ARR No. lAJ05a NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS WARTIME REPORT ORIGINALLY ISSUED November I9M as Advance Restricted Report lAJ05a EXPERIMENTAL VERIFICATION OF THE RUDDER -FREE STABILITY THEORY FOR AN AIRPLANE MODEL EQUIPPED WITH RUDDERS HAVING NEGATIVE FLOATING TENDENCY AND NEGLIGIBLE FRICTION By Marion 0. McKinney, Jr. and Bernard Maggin Langley Memorial Aeronautical Laboratory Langley Field, Va. UNIVERSITY OF FLORIDA DOCUMENTS DEPARTMENT 120 MARSTON SCIENCE LIBRARY P.O. BOX 117011 32611-7011 USA ■ WASHINGTON NACA WARTIME REPORTS are reprints of papers originally issued to provide rapid distribution of advance research results to an authorized group requiring them for the war effort. They were pre- viously held under a security status but are now unclassified. Some of these reports were not tech- nically edited. All have been reproduced without change in order to expedite general distribution. L - 18U UT V l " r w> you* NACA ARE. No. il+JC^a NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS ADVANCE RESTRICTED REPORT EXPERIMENTAL VERIFICATION OF TEE RUDDER-FREE STABILITY THEORY FOR AN AIRPLANE MODEL EQUIPPED WITH RUDDERS HAVING NEGATIVE FLOATING TENDENCY AND NEGLIGIBLE FRICTION By Marion 0. Mc Kinney, Jr. and Bernard Maggin SUMMARY An investigation hap been made in the Langley free- flight tunnel to obtain an experimental verification of the theoretical rudder-free stability characteristics of an airplane model equipped with conventional rudders having negative floating tendencies and negligible friction. The model used in the tests was equipped with a conventional single vertical tail having rudder area if.0 percent of the vertical tail area. The model was tested both in free flight and mounted on a strut that allowed freedom only in yaw. Measurements were made of the rudder-free oscillations following a disturbance in yaw. Tests were made with three different amounts of rudder aerodynamic balance and with various values of mass, moment of inertia, and cent er-of -gravity location of the rudder. Most of the stability derivatives required for the theoretical calculations were determined from force and free-oscillation tests of the particular model tested. The theoretical analysis showed that the rudder- free motions of an airplane consist largely of two oscillatory modes - a. long-period oscillation somewhat similar to the normal rudder-fixed oscillation and a short-period oscillation introduced only when the rudder is set free. It was found possible in the tests to create lateral instability of the rudder-free short-period mode by large values of rudder mass parameters even though the rudder-fixed condition was highly stable. The results of the tests and calculations indicated that, for most present-day airplanes having rudders of negative floating tendency, the rudder-free stability 2. NACA ARR No. liiJ05a characteristics may be examined by simply considering the dynamic lateral stability using the value of the directional-stabil5 ty parameter CL> for the rudder-free condition in' the conventional controls-fixed lateral- stability equations. For very large airplanes having relatively high values of the rudder mass parameters with respect to the rudder aerodynamic parameters, however, analysis of the rudder-free stability should be made with the complete equations of motion. Good agreement between calculated and measured rudder-free stability characteristics was obtained by use of the general rudder-free stability theory, in which four decrees of lateral freedom are considered.. When the assumption is made that the rolling motions alone or the lateral and rolling rrotions may be neglected in the calculations of rudder-free stability, it is possible to predict satisfactorily the character- istics of the long-period (Dutch roil type) rudder-free oscillation for airplanes only when the effective-dihedral angle is small. Tith these simplifying assumptions, however, satisfactory prediction of the short-period oscillation maj bj obtained for any dihedral. Further simplification of the theory based on the assumption that the rudder moment of inertia might be disregarded was found to be Invalid because this assumption made it impossible to calculate the characteristics of the short- period oscillations. INTRODUCTION Some military airplanes have recently encountered dynamic instability in the rudder-free condition. Certain other airplanes have performed a rudder- free oscillation called "snaking" in which the airplane yaw and rudder motions are so couoled as to maintain a yawing oscillation of constant amplitude. These phenomena have been tha subject of various theoretical investigations, and the factors affecting the rudder-free stability have beer explored and defined in the theoretical analysts of references 1 to 3. In reference 1 the most complete set of the three sets of equations of the rudder-free- motion is developed. The equations of reference 1, however, are very involved and rather unwieldy, and use of these equations to NACA APR No. llf-JO^a determine the rudder-free stability characteristics is consequently laborious. Such equations are usually simplified by neglecting certain degrees of freedom or certain parameters ani thus obtaining approximate though satisfactorily accurate solutions. In reference 2, the equations were sixrrolified by neglecting the rolling motions of the airplane. In reference 3 s which supersedes reference 2 for the rudder- free theory, further simplification was obtained by neglecting sidewise motion as well as rolling motion. An additional simplifying assumption of reference J is that the rudder moment of inertia might be neglected. It was realized that these simplified equations were not applicable throughout the entire range of the variables that could be obtained, but the results were believed to be generally applicable to airplanes of that period. In order to obtain an experimental check of the general and simplified equations, an experimental program is being conducted in the Langley free-flight tunnel. The results of the first part of this program are reported herein and are concerned with tr.e rudder-free dynamic stability of a k— scale airplane model in gliding flight equipped with rudders having inset-hinge balances and negligible friction. The rudder-free stability characteristics of the model were investigated for varying amounts of rudder aerodynamic and mass balcnce. The model was tested both in free-flight and nounted on a strut that allowed freedom only in yaw in order to determine experimentally the differences caused by neglect of the rolling and lateral motions of an airplane with rudder free. In order that the results obtained by theory and experiment might be correlated, calculations were made of the theoretical rudder-free stability of che model tested by equations involving four degrees of freedom, and by equations involving fuwer degrees of freedom. In addition, the rudder-free stability of the model was calculated by an approximate method that neglected all of the rudder parameters except those causing a reduc- tion in the directional-stability parameter C_, for the rudder-free condition. Ji FACA ARR No. Lb. JO 5 a Various force, hinge-moment, and free-oscillation tests were run in order to determine as many as possible of the stability derivatives required in the calculations of rudder-free stability. SYMBOLS S wing area, square feet V free-stream airspeed, feet per second b wing span, feet c wing chord, feet by, span of rudder, feet m mass of model, slugs m r mass of rudder, slugs ky radius of gyration of model about longitudinal (x) axi s , f eet 1<2 radius cf gyration of model about vertical (Z) axis, feet radius of gyration of rudder about hinge axis, feet x r distance from center of gravity of rudder system to hinge axis; positive when center of gravity is back of hinge, feet I distance from model center of gravity to rudder hinge line, feet ' ri \ D differential operator : ._• V" s / s distance traveled in spans (Vt/b) P oex>i od of oscillations, seconds T time required f:>r motions to decrease to one -half amplitude, seconds t time, seconds NACA ARR No. L^JO^a 5 A, B, C, D s E coefficients of stability quartic for rudder- fixed lateral stability Ap Bp Cp Dp E-j, fp G,, E, coefficients of stability septic for rudder-free lateral stabi lity A 2> B 2' c 2' D 2' E 2' F 2 coefficients of stability quintic for rudder-free lateral stability ■ A, j B-,, C,, D,, E 7 coefficients of stability quartic for ^ * -' ? rudder-free lateral stability Al 3),, Ci, D15 coefficients of stability cubic for ■ rudder-free lateral stability ^ root of stability determinant (\ = a' + ib') ib' imaginary oortion of complex root of stability quartic a' real root or real portion of a complex root of s t abi li ty qu ar t i c q dynamic pressure, pounds per square foot (jPV ) p mass density of air, slug per cubic foot \x model relative-density factor (m/pSb) [i rudder relative-density factor /m r /pb r c~ r ^) c r root-r^e an- square chord of rudder, feet a angle of attack, radians unless otherwise defined 6 angle of sideslip, radians unless otherwise defined /> angle of roll, radians unless otherwise defined \|/ angle of yaw, radians unless otherwise defined 5 rudder angular deflection, radians unless other- wise defined Y flight-path angle, radians unless otherwise defined 6 FACA ARR No. liiJO^a p rolling angular velocity, radians per second r yawing angular velocity, radians per second v lateral component of velocity, feet per second fU f tA T lift coefficient \~7^~J 'n C h '0 'P fDrag\ n drag coefficient ^-rl? . . , . , „„. . . / Ti t c hi rig mome nt \ C pitching -moment 3oei ncient ! s ) m ; .: ■ v qSc / , . , *, r.^.. . .. /'Lateral force A Cy lateral-force coefficient ^ ; j qS .-. , „,,. . . ^Rolling moment\ Cj rolling-moment coefficient \^ — E ) qSb , „„. . , Rawing moment \ yawing-momsnt coefficient ^ a=i rr- J qSb , „„. . . ( Hinge moment \ hinge -moment coefficient I E — 3-5 ) Cy rate of change of lateral-force, coefficient with 3 angle of sideslip (dC Y /df) C 7 rate of change of rolling-moment coefficient with angle of sideslip (dCj/dB) Oj rate of change of rolling-moment coefficient with rolling angular- velocity factor (dcy&ij d rate of change of rolling-moment coefficient with / rb\ yawing angular-velocity factor (cCj/dgrr) C n " te of change of yawing-moment coefficient with P ldeali p (6C n /d0) NACA ARP No, LJiJO^a 7 "> n v rate of change of vawing-moment coefficient with /oO n \ an C le of yaw ^— = -c np J rate of change of yawing-moment coefficient with rolling angular- velocity factor {$Z n /&^7j'j 'n. 'n« rate "of change of yawing -moment coefficient with / rb\ yawing angular-velocity factor IdC /dpy] rate of change of yawing -moment coefficient with rudder angular deflection (6C n /6 5) C h, n. rate of change of rudder hinge-moment coefficient wi th ang le of s i de s 11 p (6 Ch/6 P ) rate of change of rudder hinge-moment coefficient /6C h \ with angle of ya^ o\]/ C h P, Cv, rate of change of rudder hinge -moment coefficient n r rV with yawing angular- velocity factor (dCfo/b^y 'h 6 rate of change of rudder hinge-moment coefficient with rudder angular deflection (60^65) n D6 rate of change of rudder hinge-moment coefficient with rudder angular-velocity factor d(Vd /d_5 (dt_ APPARATUS The tests wore run in the Langley free-flight tunnel, a complete description of which is given in reference i|. The model used in the tests was a modified l/7-scale model of a Fairchild XP2K-1 airolane with its center of 8 MAC A ARR No. Lb. JO 5 a gravity located 23 • percent of the mean aerodynamic chord. Figure 1 is a three-view drawing of the model. The mass and dimensional characteristics of the model are given in the following table: Weight, pounds 5*75 Radius of gyration, k , foot 0. 734 Wing area, square feet 5«U7 Wing span, feet ■75 Wing cnord, foot O.785 Distance from airplane center of gravity to rudder hinge line, feet 2.07 Height of rudder, foot O.667 Root-mean- square chord of rudder, foot O.185 The vertical ts.il of the model was a straight-taper surface with a rudder of the inset-hinge type. The area of the rudder behind, the hinge line was [;0 percent of the vertical tail area. Three different nose balances were attached to the rudder in order to vary the amount of aerodynamic balance. Sketches of these surfaces are given in figure 2. The mass characteristics of the rudder were varied by moving weights within the rudder or along a thin metal strip that protruded at the base of the rudder trailing edge.' The rudders were mounted on ball bearings to reduce friction to a minimum. The yaw stand used in the tests was fixed to the tunnel floor and allowed the model complete freedom in yaw' but restrained it from rolling or sidewise motions. A photograph of the model installed on the yaw stand is shown as figure 3. T2'~ Tests -ore made to determine the period and damping of the rudder-free lateral oscillations of the model during free gliding flight and when mounted on the yaw stand. No tests were performed to determine the effect upon the rudder-free stability of eliminating only rolling motions . o 1 T ACA APR w . ll|. JO 5 a 9 Scope of Tests The range of rudder aerodynamic and mass character- istics covered in the tests is given in table I. The test range investigated was obtained by altering the mass characteristics of the rudder by addition of weights at various locations. In this manner the mass, center of gravity, and radius of gyration of the rudder were varied simultaneously. This procedure was followed for the rudder equipped with each of three different amounts of aerodynamic balance. All tests v.ere ran at a dynamic pressure of l.°0 pounds per square foot, which, corresponded to an air- speed of approximately liO feet per second. The lift coef- ficient ' c 9s approximately 0.6. Flight Tests Flight tests were made tor the model test conditions 1 to 3 anr '- 10 to 13 of table J. These tests were made by flying the test model freely within the tunnel as explained in reference h.. During a given flight, a mechanism within the model was so activated as to free the rudder after an abrupt rudder deflection of about 15°- The rudder-free lateral oscillations resulting from the rudder disturbance were recorded by a motion-picture camera. The period and damping characteristics of the flight oscillations were obtained from the motion-picture record and were corre- lated with corresponding records from the yaw- stand tests and with calculated characteristics. Several runs were made at each test condition and showed a variation of period of about 2 percent and a variati on of damping of less than 10 percent. Typical flight oscillations are shown in figure L(a) for a stable condition and in figure l;.(b) ■f'or an unstable condition. Yaw- St and Tents The yaw-stand tests v.ere made for all test con- ditions listed in table T. These tests were made under conditions reproducing those considered in the analytical treatment of reference 3, in which the rolling and the lateral motion of the airplane center of gravity are neglected. or the yaw-stand tests, the model was attached to the stand and the rudder was' deflected 15°. At the given best airspeed, the rudder was abruptly released and the resulting oscillations were photographed by means 10 L T ACA ARR No. li+J05a of a motion-picture camera installed above the model. Records of the period and damping of the yawing oscilla- tions were then obtained in the same manner as for flight tests. Approximately the same scatter of period and damping values was obtained in the yaw-stand tests as in the flight tests. Plots of representative yawing oscilla- tions obtained from the yaw-stand tests are shown in fig- ure ii(a) for a stable condition and in figure li(b) for a neutrally stable condition. Method of Analyzing Test Data Stability theory indicates that the rudder-free lateral oscillations are composed of two superimposed oscillatory modes, one of which has a shorter period than the other. The test oscillations, however, after a short interval of time represented only one of these modes - the one that subsided later because of the ocriod or damping. Tn general, the stability calculations showed that the short-period mode damned to one-half amplitude in roughly I/50 the period of the other mode. The test oscillations therefore represented the long-period mode for most of the test conditions. Measurement of Stability Derivatives The stability derivatives necessary for the calcula- tions are given on table II and were obtained by the following procedures; The partial derivatives of yawing- moment coefficient with respect to angle of yaw and rudder deflection, C~ . and Q„ , were determined from force n^ n^ ' tests of the model on the six-component balance of the Langley free-flight tunnel described in reference 5. ' 'T* 16 results of these tests are presented in figures 5 to 8. The hinge-moment derivatives due to angle of yaw and rudder deflection, Cy. . and C>. , were determined from rWr SIR ' hinge-moment tests of the model rudder, the data from which are presented in figures 9 to 11. The rudder hinge-moment derivative due to yawing angular velocity Cft was then calculated by the relationship C h r = b~ C lty ^ NACA ARE No. ihJO^a. 11 The yawing -moment derivative due to yawing angular velocity Cn^ (fig. 12) was determined by the free- oscillation method described in reference 6, and the rudder hinge -moment derivative due to rudder angular velocity c>, was similarly determined. The measured n D5 values of the oaram^ter Cb (-0.02J6 for rudder 1 and -0.0!;.2k for rudders 2 and J) did not agree with the value of -O.lli.2 as calculated by the method presented in reference 7 except that the frequency of the oscillation was neglected. The cause of this discrepancy was not determined but is believed to have been the. high oscillation frequency at -which the tests were run (about 6 cycles per second at an airspeed of h.O feet per second). This frequency corresponded approximately to the calculated frequency of the rudder in the rudder- free tests of conditions 1 to 9 in table TT. Measurements indicated that the frlctional damping of the rudder was about one-tenth of the air damping. This value was considered negligible and no attempt wc made to introduce friction derivatives into the Calculations. Four runs were made with each rudder and the scatter of values of Cv, was less than 10 percent. n D6 The partial derivative of the rolling-moment coef- ficient with resoect to the rolling velocity parameter G 7 was determined from the charts of P reference 8. The derivatives C 7 and C~ were determined from the formulas given in reference 9- jalculations Scope Calculations were made of the damping and period of the rudder-free lateral oscillations of the model for the range of airplane and rudder parameters given in table T. These calculations were made by equations that provided four degrees of freedom as well as the fewer degrees of freedom which resulted from the neglect of rolling or the neglect of rolling and lateral motions. Other calculations were made to determine the effect of n effective-dihedral parameter C, upon the 6 rudder-free stability characteristics. r 12 FACA ARR No. Ti;J05a Me thod The customary methods of stability calculation (outlined in reference 3) were employed in the present investigation. The equations of motion were set up, rendered nondimensional, and so treated as to obtain the stability equations defining the period and damping of the lateral-stability modes. Equations of motion .- The nondimensional equations of motion used in the calculations are given in the following paragraphs. The equations used for the rudder-fixed condition are /2uD - C Y )P + (-C L )0 + (2u.D + C t tan yW \ p. _ ■Hi-,/ L - oC ("%) P + 2" I- D ? + (-K. D )* = ° ^(■f) ° 2 - k, D \k i > (2) Equations (2) yield the familiar lateral-stability equations of the form AM K BV \ 2 + D\ + E (3) The general equations of motion for the rudder- free condition (four degrees of freedom) are NACA ARR No. LLjO^p 13 (2uD - Cy s )P + (-C L );2f + (2uD + C L tan f)ty = ( / N l''Xj\^ ? 1 \ - I 1 N / 1 ^ r /y \ 2 1 = ; •U r b > 00 4 ^r\.2 ^\b/ ~ 4 ^ b J • r b 2 u ' 2°h r D * ! fef 2 ^r^D./ 'h* :<"!. n o 2 ^D6 o = J Equations (lj.) yinld the rudder -free lateral-etabl.il ty equations of the form I A 1 \ l + Bj^k- + C^ + D 1 X^ + F n A^ + FA. + Q 1 \+ H 1 = (5 For the rudder-free condition, when rolling is neglected (three degrees of freedom) s the equations are ^2aD - Cy^ fk„\ 2 1 2 ^ D2 -|^n r ^ +'(-Ori f -2a r -rf-D - G hp l V3 2 a L7, L -] D^ - £a^ --D + 2u,. — rlr - - D 2 v \h~l ° " ^r ~b~ 2|4.(£) d 2 ^ h 5 2 C ' i " J D5" 5 rr V C £ (6) Ill NACA ARR No. Li; JO 5 t Equations (6) yield stability equations of the form K ? h? + B ? ^' + C ? ^ + D^ 2 + Eg\ + F 3 = (7) When the rolling motion and the lateral motion of the center of gravity are neglected (two degrees of freedom), the equations are faV 2 1 2 ^W ° ' 2 n r D - C n vU * + (-°n 5 ) 5 5 = k \2 1 D 1 - + 2u, fry t.x b 2 D2 - Ic h D 2 n r '1-1,1, * r L I n /M^2 1 On ri 1 - ^rib-y w 1. h 5 ' 2'^D5 D = (8) Equations (8) yield stability equations of the form A,V' + B^>-' -i- C,X 2 + D,\ + E, = 3 ;> 3 3 3 (9) If, in addition to neglect of rolling and lateral motion of the center of gravity, the rudder moment of inertia is also neglected, the equations are 2^-57 d2 - 1%°- sr.'N) 5 = 2u r — f- D< ? b' 2°*^ " C hJ* ^ n D5 D '-h, 5 = J 10) NACA ARR No. Lb jO^a 15 Equations (10) yield stability equations of the form Ai A. 5 + EjX 2 + G|\ + D^ = (11) Determination of pe riod en d rny' nr of 1* ' ' osc 1 l lati oni . - T he ro o t s of' e q u a t i ens ~~ ( "3 "] '" (31 7~ (?) s P)', and (11) are of the form k = a' or \ = a' + ib'. The roots are used in the following equations to deter- mine the period and the time to damp to one -half amplitude : P - &- - n - (12) b '• V v and -loe 0.5 ] a' v ( 13 uLT3 AND DTSOUSSTON The results of the tests ^nj calculations are presented in table II, which lists the period, and the reciprocal of the time to damp to one-half amplitude for sach condita investigated. Tee reciprocal of the time to damp to onj- haif amplitude was chosen to evaluate the damping, bee au this value Is a direct rather than an inverse of the degree., of stability. Negative values of the reciprocal of the time to damp to one-half amplitude refer to the time to increase to double amplitude. eneral Equations Calculate ons . - The stabilitj calculations made with the general equations of motion indicated that the motions of an airplane wi th rudder free consist of two aperiodic modes (convergences or divergences) and two oscillatory modes, one of which is oi a period 2. to 10 times the other, As shewn bv the results presented in table II, these calculations indicated that, as lone s -s the rudder radius 16 FACA ARR No. ll^JO^a of gyration and mass unbalance wore small (conditions 1 to 9)5 the short-period mode was very heavily damped arid, consequently, the characteristics of the more lightly damped long-period mode determined the nature of the rudder-free oscillations. Table II indicates also that the characteristics of the long-period mode were only slightly affected by the rudder parameters as long as the rudder mass parameters were low (conditions 1 to 9)- When the radius of gyration and the mass unbalance of the rudder were large (conditions 10 to 15 ) , the calculated period of the short-period mode increased consideraoly and the damping decreased. At high negative floating ratios (conditions 12 and 13)? the calculations indicated that the destabilizing effect of high rudder radius of gyrati on and irass unbalance was sufficient to cause lateral instability. F light tests .- Vac results of the flight tests are presented in table TI . These data indicate that, for low values of the rudder mass parameters (conditions 1 to 9), the less damped and hence the apparent mode had a oeri od of about 1.5 seconds 3 which corresponded to that calculated for the long-oeriod mode. For high values of the rudder mass parameters (conditions 10 to 13 ) , either the long- or the short-period mode was the less damped of the two modes and hence determined the characteristics of the apparent motion, depending uoon the magnitude of the rudder aerodynamic parameters Zy~ and Cu'. For the condition of high rudder aerodynamic parameters (condi- tions 10 and 11) the long-period mode was the less damped. Condition 12, however, showed the short-period mode to be the less damped at somewhat lower values of the rudder mass parameters than those of condition 11, and condition 13 gave an unstable short-period oscillation for even lower values of rudder mass parameters. Compari so n o ft h e oreti c a 1 and calculate d results . - The tests confirmed the results predicted by the theory inasmuch as an unstable lateral oscillation was obtained for test condition 13. The quantitative correlation of measured values of period and damping with corresponding values calculated by the use of the general equations is shown in figure 13 . These data show that the agreement between measured and calculated values of period, was excellent for all conditions tested. This agreement was also shown in the correlation between measured and calcu- lated values of damping except for conditions 12 and 13 • NAG A A 17 ,?. No. 14 JO 5a 17 For conditions 12 and 13, the calculations indicated a larger degree of instability than that encountered in the tests. This apparent diserepancj was explained i further calculations which showed that the rudder-frei stability was critically dependent upon the ruddj r tss characteristics for these test conditions. The results of the further calculations are given in figure 1 show that the degree of instability encountered in test condition 1J was indicated by theory to occur at some- what smaller values of n ■.■- unbalance than that used for the tests. Another possible sxplanatio.n of the discrep- ancy between tests and theory for conditions 12 and 13 'or these 1ovj-j 'e c ue ncy is thrt the actua] value of H- n D5 conditions might have been higher than the values used in the calculations and obtained from high-frequency measur - rents. fore information is required to determine t. effect of frequency of the oscillation on this para: eter. Physical interp ret ation .- T n order to explain no re clearly the physical aspects o t ~' the long- and short-period modes of the rudder-free oscillations, additional calcu- lations were made of the rudder-free and rudder-fixed stability characteristics over 1 a vide range of dihedral. In this way, it was possible to observe analytically the change of lateral stability gvi th ral when the rudder was free. A comparison and rudder- free ca period mode of the p art i n run d e r- f i x • of the results of the rudder-fixed. lcuDetions indici I v that th si :>rt- rudder-free oscillations aad no counter- ed fl This mode s therefore e r - f r e e s c 1 1 1 a 1 1 n s ha d n c .ight (table TT , condition ll±). an created bv the new the rudder was fre lation 0" the rudd lations also showe short-period mode effect! ve~dih 3 ivi '< I r?ly new oscillation :"S3 of dom that occurred 3 d a n er a 3 i n r were : robablj represents the 03c out its own ^ici^^ line. Th t th characteristics of the virtually in it of the or 11- alcu- s-r r f • e variation of the calculated v a 1 1 . s n period oscillation f dihedral oarametei L; i n ■■ c 3 nd : t i n 2 w5 t h s folio:. st r~ P 1/1 f th 1 ort- 3 ffective- 18 NACA ARR No. lliJOSa The additional calculations indicated that the characteristics of the long-period mode of rudder-free oscillations varied with the effective-dihedral parameter C 7 in a manner similar to the variation of the rudder-fixed oscillations with this parameter. The results of these calculations are presented in figures 15 and l6 and indicate that the characteristics of the long-period mode for the rudder-free condition were o£ the same order as those of the rudder-fixed oscillation hut wore of lower damping and higher period. Inasmuch as freeing rudders of negative floating tendencies is known to decrease the directional-stability parameter C n and a decree: 3 Ln this factor is known to decrease the damping and to increase the period of the lateral oscillation (references 10 and 11), the long-period mode of the rudder-free oscillations appears to he a modifica- tion of the familiar Dutch roil oscillation normally encountered in controls-fixed flight. The characteristics of the long-perloc rudder-free mode may then he concluded to be largely dependent upon hie. same parameters as the rudder-fixed oscillatory mode. For airplanes having rudders of n-jgative floating tendency, then, instability of the long-perio^ rudder-f ree mode should occur at smaller values cf effective dihedral than for the corresponding rudder-fixed condition. Correlation of Tests and Cimplified Equations ' le c t of r udder para meters.- Inasmuch as most present-day airplanes have low values cf rudder mass parameters, the long-period mode is the predominant factor affecting the rudder-free stability character- istics for airplanes having rudders of negative floating tendency. Consideration of the rudder mass parameters would therefore not seem necessary for these airplanes. An approximate solution for the rudder-fr~e stability has been obtained by simply considering the controls-fixed dynamic lateral-stability equations (2), in which the value of C„ for the rudder-free condition is used. Calculations were made for the test conditions by this approximate method, and the period and damping results are prerented in table II and, for condition 7> as points on figures 15 and 16. The value of C n for H NACA A^.R No. Ll+J05a 19 the rudder-free condition was calculated by the following relation from reference 1: n - r, P H ( rudder free) p ( rudder fixed) h 5 ° The values of period and damping obtained with this method are in good agreement with the values calculated by the general equations for test conditions 1 to 9° For conditions 10 and 11 the correlation is rather poor, as was expected, because of the high values of rudder mass parameters at these two conditions. Because the approximate method cannot predict a short-period oscil- lation,, this method failed completely to predict the important features of the rudder-free motions for con- ditions 12 and 13 , for which the short-period oscillation was about neutrally damned. These calculations indicate b j although the predictions yielded by the approximate method are good st low values of the rudder mass parameters, s more complete analysis is necessary at high values of the rudde r mass p ar ame tens. _ "^-sct of ro l ling notion .-- The simplification obtained by neglecting rolling motion was investigated for the present report by comparing the results obtained by the ra] equations with those obtained by equations r i.eotlrig rolling (equation 6), The values of period a... damping for the test conditions as calculated by equation (h) are presented in table IT. In figure 17 the characteristics of the short-period mode obtained by this method are compared with those calculated by the general equations. The correlation of the characteristics of both the long- and short-period modes calculated by the modified equations with those obtained from flight tests or from calculations by the general equations is fair except for conditions 10 and 11. This fact might indicate that the s ipllfied theory gives noor correlation for the case of nes - Ltral stability of the short-period mode when, the period of the long- and short-period, modes is nearly . ! 1 . Equation? (6) show that neglect of rolling eliminates all c " 3 rivatives involving rolling moment as well as thos-e involving rolling motions. The effect of C 7 00 6p 20 NACA AER No. li+J05a on the lateral stability cannot, therefore, be predicted by this simplified method. The effect of dihedral on the long-period mode is to reduce both the period and the damping, as is shown in figures 15 and l6. For dihedral angles less than c j° /'C, < 0.06 N . ,•• however, neglect of rolling in the equations gives conservative results, because these equations indicate less damping than the general equations for low values of rudder mass parameters (conditions 1 to Q)« This result is obtained mainly because with low dihedral the rolling component of the motion is small. The effect on the stability of neglect of rolling with rudder fixed has also been investigated. The results of these calculations are given in table II under condition lU and show reasonably good agreement with the results obtained by the general theory for the rudder- ftaed condition. Tils agreement is further proof that, for low values of dihedral, rolling may be neglected in making these calculations. On the other hand, for air- planes having high dihedral and large values of relative density and radii of gyration, the long-period oscilla- tion might become unstable, as shown in references 10 and 11. The neglect of rolling for these conditions would invalidate the results for the condition with the rudder either free or fixed. Negle ct, of r o lling and lateral moti on. - In the theoretical analysis of rudder-free stability published 1n reference 5, the equations were further simplified by neglecting lateral motion of the airplane center of gravity as well as the rolling motions. These simplified equations also predicted a long-period and a short-period oscillation. Tests of the model in the rudder-free condition were made en the yaw stand in order to reproduce the theoretical assumptions made in reference 3 (freedom in yaw about the airplane Z-axis and freedom of the rudder about its hinge line). The results of these tests are presented in table II and indicate that, for low values of the rudder mass parameters (conditions 1 to 9 ), the long-period mode was the less damped and hence determined the characteristics of the apparent motion. For higher values of the rudder mass parameters (conditions 10 to 13), either the long- or short-period mode was the less damped, depending upon the magnitude of the rudder aerodynamic N.ACA ARR No. lJij05a parameters. For conditions 10 and ll~, the long-period mode was the less damped: whereas s for ' conditions 12 and 13, neutral damping of the short-period node was obtained at lower values of the rudder mass parameters. It may be of interest to note that in uhpublishi data from yaw-stand tests., made at higher values of mass imbalance of the rudder than there present herein, an unstable short-period oscillation was obtained. This unstable oscillation could be started with •: verv small disturbance and vouid increase in amplitude until it became a const ant -amplitude oscillation of about ±15° yaw. The results of calculations made by utilizing the equations of reference 5 are listed in table II and have been compared in figure 18 to measured values obtained from the yaw-stand tests. The data presented in figure 18 show that the equations of reference 5 closely predicted the rudder-free data obtained in the jaw-stand tests for stable conditions. Li >i the general theory, however, the simplified equations predicted instability cf the short- period oscillation at lower values of rudder mass parameters t h an did th e yaw ■ s b and tests. A comparison of the yaw-stand and free-flight test results shows that the elimination of the rolling and lateral motions results in somewhat longer period, and less damping than is obtained in flight, as long as the long-period oscillation is ''be controlling factor in the apparent motion in flight. When the character] sti us of the short-period oscillations are apparent in flight, tests on the yaw stand give nearly identical results with those from flight : ' sti . These data indicate that for small effective dihedral angles neglect of the roiling and lateral motion yields conservative values for the long-period, rudder-free oscillation and accurate values for the short -period oscillation. r ::.is fact confirms the conclusion, drawn from the analytical invest j gati on concerning the affect of d1 al, that th: characteristics of the short-period mode are relatively independent of dihedral, which is a basic roiling derivative. The data of table IT indicate that the simplified theory of reference 3, which neglects rolling and lateral motion, predicted the characteristics of the short-period mode lust as well as did the general theory and that use of the simplified theory was therefore justified in this re spe c t . 22 NACA ARR No. lij.j05a Negl e c t o f rolling, 1 atera 1 met ion, a nd r udder moment of ih ertTaT^ A further assumption suggested in reierence 3 is that the moment of inertia of the rudder, in addition to rolling and lateral motion, might he disregarded in the calculation of rudder-free stability. The results of calculations made with the 'rudder moment of inertia neglected are presented in table II and indicate that, for airplanes with small amounts of effective dihedral, application of the theory gives a reasonably accurate oredicticn of the rudder-free stability characteristics as long as the long-period oscillation is the controlling factor in the apparent motion. For conditions in which the short -period oscillation is the less damped mode, however, the assumption must be con- sjdered wholly invalid for rudders of negative floating tendencies, because the calculations indicate that, when the rudder moment of inertia is neglected, the short - period mode is replaced by a heavily damped convergence. CONCLUSIONS The following conclusions were drawn from an Investigation in the Lang ley free-. flight tunnel of:the rudder-free stability characteristics of an airplane model-equipped with rudders of negative floating tendencies and having negligible friction: 1. For most present-day airplanes, consideration of the rudder mass parameters ia no' -sary in an analysis of th Lity " ;ios. These chaj srics may be examined ily by considering the • r, lateral stabilil by using the value of the directional-stability iter &' for the rudder- ' - np free ceo lit: on in the conventional controls-fixed lateral- stability equations. 2. Analysis of the rudder-free stability of airplanes having relatival;/ high values of the n ider mass parame- ters with re to + he rudder aerodynamic parameters (such as would be encountered in very large airplanes) should be made with the corn], iete equations of motion for the rudder-free condition. 5- The rudder-free stability characteristics of the model tested were satisfactorily calculated when all four NACA APR ITo. li;J05a 23 degrees of lateral freedom were considered in the calcu- lati ons . k. The rudder-free characteristics of the model tested were predicted fairly well when rolling motions or rolling ard lateral motions were neglected in the calculations. Instability of the rudder-free Dutch roll t y p e o s c i 1 1- a t i o n , h o we v e r- , 2 o uld net be predicted b y this method. 5- Large amounts of rudder mass unbalance caused an unstable short-period rudder-free oscillation for the model tested. 6. The characteristics of the short-period oscil- lation are found to be independent of the airplane effective dihedral and were satisfactorily predicted for the model tested by either the general stability equations in which all four degrees of lateral freedom are considered or by the modified stability equations in which either the effects of rolling motions alone or of rolling motions and lateral motions of the airplane center of gravity .are neglected. 7 = When the rudder moment of inertia was neglected in the calculations, the characteristics of the short- period rudder-free oscillations for rudders having negative floating tendencies could not be predicted. Langley Memorial Aeronautical Laboratory Natl onal Ad vi s ory Commit t e e f or Ae ronaut i c s Langley Field, Va. 21; NACA ARR No.. L^JO^a REFERENCES 1. Bryant, L. W. , find Gandy, F. W. G. : An Investigation of the Lateral Stability of Aeroplanes with Rudder Free, ij 30J4., S. & C. 1097, British N.P.L. , Dec. 18, x j j ■> • Z. Jones, Robert T. , and Cohen, Doris: An Analysis of the Stability of an Airplane with Free Controls. NACA Rep. No. 709, I9I4.I. 3. Greenberg, Harry, and Stsrnfield, Leonard: A Theoretical Investigation of the Lateral Oscillations of an Airplane with Free Rudder with Special Reference to the Effect of Friction. NACA ARR, March I9I4.3. I4.. Shortal, Joseph A., arid Osterhout, Clayton J.: Pre- liminary Stability and Control Tests in the NACA Free-Flight Wind Tunnel and Correlation with Full-Scale plight Tests. NACA TN No- 8l0, 191+1. 5. Shortal, Joseph A., and Draper, John W. : Free-Flight- Tunnel Investigation of the Effect of the Fuselage Length and the Aspect Ratio and Size of the Vertical Tail on Lateral Stability and Control. NACA APR "0. 3D17, 19U3. 6. Campbell John P., and Mathews, Ward 0.: Experimental termination of the Yawing Moment Due to Yawing Contributed by the Wing, Fuselage, and Vertical Tail of a Midwing Airplane Model. NACA ARR No. 3F28, I9I1.3. 7. Theodorsen, Theodore: General Theory of Aerodynamic Instability and the Mechanism of Flutter. NACA Rep. No. lj.96, 1935. 8. Pearson, Henry A., and -Jones, Robert T. : Theoretical Stability and Control Characteristics of Wings with Various Amounts of Taper and Twist. NACA Rep. Mo. 635, 1938. 9. Bamber, Millard J.; Effect of Some Present-Day Airplane Design Trends on Requirements for Lateral Stability. NACA TN No. Slu, 19iil. MAC A ARR No. L^ JO 5 a 25 10, Campbell, John P., and Seacord, Charles L. , Jr.: Effect of Wing Loading and Altitude on Lateral Stability and Control Characteristics of an Airplane as Determined by Tests of a Model in the Free-Flight Tunnel. NACA ARR No. 3F25, 19li-3. 11. Campbell, John P., and Seacord, Charles L. , Jr.: The Effect of Mass Distribution on the Lateral Stability and Control Characteristics of an Airplane as Determined by Tests of a Model in the Free-Plight Tunnel. NACA APR No. $H$1, I9U5. FACA AFP. Fo. Li|J05a 26 co 8 8 o o EH CO W Eh CO O H E-i ° g 3 <: « o IN- w Eh En H CM LTN N~\ LO. N-\ fo> UN LT\ VO r - C-- fO, .-1 -j" O O o o o o o o o o o o o o o o o k"n ro\ k"n k"n UN r— C~- r— f- OJ O O O O C\) O O O O rH o o o o o o o o o o O UN CM i>-co c~- nO [-- CM i — 1 I — 1 r-H o o o o o o a; h IB o CO C~- nO m un O VO ^) CO CO J" CM -J" rH r-H UN UN UN LO, l_TN CO CO CO CO UN rH rH r-H r-l C— OJ o ON QT UN co a- r- a OJ KN CM rrs CO C '- CM UN CM 0--4 OJ O O r— O o o o o o o C— K^ O- LO, o u',o4 UN OJ LO, LO, O O O CM o o o o • • • • ° Q.9 CO -CJ-nC CJ CO r-H. r/-\ CM CM O O O • e a i u r-j C--CO in nO O O UN O rH rc- O O O O LfN O O F-! 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ON ON ONNlN co co a'i co on t- o- c-- C---J- O O O O rH 1 1 1 > 1 nO On On OvtO CO rH O O 1 1 1 ck; a o nO vO vO _rj_t KN KN K> CM 0- 1 CM OJ CM _ -H- O O G O c. i O ! 1 1 1 1 OJ oToJ "CM kn C5 o c o a 1 1 1 1 1 NO J--3 rs^W CM OJ _h- _H; o p O 1 ! 1 CM OJ OJ OJ OJ C— D— O- n ON r- 1 rH r-H O O O' CM CM OJ CM O ON O ON f- C O O O r-H Oi CM OJ rH O O O 1 1 1 1 t 1 1 1 1 1 1 1 1 re o o o J--3 On 0" n0 nO l>0 KN rO OJ CM O 1 III 1 -f OJ OJ CM O v; l>- 0- 0- On OJ rH rH rH [O, 1 1 1 1 1 g.N^ rTN CJ rH 1 1 1 ^N O- nO rH Q C -4- O I o I CO. o crt CM -0 NO -p -J CM OJ OT co rH -Cj rt o rH NO o o • • • o o 1 o o o »■ Tl rH II II II II to rC -> u CO. (i) o 1- c? r-o s- (1) ^ w S-, - rC / — ^s /^ X -P i Afi' js*k po s NACA ARR No. L4J05a 27 TABLE II.- COMPARISON OF PERIOD AND DAMPINO FROM FLIGHT AND YAW-STAND TESTS AND CALCULATIONS Teata Calculations Teat oondltlon Flight Yaw stand General theory Rolling neglected Rolling and aldeallp negleoted Rolling, sideslip, and moment of Inertia negleeted Rudder parameters neglected Long-period oscillation 1 Period, P Damping, 1/T 1.72 1.17 1.80 .90 1.69 1.12 H* 1.66 • 92 1.67 .95 1.60 1.39 2 Period, P Damping, 1/T 1.61 1.09 I.78 .90 1.60 . 1.10 1.60 1.27 Mi ME 1.60 1.59 5 Period, P Damping , 1/T l.Jifi 1.06 1.66 1.00 1.1)0 1.10 1.70 1.32 1.60 .98 1.61 1.00 1.60 1-39 U Period, P Damping , 1/T 1.69 .90 1.60 1.38 1.70 1.26 1.68 •95 1.68 • 93 1.60 1-59 5 Period, P Damping, 1/T I.65 .90 1.56 1.39 1.60 1.50 l :.1 1.66 • 96 1.60 1-39 6 Period, P Damping, 1/T 1.62 1.00 It 1.60 1.26 1.62 .99 1.62 • 99 1.60 1-39 7 Period, P Dampl ng , 1/T 1.86 .90 l.7lt 1-33 1.80 1.20 x :fi 1.82 .88 I.7I1 1-39 8 Period, P Damping, 1/T 1.79 .90 1.63 1-35 1.74 1.26 1.78 .92 1.77 .99 1.71) 1-59 9 Period, P Damping, 1^ 1.72 1.00 1-53 1.30 1.60 1.30 .97 1.72 .97 1.71* 1-59 10 Period, P Dampl ng , 1/T 1.30 .90 1.60 1.05 1.20 .92 1.S0 i.Hj 1.U9 1.05 1.50 l.lfc 1.60 1-59 11 Period, P Damping, 1/T .87 1.50 1.20 I.05 .90 1.20 1-53 1-55 1.23 1.1)1 1.50 1.60 1-59 12 Period, P Damping, 1/T .... .... 0.96 5.30 0.99 5. BO 1.08 5.85 1.16 1-71) 1.60 1-59 13 Period, P Dampl ng , 1/T — 0.97 5.22 O.98 5-19 5^60 1.2I) 1.60 1.7U 1-59 ll> Period, P Dampl ng , 1/T 1.56 1.00 1.30 1.87 1.66 1.1* I.50 1.05 Short-period oscillation 1 Period, P Dampl ng , 1/T ..... .... 0.16 5.66 0.15 li.92 Hoo 2 Period, P Damping, 1/T 0.09 15.15 0.09 14.55 0.08 11). 16 58U 5 Period, P Damping, 1/T 0.20 It. 1*3 0.18 3.1)0 0.12 5-1)6 570 1* Period, P Damping, 1/T 0.10 32.30 0.10 30.80 0.10 52.50 152 5 Period, P Dampl ng , 1/T 0.10 32.50 0.10 50.80 0.10 52.20 IU7 6 Period, P Demplng, 1/T 0.10 32.30 0.10 50.80 0.10 52.20 lil 7 Period, P Dampl ng , 1/T 0.12 32.30 0.12 50.80 0.12 52.20 99 8 Period, P Dampl ng , 1/T 0.13 32.30 0.15 50.80 0.15 52.20 "94 9 Period, P Damping, 1/T 0.15 52.50 0.15 50.80 0.15 52.20 "88 10 Period, p Damping, 1/T 0.80 1.90 °:U O : 5 7o 522 11 Period, P Dampl ng , 1/T 0.87 1.56 °:$ °:ft 278 12 Period, P Dampl ng , 1/T 0.83 .10 0.88 .00 0.88 -3.31 0.91 -2.75 0.90 -1.62 71 13 Period, P Dampl ng , 1/T 0.92 -.15 0.90 .00 0.86 -2.76 0.92 -2.19 -91* 'hi lit Period, P Dampl ng , 1/T :::: '-'-- P and T given in seconds. ^Approximate method, all rudder parameters neglected except those affecting C n NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS NACA ARR No. L4J05a Fig. 1 NATIONAL ADVISORY COHMITTEE FOR AERONAUTICS Figure/.- Three -view drawing of the modified -£-sca/e model of the fdirch/ld XR8K-I airplane. NACA ARR No. L4J05a Fie. 2 Rudder / Rudder 3 Sections A~A Rudder Rudder area 1 ft percent yert/caf-taif area Balance inpercent ruc/der area / 40 2 40 3d 3 40 3f NATIONAL ADVISORY COMMITTEE FOR AERONAUTIC! R/gure Z. , — Sketch of rudders used //; //?e rudder- free stabd/ty /nvestigat/on in the lanpley free-flight tun/ie/ '. NACA ARR No. L4J05a Fig. 3 CJ u . • H r-l id CU Cr, c c T3 3 CU +J • H <*-! +J • H -C T3 ho O • H e rH V-i <« 1 o CU CU rH S-i CU <*H T3 O >> 6 CU rH CU no rH C o3 X) C >i U c nJ o cu u T=t CU S-i *J 0) c <-> 3 U o cd e 3 cr cu i c cu ti CD rH tj Pu -C Ul E-i • H trJ 1 rH to 1 :*: cu CM ^ a 3 X bo NACA ARR No. L4J05a Fig. 4a * o £ D in 4 Q t— U E E 3 #□ § ^ o a tI S^ ^ * d § >0 co o <\j 00 >i- 5 I VO ! 1 i i x o o a if wo «k •V CO ^ o , ^- CO I NACA ARR No. L4J05a Fig. 4b ! 8 ^ 8! 6ap r .-Conc/uded, JO NACA ARR No. L4J05a Fig. 7a C I 8" 5 -JO -20 -10 tO 20 Angle of yaw, ^> } deg F/gure 7— Rudder effect J /e/iess for rudc/er 2, NACA ARR No. L4J05a Fig. 7b \ ffO- o -./ -.2 -.3 .03 .02 .0/ -oi ^I§ Jn ~^&r ^flSr •^a &Z$a%A ^j i^^E &y$&® 6 idea) A o a - CT 5 s, o A - /O - /J rf ^5 --— - 20 25 gv>^ 1/ - 30 - -S - -10 - -tf < - -20 --23 V, rr _3/~i ,y^i s c ) r ^^ ^ NATION 11 ADVIS ORV o ' c J -30 -20 -to O /0 20 Angle of yaw } % deg JO F/gare 7. - Concluded '. NACA ARR No. L4J05a Fig. 8a I % 8^ .05 .04 .03 .02 .0/ 5 -01 -.02 .03 ■04 -/o o io ao Angle of yaW) ^jdeg figure 8. — f?udder e/Tect/veness for rudder 3 . NACA ARR ho. L4J05a Fig. 8b ^n C\J O c\J c o o I k 3 ft/su/ouj-sbuiH ! I I r I vO CQ $ o 5 8 i r I •o 1 ! I I V. ! NACA ARR No. L4J05a Figs. 11,12 n c o i o 3 c 3 ft □ 1 * 3 < i s 3 O D ( ' l ^J Q -o Q O g o «*.* « c ■ ^ OJ NJ i 5s ^ i*5 Si £ S k i ^ 5 ft i o ,§> ^« «: o I lb I, Vp ' ./. us>io/// soy / usujotu - abuif-/ NACA ARR No. L4J05a Fig. 13 *o \ ll J> Q \ ">> fo £ c i -t i 1 . §?■* ^; It NACA ARR No. L4J05a Fig. 14 fe & -2 -J ^Calculated \ Measured—^ (condtt/on /3j NATION; \V ADVIS ORY \ con IMITTEE FOR AERl JNAUTICS \ \ ./ .a f?udder c&?ter-of-&roz/ty focot/o/?^ * r /c r F/gure /<$. ~~ Camper/son of ca/cu/ated &/?d reasoned damp//yg of trie short-per/od rudder- free asa//at/on. NACA ARR No. L4J05a Figs. 15,16 1 Si 4 1 $ s si si 1 ! 1 3 ll -*/ 1 / 1 / v5 ^ / L — -t w / / / / / / 31 V / 1 i / / / / Cn o ■o P ' tC\ <\ \ 00 rvi 1 <^ _ 6 7 ^- i \ R $ o s? 8 O -o p o -Si §8 & ( 6uf/fOj du/pdjbdu 'buidtuvQ I* I 8 SB k NACA ARR No. L4J05a Fig. 18 O ^ CM cq > DS>£ r c/ c pO/J9Cf P9/U/77J/UJ \S> 1 ; A 15 r h f f i >* f\J CO o <\| CQ ^ o ^5 sy Is 1 1 1° 1^ Pi ,8| oo § tn 06 ■>£// r j/ f '6u/cfuxp psfopofoo UNIVERSITY OF FLORIDA 3 1262 08106 559 UNIVERSITY OF FLORIDA SSK 32611-7011 USA