of a circular arc it would have been necessary to construct a template of the desired form of the toe and to move it out radially the desired distances on each of the radial construction lines H2, 1 2 • . ., keeping the template always in the same relative position with each of the radial lines. At each of the four adjustments of the template, arcs would have been drawn against the template edge and the work then continued as described in the preceding para- graph. 313. The pressure angles in the toe-and-wiper cams are quite different for flat and curved toes. In Fig. 133 the line of pressure is always parallel to the axis Y Y\, of the follower rod, as illustrated by the vertical line at W Vs ; and the maximum leverage with which it acts on the bearings is Y W. With the curved-toe wiper the line of pressure is an inclined line and the pressure angle at the top of the stroke is V V\ V±, Fig. 134, and when the follower is half way up the pressure angle is 1 1\ I±. 314. Exercise problem 30a. Required a wiper cam to oper- ate a curved-toe follower which shall move: (a) Up 3 units with uniform velocity while the cam turns 60° in a counter-clockwise direction with uniform angular velocity. (b) Down 3 units with uniform velocity while the cam turns 60° in a clockwise direction with uniform angular velocity. 315. Rate of sliding of cams on follower surface. The rubbing velocity of cams which are in sliding contact with the fol- lower, may be readily determined by constructing simple velocity diagrams at each of the construction points, as explained in the fol- lowing paragraphs. 316. Problem 31. Rate of sliding between cam and flat follower surfaces. Find the curve of rubbing velocity between surfaces in a toe-and-wiper cam mechanism where the follower toe is MISCELLANEOUS CAM ACTIONS AND CONSTRUCTIONS 165 a flat surface. Assume that the wiper oscillates with uniform angular velocity. 317. In Fig. 135 let the angle I± h represenl the uniform angular velocity of the wiper cam. Then the point li on the cam will have the linear velocity I± 1 5. Laying this value off at h h, where 1 2 Fig. 136— Diagram of Velocities Fig. 135. — Problem 31. — Sliding in Toe-and-Wipeb Cams comes into action, and taking the component h It, the actual rubbing velocity is obtained. This may be transferred to h h in Fig. 136 and, finding other ordinates, the complete sliding-velocity curve A\ V7 is obtained. The ordinate A A\ is quickly obtained, for it is obviously equal to the linear velocity line A A\ in Fig. 135. In Fig. 135 the detail construction for obtaining the velocity of the fol- lower is shown only at one point, h, the construction at the other 166 CAMS points being the same. Also all lines pertaining to the construction of the cam are omitted, as they are fully given in Fig. 52. 318. The actual rate of sliding in feet per second may be readily found at any position by means of the velocity diagram in Fig. 136. For example, if the cam shaft 0, Fig. 135, is consid- ered to oscillate back and forth through 45°, 100 times per minute with uniform angular velocity, and if the radius I± is 14 inches, the line I4 1 5 will be drawn to represent a velocity of 2 X 14 X 3.14 X 2 X 100 . A . . . , 8 X 12 X 60 " = 3 ' 04 feet Per SeCOncl This value, laid off as the resultant velocity at 1%, gives a component or sliding velocity I3 I7 which is laid off at 73 I7 in Fig. 136. Other ordinates, found in the same way, will give the curve A\ V7, showing the sliding velocity between cam and toe in feet per second. The minimum rate of sliding will be A A\ shown in both Figs. 135 and 136, and will be 1.6 measured on the same scale that was used to lay out hh. 319. The velocity of the follower, in feet per second, may also be readily found by simply taking the vertical component ^3 Is, Fig. 135, and laying it off at h Is in Fig. 136. Taking the vertical components at other points the line A Vs, showing the linear velocity of the follower will be obtained. The line A Vs is a straight line in this problem because this cam illustration was taken so that the follower would have uniformly increasing velocity. In general, where the cam curve A V2 in Fig. 135 is assumed, the line A Vs in Fig. 136 will not be straight. 320. Exercise problem 31a. Sliding velocity between cam and follower. Assume a flat-toe follower with a rise of 3 inches and a cam wiper of minimum radius of 4 inches which oscillates with uniform angular velocity through 150 cycles per minute. Construct the toe-and-wiper surfaces and find the curve of sliding velocity between them in feet per second. Find also the curve of linear veloc- ity of the follower toe and state the maximum velocity in feet per second. 321. Problem 32. Sliding velocity with curved toe fol- lower. Find the curve of rubbing velocity between surfaces in a curved toe-and-wiper cam mechanism, assuming that the wiper oscillates with uniform angular velocity. MISCELLANEOUS CAM ACTIONS AND CONSTRUCTIONS 167' In curved-toe followers the general principle of obtaining the rubbing velocity is the same, although the del ail of drawing the velocity diagram differs slightly. In Fig. L34 the linear velocity of the point Hz on the cam is //.-, He and this value is laid off at II \ II-. The direction of sliding at this phase must be that of the common tangent line to the two surfaces, and its length, which represents the velocity of sliding, is found by drawing the line H7 Hs parallel to the direction of motion of the point 1I± on the follower. The length of H4 Hg is thus found and is laid out as shown in Fig. 137, directly over H± of Fig. 134. Other lines representing the rubbing velocity are similarly found and laid out in Fig. 137, thus obtaining the rubbing velocity curve AiUs Vg. 322. In the case of the curved-toe follower it will be noted that that portion of the toe from V4 to Yi, Fig. 134, will be traversed twice as often as the portion from V to F4, and in addition the rubbing velocity will be much greater. In the flat-toe follower, Fig. 135, the point of contact travels regularly forth and back the full distance on each stroke, but the wear as in the curved-toe follower will be irreg- ular, due to the variable rubbing velocity, which in the case illustrated in Fig. 136 is a maximum at the tip end. 323. Exercise problem 32a. Sliding velocity with curved- toe follower. Find the curve of rubbing velocity between cam surfaces in Problem 30a, assuming that the wiper cam oscillates through a cycle 90 times per minute. Show scale for curve. 324. Problem 33. Sliding velocity where cam has variable angular velocity. Find the curve of rubbing velocity between surfaces in a flat toe-and-wiper cam construction, assuming that the wiper cam oscillates with a variable angular velocity. 325. When an oscillating cam has variable angular velocity, as in Fig. 133, the extent of the sliding action between cam and follower may be found as in the present example. In Fig. 133, the length of crank represented by C E is 4 inches and the crank is assumed to be turning 120 revolutions per minute. The velocity of the crank pin ... . 2X 3.14 X4 X 120 will then be r ^ f - = 4.19 leet per second. 1Z X. OU 326. The velocity just obtained is represented by the line K U, Fig. 133, laid off to any convenient scale. Its component K L\ along the rod is found by dropping from U a perpendicular to the con- necting-rod position KK\. The component K U\ is then trans- ferred to the other end of the rod at K\ Uo- This component gives a 168 CAMS resultant linear velocity of K1U3 to the cam crank pin at the phase K\. At the radial distance K3, which is equal to the radii 01 2, and 1 3 the linear velocity will be K3 U± and this transferred to I3 will give I3 U5 as the resultant linear velocity of I2 when it becomes the driving point. The line 73 Uq is the component in the direction in which sliding must take place and this is laid off at I3 Uq in Fig. 138. If K U represents 4.19 feet per second, I3 Uq, will represent 1.30 feet per second to the same scale and the maximum velocity of sliding, which is represented at A3 Aq, will be 1.87 feet per second. 327. Exercise problem 33a. Sliding velocity where cam has variable angular velocity. Assume crank C E, in Fig. 133, to be 5 units long and turning at rate of 100 revolutions per minute; also, take the angle B C D = 150° symmetrical about C E, the con- necting rod B Bi = 30 units, the cam arm Bi = 7 units, the mini- mum cam radius A = 4 units and the cam lift 3 units. Construct the cam and follower and draw the curve of sliding velocity to scale. 328. Elimination of sliding friction where flat or curved surface followers are used. The ordinary toe-and-wiper cam mechanism operates with more or less sliding action as shown in the preceding paragraphs. Cams resembling the toe-and-wiper type may be constructed so as to eliminate all sliding friction by using special curves and lines for the wiper and toe surfaces as will be explained in succeeding paragraphs. Fig. 139 shows a straight sur- face toe moving up and down in a straight line while in Fig. 146 a similarly moving toe has a curved working surface. In both there is pure rolling action. Likewise, in Figs. 142 and 145 the working surface of the follower arm is straight in one case and curved in the other, yet in both cases there is pure rolling action. In all cases of pure rolling action on flat or curved surfaces it is impossible to assign various intermediate velocities to the follower as part of the data of the problem. 329. The principle of pure rolling action between cam surfaces. It is a fundamental principle of pure rolling action between two rotating surfaces that the point of contact between them must always be on the line of centers. This is illustrated in Fig. 141 where the point of contact, C, is on the line of centers A B, and where the contact point between the curves C D and C E will always be on the line of centers. This principle also applies in Fig. 139, where the follower toe B D is moving up and down in a straight line and where it must be considered that the toe is turning about a point on the line MISCELLANEOUS CAM ACTIONS AND CONSTRUCTIONS 109 A B F a.t an infinite distance. Then A F becomes the line of centers and the point of contact between B C and B D will always be on the line B F. 330. Well-known curves that lend themselves readily to pure rolling action in cam work are the logarithmic spiral and the ellipse. Examples of these will bo given in following paragraphs, where the solutions are entirely graphical and comparatively simple. The parabola and the hyperbola may also be readily used for rolling cam surfaces. Any line or curve that may readily be expressed by a mathematical equation may also be taken as one surface and the equation for the other curve that will work with it in pure rolling action may be derived. An example of this is given in paragraph 3 1<>. The use of the logarithmic curve for pure rolling action in the toe- and-wiper type of construction where the follower toe has a straight- edge working surface and moves in a straight line is given in the paragraphs immediately following. 331. Problem 34. Pure rolling with flat surface follower. Required an oscillating logarithmic cam arm that will give a straight- line reciprocating motion to a flat-surface follower arm, with pure rolling action: (a) The follower to move up 434 units, while the cam turns 30°. (b) The pressure angle to be 20°. 332. This problem is illustrated in Fig. 139 where the flat-surface toe B D is moved from the solid-line position to the dash-line position while the cam ABC swings through the angle C A F. The method of constructing the problem is as follows : Draw the horizontal line A F, Fig. 139, and from any point B draw a line B D making an angle with B F equal to the assigned pressure angle. Continue B D until the vertical distance between it and B F is equal to the assigned lift of the follower, 434 units in this problem as measured at D F. Mark the point F. Assume the dis- tance A B sufficient to allow for the cam shaft and cam hub. A B is taken as 4 units in this problem, and A F is found upon measuring, to be 1G units. Substitute these values in the following general equation : _ 180° X tana R = 180° X .364 _ ~ tt X .434 l0 ^ r 3.14 X .434 X A) ° 2 " 28 ' 8 in which r = 4, R = 16, a = 20°, and in which 6 gives the angle whose limiting radial line A C is equal in length to A F : 170 CAMS 333. The angle of 28.8° is then laid off at F A C as shown in Fig. 139 by means of a protractor. If a protractor is not at hand this angle may be readily constructed with the aid of a trigonometrical table from which the tangent of 28.8° is found to be .55. Lay off A E equal to one unit on any independent scale and draw a perpen- dicular line E H at E. On this line lay off .55 of this unit thus obtain- ing the point H. The angle E AH will then be 28.8°. Draw A H and continue it to A C making A C = A F = 16 units on the scale Fig. 139. — Problem 34, Oscillating Cam with Pure Rolling Action on Flat Sur- face Follower of the cam drawing. The logarithmic curve through B and C will be the one which will work in pure rolling action with the straight line B D. 334. To obtain other points on the curve B C as at J, assume intermediate values for R in the above formula, r remaining the same as before. Taking R at 14 units and again solving the equation, is found to be 26° and this angle is laid off at F A J. A J is made 14 units in length. In like manner other points shown by dots between J and B may be found by taking R equal to 12, 10, 8, and 6 MISCELLANEOUS CAM ACTIONS AND CONSTRUCTIONS 1, M and K\ L are obtained. 406. With the above type of cam, extreme accuracy is necessary in manufacture to overcome any binding action of the rollers on the cam disk. To overcome this a cam construction has been devised in which the two arms B C and B D, Fig. 163, are entirely separated, the former being keyed to the shaft B and the latter free to turn on shaft B. The two arms are then connected by a spring, as illustrated in Fig. 166, which keeps them drawn to each other, and both having 204 CAMS the desired pressure on the cam surface. To prevent too great a pressure of the arms on the cam surface, should too heavy a spring be used, a stop pin is cast on each arm and these stops come together just as the follower rollers touch the cam surface when newly adjusted. 407. Cam shaft acting as guide. A special form of construc- tion for guiding the cam follower is frequently used as illustrated in Figs. 164 and 165. The cam B in Fig. 164 is the simple radial cam and is constructed for any given data as explained in paragraphs 49 et seq. It moves the roller C, which is attached to the forked arm R D, back and forth in approximately a radial line the distance A M minus A L which is equal to the chord of the arc E D. The arc E D measures the swing of the shaft F. The follower rod D R is under Fig. 164. Fig. 165. Fig. 164. — Cam Shaft Guide Takes Place of Crosshead Guide Fig. 165. — Positive Drive with Cam Shaft Guide definite control all of the time, although its form of construction is extremely simple and the number of parts a minimum. The forked end R R of the follower rod bears with a snug fit against the two sides of the cam shaft, or against adjustable collars attached to the shaft. In Fig. 164 the follower shaft F is returned to its initial position by means of the spring H. 408. Positive dkive with cam shaft as guide. A cam giving positive motion where the cam shaft is used as a follower guide is illustrated in Fig. 165. The cam itself is a face cam and is con- structed for any given data as directed in paragraphs 96 et seq. The pin C is attached to the follower rod D R and is moved back and forth in approximately a radial position by the amount A M minus A L, equal to the chord of D E. The forked end of the follower rod, MISCELLANEOUS CAM ACTIONS AND CONSTRUCTIONS 205 bearing against the sides of the cam shaft A, together with the guided end D of the rod give it a motion that is under control at all phases, and this with a minimum amount of mechanical construction. The shaft F is under positive cam control all the time on account of the use of the face cam, and no return spring is necessary as in Fig. 104. 409. Positive drive double disk radial cam w t itii swinging follower. A special form of cam and follower construction, where positive action is desired, is shown in Fig. 166 where the following data are so taken: (a) That two follower arms 10 units long shall each swing through an angle of 20° with uniform acceleration and retardation while two corresponding radial cams turn through 135°, the drive to be positive with each roller having a single point of contact. (b) That the follower arms shall be returned with positive action while the cams turn through 225°. (c) That the angle between the two radial follower arms shall be 50°. 410. The follower shaft A, Fig. 166, is first laid down, the angle of follower-arm swing of 20° then drawn as at B A C, and finally the 10 units for follower-arm length laid off at the initial position A B. The horizontal centerline E O for the cam shaft is then drawn across the arc B C so that the midpoint D is as much above it as the end points B and C are below. The radius D of the pitch circle is com- puted in the usual way, taking the chord B C for the distance moved u +u f ii • + TU n ^ BCX 3.46X360 _ _ xl by the follower point. Then D O = = 5.16, thus locating the cam center 0. The circle represented by A A\ is then drawn and the pitch surface, indicated by the short portion B Bi is constructed in exactly the same manner as explained in Problem 8. The size of the roller is assumed as shown at B F and the working surface of the operating cam F G is drawn. The operating arm A B is keyed to the shaft A . 411. The return arm A H, Fig. 166, is not keyed to the shaft A but turns freely on it instead. The motion of this arm should be identical with that of the arm A B and, therefore, the swinging arc H J of the center of the follower roller is made the same as the arc B C, and it is similarly divided. The pitch surface of the return cam is represented in part at H Hi and is found in exactly the same way as directed in the preceding paragraph, and the working surface 206 CAMS K L drawn. The spring at M exerts more pull than is required to return the follower, and, therefore, it holds the two follower arms against the cams with practically uniform pressure, even should there be slight inaccuracies in workmanship, or wear in the contact surfaces. A lug is attached to each follower arm as shown at P and Q, Fig. 166. — Oscillating Positive Drive Double Disk Cam and these act as stops in preventing excessive pressure of the rollers on the cam surfaces. Cams used in this way have been called duplex cams. 412. Rotary sliding yoke cams giving intermittent har- monic motion. A yoke cam driven by sliding contact instead of roller contact is shown in Fig. 167. The cam, in this figure, is in the form of an equilateral triangle bounded by equal circular arcs having a radius equal to the straight sides of the inscribed triangle. The cen- MISCELLANEOUS CAM ACTIONS AND CONSTRUCTIONS 207 ters for the circular arcs are at the apexes of the i riangle. One of the apexes of the cam is at the center of the driving shaft. The motion given to the follower yoke will be an intermittent one, dwelling at the ends of the stroke, and the total travel will be equal to the radius of the cam surface. The follower will travel from one end of its stroke to the other with a simple harmonic motion the same as with crank and connecting rod where the connecting rod is assumed to be of infinite length. Or, the motion during the stroke will be the same as with the Scotch yoke, or crank and slotted crosshead, where the radius of the crank is one-half the radius of the present cam sur- face. 413. A diagram of the motion of the yoke follower in Fig. 167 is shown at M N S. With the cam turning as shown by the arrow, the Fig. 167. — Sliding Yoke Cam Giving Harmonic Motion follower H K will move the distance M N while C on the cam turns 60° to D, and the cam edge at C will do the driving with a scraping, sliding action. While C is turning 60° from D to E, the follower will remain at rest; while C is turning 60° from E to F the curved surface A C of the cam will be driving the follower the distance P with a rubbing, sliding action and increasing velocity; while C is turning from F to G the cam edge C will again be driving, the follower moving the distance P Q with a scraping, sliding action and decreasing velocity. The smooth working surface of the follower yoke is shown from FtoB, while the recessed surface as at W may be left rough cast. The velocity and acceleration diagrams for the equilateral sliding yoke cam here described have the same characteristics as those shown for the ordinary crank curve cam, illustrated in Figs. 88 and 89. 414. Rotary sliding yoke cam giving reciprocating har- monic motion. A circle passing through the points A B C, Fig. 167 208 CAMS Fig. 168. — Sliding Yoke Cam General Case would represent the surface of a cam attached to the crank A J, and such a cam, instead of the equilateral curved side cam, which is shown, would give a simple harmonic motion to the follower yoke without finite periods of rest at the ends of the stroke. Such a circular cam would be an equivalent of a crank and slotted crosshead where the radius of the crank would be equal to the radius of the cam circle. 415. A ROTARY SLIDING YOKE CAM, GENERAL CASE, With the Cam surface entirely surrounding the shaft is shown in Fig. 168. To lay out this cam for a definite range of motion, say 2 units, draw the indefinite circular arc B C with any desired radius and the arc E D with the same center and with a radius 2 units larger. Then with a radius equal to A B plus A E and a center anywhere on the arc E D draw the arc C D until it intersects E D as at D. With Das a center and the same radius as before draw the arc E B com- pleting the cam. The student should be able to determine the angles traveled by the cam while the follower is at rest, the angles of motion, the range of motion of the follower, and the exact portion of the follower working surface which has to resist the wear due to sliding action. 416. Cam surface on reciprocating follower rod. In some special forms of cam construction it is more convenient to place the cam curve on the follower than on the driver. Such a case is illus- trated in Fig. 169 where the cam curve E F E' is on the sliding fol- lower bar K G. The driving crank A F carries a pin at F which slides in the cam groove. The mechanism here shown is a modification of the Scotch yoke, or " infinite connecting rod." The motion in this case is such that the follower remains stationary, while the driving shaft turns through the angle C A C. The curve C F C is an arc of a circle with A as a center. The follower then picks up motion comparatively slowly, the point G being at the points 1,2, 3, etc., when the crank pin F is at the points which are correspondingly represented in Roman numbers. When the crank pin is at J, G is at N and it then moves very rapidly from N to P while the crank pin travels from J to Q. Very often, in cam work, the driving shaft A has only an oscillating motion through 90° or less. If the curve C F C is changed slightly so as not to be an arc of a circle with A MISCELLANEOUS CAM ACTIONS AND CONSTRUCTION 2()<) as a center, the end G of the follower bar will not come to rest for a definite period at the end of the stroke, but it will have a slow, power- ful motion which may be made use of in manufacturing processes where compression is required. 417. Problem 42. -Definite motion where cam surface is on follower rod. In Fig. 169 a follower rod G K has a cam surface formed at the left-hand end from E to E' , and it is driven by a simple crank pin represented at F so as to secure a desired or known motion. In the illustration let it be desired: 1st. That the follower rod shall remain at rest at the head end of the stroke while the driving crank pin turns 45° (22J^° on each side of the centerline A F). D /niv UI J\E %-fT vf\_ h mvif P 6 — 1 — 1 — 5 4 4' N 3 2 1 hri'l k' 1' Ljcr 1 Ik \\e' Fig. 169. — Cam Surface on Reciprocating Follower Rod 2d. That the follower will be moved to the left a distance G N with uniform acceleration while the crank pins turns 67^2- 3d. That the follower shall move the remainder of the stroke from N to P while the crank pin turns 90°. 4th. That the follower rod shall move in reverse order on the return stroke from P to G. 418. Before starting the solution of this problem it should be stated that, one cannot, because of either theoretical or practical considerations, or both combined, always secure desired results in cams of this type where arbitrary distance and motion assignments are given as in this illustration. It is nevertheless advisable to complete the solution of the problem, if possible, on the basis of the desired data, because one can then make the necessary modifications 210 CAMS with a sure knowledge that the least departure has been made from the theoretical or assigned conditions. 419. The method of solution for the above data is as follows: Assume the driving crank length A F and draw the crank-pin circle F J M. Lay off the angle F AC equal to 22^°. The circular arc F C will then be part of the pitch line of the follower cam head, and while the crank pin F is moving through this arc the follower rod will not move at all. To secure uniform acceleration of the follower for the distance G N, divide G N into 9 equal parts and mark the 1st, and 4th division points as indicated at V and 2' in the figure. This will be the first step in securing the uniform acceleration called for because the distance from Gtol' will be one unit, from 1' to 2' will be three units, and from 2' to 3' will be five units. By dividing G N into three parts as here described, three construction points will be secured on the cam curve. If more construction points are desired, G N may be divided in 16 equal parts and the 1st, 4th and 9th inter- mediate division points taken, thus obtaining four construction points on the part of the pitch surface of the cam from C to E. Like- wise, if five construction points are desired, G N would be divided into 25 equal parts, and the 1st, 4th, 9th, and 16th division points taken. 420. Since the motion from G to N is to take place while the crank pin moves 67^° as called for in the data, and since three construction points have been used in this illustration, the 67^° arc from C to J is now divided into 3 equal parts as indicated at I and 77 in Fig. 169. At I draw a horizontal line and make the distance I-R equal to l'-G; at II make the distance IIS equal to 2 f -G; and at J, make the dis- tance III-E equal to S'-G. A curve through the points C, R, S and E will be the pitch line for the cam surface on the follower rod for uniform acceleration from G to N. The p oint 8' coincides with N. 421. The part of this pitch curve from R to E is shown by a dash line and is not practical because of the sharp curvature from S to E, which would produce too large a pressure angle and this in turn would give a large bending moment on the follower arm and large side pressure in the bearing H. This part of the curve should, therefore, be modified, and a good plan on which to effect the mod- ification is to start by making the pressure angle as large as is prac- tically allowable and then to keep the new curve as near to the old as possible. A maximum pressure angle that is safe under all ordinary circumstances is 30° and, therefore, the first step in the modification MISCELLANEOUS CAM ACTIONS AND CONSTRUCTIONS 211 will be to draw a vertical line through E, the end of the theoretical curve, and make an angle of W E V equal to the maximum practical pressure angle of 30°. The line V E is then produced until it crosses the dash curve, and a smooth curve is next drawn so as to connect the straight line and the original curve. This will leave, in this case, E T as a straight 30° line, T R as a new assumed part of the pitch curve, and R F as the portion of the original curve that remains. If the cam is to turn slowly, or if the load on the cam is not large, a greater pressure angle could be taken at W E V and then the arbi- trary new curve would come closer to the original or theoretical curve. 422. The practical pitch line of the cam is now found to be FCRTE. The cam will run smoothly and the variation in the motion of the cam from the originally desired motion may be par- tially indicated by pointing out that the end G of the follower will be at 2, and at 4 instead of 2' and 4' as originally intended. This varia- tion may be most completely shown by a velocity diagram which will be taken up in a succeeding paragraph. 423. The pitch curve F T E, it will be noted, has been con- structed to give a definite practical action to the follower from G to N. Since the curve F E is now determined, and since the crank pin must drive through the same cam slot from E to F while it turns through the remaining arc J Q, it follows that the motion of the rod from N to P cannot be assigned, and that it must be taken as it comes. To find out in a general way what this motion will be it is only necessary to pursue in reverse order the methods already used; i.e., to lay off the distance T-IV at G~4, the distance R-V at G-5, etc. By noting the distances N~4, 4~5> etc., which the follower rod travels in uniform periods of time, some useful idea of the retardation, and consequently of the smoothness of action of the cam may be obtained as it approaches the inward end of its stroke. In the illustration the follower rod will slow down perceptibly from N to 4, and have slightly higher but a fairly uniform velocity from 4 to 5, and from 5 to 6. It will retard rapidly from 6 to the end of the stroke. 424. The lower part of the pitch curve from F to E' will be made symmetrical with the upper part from F to E in this problem thus making the action of the follower on the return stroke the reverse of what it is on the forward stroke. If it were desired, the curve F E' could be constructed, by the methods described above to give 212 CAMS the same characteristic motion to the follower on the return stroke as it did on the forward stroke. 425. An exact knowledge of the effect of arbitrarily changing the theoretical curve R S E, Fig. 169 to R T E may w y Fig. 169. — (Duplicate) Cam Surface on Reciprocating Follower Rod be readily obtained by a time-velocity diagram construction as illustrated in Fig. 170. In the latter figure let the length of the base line F Q represent the time necessary for the crank pin to make a half revolution from F to Q, Fig. 169. Since the crank pin is assumed to travel with uniform velocity, the line F Q, Fig. 170, is divided into eight equal parts the same as is the semi-circle F Q in Fig. 169. The velocity of the follower at each of the construction points is then found as indicated in the following paragraph. 426. At the point II, for example, in Fig. 169, draw the tangential line II-B of any desired length. This line will represent the velocity of the crank pin in feet per second, which may be readily computed, for, if the crank A F is 4 inches long and makes 120 revo- lutions per minute the point F will be moving with a velocity of 4 120 ^ X 2 X 3.14 X w = 4.19 feet per second. Q6 5 4J21CF Fig. 170. — Problem 42, Time- Velocity Diagram for Re- ciprocating Follower Rod Shown in Fig. 169 MISCELLANEOUS CAM ACTIONS AND CONSTRUCTIONS 213 Through the point B draw a line B D parallel to the line that is tan- gent to the cam pitch curve at T. The line V E, continued, is tan- gent to the cam curve at T because it will be remembered that the practical curve from R to T was taken so as to be tangent at its upper end to the straight line E T. The distance II-D will represent the velocity in feet per second with which the follower rod is sliding through the bearing at H. This velocity is laid off in the time-velocity diagram in Fig. 170 at 2-D. In a similar manner other points on the solid-line curve C D Q may be found. This curve shows at a glance just how fast the cam follower is moving at every phase of its stroke. 427. The dash line construction in Fig. 170 shows the follower velocities called for in the original data, but abandoned, as explained above, because of the large pressure angle involved. The point on the dash curve is found by drawing the line B 0, Fig. 169, through B parallel to the short straight dash line which is shown tangent to the theoretical curve at S. Then II-O would represent the velocity of the follower bar at phase 77 if the original data were used. As a check on the accuracy of the construction the points C, L, and X, Fig. 170, should all be on a straight inclined line, because C X is a velocity line and it must show uniformly increasing velocity for the follower in order that there may be uniform acceleration as called for in the original data. 428. The difference between the solid and dotted parts of the velocity diagram in Fig. 170 shows the effect on the velocity of the follower of arbitrarily changing the theoretical cam curve R S E, Fig. 1G9, to the more practical cam curve R T E. 429. Problem 43. Cam surface on swinging follower arm. When the cam surface is on the follower and it is desired that the follower shall have a swinging motion instead of a rectilinear recip- rocating motion as it had in Fig. 1G9, the method of construction will vary in detail as illustrated in Fig. 171. The data for Fig. 171 are, that the driving crank A C with a crank-pin roller at G shall swing the follower shaft B through an angle of 30° counterclockwise with uniformly increasing and decreasing angular velocity while the driving shaft turns through 60° with uniform angular velocity in the same direction. 430. The method of locating points on the curve C F of the fol- lower cam pitch surface, Fig. 171, follows: Divide the assigned 30° arc, C E, into any number of parts, say six, which are as to each other as 1, 3, 5, 5, 3, 1. This will provide for the uniformly increasing and 214 CAMS decreasing motion to the shaft B. Divide the assigned 60° driver arc, C D, into six equal parts. The method of locating the point L, which is the second construction point on the cam curve, will be taken for explanation purposes. Other points are found in the same way. Draw a radial line B 2 through the second construction point, con- tinuing it to J which is on an arc which passes through 77 on the arc C D . Lay off the arc J K at II-L thus obtaining the point L on the cam curve. This form of cam has positive action. When it is Fig. 171. -Problem 43, Cam Surface on Swinging Follower Arm Having Uniform. Angular Acceleration and Retardation allowed to reach a dead center position as shown in Fig. 171, auxiliary action will be required in starting. 431. Effect of swinging transmitter arm between ordinary radial cam and follower. In Fig. 172 let B C D E F be an ordi- nary radial cam with straight sides as at B H rounded off by circular arcs with center as at G. Let I J K be the swinging transmitter arm with the working surfaces at J and K as arcs of circles with centers at L and M respectively. Let N N f be the centerline of the follower rod which moves straight up and down. 432. In order to reach a useful understanding of the action of this type of cam construction it will be necessary to learn the rate of MISCELLANEOUS CAM ACTIONS AND CONSTRUCTIONS 215 change of velocities in the follower parts so as to judge the accelera- tions and retardations which cause the most trouble at high speeds, also to learn the rates of sliding at J and K, and then to balance t hese against the pressure angle produced by the same radial cam with an ordinary direct roller-end follower. Fig. 172. — Swinging Transmitter Arm with Sliding Action 433. The method of analyzing the cam action in Fig. 172 will be pointed out by using six equally spaced construction points during the period that the surface B C is in action, the cam turning as shown by the arrow. To obtain the positions of the six points for analysis one cannot divide the subtended arc B P of the working surface arc B C into six equal parts where a swinging follower arm is used, as may be recalled from Problems 8 and 11. Instead, it is convenient foi analytical construction purposes to revolve the swinging follower arm around the cam with uniform angular velocity while the cam 216 CAMS remains stationary. The detail work necessary to accomplish this is done first by drawing an arc of a circle through I with A as a center, finding where I is on this arc at the beginning and end of action while the arm I J slides on B C, and then dividing the arc of swing of I into six equal construction parts. 434. The initial position of I, Fig. 172, is found by laying off the distance L J from the point B on the radial line A B, thus obtaining the point ; then using as a center and a radius equal to L I draw a new arc to intersect the arc through / at 7i. This will be the posi- tion of I when the swinging arm is tangent to the cam at B\ in a sim- ilar manner 76 will be found to be the position when the arm is tan- gent at C. With the arc I\ Iq obtained and divided into six equal parts, it is no longer necessary or convenient to consider the center I as revolving about A, and it will, therefore, be considered as fixed in further work, the next step of which will be to find the six corre- sponding positions of the point L. This is readily done by drawing an arc through L with I as a center and then taking I L as a radius and the point I*, for example, as a center and drawing an arc such as one of the short ones shown at O4. Then with a radius equal to L J, find by trial, a point on the arc just drawn which will be a center for an arc that is tangent to B C of the cam. This center is shown at O4 and the tangent arc is shown at B4. With A as a center draw an arc through the point O4 until it cuts the arc through L already drawn, as at L4. In the same manner the six points on the arc through L. are found, and the corresponding points of tangency on the cam outline B C are obtained as shown from B to C. 435. The locus of the point of contact may now be found, as at R J Re, Fig. 172, as follows: To find, for example, the point #4, draw two intersecting arcs, one having L J for a radius and L\ for a center and the other having A B± for a radius and A for a center. Similarly other points on R J R§ are found. 436. The angular velocity curve for the swinging follower arm may now be readily found and its acceleration and retardation judged. Let S T represent the linear velocity of a point S at radius A S on the cam. Then a point at B± on the working surface of the cam has a linear velocity of S' T' and this value is laid off at R± T2 where the point B± is in action. The component of R± T2 that pro- duces rotation in the swinging follower arm is R± T3, perpendicular to I #4, and this reduced to a radius of I $4, equal to A S, for purpose of comparison with the cam rotation, is $4 T4. This value is laid MISCELLANEOUS CAM ACTIONS AND CONSTRUCTIONS 217 off on the 4th ordinate in the velocity diagram in Fig. 173, as at #4 7 7 4. In a similar manner other values are obtained in Fig. 173 and the curve B Q C drawn. This curve shows the rate of change of angular velocity in the transmitting follower arm while the straight horizontal line D E shews the uniform angular velocity of the driving cam. The length B C of the base line of the velocity diagram may be taken any length and then divided into six equal parts to locate the various ordinates of the velocity diagram. The length D B in Fig. 173 equals S T in Fig. 172. 437. The amount of sliding of the cam may readily be found for example, by first breaking up the velocity R± T2 of the point R± on the cam in Fig. 172 into its normal and tangential components — the former being shown at R± T5 and the latter at R± Ta — and, second, by breaking up the velocity R± T3 of the corresponding point on the swinging arm into the components Ra T5 and R± T7. The difference Tq T7, in the longitudinal components will be the rate of sliding at that phase and this difference is laid off at S T in Fig. 174. Sim- ilarly other points on the curve D T F are found. The rate of sliding when the circular surface of the cam B F E is in rction, providing there is no stop rest for the follower arm, is B D, equal to S T in Fig. 172; and when the surface C D is in action it is C 1\ Fig. 174. 438. A VELOCITY CURVE FOR THE FOLLOWER ROD N N' , Fig. 172, will give some indication of its acceleration and retardation and the relative strength of spring required to operate it in comparison with the results secured by an ordinary roller-end follower. The first step in this construction consists in finding the six positions of the center M of the upper curved surface K' K" of the swinging arm. This is readily done because the points M, L and / are fixed relatively to each other, and, therefore, the point M4, for example, is found by taking / M as a radius, / as a center, and drawing an arc at Mo M&. Then with L M as a radius and L4 as a center draw another short arc intersecting the first, as at M4. With M K as a radius and M± as a center draw the arc passing through ivV The point K4 is on a ver- tical line through A/4. The horizontal line tangent to the arc at K4 will have the position of the bottom of the follower rod at phase 4- In a similar way other points on the curve K$ K$ which is the locus of the point of tangency, is obtained. The distances between the horizontal lines drawn through the points Kq, K\, etc., will show the amount of vertical travel of the follower during each of the six equal time periods. 218 CAMS 439. The velocity diagram for the vertical follower rod is quickly obtained by first laying off the same angular velocity for the swinging arm at K4, Fig. 172, as was found at R± and finding the vertical com- ponent of the velocity of the point K±. This is done in detail by tak- ing the unit radius I $4 together with the linear velocity $4 T± at this unit radius, both of which have already been found, and trans- Fig. 172. — (Duplicate) Swinging Transmitter Arm with Sliding Action ferring the distance $4 T4 to £5 TV This will represent the linear velocity of the point £5 on the radial line I K±. The resultant linear velocity of the contact point K± on the swinging arm is found, as shown, to be equal to K± Tg. The vertical component K4 T\o of this resultant velocity for the arm gives the actual upward velocity of the rod TV N'. This value is laid off at S Tio in Fig. 175 and is an ordinate on the velocity curve B Q C. MISCELLANEOUS CAM ACTIONS AND CONSTRUCTIONS 219 440. The corresponding ordinate S U3, Fig. 175, for the velocity curve of the follower rod, if it had an ordinary roller end with a roller radius equal to B B' of Fig. 172, may be found 1st, by drawing the pitch surface line B' C of the cam; 2d, by dividing the arc B P into six equal parts; 3d, by drawing a radial line through the fourth point P4 to B\\ 4th, by revolving B\ to N4 and obtaining the full linear velocity N4 T± and laying it off at B\ U; 5th, by find- ing the radial velocity B\ U2 by drawing the line U U2 perpendicular to the normal B\ JJ\. The length B\ U2 will then represent the velocity of the follower bar if it had a roller end and this length is laid off at S U3 in Fig. 175. Similarly other ordinates of the curve B UsC are found. 441. Comparing the velocities of the follower rod N N', Fig. 172, when a trans- mitting swinging arm is used and when an ordinary roller end is used, it will be seen that the follower rod attains a higher velocity in the former case as shown by the greater height of the curve B Q C over the curve B U3 C. Also the acceleration of the follower rod N N' on the upstroke will be greater with the swinging arm as is indicated by the greater steepness of the curve from B to Q over that of the curve from B to U 3 . 442. The sliding action of the surface K f K", Fig. 172, of the swinging follower arm on the bottom of the rod A7 N' has a max- imum value of about one- fifth of that of the cam surface B C on the lower face of the Fig. 173. — Angular Velocity Diagram for Cam and Swinging Arm Fig. 174. — Sliding Velocity Diagram of Cam on Swing- ing Arm and of Arm on Follower Rod Fig. 175. — Linear Velocity of Follower Rod, with Trans- mitting Arm and with Ordinary Roller Follower Fig. 176. — Pressure Angle Diagram, with Ordinary Roller Follower £ 900- ^760- Seoo- eo *450 O F 300- s k B S 4 Fig. 173. 600n Fig. 175. Fig. 176. 220 CAMS swinging arm. This is readily determined by making use of work already done, as, for example, by simply measuring the line T 9 Tio, in Fig. 172, which is the horizontal or sliding component of the resultant velocity K± Tg when the point of driving contact is at IQ. The distance T 9 T 10 is laid off at S T n in Fig. 174. Other points of the curve B Tn C are found in the same way. The ordinates of this curve added to those of the curve D T F would give a measure to the total sliding action at any instant when a swinging transmitting arm is used. 443. If an oedinary roller follower instead of a swinging transmitting arm were used the pressure angle which would exist, with a cam of the size used in Fig. 172 and with a radius of roller equal to B B', may also be readily determined from work already done. For example, when the center B\ of the roller is in action the roller will be pressing against the cam in the direction of the normal B\ L\ relatively to the cam and the follower rod will be moving in the direc- tion of the radial line B\ U2 relatively to the cam. Therefore the pressure angle at phase 4- would be a, which is equal to 29°, and this value is laid off on the fourth ordinate as at S V in Fig. 176, thus ob- taining a point on the pressure angle curve which, it will be noted, has a maximum of about 31° — a very easy angle for general use. 444. If it is desired to know the actual rubbing velocities in feet per minute of the cam on the swinging arm, and of the arm on the follower rod; and the linear velocity of the follower rod N N f , Fig. 172, it may quickly be obtained from the velocity diagrams now drawn, for any given problem. For example, let it be assumed in this problem that the short radius A B of the cam in Fig. 172 is % inch and that the cam is making 900 revolutions per minute. 445. For the data just assumed the point B on the cam will 1 - + t . 75 X 2 X 3.14 X 900 be moving with a velocity 01 r~ = 006 teet per minute. This then would be the velocity represented by the line S T in Fig. 172. Since all the velocity lines shown in the drawings have been found and laid down without any change in the scale of the drawing, it is only necessary to compute the distance on S T that represents 100 feet per minute, and to make that distance the unit for the velocity scale for measuring the curves in Figs. 174 and 175. If A S measures % inch, S T will be found to measure .98 inch to the same scale. Then .98 inch represents 353 feet per minute, or, in other words, .28 inch represents 100 feet per minute. In Figs. 174 MISCELLANEOUS CAM ACTIONS AND CONSTRUCTIONS 221 and 175 the distance BA is .28 inch to the same scale on which A B was measured in Fig. 172, and this distance becomes the unit measurement for 100 feet per minute in the velocity diagrams. 446. By drawing the scales as above described it will be noted, in Fig. 174, that the maximum rubbing velocity of the cam on the lower face of the swinging arm is about 560 feet per minute and that the maximum rubbing velocity of the upper face of the swinging arm on the bottom of the follower rod is about 110 feet per minute. These considerations would affect the design in so far as lubrication and wear arc concerned. 447. The maximum velocity of the follower rod N N' in Fig. 172, may also be read off directly in Fig. 175, after the scale has been laid down as above described. This maximum velocity, it will be noted, is about 460 feet per minute. Had an ordinary roller follower been used on the end of the follower rod, the maximum velocity of the rod would have been appreciably less, or about 380 feet per minute. This consideration has an important bearing on strength of the moving parts in the general design of cam work. Its comparative effect, as for example in the strength of spring required to return the follower parts, may be definitely obtained by con- structing an acceleration and retardation diagram from the velocity curves shown in Fig. 175, as explained in detail in paragraph 268, et seq. 448. Boundary of surface subject to wear. In a cam design where there is a sliding follower as in Fig. 172 it will be of advantage to know not only the rubbing velocities as found above, but also the limits of the surfaces on which the rubbing takes place and the posi- tions on the surfaces where the rubbing velocities arc highest and the pressures due to acceleration are greatest. With respect to the cam in this problem, the conditions are ideal because the accelerations of the follower parts are greatest when the rubbing velocities are least. This combination occurs on the portion of the cam surface between B\ and Bi as may be pointed out as follows: (a) In Fig. 175 the velocity curve B Q is steepest between the phases 1 and 2 and con- sequently the acceleration of the follower rod N N' is greatest; (b), In Fig. 173 where the angular acceleration of the swinging arm is greatest also between 1 and 2\ (c), and in Fig. 174 where the sliding velocity is lowest between 1 and 2. The conditions for the swinging arm I are not so good. In the first place the total wear on the lower surface of the arm on the upstroke takes place between J' and J-$ as 222 cams found by drawing the dashline arcs through the extremities R, R3 and Rq of the path of action taking / as a center in each case; sec- ondly, the portion of the surface from Jq to J3 is rubbed over twice on the upstroke, or, in other words it receives twice as much wear as the part from J' to Jq; thirdly, the rubbing velocities are highest while the doubly worn surface from J3 to Jq is in action as indicated by the higher part of the curve from Q to F in Fig. 174; fourthly, the part of the swinging arm surface just to the right of Jq is also under the most intense pressure, due to acceleration, as well as being subjected to double wear and high velocity, as may be noted by the fact that Jq lies between the phases Ji and J 2 and that between these phases the accelerations are greatest, as indicated by the steepness of the curves between the ordinates 1 and 2 in Figs. 173 and 175. The points J\ and J 2 are not shown in Fig. 172, but they may be readily found by drawing arcs through R\ and R 2 with I as a center. The point R2 is on the path of action just above the point B2. 449. Cam action different on up-and-down strokes. All of the velocity and sliding curves obtained as above for the cam with a transmitting swinging arm, it will be noted, are for the action that takes place while the follower rod N N', Fig. 172, is on its upstroke, or, in other words, while the part of the cam surface from B to C is in action. While the follower is on its downstroke the surface of the cam from D to E is in action and the velocity and the sliding curves will be different, and should be obtained by similar methods where full information for specific practical application is desired. It may easily happen, according to the forms of the acting faces of the swinging arm, that the velocities and the accelerations and retard- ations may be quite different on the two strokes. Hence the informa- tion regarding both strokes should be known in order to properly judge the friction and wearing characteristics, and also to judge the strength of parts to be used. 450. The disadvantage of the side pressure that accompanies the ordinary roller-end follower, and the disadvantage of the high rubbing velocity that accompanies the swinging transmitter arm which is illustrated at / J K in Fig. 172, may be overcome by using a roller on the swinging arm to act against the surface B C of the cam, and a roller on the end of the follower arm to act on the transmitter head at K' K" . The side pressure produced by the slope of the cam is thus taken up by a tensional strain in the swinging arm instead of a side strain in the follower rod N N', and a smoother and easier cam action MISCELLANEOUS CAM ACTIONS AND CONSTRUCTIONS 223 should result although there will be an increased number of parts in the cam mechanism. 451. Problem 44. Small cams with small pressure angles SECURED BY USING VARIABLE DRIVE. By giving the cam sliafl a variable angular velocity very quick follower action may be secured with a relatively small cam without appreciably increasing the pres- sure angle. To illustrate, the same data will be taken as were used in Problem 3 except that the follower is to move up the given 3 units in 45° instead of 90°. The complete statement of the present problem is as follows: Required a single step radial cam to move a follower 3 units in 45° turn of the main shaft with uniform acceleration and retardation; to similarly return it in the next 45°, and to allow it to rest for the remainder of the cycle. 452. Let N, Fig. 177, be the center of the uniformly rotating main shaft of the machine to which the cam is to be applied. Assume any length for the driving arm N P and draw the two 45° angles P N T and T N Q. Draw the circle whose radius is N P and divide each of the arcs P T and T Q into six equal parts. Connect the points Q and P, thus obtaining the point on N T which will be the center of the auxiliary or cam shaft. Attach a slotted arm H to the cam shaft, making the shorter working radius of the arm J equal to T, and the longer working radius H equal to N plus N P. Assume the diameter of the driving pin at P which works in the slotted arm, and make the length of the slot a little greater than J H to allow for clearance. 453. Variable drive by the Whitworth motion. From the preceding paragraph it may now be seen that the arm H, Fig. 177 and the cam shaft to which it is keyed will turn through 90° while the main machine shaft turns through 45°. The mechanism thus far described for producing this result is equivalent to the Whitworth slow-advance and quick-return mechanism, but any other type of slow-advance and quick-return mechanism that gives complete rotary motion could be used instead. 454. To construct the cam, compute the size of the pitch circle in the same manner as in an elementary problem, but using the 90° that the cam w r ill turn during the outward motion of the follower instead of the assigned motion of 45° that the main shaft will turn. Thus the diameter of the pitch circle will be found to be, 3 X 3.46 X 360 3.14 X 90 13.2. 224 CAMS Lay this value off at D S, Fig. 177, and draw the pitch circle with as a center. Lay off the assigned motion of 3 units of the follower symmetrically about D as at A V. Assuming 6 construction points for finding the cam pitch curve, divide A D into nine equal parts and take the 1st, 4th and 9th division points; do the same with V D. Divide the arc Q T into six equal parts and draw radial lines through each division point, as indicated at E and K. Carry the division Fig. 177. Problem 44, Showing that Very Small Cams and Small Pressure Angles May be Obtained by Using Variable Velocity Drive points on Fi around to their corresponding radial lines by means of circular arcs, as indicated at A K\ . Then the curve through the points A, Ki, etc., will be on the pitch surface of the desired cam. 455. The present cam does the same work in half the time of the cam that is shown in Fig. 32, and both have the same overall dimensions. The cams are of different shape, however. The cam shaft will have widely varying angular velocity, ranging between MLSCELLANEOUS CAM ACTIONS AND CONSTRUCTIONS 225 values which vary from — - to — -. At the phase of the mechan- ic J . U 11 ism shown by the object lines in Fig. 177 the driving shaft N and the cam shaft have the same angular velocity, and this is true for this phase no matter what length of driving arm is taken at the start. The cam will have its greatest angular velocity when N P is in the position N T, but at this phase the pressure angle will be zero, and it will be comparatively small while the cam is approach- ing and receding from this phase. Had a cam been constructed in Swash-plate Cam the regular way, that is without variable drive of the cam shaft, to give 3 units motion in 45° under the condition of this problem it would have required a cam with a pitch circle diameter of 3 X 3.46 X 360 3.14 X 45 26.4 units instead of 13.2 as here used. 456. Swash-plate cams differ in structural details from any thus far considered but they are, in effect, end or cylindrical cams. If in Fig. 178 a basic cylinder C Y is intersected by an inclined plane P L it will cut a flat surface from the cylinder and the form 226 cams of this surface, when viewed perpendicularly, will appear as an ellipse in which the major and minor axes will be PL and N L, respectively. The flat surface thus formed is termed a swash plate. It is shown in the top view by the elliptical curve P2 L2 and in the end view by the circle Pi L\. As the swash plate turns on its axis X X\, which is the axis of the original cylinder it gives a reciprocating motion to a follower rod F E. The range of the follower motion will be greater or less according to the true radial distance A\ D\ of the follower from the axis of the cam, and in the present illustration the range of follower motion is R S. If a sharp F-edge were used on the follower E F, the contact would be at A, instead of at T as it is with the roller, and the motion of the follower would be harmonic, giving velocity and acceleration curves similar to those shown in Figs. 88 and 89 respectively. The smaller the follower roller F T, Fig. 178, the truer and smoother will be the running of the swash- plate cam. 457. Rotary cam giving intermittent rotary motion. A cam of unusual form is shown at A B in Fig. 179. It is designed to change a uniform rotary motion in the shaft P P to an intermittent rotary motion in the shaft C by operating on the roller pins D, E, F, G. Specifically, it is desired that the shaft C shall make a 34 turn while the shaft P P makes a Y2 turn, then that the shaft C shall remain stationary while P P makes Yi turn, and finally, that shaft C shall be under positive control all the time. Such a cam would be auto- matically formed on a previously prepared blank by using a rotary cutter of the same size as the rollers, D, E, etc., and which travels in the same path as the rollers while the cam blank is turned by inde- pendent means. For the purpose of laying out the blank and of representing the form of the cam surface in a drawing, the roller is taken in several intermediate positions, one of which is shown in fine lines at H, and constructions made as follows. The method here given is reduced to simplest terms and is approximate. It is suf- ficient, however, for the cam surface will be true, because of its auto- matic manufacture, even if the delineation is not exactly so. 458. Make an end view of the roller as shown at Hi, Fig. 179. The circle through Hi represents the circle half way down the roller and the short vertical line tangent to it locates the point of tangency for the cam surface and roller assuming that the cam surface has a 45° slant when the roller has turned 45° from E to H. Projecting Hi first to H and then down to H2 on the centerline, a point will MISCELLANEOUS CAM ACTIONS AND CONSTRUCTIONS 227 be found on the centerline of the cam surface, it being noted thai the cam turns through 90° while the follower turns 45°. A straight line on the surface of the roller through H would represent approxi- mately the line of contact between cam surface and roller and would be projected down to give K and N if the slant of the cam surfaces edges were the same. The slant for the edge J K L of larger radius is a little less than that of M N 0, being on a larger average radius, and, making a corresponding allowance, H2 K is taken a little less l 'o Fig. 179. — Special Cam Type Giving Intermittent Rotary Motion than H 2 N. The points L and will be directly under the roller in position F, and the distances from the centerline P P to these points will be the same as the distances from the centerline to the corre- sponding points on the bottom of the roller. It will be noted in this construction that the width of the cam surface is less than the length of the roller. 459. The eccentric may be considered as a special type of cam. It is widely used in engine and other work where it is desired to secure a simple reciprocating motion from a rotary motion. Where 228 CAMS the initial motion may be taken from the end of a rotating shaft, a crank is the simpler device to use, but where the motion must be taken from an intermediate point on the shaft an eccentric is neces- Fig. 180. — Practical Example of Cam Shaft Carrying Eleven Cams sary. The eccentric gives a characteristic motion to the follower the same as a driving crank would give to the crosshead in an ordinary crank and connecting-rod mechanism, the equivalent crank length being equal to the distance from the center of the shaft to the center Separate End Views of Cams Shown in Fig. 180, to Reduced Scale of the eccentric circle as shown at R S in Fig. 181. The eccentric cannot be used where specific intermediate velocities are desired for the follower. The use of the eccentric as a cam in automatic machin- ery is illustrated in Fig. 180 which represents the main cam shaft of a machine devised for special manufacturing purposes. Eleven cams, MISCELLANEOUS CAM ACTIONS AND CONSTRUCT IONS 229 compactly arranged, are shown on this shaft, four of them being eccentrics, namely Nos. e, g, h, and k. All eleven cams are shown in end view in Fig. 181 with the exception of i, which is shown to en- larged scale in Fig. 179. 460. An example of a time-chart diagram for all of the cams illustrated in Figs. 180 and 181 is given in Fig. 182. Time-chart 1 2 Bunch Plunger Cam Ba«k Fold Cam "111 1 Ml 1 1 ±d= i-.-i-: _______ . . ■ ■ ■ " '" — — ] 3 Placer Cam ^ 4 Ink Roll Cam | ' 1 1 fr— y*"fl ^t4J-j 5 Seal Fold - 1_ L ! 1 1 i 1 1 ~ 6 Seal Fold Cam — — 1 ; — ~ = — — 7 End Fold Eccentric - — — — I'll 8 Creaser Eccentric - - r*^g- — — ; — 9 Main Cam =p 1C Printing Cam 11 Rock Shaft Eccentric -L Fig. 182. — Practical Example of Time Chart Diagram for Eleven Cams in Oni Automatic Machine diagrams are treated in a general way in paragraph 19, and in detail with reference to a specific example in paragraphs 143 to 147. The form of diagram here shown is specially to be commended in that the individual diagram boxes for each cam are separated from each other by a small space so that it is impossible for the heavy base lines to touch or cross each other under any circumstances. INDEX A PAGE Acceleration diagrams for different base curves 89 Acceleration diagrams. Method of determining 138 Accelerations produced by differ- ent base curves 142 Accuracy in cam construction. . . . 146 Adjustable cam defined 11 Adjustable cylindrical cam plates. . 193 All-logarithmic base curve 89 All-logarithmic cam problem 94 Angular velocity curve for swing- ing follower 216 B Balancing of cams 148 Barrel cam defined 7 Base curve defined 14 Base curves in common use 14 Base curves. Comparison of . . . . 88 Base curves. Complete list of . . . 88 Base curves. Construction of common 20 Base line defined 14 Box cam defined 8 C Cam action different in up-and- down strokes 160, 222 Cam chart applied 29 Cam chart defined : 12 Cam chart diagram defined 12 Cam considered as bent chart .... 34 Cam defined 1 Cam factor chart for all base curves 151 Cam factor chart for common base curves 19 Cam factors for all base curves .... 150 Cam factors for common base curves 18 231 PAGE Cam factors. Method of deter- mining 152 Cam mechanism for drawing ellipse 79 Cam mechanism for reproducing designs 80 Cam shaft acting as guide 204 Cam size. Effect on pressure angle 33 Cam surface on follower 208, 213 Cam with flat-surface follower. ... 45 Cam with sliding follower 57 Cams classified 1 Cams for high-speed work 148 Cams for low-starting velocities 129, 132 Cams for swinging follower arms 50, 52, 57 Carrier cam defined 11 Characteristics of base curves .... 88 Circles. Subdivision of 86 Circular base curve. Case 1 119 Circular cam problem. Case II... 129 Clamp cam defined 11 Comparison of base curves 88 Comparison of parabola and crank curves Ill Comparison of velocities and forces of different base curves 141 Conical cams defined 2 Conical follower pin for cylindrical cam 190 Construction of common base curves 20 Crank curve as projection of helix. . 108 Crank curve characteristics. . .108, 111 Crank curve construction 21 Cube base curve 125 Cube curve cam problem. Case 1 . 127 Cube curve cam problem. Case II 133 232 INDEX PAGE Cube curve cam specially adapted for follower returned by spring 144 Curved follower toe 162 Cylindrical cam defined 1,7 Cylindrical cam problem 68, 70 Cylindrical cams. Drawing of grooves in 186 Cylindrical cams. True pressure angle in 186 D Derived curve for pure rolling action 174 Diagram. Cam chart 12 Diagram. Timing 13 Disk cam defined 1 Dog cam defined 11 Double-acting cam defined 9 Double-disk positive drive cam for swinging arms 205 Double-disk yoke cam problem .... 65 Double-end cam defined 7 Double-mounted cam defined. ... 11 Double-screw cams 194 Double-step radial cam 39 Drum cam defined 7 E Eccentric as a cam 227 Ellipse. Cam mechanism for drawing of 79 Ellipse. Construction of 178 Elliptical arcs for pure rolling action 177 Elliptical base curve character- istics 123 Elliptical curve construction 23 Empirical cam design 25 End cam defined 7 F Face cam defined 3 Face cam problem 55 Factors. Methods of determining cam 152 Factors. Table of cam 18, 150 PAGE Flat-surface follower 45, 49, 59 Follower carrying cam surface . 208, 213 Follower returned by springs .... 142 Follower rollers for cylindrical cams 188 Follower roller. Size of 35 Follower velocity in ft. per sec. 165, 220 Follower with curved toe 162 Forces produced by different base curves 141 Formula for cam size 17 Frog cam defined 2 G Gradual starting of follower shaft. . 177 Graphical methods. Degree of precision in 141 Gravity curve 110 Groove cam defined 7 H Handwriting. Cam mechanism for reproducing 79 Harmonic curve 108 Harmonic motion 108, 206 Heart cam defined 3 Helix as pro j ection of crank curv e . . 1 08 High speed in cam work 148 Hyperbola for pure rolling action . 184 Hyperboloidal follower pin for cylindrical cam 190 I Infinite connecting rod 108 Interference of cams 75 Intermediate transmitter arm .... 214 Intermittent harmonic motion.. . . 206 Intermittent rotary motion 226 Internal cam defined 8 Involute cam problem 199 Involute curve defined 197 Involute curve. Construction of. 192 K Keyways. Location of 78 INDEX 233 L PAGE Length of follower surface 58, 62, 159, 164 Limited use of flat-surface follow- ers 49, 59 Limited use of single-disk yoke cams 64 Limiting size of follower roller 35 Locus of point of contact between cam and follower 58, 62, 159, 164, 216 Logarithmic-combination cam problem 101 Logarithmic curve. Construction of 101, 172 Logarithmic curve for pure rolling action 169, 171 Logarithmic curve. Properties of 171 Logarithmic spiral. Construction of 95,98 M Multiple-mounted cam defined .. . 11 Mushroom cam defined 3 Mushroom cam problem 45 N Names of cams tabulated 12 Noise from cams 147 O Offset cam defined 8 Offset cam problem 42 Omission of cam chart 31 Oscillating cam defined 11 Oscillating single-disk positive- drive cam 202 P Parabola cam characteristics 110 Parabola construction 22, 182 Parabola for pure rolling action. . . 182 Parabolic easing-off arcs 103 Parabolic curve. Property of . . . . 182 Perfect cam action 110 Periphery cam defined 2 Pins for cylindrical cams 188 Pitch circle defined 16 PAGE Pitch line defined 15 Pitch point defined 16 Pitch surface defined 16 Plate cam defined 3 Plates for cylindrical cams 193 Positive-drive cam defined 8 Positive-drive double-disk cam for swinging arms 205 Positive-drive single-disk cam for swinging arms 202 Precision of graphical methods ... 141 Pressure angle characteristics of involute curve 198, 201 Pressure angle defined 16 Pressure angle factors ... 18, 149, 150 Pressure angle relation to cam size . 31 Pure rolling in cam work 168-185 R Radial cam defined 1 Radius of curvature of non-circular arcs 38 Rate of sliding of cam on surface of follower 164, 166 Regulation of noise in cam design 147 Relative strengths of springs re- quired for different cams .... 143 Roller. Limiting size of 35 Rollers for cylindrical cams 188 Rolling action 168-185 Rolling cam defined 5 Rotary cam giving intermittent rotary motion 226 Rotary sliding-disk yoke cam . .205, 206 S Scotch yoke 207 Screw cams 193 Shaft guide for cam followers .... 204 Side cam defined 1 Sine curve 108 Single-acting cam defined 9 Single-disk positive drive cam for swinging arms 202 Single-disk yoke cam problem .... 63 Single-step cam problem 28, 31 Sinusoid 108 234 INDEX PAGE Sliding contact follower 57 Sliding friction eliminated 168 Sliding of cam on follower surface, 164, 166, 219 Slow-advance and quick-return by cylindrical cams 195 Small cams with small pressure angles secured by variable speed drive 223 Spherical cam defined 2 Springs. Use of, for returning cam followers 142 Starting velocities of cam followers 129, 132 Step cam defined 9 Straight-line base curve construc- tion 20 Straight-line base curve problem . 106 Straight-line combination base curve construction 20, 107 Straight-sliding plate cams 196 Strap cam defined 11 Subdivision of circles 86 Sub tangent of logarithmic curve... 102 Swash plate cam 225 Swinging follower arm 50 Swinging transmitter arm 214 T Table of cam factors for all base curves 150 Tangential base curve 113 Tangential cam problem. Case I. 113 Tangential cam problem. Case II 135 Technical cam design 27 Time-acceleration diagrams 139 Time chart applied 76 Time chart defined 13 Time-chart diagram for eleven cams 229 Time-distance diagrams 138 Time-velocity diagrams 138 Timing of cams. Problem 75 Toe-and-wiper cam defined 7 Toe-and-wiper cam problem 61 Toe-and wiper cam with variable angular velocity 157 Transmitter arm between cam and follower 214 Variable angular velocity in cam shaft 157 Variable speed for small cams .... 223 Varied forms of fundamental base curves Ill, 149 Velocities produced by different base curves 141 Velocity diagrams for different base curves 89 Velocity diagrams. Method of determining 138 Velocity of follower in feet per second 165, 220 W Wear. Distribution of, on follower surface 58, 62, 159, 164, 221 Whitworth motion 223 Wiper cam defined 5 Working surface defined 16 Yoke cam defined 6 Yoke cam with rotary sliding disk 206 Yoke cam problem. Double-disk . 65 Yoke cam problem. Single-disk.. 63 Wiley Special Subject Catalogues For convenience a list of the Wiley Special Subject Catalogues, envelope size, has been printed. These are arranged in groups — each catalogue having a key symbol. (See special Subject List Below). 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