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T R A N S O N I C FLIGHT FLUTTER T E S T S OF A CONTROL SURFACE
UTILIZING A N IMPEDANCE RESPONSE TECHNIQUE
L. 1. Mirowitz -McDonnell Aircraft Corporation,
S t . Louis, Missouri
Abstract "Demon", Figure 1 - a flight flutter testprogram was
conducted concurrently with the speed build-up of the
Transonic flight flutter tests of the XF3H-1
airplane. This program consisted of the transient
"Demon" Airplane have been conducted utilizing a
response technique of flight flutter testing through
frequency response technique in which the oscillating
pilot induced control surface impulse motion. How-
rudder provides the means of system excitation. These
ever, during the course of this flight flutter test pro-
tests were conducted as a result of a rudder flutter
gram, a neutrally stable empennage flutter condition
i n the transonic speed range. The technique
incident
was encountered at a Mach number of 1.04 and an
employed is presented including a brief theoretical
altitude of about 30,000 feet. Records taken during
development of basic concepts. Test data obtained
the flight indicated that the flutter condition emanated
during the flight are included and the methodof inter-
from the fin-rudder system with a frequency of 20
pretation of these data is indicated. This method is
cycles per second. A s aresultof this, a program w a s
based on an impedance matching technique. It is
initiated consisting of theoretical investigations i n
shown that an artificial stabilizing device, such as a
conjunction with flight flutter testing in order to
damper, may be incorporated in the system for test
establish the cause of the instability and to guide in
purposes without complicating the interpretation of the
the determination of corrective measures.
are
test results of the normal configuration. Data
presented which define the margin of stability intro-
This paper concerns itself with the concepts and
duced to the originally unstable rudder by design
results obtained from the subsequent flight flutter
changes which involve higher control system stiffness
test program. The theoretical concepts underlying
and external damper, It is concluded that this tech-
the approach which was utilized, and which involves
nique of flight flutter testing is a feasible means of
i n particular a frequency response technique i n which
obtaining flutter stability information in flight.
the oscillating rudder is utilized as the aeroelastic
forcing system, have been presented in R. A. Pepping's
INTRODUCTION
paper, "A Theoretical Investigation of the Oscillating
With the initiation of the first flight of the
Control Surface Frequency Response Technique of
XF3H-1 Airplane - the prototype version of the F3H-1
Figure 1.
Flight Flutter Testing", Journal of the Aeronautical system and the control system is defined as the Sciences, August 1954, Volume 21, No. 8. The idea moment applied to each system per unit deflection to behind this approach involves the concept of impedance sustain motion at any given frequency. As seen from matching as applied to dynamic aeroelastic systems.
the block diagram, the complete system consists of For convenience, a summary of the theoretical de- two feed-back loops. Loop 1, the inner loop, accounts velopment is repeated herein.
for the fact that the servo -- which could be, for ex- ample, a hydraulic actuator -- acts as a root restraint THEORETICAL BACKGROUND for the control surface, tending to return the control surface to neutral upon deflection. Loop 2, the outer In order to determine the flutter stability of a loop, accounts for the servo as part of the autopilot system which has incorporated a servo control me- system, sensing airplane motion away from the set chanism, it is important that the dynamic behavior of path with resultant corrective action.
the servo be included in the flutter investigation in combination with the structural, inertia, and aero- In this paper we will confine ourselves of the dynamic contribution of the remaining control surface- dynamic character of the inner loop, or Loop 1, since primary surface system. In addition to acting as a the autopilot of the airplane is not of immediate control surface restraint mechanism, the servo serves interest. The block diagram for the inner loop, also as a control actuation system which receives its Loop 1, is again shown in Figure 3 where z z is the signal either from the pilot, a radar beam, or from a impedance of the servo mechanism, i.e., the hydraulic sensing element in the fuselage in which case it be- actuator, obtained from calculations or from measured comes part of the autopilot system of the airplane.
frequency response data, and where MZ is the moment A method showing the interaction of the servo-control surface-airplane system is the block diagram repre- FLUTTER SYSTEM sentation used frequently in the theory of servo me- chanism analysis which schematically traces through CONTROL /RAED ,M_ TR TANCEo F the events which take place when a signal is received by the servo resulting in motion of the airplane from its predetermined or pilot-set path.
IMPEDANCE /IMPEDANCE) Block Diagram Representation of the Aeroelastic System The system analyzed consists of the control surface-airplane flutter system, the servo system, BLOCK DIAGRAM OF INNER LOOP - LOOP I and the return loop from the airplane fuselage sensing element (gyro) back to the servo. In block diagram form, this feed-back system may be represented as Figure 3.
shown in Figure 2. The impedance of the aeroelastic REFERENCE CONTROL /-FLUTTER SYSTEM ADMITTANCE S SIGNAL IMPEDANCE /(RECIPROCAL OF IMPEDANCE) R'_"" °ti=R-Ct° I-_ " SERVO _ '8':'8i- LOOP I LOOP rr Of o ! !
BLOCK DIAGRAMOF COMPLETE AIRPLANE LOOP
Figure 2.
required to deflect the control surface through an angle /3 measured at the point of moment applicRtion.
This can be measured in flight or can be idealized h(BENDING) mathematically by a number of degrees of freedom such as surface torsion, surface bending, fuselage __.a (TORSION) bending, and control surface rotation. The equations of motion for the inner loop are given in Figure 4 in terms of the ratio of output over input. From tile equations of motion the characteristic equation defining ,6' (CON TROL the frequency parameters of the system and its SURFACE ROTATION) stability is given by the denominator of equation 1.0 IDEALIZATION OF FLUTTER STSTEM as equation 1.1. Neutral stability is determined ff IMPEDANCE OF there exists a finite frequency which satisfies equation ACTUATION SYSTEM (HYDRAULIC SERVe) 1.1 or, as stated in 1.2, there exists a hinge moment impedance at some finite frequency which is equal and opposite to the control system impedance.
_ i M B IDEALIZATION OF CONTROL SYSTEM EQUATION OF MOTION :
OUTPUT zp '
1.0 Figure 5.
INPUT _i Z_ 4- I 1 4- CHARACTERISTIC EQUATION: !.1 ,
_--; = o
O= Eh, e. ,8 -I- Ehh. H 4- Ehc,. ot O= E.,9" _ 4- Eah-H 4- Eaa. a NEUTRAL STABILITY CRITERION: 2.1 Where: Ejk=Rik +ilik-O'oik for j _k
_ (o
,%_ -1 ) I
1.2 (Eli) =E ii -Ooii _"_-I"_ ( ' +ieii) = eii- Zi
:- From 2.1
BASIC STABILITY CRITERIA _pb_ ] Eh_ Ehh , z, Eha I Mp_ DO _ t(w) - J Ea_ Eah Eoa 2.2 ,8 N#8 g(w) Figure 4.
Eh_
I" I
Eah Ea_ Idealization of Aeroelastic Impedance ANALYTICAL EXPRESSION FOR IMPEDANCE - M_/,8 From the theoretical standpoint the aeroelastic impedance of the control surface can be idealized Figure 6.
s c h e m at i c a 1ly by the three-degrees-of-freedom: primary surface bending, primary surface torsion, and control surface rotation as shown in Figure 5.
This idealization is the minimum required to cover system with free-floating or unrestrained control sur- face and the denominator is the flutter stability de- all the concepts of the approach utilized herein -- terminant for infiniterestraint in rotation or, inother additional degrees of freedom may be added as neces.
sary without invalidating any of these concepts. The words, rotation is not a degree of freedom. The upper diagram is the idealization of the flutter system. stability determinant for the denominator would be The lower diagram is the idealization of the control the primary surface flutter stabilitydeterminant.
system. The equations of motion of the flutter sys- Stability Criterion tem as actuated by the driving hinge moment M_ is Shown in Figure 6, equation 2.1. Solving from 2.1 for Figure 7 again shows the characteristic equa- the impedance /4_/_ , equation 2.2 is obtained. As tion of the. inner loop as equation 3.1. Substituting seen, the rudder hinge moment impedance is the equation 2.2 into equation 3.1, the characteristic ratio of two stability determinants: the numerator equation is rewritten as equation 3.2 in terms of the is the stability determinant of the aeroelastic flutter stability determinants of the aeroelastic flutter sys-
14S
CHARACTERISTIC EQUATION " infinity to plus infinity. For stability, none of the roots of the transformed characteristic equation -- ZB+_-_ =O 3.1 the numerator of 3.4 - may have a positive real part.
FROM 22: M_ Do By the use of a modified Nyquist approach, this 3.2 is established by observing the behavior of the re- Z 8 + ,_ -- Ni;r: sponse vector (Mz/9 + z_) as the frequency is FROM 21: varied from minus infinity to plus infinity. If there Do*Z_N =D =STABILITY DETERMINANT WITH CONTROL are any roots with positive real part -- which denotes _ SURFACE RESTRAINED BY Z_ 3.3 instability -- the vector (,vZ/_ + Z z) will perform as FROM 3.3 AND 3.2(b): many clockwise rotations about the origin when M_ D plotted on a complex plane as there are roots with Z,+ -'_- _ _r: 3.4 positive real parts. It is possible, however, also to 3.4 STATES THAT BOTH(Zr;+ _)AND D DEFINE THE STABILITY OF THE have roots in the denominator of the vector equation FLUTTER SYS/EM WITH THE CONTROL SURFACE RESTRAINED BY 3.4 with positive real parts. Roots in the denominator Zp. THUS THE STABILITY IS RELATED TO THE MEASURABLE Np_ , are denoted as poles and are the solutions of DRIVING HINGE MOMENT.
the flutter system with infinite restraint in the control STABILITY EQUATIONS system. In that case, the vector equation 3.4 will perform as many counterclockwise rotations about the origin as there are poles with positive real parts.
Figure 7.
Since both conditions can exist simultaneously, then for the system to be stable -- or no unstable roots in tem. The numerator of 3.2 is the stability determin- the numerator of 3.4 -- the direction of rotation of ant with the control surface restrained by the control the vector (,v_/2 + z z) about the origin must be system impedance, zz . This determinant is denoted counterclockwise and the number of rotations must as D. Substituting 3.3 into 3.2, the characteristic be equal to the number of unstable poles.
equation 3.1 is transformed into 3.4 which states that both (Z_ + Mnl/_) and D define the stability of the If there are no unstablepoles, or inother words, there are no unstable roots in the denominator, and flutter' system with a control surface restrained by therefore, the primary surface is flutter-free with an Zz . Thus the stability is related to the measurable infinitely restrained control surface, then for the driving hinge moment. The characteristic equation system to be stable, the vector (MZ/,3 + Z;_) must 3.4 will determine the system stability. In general, not envelop or rotate about the origin. In this latter the numerator of the right hand side of 3.4, D, is a case, simple energy concepts will also lead to the differential equation of rth order and the denominator same conclusions regarding the definition of stability, k'nz is a differential equation of nth order. The for example, Pepping's paper referred to previously discusses the energy approach.
problem then, is to determine if there exist any roots of the numerator which are characterized by a posi- Figure 8 presents the ground rules for applying tive exponential decay function (divergence) which the modified Nyquist stability criterion to the imped- would indicate instability or by a zero decay function ance stability plots. It should be noted that in actual which would indicate neutral stability. If the equa- practice the frequency variation can be limited to a tions are written in differential equation form, then reasonable range enveloping the suspected flutter the roots of the stability equation or of the stability frequency. Figure 9 indicates aparticularapplication determinant, D, may be solved for directly. This is of the impedance plots and shows a speed which would roughly the case of theoretical flutter analysis. How- be unstable and a neutrally stable speed and relates ever, equation 3.4 states that stability may also be this to the well known flutter stability, plot of velocity determined as a measure of the driving hinge mo- versus control surface rotational frequency.
ment impedance, Hz/9 , for this is a measurable quantity.
Quite often instead of utilizing the complex In the theory of servo mechanisms, a relation- plane plots of the impedance vector an alternate ship is drawn between the determination of system method is applied which makes use of the so-called stability from the solution of the differential equa- phase margin plot. This is shown in Figure 10.
tions (transient stability) and the results from the frequency response behavior of the dynamic system. The significance of the impedance matching ap- The frequency response technique is denoted as the proach is as follows. The aeroelastic impedance, Nyquist approach. In this case the differential eaua- M_//_ may be calculated or measured. It is then a tions of motion are written in transformed form and given known quantity for a given set of conditions the system is analyzed without solving for the roots independent of the restraint, Z_. The restraining by an examination of the behavior of the response (Mnl;_ + z z) as the frequency is varied from minus impedance, Z_, may be measured or calculated. The MODIFIED NYQUIST STABILITY CRITERION: 1. For each forward velocity and a reasonable frequency range, measure or calculate M_//fl vs. c_.
2. Add to M_/_ the measured or calculated control system impedance Z_ and obtain M_/_+Z_ vs. co . Plot on a complex plane for each forward velocity.
3.
Calculate the flutter speed or speeds of the system with infinite control surface restraint. This is designated as Vf0" 4, For velocities less than VF0 the system is stable if the vector M_/fl+ Z_ does not envelop the origin.
5. For velocities greater than VF0 the system is stable only if the number of counterclockwise rotations about the origin of MB/_ +I#_ is equal to the number of VF0'S.
6. For any velocity the system is neutrally stable Z_ M#/_+ Z# passes through the origin.
Figure 8.
vector sum of the two determines the stabilityof the total system. Additional devices which modify the + I?) IMIGINARY restraining impedance Z_, may be evaluated without NEUTRALLY I r STABLE ,_-'-'_V ; ¢ further experimental work in flight once the aero- elastic impedance, M_/_ has been established VIC VffiC+ i uniquely.
V=( TEST CONFIGURATION The flutter testing of the XF3H-1 rudder is j UNSTABLE !
based on the theory presented in the previous discus- sion. The hinge moment M_ was supplied by the hydraulic actuating _ylinder through sinusoidal opera- FLUTTER STABILITY PLOTS-IMPEDANCE tion of the actuator valve. This is shown in Figure 11 which presents a schematic of the shaker system.
Figure 9.
I_RUDDER HL BELLCRANK 19-61390- _ INSTRUMENTED WITH ACCELEROMETER I_F,x _J_PUSH RO D i_ p,L--,_/f 19.6130 S. 905 VISCOUS DAMPER.,_ _--_ 7 -RUDDER HORN J[ I " INSTRUMENTED /_j PUSH_ROD_
MAROI,-6 _ I / ,_L__
_"- 19-61392-1 PILOT VALVE ._.)[//\ _"RUODER POWER FLIGHT TEST BELL _ _ CYMNDER
• , :o ,/ •
CRANK 19-04186 \ II _-/
= I V
& ECCENTRfC _"_'JL,"_/_.,._ /.FLEX SHAFT co--GEAR 80X & COUNTER
I / u'sTA't'
FLEX t EX ,,ER O'OR
=__,., i/
SHAFT COVER _ _--_" bJ_ FLUTTER STABILITY PLOTS-PHASE MARGIN RUDDER EXCITER INSTALLATION Figure 11.
Figure I0.
Also added to the system was a viscous damper for • VERTICAL ACCELEROMETER stability reasons. Figure 12 indicates the type of • LATERAL ACCE LEROMETER • AUTOMATIC CUT-OFF SWITCH instrumentation employed on the airplane for these A POSITION (NDrCAFOR tests. One of the strain gages shown in this figure o STRAIN GAGE was installed on the control rod leading directly into the rudder and gave a definition of the driving hinge moment, M_, and the other was installed on the control rod just upstream from the damper to define the hinge moment of the rudder as restrained by the damper. The damper was a non-linear velocity squared damping device.
TEST PROCEDURE The general technique of testing consisted of stabilizing the airplane at a constant Mach number at _ENTATION SET-UP about 30,000 feet altitude (this was the altitude at which all the test data was obtained and varying the fre- Figure 12.
IbblqllqLhl IIILqklqqlllklllilqhlllllllllllllllllllll
CHAN. 16 UPPER RUDDER 11 Nz PILOT'S B H 2 CHAN. 14 LOWER RUDDER POS. 13 LOWER RUDDER POS CHAN. 19 CORR." J. 9 Lz UPPER FIN AFT 15 RUDDER BELLCRANK POS.
CHAN. 2 RUDDER PUSH ROD _CHAN. 17 VOLTAGE MONITOR
TYPICAL OSCILLOGRAPH TRACE
Figure 13.
quency of oscillationthrough the range of about 5
RUDDER FLUTTER IMPEDANCE M=.83-.86
through 35 cps. This rangewaschosen as being
sufficient to envelop the flutter frequency previously encountered.
Oscillations were introduced through hydraulic actuator displacement of the rudder through suitable valve motion. The rudder driving hinge moment • 444-72 FLIGHT NUMBER [ I a 444-74 upstream and downstream of the damper was measured as well as the rudder angular deflection. This pro-
.,44. t |
cedure was repeated at successively higher values of Mach number to establish the trend of stability
!
with increasing Mach number.
/
O t Z $00C Rudder static deflections of approximately a
!
Q degree and a half were employed. The frequency {3 4001 variation was accomplished by an automatically oper- K ating rotary switch located in the cockpit and aDproxi-
/
Z w 3000 mately 3 seconds were devoted to each frequency r.
o point. Sufficient fatigue strength was provided in the power cylinder back-up structure and in the connecting links so that these components were good for 50,000 Z
/
cycles of rudder limit hinge moment. Strategically located acceleration sensing devices were tied into o a the circuitry of the drive motor which were set to turn off the drive motor whenever excess accelera- tions were encountered. Dynamic measurements throughout the airplane were taken during these tests and the data were recorded on the airplane oscillo- graph. wO 150
"l
12¢
\
TEST RESULTS
! \
< 6C A typical oscillograph trace is shown in Figure 13. The instrumentation was somewhat primitive by present day standards; however, the test was con- ducted in 1952 and much of the more sophisticated types of recording transducers and data reduction 10 15 25 30 machines were not available at that time. Several FORCING FREQUENCllLC p S problems which were encountered with this instru- mentation were: Figure 14.
(1) Rudder angles of about a quarter of adegree or less were difficult to measure and some drift in the measurements occurred. Con- The data shown in Figure 13 were manually tinuous calibration of the position indicators reduced and typical plots for various Mach numbers was required in order to hold down the of the test results are shown in Figures 14 and 16 errors from this source through 18. The plots cover the Mach number range (2) Aecelerometers in the tail assembly were of .85 to 1.16 and are representative of the impedance not temperature compensated and no ac- data taken through M = 1.26 for the system with and curate definition of the characteristics of without damper. Data for each Mach number plot the tail oscillations could be obtained. Tem- were obtained during several flights as indicated.
perature measurements were recorded for Positive phase margins indicate stability. Negative several locations in the tail assembly and phase margins indicate instability. To determine the this data was used to correct the measured stability of the actual restrained rudder, this data test results must be combined with the control system impedance, [z_]. Stability is determined by the phase margin
(3)
Power cylinder valve displacement and power existing when [/4Z//31 is equal to [zzl. The hinge cylinder output displacement could not be obtained correctly.
moment data obtained at M = 1.04 were utilized in a As noted, this data does not indicate a zero "RUDDER FLUTTER IMPEDANCE M"1.04 phase margin point at M = 1.04 which was the Mach number of the flutter experienced during the initial _J <• stages of the flight flutter testing of the airplane.
•5
However, the frequency of flutter is correlated.
Tests of the hydraulic actuator impedance in- 800_ FLIGHT NO.
dicated that in this frequency range the system was Q • 444-88 not acting as a pure spring but that some negative ,_ 7ooc • 444-93 -d 8 444-94 phase margin was contributed by the actual impedance • 444-9S
/
of the hydraulic power cylinder. A measure of this m, 444-98 loss in phase margin is indicated as the shaded area • 444-100 around M = 1.04. An increase in control system ¢ 444-101 stiffness to 1740 in. Ibs per degree which was basically
_z s_
< obtained by a more powerful power cylinder is shown
L
i/
It by the dashed curve. This increase in stiffness in- Q 400{ creases the stability of the rudder around M = 1.04; u however, it also indicates that the second unstable
/
Z 301_ o _
/
COMPUTED RUDDER FLUTTER IMPEDANCE BASED ON TEST RESULTS AT M=1.04 Z
/
m,, 100¢ IOO Q O u C
Z /
/
Z Q,
z 0
_ 9o < I Z 4°°( 6O
1- /
• I
_ 3c _ 30OO
•
C
/
10 15 20 25 30 FORCING FREQUENCY-CPS
__ /
100( ..d Figure 15.
Z C 18C theoretical study in order to determine which of the possible critical degrees of freedom of the aeroelastic _SC system are influential in the flutter system. It was found that utilization of the two degrees of freedom, rudder rotation, and rudder torsion, was sufficient
\
to describe reasonably well the variation of the hinge moment and phase margin with frequency atthis Mach
/
\
number. This is shown in Figure 15.
/
\
The measured test results have been plotted
_J
for two restraint conditions of the rudder; one, the I0 IS 20 25 30 degree, and the other, that of the final control system - FORCING FREQUENCY--CPS 1740 in. lb/degree. This is shown in Figure 19, for the idealization of the impedance of the control sys- tem as a pure spring without hydraulic damper.
Figure 16.
RUDDER FLUTTER IMPEDANCE M'1.15-1.16 RUDDER FLUTTERIMPEDANCE M=1.11-1.12 Z ° O• Z_ |OO¢ WITHOUT WITH FLIGHTN DAMPER DAMPER WITHOUT WITH t_
o
DAMPER DAMPER FLIGHT NO.
444-112 o _e 444-169 a o * 444-170 a .=J .,J o *b 444-170 o • 444-172 • 444-172
!
@ 444-192 _ 444-174 444-177 T !
• • 444-197 Z Z • 444-201 " 444-200
o
_-.
sooc ¢.
/
i a l 400¢ a Q Q
\
/
ig m _ -_ 300_ i- Z O 20OC D Z _
;ooc #
.r z s o it m IRC IIC
J
i i 12¢ 12C
-\
a
,!
.A,
\
_, 6c
/
Q
_, o
I
/
-18¢ 10 IS 2O 2S 30 FORCING FREQUENCY-C P S FORCING FREQUENCY-C.P.S.
Figure 18.
Figure 17.
control system (final configuration) is indicated in Mach number range around M = I.I is not stabilized markedly by this stiffness change even though some Figure 20. An additional gain in phase margin is shown with adequate stability existing throughout the improvement inphase margin is indicated. From this data, it was concluded that stiffness alone willnot applicable Mach number range. Similar plots were constructed for various combinations of damper and eliminate the flutter instability on the rudder. The stability of the system with the damper and the stiffer power cylinder impedance characteristics.
e_ORIGINALCONTROL SYSTEM • STIFFER CONTROL SYSTEM Z,e = KB= 880 IN. LB./DEG.
ZB-K_-1740 IN. LB./DEG.
WITH DAMPER e--- STIFFER CONTROLSYSTEM 30-- Zm=KB = 1740 IN" LB'/DEG" ® FLUTTER INCIDENT _e ''_'l ,,_..6 A_F. d _.,_,..A u 2e
J u
2O z
I r"_ 0 10
7'
IO
ii
',1
-8 • 9 1.0 1.1 1.2 1.3 O 1.2 1.3 I I I 12(_
il
!!
O STABLE STABLE ,, I I r_ I _ BO 8O
ol I
a Z t !
O V O DAMPER fESTIMATED EFFECT Z 4¢ 4O / OF HYDRAULIC ACl 5, TOR
J
IMPEDANCE • _&.db.A,..i -1-
ga I
o._ I'_ i.o-
UNSTABLE I "UNSTABLE I I_,_.
L..4C !.3 J 1.2 .40 / I I I MACH NUMBER AACH NO.
XF3H-1 FIN-RUDDER XF3H-1 FIN-RUDDER FLUTTER STABILITY FLUTTER STABILITY NO DAMPER WITH DAMPER Figure 20.
Figure 19.
REFERENCES CONCLUSIONS 1. Pepping, R. A., A Theoretical Investigation of the On the basis of this discussion, the following Oscillating Control Surface Frequency Response is concluded: Technique of Flight Flutter Testing, Journal of the Aeronautical Sciences, Vol. 21, No. 8, August 1954.
(1) The impedance matching technique of de- termining stability is a feasible means of 2.
Chestnut, Harold and Mayer, Robert W., Servo- conducting flight flutter testing.
mechanisms and Regulating System Design, Vol. 1, John Wiley and Sons, Inc., New York, 1951.
The interpretation of the test data is straight-
(2)
forward and revolves mainly around the 3, Mirowitz, L. I., Flutter Stability of a Wing with determination of the aeroelastic impedance Servo Root Restraint and Automatic Servo Control, vector of an oscillating control surface.
McDonnell Aircraft Corporation Internal Engineer- ing Note, 1 July 1953.
(3) The aeroelastic impedance can be deter-
mined in flight utilizing a control surface 4. Mirowitz, L. I., F3H-1 Airplane, Flutter Analysis which has been stabilized artificially and Summary Report, McDonnel Aircraft Corporation information from this data may be obtained Engineering Report No. 3333, 23 December 1953.
for a variety of artificial stabilization sys- tems. These stabilization systems may take the form of external dampers or stabilizing feed-back signals.
SYMBOLS #_ = oscillatory rudder hinge moment.
-- circular frequency.
= rudder deflection.
= air density.
#_/_ = flutter impedance of the rudder aeroelastic b o = reference semichord.
system.
-- phase margin -- 180 ° less the phase angle.
z_ = rudder control system impedance.
Subscripts: RJk + = oscillatory flow aerodynamic Ijk derivative. See o = output.
%jk = inertia derivative. Equation = input.
2.1 _j# = natural frequency of a degree of E = error or input less output.
freedom in equations of motion.