Document
N94- 35873
FLUTTER SUPPRESSION DIGITAL CONTROL LAW DESIGN AND TESTING FOR THE AFW WIND-TUNNEL MODEL Vivek Mukhopadhyay NASA Langley Research Center, Hampton, Virginia SUMMARY Design of a control law for simultaneously suppressing the symmetric and antisymmetric flutter modes of a sting mounted fixed-in-roll aeroelastic wind-tunnel model is described. The flutter suppression control law was designed using linear quadratic Gaussian theory, and involved control law order reduction, a gain root-locus study and use of previous experimental results. A 23% increase in the open-loop flutter dynamic pressure was demonstrated during the wind-tunnel test. Rapid roll maneuvers at 11% above the symmetric flutter boundary were also performed when the model was in a free-to-roll configuration.
INTRODUCTION A summary of the Active Flexible Wing (AFW) Program is presented in Ref. 1. Within the operating range of the Langley Research Center Transonic Dynamics Tunnel, the sting mounted AFW aeroelastic model had both symmetric and antisymmetric flutter modes, in a fixed-in-roll configuration, and a symmetric flutter mode only, when the model was in a free-to-roll configuration. The active flutter suppression system (FSS) test goals were to demonstrate: a) simultaneous symmetric and antisymmetric flutter suppression for the fixed-in-roll configuration, and b) symmetric flutter suppression in the free-to-roll configuration. An additional goal was to test a rolling maneuver load alleviation system along with the FSS above the open-loop flutter boundary. Since the free-to-roll symmetric flutter and the fixed-in-roll symmetric and antisymmetric flutter modes had very similar characteristics, a single FSS control law was designed and demonstrated for both the flutter test configurations, a) and b) as stated above. This paper addresses the mathematical modeling, control law design and wind-tunnel test results.
NOMENCLATURE A,B control law state-space matrices C,D control law output matrices Bo Kalman state estimator gain matrix
Co
optimal regulator gain matrix E expectation operator F,G plant state-space matrices Gw gust input matrix g gravitational acceleration constant H sensor output matrix I identity matrix M Mach number p estimator Riccati equation solution q dynamic pressure, psf qf flutter dynamic pressure, psf Q1 plant output weighting matrix Q2 control input weighting matrix Rv measurement noise intensity matrix Rw gust input noise intensity S regulator Riccati equation solution s Laplace variable T sample period, seconds t time, seconds u control input vector v measurement noise vector w gust input noise to frequency, radians/second x plant state vector Xc control law state vector y measurement vector accelerometer output, g's control surface angular position, degrees _ubscripts: LEI wing leading edge inboard LEO wing leading edge outboard TEl wing trailing edge inboard TEO wing trailing edge outboard tip wing tip AFW active flexible wing CL closed loop CPE controller performance evaluation FSS flutter suppression system LQG linear quadratic Gaussian OL open loop psf pounds per square foot RMLA rolling maneuver load alleviation rms root mean square SISO single-input single-output AFW EQUATIONS OF MOTION The description of the AFW aeroelastic wind-tunnel model and the wing-tip ballast stores, including details of the accelerometer sensor positions and multiple control surface actuation capabilities, are provided in Ref. 2. The accelerometer sensors and the control surface locations on the wing-plan form are shown in figure 1. The development of the aeroelastic equations of motion is described in Ref. 3. The equations for the symmetric and antisymmetric motion were developed separately, using ten flexible modes for each configuration. The flexible mode shapes and natural frequencies were derived from a finite- element modal analysis and were corrected using ground vibration test data.
State-space Equations A set of state-space mathematical models were developed 3 for control law design. For the aeroelastic equations, the doublet-lattice oscillatory aerodynamics approximation used four aerodynamic lag terms for each flexible mode. In addition, the state-space models included corrections for control surface effectiveness based on results from the 1989 wind- tunnel test 2, and the third-order transfer functions of the actuator dynamics derived from ground test of the unloaded control surfaces. A Dryden gust spectrum transfer function, driven by a white noise process, was used to simulate the random vertical gust of the wind- tunnel. The complete linear equations of motion at a specified dynamic pressure were expressed by the state-space equations, and dx/dt = Fx + Gu + Gww (1) y = Hx+ v (2) where x is the state vector, u is the control input vector, w is the gust input noise, y is the accelerometer sensor output vector, and v is the measurement noise vector, uations 1) and (2) were scaled such that the units of the control inputs were in degrees,Eq_ ( sensor outputs e units f the were in g's, and the gust input unit was in feet/second.
Open-loop Dynamic Pressure Root-locus Using these state-space mathematical models at six dynamic pressures, q = 100, 150, 200, 250, 300 and 350 psf, the flexible-mode root-loci with dynamic pressure were studied. The open-loop, dynamic pressure root-locus of the first four flexible symmetric and antisymmetric modes, for the fixed-in-roll configuration, are shown in figures 2 and 3, respectively. The figures 2 and 3 indicate that the second and third flexible mode frequencies coalesced to produce the flutter instability. The unstable mode was primarily wing-tip torsion, for both the symmetric and the antisymmetric motions. The sixth and seventh symmetric flexible mode frequencies also tended to coalesce (not shown in figure 2). At Mach 0.5, the analytical open-loop symmetric flutter dynamic pressure was estimated to be 248 psf at 11.2 Hz. The analytical open-loop antisymmetric flutter dynamic pressure was estimated to be 233 psf at 10.9 Hz. The closed-loop dynamic-pressure root- locus is also shown in figures 2 and 3 and will be discussed later.
CONTROL LAW DESIGN The flutter suppression design objective was to develop low-order robust digital control laws which would simultaneously suppress the symmetric and antisymmetric flutter modes of the model in the fixed-in-roll configuration with allowable control surface activity. The maximum permissible control surface rms deflection and rates were 1.0 degree (at 11.2 Hz flutter frequency) and 75 degrees/second, respectively. From the 1989 test 2 the antisymmetric flutter frequency was known to be 1.8 Hz below the analyticai value. The control law was also required to be sufficiently robust to compensate for this difference.
The FSScontrollawsweredesigned usinglinearquadraticGaussian (LQG) theoryand
involvedcontrollaw orderreduction,a gainroot-locusstudy,anduseof previous
experimental results 2. Sincethe symmetric andantisymmetric flutter modes hadvery
similar characteristics for the fixed-in-roll configuration(seefigures2 and3), a singleFSS
controllaw wasdesigned to suppress boththeflutter modes.This controllaw usedthe_tip
pair of accelerometers andtheTEO pairof controlsurfaces ontheright andleft wings.The
blockdiagramfor digital implemen tation2of thesymmetric andantisymmetric FSScontrol
lawsis shownin Figure4. The accelerometer outputsfrom theleft andright wing were
passed through 25Hz first-order antialiasing filters, modeled by the transfer function
157/(s+157) and converted into digital data at a sampling rate of 200 Hz. The digital controller separated the data into symmetric and antisymmetric components, computed the digital control law outputs and then distributed the processed feedback signals to the right and left actuators after 0.005 seconds computational delay.
Design Plant Model The 68th order antisymmetric state-space equation at q = 350 psf for the fixed-in-roll configuration was used as the design plant model, since from the analysis and the 1989 test, the antisymmetric flutter mode was found to be most critical and was encountered at a lower dynamic pressure than the symmetric flutter mode. The accelerometer sensors and control surfaces were selected based on the frequency response analysis of the open-loop system. The _'TEOand _tip accelerometer responses were predominant at the wing-tip torsion frequencies due to the excitation from TEl and TEO control surfaces. In addition, the _tip sensor exhibited relatively low response at frequencies above 25 Hz. Therefore, _'TEO and _tip accelerometer sensors and TEl and TEO control surfaces were initially studied as candidates for measurement inputs and control outputs, respectively.
Full-order LQG Design A full order LQG control law was designed using the design plant model state-space equations (1) and (2). The full-order LQG control law which is given by equations (3) and (4), minimizes a weighted quadratic cost function defined by E[yTQIy + uTQ2 u], where Q1 and Q2 are the plant output and control input weighting matrices 4,5.
dxc/dt = Aoxc + Boy, (3) u = Coxc, (4) where Ao = IF- Boll + GCo] Bo = pHTRv -1 Co = - Q2 -IGTS.
The matrices Bo and Co are the Kalman state estimator gains and the full-state optimal regulator gains, respectively. The matrices P and S are the positive definite solution of the steady state dual matrix Riccati equations, given by FP + PF T + GwRwGw T - pHTRv- II-IP = 0 SF + FTs + HTQ1H - SGQ2-1GTS = 0, where Rw and Rv denote the intensity matrices of the gust input and measurement Gaussian white noise processes, w and v, respectively. To obtain the LQG control law, full-state optimal regulator gain matrix Co was fh'st determined using a unit output weighting matrix, Q1 = I, and a control weighting matrix Q2 = 0.001 I, where I is a 2x2 identity matrix. Then the Kalman state estimator gain matrix Bo was determined using Rw = 0 and Rv = I. The final selection of these weighting and noise intensity matrices for the full order control law, and the subsequent order reduction process were determined after several design iterations, until a stabilizing low order controller was found for the nominal design plant model. The control law order reduction process is described next.
Order Reduction The full 68th order LQG control law given by equations (3) and (4) was first block- diagonalized, and then reduced to 11 th order by residualization of all the damped modes above 19 Hz. Equations (3) and (4), in block-diagonalized form, are shown in equations (5) and (6), where the vector xcl represents the retained states and the vector Xc2 represents the remaining states associated with the damped higher frequency dynamics.
&-[xc2J= A°2J[xc2J +L o2-1 y (5)
c .1;xo1
u =[Col oza[xc2j (6) In the residualization procedure, only the steady state part of the stable higher frequency dynamics in equation (5) were retained. This was accomplished by setting the state derivative dxc2/dt to zero and solving for Xc2, provided the matrix Ao2 is nonsingular 4.
The reduced state space model of the control law is given by equations (7) and (8).
dxc/dt = A Xc + B y (7) u = Cxc+Dy (8) where Xc = Xcl, B = Bol , C = Col and D = - Co2 Ao2-1 Bo2.
This procedure introduced a direct feedthrough matrix D in equation (8). The residualized 11 th-order control law was subsequently reduced to a second-order control law by balanced realization and truncation of the balanced system. The balanced realization procedure finds a linear transformation in which the control law states have equal controllability and observability properties 4. The weakly controllable and observable states are then truncated. Even with the elimination of these states, the resulting set of equations retained the most important input-output characteristics of the original system. This second- order, two-input two-output control law, is given by equations (9) and (10).
dt - 64.6 -5.2 xc + -0.45 -0.73 (9)
646i, ,95jf1
-0.4 -0.06 2.1 -0.09]f.'._.ol 3.6 -9.4]xc + [ 0.13 0.21 J[7'tip j (10)
The corresponding Bodediagrams of thefour components of this 2x2 controllaw are
shownin Figure5.This figureindicatesthatthemaximumgainof this controllaw was2.5
deg/g(8 dB) with a peakgainat 10.3Hz.The primarystabilizinggainof this controllaw
wasfrom the sensor_ip to thecontrolsurface _'rEO. Althoughthiscontrollaw stabilized
thesymmetricandantisymmetric plantmodelsat350psf, thestepresponses contained
high frequencv components. With theadditionof 25Hz antialiasing filters to each
accelerometerchannel, thehigh frequency components of the stepresponses were
eliminated. However,with the additionof T----0.005 second computational delay( modeled
by thefirst-orderPadeapproximation(2/T-s)/(2/T+s)),the systemwasmarginallystable.
It wasalsonotedthat,whenthis controllaw wasreduced to asingle-inputsingle-output
(SISO)controllaw by retainingonly thecontrollaw input _tipandtheoutput_rrEo. the
nominaldesignplantwasalsostable.This simplifiedSISOcontrollaw wastherefore,
studiedfurtherin orderto compensate for thecomputational delayeffects,andpossible
uncertainty in the actual flutter frequencies, as mentioned earlier.
SISO Control Law This simplified SISO control law (plot labeled by _3/_fip, in figure 5) was improved further via gain augmentation. The required gain level was determined using a gain root- locus analysis. The output gain feedback root-locus of the design plant model at 350 psf, as plant in ut, and ztio as plant output, is shown in figure 6. This root-locus with _q_EO the ope_-loop unstable pole (mode 3) near 11 Hz migrated into the stable left indicated that half plane, with a negative feedback gain of 1.3 deg/g from Ztip to _TEO- However, the gain of 0.75 deg/g. Therefore, a gain level actuator poles near 50 Hz become unstable at a of at least 1.3 deg/g in the 8 to 12 Hz frequency range, with subsequent gain attenuauon at higher frequencies, was necessary to stabilize the system, and accommodate the possible difference between the analytical and experimental flutter frequencies. In addition, compensation for the phase lag effects of the antialiasing filter and one cycle computational delay was also required. The total phase lag introduced by these two effects was about 40 degrees at the frequency 10 Hz.
and phase compensations were achieved by varying the three elements of C The gain SISO control law, and studying the gain and phase diagrams and the closed and D in the loop stability responses. An increase in C1 and decrease in IC21 resulted in a desirable phase increase at low frequencies. An increase in D reduced the phase (towards zero) at high frequencies, which was also beneficial. These three parameters were varied, until a gain- level near 1.3 deg/g (2.3 dB) was maintained over the frequency range 8 to 12 Hz, and sufficient phase lead was obtained. The real part of the control law complex pole was also moved from - 5.2 to - 6.0 to achieve a wider gain range. The high frequency gam was kept below 0.75 deg/g. This modified SISO control law is given by equations (11) and (12), assuming negative feedback.
dxc _I--6.0 64.6] I 1"95 _'" (11) dt -64.6 --6.0J xc + -0.73J Ztip _TEO = [ 14.4 --3.1]xc + 0.63 Ztip (12)
Thecorresponding gainandphase plotsareshownin figure5 andarelabeled8,rEo/_tip
(SISO).The complexpolesandzerosof thiscontrollaw were-6_64.6 and-30+_j56,
respectively. A second-order notchfilter, givenby the transferfunction(s2+42s+44100)/
(s2+84+44100),wasaddedto increase the symmetric modelgainmarginto 6 dB, near33
Hz. This filter attenuated a 33Hz lightly dampedoscillationdueto theinteractionof the
sixth andseventhsymmetric flexible modes.A first-orderwashout filter, givenby the
transferfunctions/(s+6),wasalsoaddedtoremoveanysteadystateinputbiasto the sensor
signal.
Discretization. The resulting 5th order SISO control law in Laplace domain was discretized using the Tustin transformation z = (l+sT/2)/(1-sT/2), where T is the sampling interval. For the 200 Hz sampling rate used by the digital controller, T = 0.005 seconds.
With the Tustin transformation at this sampling rate, the Bode diagrams in the Laplace domain and the discrete domain were almost identical below 15 Hz. Hence no frequency warping corrections were applied.
Dynamic-pressure root-locus: The open- and closed-loop dynamic pressure root-locus plots are compared in figures 2 and 3. These comparisons indicated that both the symmetric and antisymmelric models were stable, up to dynamic pressure q = 350 psf. The closed- loop frequency decoupling was due to lowering of the frequency of mode 2 to about 6.8 Hz. The frequency of mode 3 was increased to 11.6 Hz, but the damping ratio was only of the order 0.010 at 300 psf.
Sensitivity studies. The closed-loop system sensitivity was studied by perturbing the second and third modal frequencies in the state-space block-diagonalized plant model by +10% and the nominal gains by +4 dB at q = 250 psf and examining the closed-loop system step responses, for all possible combinations. These studies indicated that the design could accommodate simultaneous gain and frequency changes for all cases except when the second and third mode frequencies were perturbed to approach each other.
Sensitivity studies were also done using the state-space model with and without the 25 Hz antialiasing filters, with and without one cycle delay, with additional delays, and with + 6 dB gain perturbations at 250 psf. These studies indicated that the symmetric configurat]-on could tolerate one additional delay (or phase lag of 1.8 degrees/Hz) at half the nominal gain, but the antisymmetric configuration would become unstable with an 11 Hz oscillation. The phase and gain margin comparisons with the experimental results, described in the next section, indicated that this particular situation may have been encountered during the experiment. The gain loss was apparent from the experimental Bode diagram.
SUMMARY OF TEST RESULTS Open-loop Flutter. Based on examination of the peak-hold data obtained during the wind tunnel test with the tip ballast store coupled, the open-loop (OL) flutter dynamic pressures were as follows: The free-to-roll OL symmetric flutter was at a dynamic pressure of 235 psf, at a frequency of 9.6 Hz. The fixed-in-roll OL antisymmetric flutter was at a dynamic pressure of 219 psf, at a frequency of 9.1 Hz. These experimental symmetric and antisymmetric OL flutter dynamic pressures were, respectively, 13 and 14 psf below the predicted values, and the flutter frequencies were, respectively, 1.6 Hz and 1.8 Hz below the predicted values.
Open-loop frequency responses. Figures 7 and 8 show the OL frequency responses of z-tip, due to _'rEo from analysis and experiment at 250 psf, for the symmetric and anusymmetric (fixed-in-roll) cases, respectively. At this dynamic pressure, the OL plant is unstable. So, the OL frequency responses were computed from closed-loop (CL) experimental data, using the Controller Performance Evaluation (CPE 6,7) procedure.
Figure 7 indicates good agreement below 9 Hz and qualitative agreement above 12 Hz.
Above 12 Hz, the magnitudes differ by about 5 dB while the phase angles are nearly equal.
Figure 8 indicates fair agreement, below 7 Hz, and qualitative agreement above 12 Hz.
Above 12 Hz, the magnitudes differ by 6 to 8 dB and the phase angles differ by 10 to 20 degrees. Note, that for each phase diagram, the 180 degree crossing occurs near the respective OL flutter frequencies, and the difference between their predicted and experimental values is quite apparent.
Closed-loop Tests The active flutter suppression test results are summarized in figures 9 through 13.
Figures 9 and 10 show the wind-tunnel test dynamic pressures versus the free stream Mach number. During the wind-tunnel test, in the fixed-in-roll configuration, with both the symmetric and antisymmetfic FSS control laws operating, the CL system was stable up to q = 270 psf, at Mach 0.46. This augmented q represents a 23% increase over the OL antisymmetnc qf.
During the wind-tunnel test, in the free-to-roll configuration, with the symmetric FSS control law operating, the CL system was stable up to q = 290 psf, at Mach 0.48. This augmented q represents a 23% increase over the OL symmetric qf as shown in figure 10.
This FSS control law also suppressed the flutter when a Roiling Maneuver Load Alleviation (RMLA 8) system was tested with rapid roll maneuvers at q = 260 psf, 11% above the OL symmetric flutter boundary. This RMLA control law used LEO and TEl control surfaces, so the interaction with the FSS control law was minimal.
The rms deflection and deflection rate of the fight and left side TEO control surface were computed from the data sampled at 200 Hz at each fixed-in-roll FSS test condition, ff the value of the right and left differed, the maximum is plotted in figure 11. The maximum rms deflection and rates were less than 0.4 degrees and 25 degrees/second, respectively.
These maximum rms deflection and rate demands of the actuators were well below the maximum allowable values of 1 deg and 75 deg/sec as stated earlier in the paper.
The Nyquist-diagram-based gain- and phase-margins were estimated using the CPE technique, during the experiment. These estimates were compared with corresponding analytical quantities in figures 12 and 13, for the symmetric free-to-roll and the antisymmetric fixed-in-roll configurations, respectively. For the symmetric, free-to-roll configuration (figure 12), the analytical and experimental gain margins were above +_6 dB up to 270 psf. The analytical positive phase margins (at or below 7 Hz) were about 20 degrees, but the negative phase margins (at or above 12 Hz) were well above 45 degrees.
The analytical phase margins were close to experimental results up to about 270 psf.
For the antisymmetric, fixed-in-roll configuration (figure 13), the analytical negative gain margins were only -3 dB.The analytical positive phase margins (at or below 7 Hz) were about 20 degrees, but the negative phase margins (at or above 12 Hz) were 45 degrees. The analytical phase margins were close to the experimental data at 250 psf, because the design model was fairly accurate at frequencies below 7 Hz (see figure 8). The negative gain and phase margins at the high frequency end were primarily responsible for
preservingthesystemstability.Thesourceof additionalphase lag with increasing dynamic
pressure waspossiblydueto highly loadedactuators. Thegain losswasapparent from the
experimental Bodediagramshownin figure8 in the8 to 12Hz frequency range.
CONCLUDING REMARKS A single-input single-output control law was designed for flutter suppression using linear quadratic Gaussian theory and involved control law order reduction, a gain root- locus study and use of previous experimental results. The control law was digitally implemented and tested. Simultaneous suppression of symmetric and antisymmetric flutter modes in close proximity was demonstrated to 23% above the open-loop antisymmetric flutter boundary when the model was in a fixed-in-roll configuration. Symmetric flutter suppression system operating simultaneously with a rolling maneuver load alleviation system was tested to 23% above the open-loop symmetric flutter boundary, when the model was in a free-to-roll configuration. With this combined system, rapid roll maneuvers were also performed at 11% above the symmetric flutter boundary.
REFERENCES 1 Perry, B. III, Cole, S. R and Miller, G. D., "A Summary of the Active Flexible Wing Program," AIAA Paper 92-2080, April 16-17, 1992.
2 Perry, B. III, Mukhopadhyay, V., Hoadley, S. T., Cole, S. R., Buttrill, C. S. and Houck, J. A., "Digital Implementation, Simulation and Testing of Flutter-Suppression Systems for the Active Flexible Wing Wind-Tunnel Model," AIAA Paper 90-1074, April 1990.
3 Buttrill, C. S., Bacon, B. J., Heeg, J. and Houck, J. A., "Simulation and Model Reduction for the AFW Program," AIAA Paper 92-2081, April, 1992.
4 Maciejowski, J. M., Multivariable Feedback Design, Addison Wesley Publishing Co., Great Britain, 1989.
5 Bryson, A. E., Jr, and Ho, Y. C., Applied Optimal Control, Hemisphere Publishing Corporation, Washington, 1975.
6 Pototzky, A. S., Wieseman, C. D., Hoadley, S. T. and Mukhopadhyay, V., "Development and Testing of Methodology for Evaluating the Performance of Multi- input/multi-output Digital Control Systems," AIAA Paper 90-3501, August 1990.
7 Pototzky, A. S., Wieseman, C. D., Hoadley, S. T. and Mukhopadhyay, V., "On-line Performance of Multi-loop Digital Control Systems", Journal of Guidance, Control, and Dynamics, Vol. 15, No. 3, May-June, 1992 (TBP).
8 Woods-Vedeler, J. A. and Pototzky, A. S., "Rolling Maneuver Load Alleviation Using Active Controls," AIAA Paper 92-2099,April 16-17, 1992.
Inches SO 6O 8O 9O I tO0 20 30 40 50 Inches i Figure I.Acceleron_tcr and conu'ol surface locations on AFW wing plan form.
imaginary part 1 O0 filter mode mode 4 _--90 mode 3
: -60
D-_. p_s"suro mode 2 3 2_t NO.
--5O 100 psf ,_.xx_ - 150 " o , --40 200 " 250 " /, --30 mode 1 300 " --20 350 " open-loop --10 closed-loop t I I I I I -20 -15 -10 -5 0 5 10 real part Figure 2. Symmetric open- and closed-loop dynamic pressure root-locus at M=0.5 (arrows indicate increasing dynamic pressure).
imaginary part -100 -90 filter mode mode 4 -80 6 5 -60 5 6 No. Dyn. pressure -50 100 psf 150 " --40 3 200 " 4 250 " mode 1 --30 300 " 350 " -20 x open-loop close_-Ioop I I I -20 - 15 - 10 -5 0 5 real part Figure 3. Anfisymmetric fixed-in-roll open- and closed-loop dynamic pressure root-locus at M=0.5 (arrows indicate increasing dynamic pressure).
right and left 6TEO _ right and left Z tip ;put, degrees _ _ sensor output, g , _ .._ antialiasing L f
'+'.er' "
E
control law I _ASym FSS control law ntisym FSS ]_ Hgure 4. Digital FSS control law implementation block diagram.
agnituoe, 0_ __-" .f _,-. _ .,- j
_,_,<dm_ .... _.--. "<t-....._
-
2o _
100 6TEO f,%p (StSO_ Phase. 0 deg .100 _._ S,rEif_,lp ___ 'X_..
• , °TE/ZTEo_ _ r...._ t-- 200 _ _ I I _ I ,,,=u LTEO - i I I (' 2 4 6 8 10 12 14 16 18 20 Frequency,Hz Fibre 5. Bode diagram of reduced, second-order control laws.
imaginary pad - 350 f "lEO actuator pole mode 10 C]K mode 9 x.e e---_ - 250 mode 8 x_'-,_,-------'l P'_ - 200 mode 7 mode 6 - 150 mode 5 mode 4 ( _ gain'l'3 dveCj/g G.- ==_- _ " ..
1QI 50 mode3 ode l I I I _ _ ..I -20 -15 -10 -5 0 5 10 real part Figure 6. Gain naot-tocus plot for negative feedback from _fip to 5n/o at 350 psf, antisymmetric fixed-in-nail configuration (x = poles, o = zeros, * indicates gain increment by 0.1).
Magnitude, 0 | .,_ -_. v -20 J-- _ experiment -30_ I l I I I I I L A
- !!!!
2 4 6 8 10 12 14 16 18 L-'U Frequency. Hz Figure 7. Comparison of Y.tip / 8"rEO Bode diagrams at 250 psf, symmetric config_atiOn.
-10 -20 rime -30 r I I I I I I I • 100 _aJysis Phase.
-100 exp_kllenl -200 2 4 6 8 10 12 14 16 18 20 Frequency, Hz Figure 8. Comparison of _tip / &rEO Bode diagram at 250 psf, antisymmetric fixed-in-roll configuration.
300 - Augmented q 23 % Increase q, psf Symmetric q,//a_ove qf 220 - Antlsym/_ 180 ?
I I • I
O
0.3 0.4 0.5 Mach number Figure 9, Summary of results for fixed-in-roll FSS wind-tunnel test.
• [] sensor end control surface for FSS _) Pq sensor and control surface for RMLA 300 - Augmented _ 23 %-_Increase 260 - Roll maneuver above qf q, psf Symmetric qf I I 10% 220 -
/
180 - I I_ I 0.3 0.4 0.5 0.0 Mach number Figure 10. Summary of results for free-to-roll RMLA/FSS wind-tunnel test.
0.4 - 40 6rms 6rms deg deg/s 0.2 - 20 I I I 0.0-- 0 220 260 300 0 180 q, psf Figure 11. Maximum 5TEO control surface deflection and rates demands for simultaneous symmetric and andsymmemc flutter suppression tests.
20 m Gain margin, 0 ,,,,__k,L _o _o _ I , , dB •10 -- Dynamlcpressum, psf 2_ "2o- _...,y,. l |i *xpe*'imen t Figure 12. Gain and phase margin comparison (symmetric).
!
Gain rmirgln, S dO / ! I I I I I 200 220 -S Dynamic pressure, paf .-I 2i_ 380 300.
-10 r f-I anelyad8 • ox_ment J 4O Phase 2O mari_ln,0 *4O -SO -8O -100 i i Figure 13. Gain and phase margin comparison (antisymmetric, fixed-in-roll).