LQG FLUTTER CONTROL OF WIND TUNNEL MODEL USING PIEZO-CERAMIC ACTUATOR

Transcription

1 5 TH INTERNATIONAL CONGRESS OF THE AERONAUTICAL SCIENCES LQG FLUTTER CONTROL OF WIND TUNNEL MODEL USING PIEZO-CERAMIC ACTUATOR Tatsunori Kaneko* an Yasuto Asano* * Department of Mechanical Engineering, University of Fukui, Fukui, Japan Keywors: Flutter, LQG control, Piezo-ceramic Actuator Abstract A single element of piezo ceramics is use in this stuy, instea of conventional actuators such as a hyraulic actuator or electric motor, as an actuator in orer to control flutter. This stuy aims at confirming that the free vibration in the still air an the flutter in the win tunnel, of an aluminum plate wing, can actually be controlle by a single element of a piezoceramic actuator by LQG control. The wing moel of a rectangular aluminum plate (7. mm 9. mm.5 mm) an the aluminum s aitional mass (. mm 5. mm.5 mm) has one piezo ceramic element PZT (4. mm. mm. mm) attache on its one sie. First we use proportional control for free vibration an flutter in orer to confirm the funamental effectiveness of PZT. An use LQG control for flutter in orer to confirm a better effect of PZT. So piezo-ceramic actuator was confirme to be effective enough to control flutter an confirme to flutter spee increase. actuator for making the flutter control actively []. In this stuy we trie to improve his work paying attention to electric force characteristics of a piezo-ceramic actuator. It is assume that a wing moel is compose of an aluminum flat boar for funamental stuy. We construct mathematical moel of a wing moel an analytically estimate at how much win velocity flutter occurs an compare it with experimental value. We obtain aeroynamic forces acting on wing surface by Doublet Point Metho []. Base on the mathematical moel, we esign the control law for controlling flutter applying LQG metho. We confirm whether we can control flutter by an experiment. We also confirm how much flutter velocity will increase from open loop to close loop system. Flutter Control Moel A wing moel use in the vibration control tests is an aluminum plate wing shown in Fig.. The Introuction Recently, research of the active control technology for flying qualities improvement is avance in many countries. In active flutter control research control surface is mainly use as an actuator. Piezo-ceramics has small size an lightweight so that it can install in a small place suitable for a sensor/actuator. Therefore, its application to sensing the oscillation an controlling the vibration of structures will increase rapily in future. Kawai use the piezo ceramics, instea of a hyraulic actuator or electric motor, as an Fig.. PZT actuator place on a moel wing spar

2 T. KANEKO AND Y. ASANO wing moel of aluminum plate whose aspect ratio is 3 has a single piezo-ceramic element PZT attache on its one sie. The size of the aluminum rectangular plate was etermine as 7. mm 9. mm.5 mm an the aluminum s aitional mass was etermine as. mm 5. mm.5 mm as shown in Fig. consiering the sectional size of the win tunnel outflow of 3 mm 3 mm square an the maximum win spee of 5m/s.. PZT Actuator In orer to control free vibration an flutter by imposing the strain prouce by voltage to the aluminum plate wing moel large surface strain for PZT is neee. Piezoelectric relation of PZT attache at wing moel is shown in Fig.. The surface strain x for PZT is efine by piezoelectric charge constants ij an an electric fiel strength E i as: x x x x x x = E 33 E E3 () Fig.. Piezoelectric relation of PZT where strain x 4, x 5 an x 6 is sharing strain of rotation of axis, axis an 3 axis. The irection of polarization is towar x 3 axis since we make use of istortion along the x an x axis. Electric fiel strength is expresse by the impose voltage V by E = V t an the surface strain x can be expresse as follows. x V = 3 () x t Since the large surface strain is esirable for PZT, piezoelectric charge constants 3 shoul be large an thickness t shoul be small. Young s moulus E shoul be large in orer to get a large exerte force. Therefore, PZT of the material properties shown in table was selecte an the size of PZT was etermine as 4.mm.mm. mm. We use a laser isplacement sensor to measure the isplacement of a wing tip. The position to measure is shown as a target in Fig. where bening as well as torsion eformation can be etecte. Table. Physical constants of PZT at normal temperature PZT characteristics Constant Density 8. 3 kg/ m 3 Dielectric constants 49 Piezoelectric charge constants m/v Young s moulus 6.4 N/m Poisson s ratio.3 For simplicity an as a challenge, we ecie to use only a single element of PZT. PZT actuators can excite simple bening, simple torsion eformation, or those combinations to the aluminum plate epening on the position an the irection of the actuators. The amount of eflection also epens on them. The position an the irection were etermine such that both bening an torsion can be excite an eflection can be as large as possible. Figure 3 shows the actual arrangement of PZT on the aluminum plate. Important restriction of input voltage impose to PZT is that it shoul be positive in orer to prevent epolarization of PZT.

3 LQG FLUTTER CONTROL OF WIND TUNNEL MODEL USING PIEZO-CERAMIC ACTUATOR 3 Proportional Control experiments Fig. 3. Aluminum wing moel with PZT attache. Vibration Characteristics of Wing Moel In orer to obtain natural frequencies an moe shapes of the wing moel, we performe finite element metho structural analysis. The finite element moel use in analysis is shown in Fig. 4. In this figure, the wing is fixe at the bottom. In analytical result, the first two natural frequencies an moe shapes are shown in Fig. 5. Natural frequencies of a bening ominant first moe an a torsion ominant secon moe are 4.6Hz an.5hz, respectively. The free vibration an flutter control test were performe with the test set-up shown in Fig. 6. This test composition is a single input single output system since only single element of PZT actuator is attache on the aluminum plate. The isplacement of wing tip is measure by laser isplacement sensor an this signal is taken in the AD converter boar incorporate the personal computer through charge amplifier. A sampling frequency is khz. The signal output from DA converter boar is amplifie 4 times with power amplifier, an is impose on PZT actuator. Maximum-output voltage is 4V. Fig. 6. Schematic iagram of the test set-up 3. Control of Free Vibration Fig. 4. Finite Element Moel (a) Moe 4.6 HZ (b) Moe.5 Hz Fig. 5. Moe shape an natural frequency We use proportional control for free vibration in orer to confirm the funamental effect of PZT actuator. When the free en of a wing moel was flippe by han, the time response when not performing proportional control is shown in Fig. 7(a), while the time response when performing control is shown in Fig. 7(b). The control here uses positive voltage alone in orer not to loose the piezo-electric effect of PZT, which is so-calle control of onesie effect. With a bias ae in PZT an performing linear control is shown in Fig. 7(c). Figure 7(b) an Figure 7(c)(with a bias ae) 3

4 T. KANEKO AND Y. ASANO shows the time response of the proportional control, u = Ky (3) For the flutter control experiment was use following small win tunnel (blow-own type) which is shown in Fig. 8. where the isplacement y [mm] is fee back to the PZT comman signal u [V] with the proportional gain K [V/mm]. (a) Fig. 8. Win tunnel an position of wing moel (b) (c) Fig. 7. Time response for proportional control of free vibration We compare amping characteristics of the system without control an with performing control. The amping has increase greatly by vibration control with the PZT actuator. Though we trie to a a bias voltage to control signal, control effect was not improve so much for this free vibration control. 3. Flutter Experiments Comparison of the response before flutter an uring flutter is shown in Fig. 9. The time omain is shown in Fig. 9(a), while the frequency omain is shown in Fig. 9(b), an the response before flutter is shown in Fig. 9(), while uring flutter is shown in Fig. 9(). In time omain, vibration of bigger amplitue than before flutter has occurre at flutter. The amplitue of flutter is about 6 times larger than the amplitue before flutter an is continuously at constant value. In frequency omain, the bening moe an the torsion moe can be istinguishe yet before flutter. When flutter occurs, two frequencies have merge. (a) Time omain (b) Frequency omain Fig. 9 () Response just before flutter (win velocity 3m/s) 4

5 LQG FLUTTER CONTROL OF WIND TUNNEL MODEL USING PIEZO-CERAMIC ACTUATOR generalize forces of PZT actuator acting on the wing moel, an white noise entere in PZT actuator, respectively. The generalize forces of PZT is assume here to proportional to the impose voltage c ( as, (a) Time omain (b) Frequency omain Fig. 9 () Response uring flutter (win velocity 5m/s) We performe proportional control of flutter that is shown in Fig. (a) f p ( = P c ( (5) The system configuration is a single inputsingle output system since only single element of PZT actuator is attache on the aluminum plate. We have built a mathematics moel of moel wing base on FEM analysis followe by Doublet Point Metho for aeroynamic analysis. The state space equation takes the following general form. x &( = Ax( + Bu( + Gw( (6) where the matrices in the above equation are, I A = ( M A ) ( M A ) I B I ( C A ) I Λ (b) Fig.. Time response for proportional control of Flutter But there is a control effect neither, we trie to perform control by LQG metho. 4 Mathematical Moeling General ynamic equation of the generalize coorinate q ( of the wing moel on PZT actuator is given in the next expression. Mq& ( + Cq& ( + Kq( = f ( + f ( w( (4) a p + Where M, C, K are generalize mass, generalize umping, an generalize elasticity matrices, respectively. F a (, f p ( an w ( are generalize aeroynamic forces term, I B = ( M A ) I P G = ( M A ) (7) an where A, A, A, B an Λ =iag(-,,- ) are coefficient matrices of the following finite imensional aeroynamic moel. f a = A q&& + A q& + Aq + z z& = Λz + B q (8) Output equation of the isplacement of the wing tip can be expresse as, y ( = cx( + v( (9) where v( is a white noise entering in the laser isplacement sensor. 5

6 T. KANEKO AND Y. ASANO 4. Parameter ientification of the structural moel Since we have constructe a mathematical moel for the wing, we can ientify the structural parameters in the equation of motion for the moel. We expecte vibration tests to get a frequency response of the wing. The frequency response when we change frequency of sine wave with.5 Hz step is shown in blue lines in fig.. We impose a voltage of a sine wave to PZT an measure the wing response by a laser isplacement sensor at a wing tip target. Base on the frequency response, we can get all the parameters in the mathematical moel, generalize mass, generalize amping, generalize elasticity, as well as eigen frequencies an eigen moes for the vibration moes of the moel in still air. From now on, we will confine ourselves to two moes system retaining only first two moes because flutter occurre in the moel can well be expresse with these two moes. Assuming the linearity for the structural characteristics comprise of a PZT, input voltage c ( impose on PZT actuator an the generalize coorinate q( can be expresse in the following formula. t e iωt c ( ) = c () iωt q q e q( = = i t () ω q qe Substituting the above expressions into eqs. (4), () an (), we can get the following relationship between the amplitues an eigen frequencies, incluing amping coefficients,. ω q P = m + iωζ ω q + ω q c ω q + iωζ ω q + ω q + c m P () ω q + iωζ ω q + ω q + c (3) m Deflection of the wing y( at the target of sensor can also be expresse accoringly as, iωt y( = ye (4) by harmonic vibration. An the amplitue of the wing becomes y = f( x, y ) q + f ( x, y ) q (5) Therefore, transfer function comprise by the state space equation takes the form: P P f ( x, y ) f ( x, y ) y m m = + c ω + iωζ ω + ω ω + iωζ ω + ω (6) Ientifying the parameters in eq (6) by the ata obtaine from a groun vibration test, we coul get the following parameters as shown in Fig. ; the natural frequency of moe, moe are ω =4.5 Hz, ω =. Hz, an the amplitue of moe shapes are f (x,y )=.84 an f (x,y )=.6, respectively. The other parameters ientifie are =.75, =.7, P /m =.5 an P /m =-.5. P 4. Construction of finite imensional aeroynamic moel Fig.. Frequency response for parameter ientification Having ientifie the structural parameters, we have analyze generalize unsteay aeroynamics using Doublet Point Metho. Figure shows the analytical results for elements. The aeroynamic matrices can be obtaine by least square fitting the analytical results with a finite imensional moel. Fitting has succeee in getting a moel as shown in 6

7 LQG FLUTTER CONTROL OF WIND TUNNEL MODEL USING PIEZO-CERAMIC ACTUATOR Fig.. The resulting non-imensional aeroynamic coefficient matrices, A ˆ, Aˆ ˆ ˆ, A, B are shown below, ˆ 7.546e e + 4 A =.9739e e + 4 (7a).97e e + 5 A ˆ =.38e e + 5 (7b).e e + 5 A ˆ =.98e e + 5 (7c) where is air ensity, U is airflow velocity an b is a mean chor length. 5 LQG Control Design Having constructe a finite state aeroelastic moel, we can obtain the flutter characteristics in the form of root locus in airspee. Incorporating the velocity-epenent characteristics in the aeroynamic forces expresse in eq. (8), we can get the eigen value variation with an airspee as a root locus..e e + 3 B ˆ = (7).343e e + 3 Using the above nonimensional matrices, finite imensional aeroynamic moel, eq. (8) can be expresse as the following equation. b ( ) U b Aˆ q&& + ( ) Aˆ q& + A q + z U (8) f a = ρ U b ˆ r b ( ) r& = Λr + ρu b Bˆ q (9) U Fig. 3. Win velocity root locus of the open loop system Actually, a branch starting from the secon (torsion) moe behaves rather strange. Instea of ecreasing the frequency as the velocity, it increases its frequency an coupling is not realize with the first (bening) moe. Though at present moment, we only have checke that the change of the sign in static aeroynamic force in torsion moe can influence this coupling, we have not yet foun any ecisive reason of this phenomenon. We have continue to esign an output feeback by LQG metho. By applying the LQG metho, we have obtaine the following control laws that comprises regulator with estimate states x ˆ( by Kalman filter. { y( cxˆ( )} xˆ & ( = Axˆ( + Bu( + K t () Fig.. Generalize aeroynamic coefficients an finite state moel fitting 7

8 T. KANEKO AND Y. ASANO the control as shown in Figure 6. When LQG controller is applie, flutter spee is increase to.8 m/s, i. E., 4. % increase. Since the preicte increase of flutter spee is.7 %, the actual increase is not sufficient. Possible cause of the iscrepancy may exist in insufficiency in the mathematical moel. Fig. 4. Win velocity root locus of the close loop system for LQG controller ( : open loop, : close loop) u( = Kxˆ( () Eigen value analysis was carrie out for a close loop system an figure 4 shows a win velocity root locus of a close loop system compare with an open loop system. The results show that the flutter velocity 5. m/s preicte for an open loop system is increase to 8.5 m/s with.7 % increasing. 6 Flutter Control Win Tunnel Experiment Win tunnel tests were performe to verify the control law effectiveness. We first checke the frequency change of two moes that may contribute flutter accoring to velocity change. Figure 5 shows the results that show the coalescence of two moes. Flutter spee without control is estimate as.3 m/s. Fig. 6. Win tunnel test ata for successful flutter suppression by PZT 7 Conclusions We examine a free vibration control, flutter proportional control an LQG flutter control using piezo ceramic actuator. In case of flutter control we examine the effect of intentional shift that moves an equilibrium state to positive state (bias). In the result, flutter can be controlle better by a linear control with a bias ae than a non-linear control of a one-sie effect. Since the increment of the flutter velocity in the test was not enough compare with the analytical preiction, improvement of the mathematical moel is necessary in future work. References Fig. 5. Frequency coalescence of two moes The control law esigne by LQG metho base on a mathematical moel expresse in eq. (6) an showing unique tren of frequency was installe in PC. The flutter was suppresse by [] Kawai, N., Flutter Control of Win Tunnel Moel Using a Single Element of Piezo-Ceramic Actuator, Proceeings of the 4th International Congress of the Aeronautical Sciences, Yokohama, Japan, , 4 [] Uea, T. an Dowell, E. H., A New Solution Metho for Lifting Surfaces in Subsonic Flow, AIAA Journal, vol., March 98, pp