National Advisory Commitiee for Aeronautics
mCD r------O":l N Z f-i ~ u NATIONAL ADVISORY COMMITIEE ~ Z FOR AERONAUTICS
TECHNICAL NOTE 2985
A FLIGHT INVESTIGATION OF THE EFFECT OF STEADY
HOLLING ON THE NATURAL FREQUENCIES
OF A BODY- TAIL COMBINATION
By Norman R . Bergrun and Paul A. Nickel
Ames Aeronautical Laboratory Moffett Field, Calif.
Washington August 1953 ERRATA
NACA TN 2985
A FLIGHT INVESTIGATION OF THE EFFECT OF STEADY ROLLING ON THE NATURAL FREQUENCIES OF A BODY-TAIL COMBINATION By Norman R. Bergrun and Paul A. Nickel August, 1953
1. In figure 8, sublegend 8 (a ) should read IIshort-period oscillationll and sublegend 8(b) should read IIlong-period oscillation. 1I
Iy I z 2 . I n the title of figure 9, -- = -- = 7 instead of 1/7. SC Sc J lY NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS
TECHNICAL NOTE 2985
A FLIGm' INVESTIGATION OF THE EFFECT OF STEADY
ROLLINc:J. ON THE NATURAL FREQUENCIES
OF A BODY-TAIL COMBINATION
By Norman R. Bergrun and Paul A. Nickel
SUMMARY
Flight experiments with a freely falling model have been conducted to observe the effects of steady rolling on the pitching and yawing motions of a body-tail combination. Time histories of the pitching and yawing motions show that steady rolling introduces into the pitch oscil lations the yawing motions, and into the yaw oscillations, the pitching motions. The observed effects of steady rolling on the values of fre quencies are compared with those predicted by theory for damping and for no damping. When damping is included, the calculated frequencies are in good agreement with the observed frequencies, although a good first-order approximation of the frequencies may be obtained by neglect ing damping.
Static stability derivatives are assumed for a miSSile, and it is found that at low speeds and high altitudes, the rolling frequency tends to approach the lower of the body natural frequencies, thereby reducing effective static stability. In addition, graphical relations are presented which permit rapid estimation of changes in the effective static stability derivatives due to steady rolling.
I NTRODUCTION
Rolling of an airplane or missile introduces an iner~ial coupling between the pitching and yawing motions and tends to destabilize the body. This destabilizing effect of rolling becomes more pronounced for aircraft having most of the weight concentrated in the fuselage. Thus, the trend toward use of low-aspect-ratio wings causes the influence of roll on the stability of airplanes and missiles to assume importance.
An analysis of the effect of rolling on stability has been made by Phillips (ref. 1). The principal results are that: (a) For a nonrolling 2 MeA TN 2985 body, there will be only one frequency of oscillation in the pitching motion and only one frequency in the yawing motion, and the frequencies in pitch or in yaw may be the same or different, depending on the aero dynamic properties of the body; and (b) for a rolling body, there will be two distinct frequencies of oscillation in the pitching motion, and these same two frequencies will appear in the yawing motion; in addition, the faster of the two frequencies will be greater (and the slower of the two will be less) than either the nonroll pitch or yaw frequency. As the roll rate approaches zero, the fast frequency approaches the faster of the nonroll pitch or yaw frequencies, and the slow frequency approaches the slower.
The analysis of reference 1 provides equations for predicting the effect of steady roll on the no- roll natural frequencies in pitch and in yaw. When damping i s considered, the equation for the frequencies of oscillation is in the form of a fourth-order polynomial which,ordi narily, is solved by trial and error. When damping is neglected, the fourth-order polynomial derived for the case of damping simplifies to the form of a quadratiC equation which can be solved explicitly.
The present investigation was undertaken to provide a free-flight observation of the influence of steady rolling on the frequencies in pitch and yaw of a fuselage- tail combination. The results obtained, when compared with reference 1, would establish the validity of the analysis for the prediction of the effects of rolling on body frequencies.
The fuselage selected for the present investigation had a fineness ratio of 12 . 4 . A roll-stabilization system was employed to maintain the rate of roll at a selected value . The frequency data were obtained by disturbing the model abruptly in pitch and recording the resultant oscillations . The Mach number range of the tests was 0.9 to 1.1.
The t ests were conducted at Edwards Air Force Base by personnel from Ames Aer onauti cal Laboratory of the NACA .
SYMBOLS a ,b,c,d,e dimensionless coeffi cients of a polynomial having the form 4 3 2 aD - bD _ cD - dD - e in which -a 1 W b - 2Ct t + 2~ewa 2 2 c _"0 2 + wt + we + 2 ~ewe 2 StW t 2 d =- 2 ~eWe (.u t2 + 2Cewe + 2C1(!Wt We + 2 Stw ~r 2 2 2 e -- .1 + we :;Jt - Wt - we "" f- 2~ tWt 2 Ce we
D i s a differential operator NAeA TN 2985 3
ax,ay,az l ongitudinal, l at eral, and nor mal accelerations, g 's c mean aerodynamic chor d of a r eference wing, ft
fe frequency in pitch under conditions of no roll, cps
fe' fre quency applicable to a mode of oscillation which approaches the pitching oscillation as the rate of r oll approaches zero, cps
f W frequency in yaw under conditions of no roll, cps
fW' frequency applicable t o a mode of oscillation which approaches the yawing os cillation a s the rate of roll approaches zero, cps
moments of inert i a about the x,y, and z body axes, r espec tively, slug- ft 2
M Mach number p rol ling vel ocity, radians/sec
tot al pressure, in. Hg
2 dynamic pressure, l pv , lb/sq ft 2 q pit ching velocity, radians/sec r' ordinate of fuselage, in. r yawing velocity, radians/sec s exposed area of a reference wing, sq ft t time, sec v free-stream velocity, ft/sec a.. angle of attack, deg
angle of sideslip, deg
horizontal- tail deflection, deg
vertical-tail deflection, deg p atmospheric density, slugs/cu ft 4 NACA TN 2985
fraction of critical damping in pitch of nonrolling body, Mq dimensionless
fraction of critical damping in yaw of nonrolling body, N r dimensionless
ratio of no-roll natural frequency in pitch to steady rolling f frquency, __e__ , dimensionless p/21C ratio of no-roll natural frequency in yaw to steady rolling frequency, ~, dimensionless P/21C pitching-moment coefficient of complete model about center of gravity, mo~nt, dimensionless q c (21Cfe )21y longitudinal stability derivative for no roll, - ----~~~ CIDa. qoSc ' per radian
C t effective longitudinal stability derivative applicable to a IDa. mode of oscillation which approaches the pitching oscilla- (21Cfe t )21 tion as the rate of roll approaches zero, - _ y per radian qoSc dC ------,m per radian d(qc!2V)
Cn yawing-moment coefficient of complete model about center of gravity, moment_ , dlmensionless. qoSc (21Cf1jr)2 1 lateral stability derivative for no roll, z , per radian qoSc
effective lateral stability derivative applicable to a mode of oscillation which approaches the lateral oscillation as the (21Cf1jrt)2 I rate of roll approaches zero, z, per radian qoSc
dCn per radian d(rc!2V), NACA TN 2985 5
C1Ilq (~~) qoSc, Ib-ft-sec/radian
Nr Cn qoSc, Ib-ft-sec /radian r (c)2V
DESCRIPTION OF EQUIPMENT
Test Body
The test body was a wingless configuration having a fuselage of fineness ratio 12.4. Figures 1 and 2 show, respectively, a drawing and a photograph of this configuration. Specification of other physical properties is given in table I.
Both the horizontal- and vertical-tail surfaces were all-movable, pivoting on axes perpendicular to the fuselage axis. For horizontal tail movement, a schedule could be preset so that the tail-fins would deflect and return to trim position in rapid pulse-type movements at designated time intervals during the test phase of the drop. The vertical-tail surfaces were actuated differentially to provide roll control. All tail surfaces were constructed of solid aluminum alloy.
Roll Stabilization
For the tests of this investigation, two different methods were used to obtain the desired roll stabilization. One method consisted of roll-attitude control of the body for the tests in which the roll rate was intended to be nOminally zero. The other method consisted of roll rate control and was used for those tests in which roll rate was to be different from zero. Because of mechanical difficulties, this latter system unavoidably permitted a slight increase in roll rate with increased speed. In addition, the system permitted small variations around the nominal value desired. At most, the magnitude of these variations was about ±O.2 radian per second.
Instrumentation
NACA continuously recording flight instruments were used to record the various quantities measured. A listing of the quantities and of the instruments used to measure them is presented in table II. 6 NACA TN 2985
TEST PROCEDURE
The results presented in this report were obtained during a series of free-fall drops of the test model in which the model was allowed to fall freely at approximately 0 0 angle of attack. During the test period of the drop, pitch disturbances of the model were produced intermittently over intervals of time varying from 3 to 8 seconds. At the conclusion of the test period for each drop, the model was recovered by the use of a dive brake and parachute.
Five drops of the body-tail combination comprised the test program. Three of these drops were conducted at constant roll r ates of nominally 2.0, 3. 5, and 4.0 radians per second. The other two were drops at nomi nall y zero "roll rate and were conducted t o obtain the nat ural f r equenci es of the test configuration i n pi tch and in yaw.
RESULTS AND DISCUSSION
Experiment al Effects of Steady-Roll on No-Roll Natural Frequencies
Reference I has indicated the possibility of having coupled motions due to steady rolling. One result of the investigation reported herein has been an experimental verification of the existence of these coupled motions, as shown by the example time histories of angle of attack and angle of sideslip in figure 3. The angle-of-attack and sideslip records presented in this figure are representative of the type of curve result ing when a short- period oscillation and a long-period oscillation are superimposed. The long- period oscillation has been represented by a dashed line in figure 3 to make it more readily discernible. The appear ance of both oscillations in each angle-of-attack and Sideslip record is a result of coupling due to rolling. The long-period oscillation should not be confused with the phugoid oscillation which, for the test model, would have a period much larger than those of figure 3 (of the order of 25 sec).
As predicted by the analysis of reference 1 and as shown by the time histories in figure 3, the presence of steady rolling causes two frequencies instead of one to appear in the pitching motion, and the same two frequencies also appear in the yawing motion. In the non rolling case, the pitch and yaw frequencies are known as the "body nat ural frequencies," and, in order to make the following discussion succinct, the fast frequency will be designated arbitrarily the "pitch" frequency, and the slow frequency, the "yaw" frequency. Similarly, in the subse quent analysis of the rolling case, the fast component of the pitching and yawing frequencies will be referred to arbitrarily as the "pitch" NACA TN 2985 7 frequency and the slow component as the "yaw" frequency. In tills sense, the presence of steady rolling can be said to have an effect upon the body natural frequencies. This effect is confirmed by experimental data presented in figure 4. Figure 4(a) indicates that as roll rate increased, the frequency in pitch also increased, and figure 4(b) indicates that the frequency in yaw decreased when roll rate was increased. In fact, in one instance, the yaw frequency decreased to zero at some roll rate ranging from about 3.0 to 4.0 radians per second for a eondition of low Mach num ber (M RI o. 86) .
The instance of body instability indicated in figure 4(b) is shown in time illstory form in figure 5. For tills test, the roll-control system was programmed to stabilize the model at zero roll rate for 24 seconds, and then to increase the roll rate to a constant value of 4.0 radians per second. At t = 20.6 seconds, the body was pulsed in pitch, and, while the body was still oscillating because of the pulse, a change to the 4.0 radian per second roll rate was initiated. At a roll rate in the neighborhood of 3.5 radians per second, an apparent divergence began in the long-period component of pitch and yaw oscillations. Since roll rate was the only factor sensibly changed at the approximate time of divergence, the resulting instability is attributed to the equality of the roll frequency with the lower of the two body natural frequencies. For example, figure 5 indicates that the cyclic roll frequency is approx imately 0.56 cycle per second (3.5/2~) for the Mach number at which divergence occurred. In comparison, the lower of the two natural fre quencies at the same Mach number is estimated to be about 0.5 cycle per second (fig. 4(b)).
Comparison of Measured and Experimental Effects of Steady Rolling on the No-Roll Natural Frequencies
The method developed in reference 1 for computing oscillation fre quencies, neglecting damping, was employed to compute the frequencies for the test model for the two rates of roll, 2.0 and 3.5 radians per second. The particular equation developed in reference 1 for the calcu lation of frequencies for no damping (S8 = Sw = 0) is
o (1 ) where the roots of the equation, after evaluation of the coefficients, provide the results sought. Figure 6 compares the calculated values with the experimental values previously presented in figure 4. As is apparent from the comparison in figure 6, some difference exists between calculated and experimental r esults, these differences being greatest for the long period oscillations . Because of these apparent differences, the question is raised as to whether the differences are due to the omission of damping.
J 8 NAeA TN 2985
The method of reference 1 can be used also to calculate the fre quencies of oscillation for the presence of damping. The particular equation developed in reference 1 to calculate oscillation frequencies for presence of damping is a quartic
(2 )
Roots of the quartic, after evaluation of the coefficients, provide values of the frequencies sought. The results of calculations using equation (2) are presented in figure 7 where a comparison is made with the experimental data for roll rates of 2 .0 and 3.5 radians per second. By comparison of figure 7 with figure 6, the effect of including damp ing in the calculations is seen to have improved, in general, the agree ment between calculated and experimenta l frequencies. This improvement appears particularly pronounced in the long-period oscillation for a roll rate of 3. 5 radians per second. Values of damping used in the computations were small, being in the order of se = 0.06 and So/ = 0.04 for a roll rate of 2 .0 radians per second, and se = 0.09 and So/ = 0.06 for a roll rate of 3.5 radians per second.
Effects of Roll as Applied to the Design of Missiles
The designer usually thinks of the natural frequencies of an air craft in pitch and yaw in terms of the static stability parameters emu and Cnf3 where (2rrf )2 C e loY: IIlu qJ3c
and (2rrfo/ )2 l z Cnf3 (4) qJ3c
The effect of steady rolling on these parameters may be considered to modify the nonrolling frequencies (fe and fo/) to new frequencies (fe ' and fo/') having values dependent upon the rate of roll. The stabil ity parameters which are considered as being effective for the rolling ' case, therefore, are Cma and Cnf3 '. These effective stability param- eters are obtained by replacing fe and fo/ in equations (3) and (4) with fe' and fo/ '. A factor of interest to the designer is the ratios Croa'/Croa and Cnf3'/Cn~) which represent the ratio of the stabilities dur- ing steady rolling to those prevailing at no roll. NACA TN 2985 9
In order to provide a rapid method for the determination of the ratios Cma'/Cmn and Cn~'/Cn~ for a wide range of pitching, yawing, and rolling frequencies, two charts based on equation (1) have been prepared, and these are presented in figure 8 for Ix = 0 and Iy = I z • Equation (1) neglects damping but provides the first-order effects of steady roll on the no-roll frequencies as evidenced in the present tests. Use of the charts of figure 8 requires knowledge only of the parameters roe2 and ~2 which, for a particular body and roll rate, define a single point in each of figures 8(a) and 8(b). In this regard, the curves of figure 8(a) are drawn for short-period oscillations, and the curves of figure 8(b) for long-period oscillations; that is, when fa > f1j.r (short period oscillation in pitch, long-period oscillation in yaw), values of ~'/C~ are given by figure 8(a) and values of Cn~'/Cn~ are given by figure 8(b). Conversely, when fa < f1j.r (long-period oscillation in pitch, short-period oscillation in yaw), values of ~'/Cmn are given by figure 8(b) and values of en '/Cn are given by figure 8(a). Important regions of instability~are ~abeled in figure 8(b).
The charts of figure 8 have been used to estimate the effects of roll rate upon the static stability of two hypothetical missiles, one symmetric in pitch and yaw, the other asymmetric. The results are shown in figures 9 and 10, respectively, as functions of Mach number and alti tude. For the symmetrical missile, the assumed static stability deriv atives for no roll are given in figure 9(a) as a function of Mach number. 2 These values were converted to frequencies, and then the parameter, ae , was formed for roll rates of 2.0 and 4.0 radians per second. This param eter was used to obtain, f r om figure 8 , values of the factors C~'/C~ and C~ I/Cns ' applicable t o the short- and long-period oscillations for steady rolling. The factors obtained are shown in figures 9(b) and 9(c) for the short- and long-period oscillations, Cmu and Cn~' respec- tively. For ~,all values of the factors obtained are greater than unity; but, in general, the reverse is true for C~ with values of the multiplying factor Cn~'/Cn~ becoming as small as zero.
A set of curves similar to those in figures 9(b) and 9(c) are plotted in figures 10(a) and 10(b) for an asymmetric missile. For this missile, values of Cmu and Iy/Sc are assumed to be the same as those presented in figure 9(a), but the values of Cn~ have been taken arbi trarily as half the values presented in figure 9(a). The factors obtained for the unsymmetrical missile, in general, follow the same trends as the factors obtained for the symmetrical missile. However, for the unsymmetrical missile, values of the factor C~'/Cn~ can become negative. This condition arises when the solution to equation (1) contains a pair of real roots, and the motion will diverge. Thus, for both symmetric and unsymmetric missiles, steady rolling can be an impor tant factor in reducing static stability, Solutions to equation (1)
J 10 NAeA TN 2985
indicate that the greatest reduction in static stability is experienced when the roll frequency approaches equality with the lower of the two body natural frequencies. Instances of these reductions in stability are apparent in figures 9(c) and 10(b) (Cn~'/Cn~ ~ 0). From these fig- ures it appears that for a given roll rate, the Mach number at which the most reduction in stability occurs increases with increased altitude.
CONCLUSIONS
The results of a series of free-fall-drop tests of a body-tail combination under conditions of systematically varied rates of roll indicate the following conclusions:
1. The tests reported herein have verified experimentally that steady rolling produces coupling of the pitching and yawing motions. In addition, the frequencies of these motions differ from the body nat ural frequencies.
2. For body-tail combinations having a steady roll rate and body natural frequencies of approximately the same order of magnitude as those measured in the tests of this report, the calculation technique of NACA TN 1627 for no damping may be applied to estimate the first order effects of steady rolling on body natural frequencies. For a more exact appraisal of this effect, the equations of NACA TN 1627 which include damping should be employed.
3. In accord with theory, a roll frequency which approaches the lower of the two body natural frequencies can cause an unsymmetrical body-tail combination to become statically unstable.
Ames Aeronautical Laboratory National Advisory Committee for Aeronautics Moffett Field, Calif., May 27, 1953
REFERENCE
1. Phillips, W. H.: Effect of Steady Rolling on Longitudinal and Directional Stability. NACA TN 1627, 1948. NACA TN 2985 11
TABLE I. - PHYSI CAL PROPERTIES OF THE TEST MODEL
Body
Gross weight, 1b • • • • 1365 Iy and I z , slug-ft2 · . . 800 Ix, slug-ft2 • • • • · . . 5 Center of gravity Station 85.6
Horizontal Tail
Area (plan form projected to body center line), sq ft 6.0 Aspect ratio (based on plan form projected to body center line), dimensionless . . . . . • . • 4.4 Taper ratio, dimensionless . . • • • • • 0.2 Sveepback, quarter~chord line, deg 45 Span, feet . • • • ...... · .. 5.12 Airfoil section, parallel to stream •. NACA 65006
Vertical Tail
Area (plan form projected to body center line), sq ft . • . . 3.3 Aspect ratio (based on plan form projected to body center line . • • • • • • . . 5.0 Taper ratio • • • • • . 0.22 Sweepback, quarter-ehord line, deg . 45 Span, ft • • • • • ...... 4.05 Airfoil section, perpendicular to quarter-ehord line • NACA 65009 12 NACA TN 2985
TABLE 11.- MODEL INSTRUMENTATION
Classification Quantity Instrument of measurement measured used Slave selsyns record- a, ing movements of vanes i3 mounted on boom ahead Angles of body % NACA 2-component con- Bv trol position recorder p Angular q NACA single-component velocities r turnmeters
a.x Accelerations a NACA 3-component y accelerometer a z
Po Pressures Pt NACA 6-cell manometer Station Station o~ o 85.6 :x> Station ~ I\) \0 -25./ OJ \.Jl
Angle-Of-OffOCk I r' . CG __ - t vane ~ r ~ ____ 62.5
Equation for body ordinates between x = 8 and Station /48.25 ~ 27~ x = 139.4 IS: r': 8.5 1 - (jtf2 I) L - J Station Station 2/0.5 /50.5 Station Station ' -2/.7 Station /39.4 8.0 I \ - -- - - 48.7 Angle-of-sideslip~ vane t Note: All dimensions are in inches. ~
Figure 1.- Geometry and dimensions of test model.
f-..J W L f--' +=-
s; ~
~ II) \0 co Figure 2.- Test configuration immediately after release from carrier airplane. \Jl
L NACA TN 2985 15
8r---~--~----~--~--~----~--~--~
-2R-----+-~-_+_-___4--__+_--~-__+--_+_-___f Short-period oscillation
Tim~# s~conds
(a) Roll rate, 2.0 radians/sec.
8r---~--~----~--~--~----~--~---,
-2~~~~-4----+---~--~----+---~--~ oscillation~ Short-period NACA
31 32 33 34
(b) Roll rate, 3.5 radians/sec.
Figure 3.- Time histories of angle of attack and sideslip for comparable Mach number ranges showing existence of coupled frequencies due to rolling. f-' ~ 3rl------4 .. -0- '(}(Ro/l raf~,0 rad/s,c) ...~ ..... -
..... CI) ~~~"
.~~ II T Co) ~ ~ ..~ ~ 0 1 1 1 1 ~ , .8 .9 1.0 1.1 1.2 1.3
Mach numb~r
(a) Pitch frequency (short-period oscillation).
Figure 4.-Effect of steady rolling on natural frequencies of the test body. ~ &;
~ f\) \0 co \Jl
~ ~ -
~ &; 3rl------~------~ ... -0-- f; (Roll rate, 0 rod/sec) ~ .. ~ f\) -..: \0 -0-- f~(Roll rate, 2.0 rod/sec) OJ 'b \Jl c::: tI -<>-- f~(Roll rate, 3.5 rod/sec) 2 --6-- f~(Roll rate set for 4.0 rod/sec i model ~ ~ ...... ,= instobility occurred at M ~ 0.86, see tI CI) ~ ~ figure 5) ~ .~ ~ /I j;;O<"'3l'--""-'-tI---i---ir----Ir-----i ~ ' :,~ b ~ l&: '~ 0' '8~~----~--~--~----~--~--~----~--~ .8 .9 1.0 1.1 1.2 1.3 Mach number
(b) Yaw frequency (long-period oscillation).
Figure 4.- Concluded.
t-' -...:] -10 10 I-' a--.....,. OJ -5 5 p- ~ ir\ 2- r'\ .co ,r '" ~ b, ~ c::::: J ~--I-~\~L \ L V ~ ~ ~o ~ 0 - '- ... Starting time ... ~ Divergence c~nside~edI-H~ .~ of .1!..ulse ./v r--'"'\ -;; 5 ~... -5 fo begin dUring this ~ :t ...~ interval of time .~ ... \ ..l::!;,. .... 10 tJ -10 I~ C) ..... C) ~ ~ /j ~ 15 ~-15 Uv c:: ~ ~ ~ , 20 -20
25 -25 10------~
~ 10 I I I I I I I ,,'" ...~ Range In roll rate values for time - h I I / ~ ~ 5 ~ '\..... interval of Inltlol divergence (M ~O.86) ::::::' 1'7 ~~0" LY 0 I I I I I I I '" 19 20 21 22 2~ 24 25 26 27 Time, seconds ~ ~ &;
Figure 5.- Time hi s tories for a drop of the test body showing the occurrence of divergence at a ~ roll rate of approximately 3.5 radians/sec. !\) \() OJ \Jl MCA TN 2985 19
3 ----Experil77ental - -NA CA TN 1627 Roll rate ,,/ 1- 3.5 rod/sec ~ I .1 ~ } - 2. P rod sec ~ ~ ~ f- --- 1---V
(a) Pitch frequency (short-period oscillation).
'foil role
." } -2.0 rod/sec .... /-" ~ ~-- } -3.5 rod/sec - -- -~- -- ..- - ~ o -- p -- .8 .9 lO 1.1 l2 l3
(b) Yaw frequency (l ong-period oscillation).
Figure 6.- Comparison of experimental frequenci es with thos e calculated by the method of NACA TN 1627 for steady rolling and no damping. 20 NACA TN 2985
3 --E xperimenfal - - N AC A TN 1627 .s:~ ~ Roll rate ~ 2 -3.5 rad/sec ~ ~ - 2.0 rad/sec ~ .....CIa ~ ~V CI) -V ~'" I CI) c:: .0 ~ ~ ..... ~ - CI) ~ -CI) " 0 CI) 0 ~" ~ ~ .~ ?\ ~ ~ ~ (a) Pitch frequency (short- period oscillation). ~ .. ~ ... ~ c:: .... ~~ ~c:: .s: ~ .c:::: 2 ~ ... ~ :t;: ...... ~ 0 ~ Roll rote CIa I ...... ~ } -2.0 rod/ sec ~ /.----/-::::-=-==:- ~ J-3.5 rad/sec ~ ~ ~ r L..--- ~ I ~ o .8 .9 lO 1.1 l2
(b) Yaw frequency (short- period oscillation).
Figure r.- Comparison of experimental frequencies with those calcul ated by the method of NACA TN l62r for steady rolling, by use of experi mentally det ermined values of damping. NAeA TN 2985 21
1000 , Cma C 1/ ma i0'
/ V/ VV Cn,' 100 / Cn,
L-----" I- ./ 'i VI I' ~ V? V; 1J "1 l-I-- V 1.2 ~ / v V; "',2 10 1.3 ~ I r - 1.4 1.5- . -- v 17 V / I I I / I 2.0 f- I------V; / II I 25-1-1------II II 1-1- I 30 I- F77 = 35 4.0 ~. o -l-I-- T -1 ,t ~. O I--17 II / 70 II II I I
~Ll ./ I 10 100 1000
(a) Long-period oscillation.
Figure 8.- Static stability derivative ratios as a function of body natural frequency and roll rate; Sa = S~ = O. 22 NACA TN 2985
1000 1-- - T
J-- r- Region of Cn) J-- Instablllfy J-- Cn,e V J-- (Neqative [7 / factors) 100 .85 ~ I ...... I~ r- Cm/r J C - :::..---::: ~ ' J ma .4. ,\ 1\ L----::; /;/ V. t'-., \ '-. If ~ 2?\\ ~ [\ tiI,Z 10 l\~ Ffi~ . v ..... I \, \.. "- \ \ " .TO \ \ \ ...... '\ "- _" ' .bV \"'- \ .\ ~- .50 .30 ~~'-: "./0 y I .05: "- "- :. 01 ,,- \... T T :"\ \. '" 1\ [. 60 -> 1\ Reqion of InstabIlity 1--'" ,,- " {t~ n rve:-j(v rye:- factors) /-~ 1\ , 1\~~20 1 1 t-"...... "'\"1\ \ 1\ i' I!~ ~ .I I 10 100 1000
w Z f
(b) Short-period osci llati on.
Figure 8.- Concluded. -.16 .. ~ &; ~ :;:: t) ~ ~ f\) .~ \0 co '5~ -.12 \Jl
I .~~ , :::: v ~ 1~ 1;; ~ "'
• (.) CI) .t;:-..;.."... -.08 " ~ ~ ~ fI) :::i . ::::: .. ~ "' """...... e1(.) ~ r--.. ~ I -.04 ....a a\,.
!: ~ ~J o ~ o .5 1.0 1.5 2.0 2.5 3.0 Mach number
(a ) No-roll sta bi li t y deri vat i ves for lat eral and pi t ch plan es.
Fi gure 9.- St at ic sta b i l i t y der i vativ es for a symmet r i c mi ss i le as affe ct ed b y Mach number, rate I I 1 of ste ady rolling , and al t i tu de ; ~ = ~ = -. Sc Sc 7 f\) w
L 8 [\) -l=" \ \ --10,000 ff " -- 40000, ff 1', "- 6 "- ...... 'Roil rate ...... rod/$~c \ ------y ~P.:4 I '\\~ 4 CmQ C- , mQ ~ - / ------V ~ - I~ p=2 2 ------/'
o ~ , o .5 1.0 1.5 2.0 2.5 3.0 Mach numbllr
(b) Effect of steady rolling on Cma.when missile is disturbed in pitch; se S1jr= O. ~ &; Figure 9.- Continued. ~ [\) \.() (Xl Vl ~ ~ l
1/.4 - 10,000 ft ~ - -40000, ff &; 1.2 ~ f\) \ \0 \ 0::> 1.0 \Jl \ .8 ,, Roll raf~ C"~ \ rad/s~c -Cn .6 \ ~ \ '", .4 \. """ ---- p=2 ~ ~, V- .2 / -, - &. _ '~ -- -- ~[>- p=4 t--_I-_~_ o - --- ~~ -- o .5 1.0 1.5 2.0 2.5 3.0 Mach numb,r ~
(c) Effect of steady rolling on Cn~when missi l e is disturbed in pitch; Se Sw= O. Figure 9.- Concluded.
f\) \Jl 8 I\) , -- 10,000 ff ~ \ - -40000, ff " , 6 \ ,
" "-
~ ... " ...... Roll raf, -- rad/s,c f\ ~--- c' 4 ... ma ~ 1- -C '~ ma ~ ~ ip z 4 " ~ V I , ~ ~ / 2 ~ ------l>- p.2
o ~ o .5 1.0 1.5 2.0 2.5 3.0 Mach /lumlJ,r
(a) Effec t of ste ady ro l ling on emu when missile is disturbed in pitch. ~ ~ Figure 10.- St at ic s tab i lity der iv at ives f or a hypoth et ical unsymmet ri cal mi ssi l e as affected ~ by Mach number, r ate of eady l g , and de; ~e = ~w O. st rol in altitu = I\) \0 co \Jl 1.2 /0,000 ft ~ ~ -- ('J f;; > - 40.000 ft 1.0 \ ~ \ ~ 'a I\) ~, \0 , \ ():) '" \ VI CD, .8 , '" '", \ Roll rot~ 0 \ 0 -0 rad/s~c .6 \ \\ ...... ,-- I - ~- p=2 C;,~.4 ~ ---- ~ "- 1\ " ',- en".2 V "- / ~.L1------:;:>- p=4 0 V\ -.2 -.4
-. 6 ~ o .5 /.0 1.5 2.0 2.5 3.0 Mach numb.r
(b) Effe ct of steady r ollin g on Cn~wh en missile is distur bed in pit ch.
I\) Fi gur e 10.- Concluded. -.l