Recent Advances in Aerospace Actuation Systems and Components, June 13-15 2007, Toulouse, France

ENHANCING WING DYNAMIC BEHAVIOR BY USING PIEZO PATCHES

Lucian IORGA Mechanical and Aerospace

Engineering Piscataway, New Jersey 08854

e-mail: liorga@rutgers.edu liorga@aero.incas.ro

Phone: +1 732 445 2718 Fax : +1 732 445 3124

Eliza MUNTEANU Advanced Studies and Research

Center Bucharest, Romania

e-mail: elizamun@gmail.com

Ioan URSU “Elie Carafoli” National Institute

for Aerospace Research Boulevard Iuliu Maniu 220 Bucharest 061126, Romania

e-mail: iursu@aero.incas.ro Phone :+4021 434 00 97 Fax : +4021 434 00 82

ABSTRACT The present work presents the development of a robust modal control law based on ∞H synthesis, for a piezoelectric actuated full wing. More specifically, piezoelectric PZT layers are embedded in the wing skin, performing as actuators to control selected elastic deformation modes. The use of piezoelectric actuators offers the possibility to maintain aerodynamic conformability and reduce the weight, compared to the classical approach that uses a flap mechanism, often requiring conventional electric or hydraulic actuators. The piezo actuators input voltages are computed using a robust control law which accounts for uncertainties due to modeling errors or unmodeled sensor and actuator dynamics. The numerical simulations performed show that the designed control law ensures good vibration reduction when compared with the corresponding passive wing. KEYWORDS Aeroelasticity, smart structures, FEM, piezoelectric actuators and sensors, active wing control, uncertainty, robust control. I INTRODUCTION Over the last two decades, techniques based on the use of embedded piezoelectric materials have emerged as viable solutions for a large number of structural control problems and, consequently, have received considerable attention (from the earlier paper of Crawley and Luis, 1987, to many others, such as, for example, the work of Chopra and Sirohi, 2000). Recently, an increasing interest is manifested in the field of active aeroelastic control of wings, both rotating and fixed, by using piezoelectric materials (Wilkie and Park, 1996; Nam et al., 2000; Rocha et al., 2007) as an alternative to the classical one in control of aeroelastic phenomena. The

classical approach requires the use of conventional trailing edge surfaces (see e.g. Block and Strganac, 1998) and a conventional electrohydraulic actuator (Ursu and Ursu, 2004), accomplishing a double function, that of primary flight control actuation and that of aeroservoelastic active control (of vibrations). The objective of the present paper is to investigate the capabilities of piezoelectric actuators as primary aeroservoelastic control devices for full size wing structures. Our work is inspired by the numerical and experimental results described by Abramovich et al. (2005), who used Finite Element Method (FEM) modeling to perform a modal analysis of a high aspect-ratio (AR=16.7) composite wing. For each of the first 5 natural modes, a model wing equipped with different configurations of PZT patches was excited, thus demonstrating their capability in a potential aeroelastic vibration control. Based on the theoretical results obtained in (Iorga et al., 2007), we propose the use of robust ∞H tools (Zhou et al., 1996) for the control of a similar wing equipped with a different configuration for the PZT patches. Consequently, the organization of the paper is as follows. In Section 2, the structural mathematical model of the wing is obtained based on the data furnished by Abramovich et al. (2005). The ∞H robust control procedure is presented in Section 3 and numerical applications are presented in Section 4. Conclusions and future work are discussed in Section 5. II WING STRUCTURAL MODEL The model considered is that of a straight composite wing with a semi-span of 3000mm and aspect ratio 16.7 (Abramovich et al. 2005). The wing has a symmetric NACA 0012 airfoil and two spars located at 30% and 70% chord from the leading edge. The material for the wing skin is a unidirectional E-glass/vinyl ester composite with 7 layers, each 0.16 mm thick, while the spars are formed by a sandwich of two groups of 4 E-glass/vinyl-ester composite

Recent Advances in Aerospace Actuation Systems and Components, June 13-15 2007, Toulouse, France

bordering a 8 mm thick PVC foam core. The active region is located at the wing root, between the spars, both on the upper and lower skins, and covers 500 mm in the spanwise direction. In the active region the wing skin includes two additional 0.5 mm thick PZT-5A (see e.g. www.efunda.com) layers on the top and bottom surfaces. The piezo layers on the upper and lower skins are considered to act in phase opposition, such that the induced bending moment is maximized. Thus, the system is, effectively a Single Input one. The geometry of the wing equipped with PZT actuators is given in Figure 1.

Figure 1. ANSYS model for wing equipped with PZT

patches and main geometric dimensions The commercial FE package ANSYS is used for modeling and all structural components are discretized using SHELL99 laminated plate elements. A total of 8201 nodes, i.e. 49260 degrees of freedom are used to model the active structure. We note that the SHELL 99 element does not include electric degrees of freedom but has thermal expansion capabilities. Thus, to calculate piezo actuator influence, it was necessary to use the analogy between thermal and piezoelectric equations (Mechbal, 2005), so introducing a thermal model for piezo material. The linear constitutive relation for the inverse piezoelectric effects is written as

E TC e Eσ = ε − (1)

where σ - the stress tensor, ε - the strain tensor, EC - the elastic stiffness matrix evaluated at a constant value of the electric field E, which depends on the voltage V applied to the PZT patches of h thickness:

[ ]0 0 TE V / h= (2)

The thermal equation, justifying the thermal-piezoelectric analogy, is

( )0E T

t tC C C e E, T Tσ = ε − ε = ε − ε = α − (3)

where α is a vector of thermal dilation coefficients, 0,TT are the current and reference temperatures and e~ is the matrix of piezo coupling coefficients. Following FEM modeling and assuming proportional, structural type damping the structural equations of motion can be written

ffKxxCxM u +=++ (4)

where x is the vector of nodal displacements, uf and f are, respectively, the vectors of induced piezo loads and externally applied loads and M, C, K are the mass, structural damping and stiffness matrices. Following a modal analysis using the full ANSYS model, the first 5 natural modes (see figure 2) and frequencies (Hz) – 3.8; 20.9; 22.9; 34.6; 56.1 – were found similar as deformation type and, respectively, relatively close to the values given by Abramovich et al. 2005: 3.2; 17.9; 19.6; 37.7; 51.8. The small differences in the natural frequencies may be justified by the lack, in the referenced work, of exact data on the spar locations.

Figure 2. Natural first 5 modes of the wing:

a) 1st bending; b) yaw; c) 2nd bending; d) 1st torsion; 3) 3rd bending

In a second step, uf will be dissociated as

Bufu = (5)

where u is the magnitude of the electric field – the control applied to different actuator patches – and B is the piezoelectric influence matrix. This operation assumed the static interaction (Tralli et al., 2004) cause-effect

kk BuKx = (6)

Recent Advances in Aerospace Actuation Systems and Components, June 13-15 2007, Toulouse, France

where kx is the displacement vector corresponding to an electric field of 1 V/mm applied to the k PZT piezo patch. III ROBUST ∞H SYNTHESIS In fact, the main objective of the previous Section was to provide the basic structural equation of control synthesis

fBuKxxCxM +=++ (7)

herein with added external, aerodynamic type, forces f . Through the transformation to modal coordinates η

( ) ( )tVtx η= (8)

where V is the matrix of normalized eigenvectors iv corresponding to eigenvalues iω and satisfying well known

relations ( ) ,..., 102 ==ω− ivMK ii , IMVV T = , the basic modal equations of motion can now be written

fuBK~~~~

+=++ ηηCη (9)

The modal damping and stiffness matrices C~ and K~ are diagonal but, in general, the modal equations (10) remain coupled through the components of the matrix B~ . For control synthesis, the system (10) is recast by introducing the state vector [ ]Tx ηη= . A subsequent model reduction is performed by ignoring residual modes rη in the

vector [ ]TTr

Tn η,ηη = . Thus the state will be dissociated in

[ ]Tr

Tn xxx ,= , where the representative and residual states

are selected using the Modal Hankel Singular Values procedure (Chant et al. 2002). Now, we can write the system in control synthesis specific form:

uBfBxAx nn nnnn 21~

++=

uDfDxCz nnn nn 12111~

++=

uDfDxCxCy nnrn nrn 222122 +++=~

(10)

where y and z are the measured and, respectively, quality, regulated output. Numerous types of control laws, ranging from the classical velocity-feedback to “post-modern” LMI-based control, fuzzy logic control or neural control, have been employed to achieve performance and robustness objectives. In principle, ∞H optimization of control systems as the one shown in figure 3a) is concerned with finding a controller ( )sK that minimizes the ∞H norm of the input-output closed

loop transfer function nfzT ~ , which represents the maximum

input to output energy gain of the system

( )

( )( )

2

20~~ ~max

tf

tzT

ntffznn ≠∞

= (11)

The solution of this standard problem is well known (see e.g., Zhou et al., 1996). But in the present work we are interested in the robust paradigm of ∞H synthesis: the controller synthesis will be performed for an extended system which corresponds to the physical system with the uncertainty blocks [ ]map ∆∆∆∆ ,,= involving parametric, neglected dynamics and sensor uncertainties (figure 3b). Thus, recent results presented in Iorga et al., 2007, concerns a general case of a robust synthesis for a complex uncertainty block ∆ , providing a relationship between the bound of the transfer ˆzwT , the size of the uncertainty and the stability and performance (in an ∞H bound sense) for the “real” I/O

transfer edT ; see the two theorems below and figure 4 with the notations involving the insertion of well defined fictitious vectorial input and, respectively, output

{ }{ }TTTTTT

TTTTTT

eez

nfw

21321

321

ˆ

~ˆ

ζζζ

ννν

=

=

(12)

and weighting iW matrices.

Figure 4. Open loop uncertain plant

Figure 3. a) ∞H paradigm; b) uncertain closed loop

G

∆

K

f~ z

yu

b) a)

G

K

f~ z

yu

Gδ

W2 ∆a W1 W4 ∆m W3

∆p ς1

ς2 ς3

ν1

ν2 ν3

nf~

u

z n y

Recent Advances in Aerospace Actuation Systems and Components, June 13-15 2007, Toulouse, France

Theorem 1 If γ<∞11T , with γ > 0, then ( )∆,ˆˆwzued TT F=

is stable for all ( )s∆ stable and with γ<∞

/1∆ .

Theorem 2 If γ<∞wzT ˆˆ , with γ > 0, then

( )∆= ,ˆˆwzued TT F is stable and γ<∞edT for all ( )s∆

stable and with γ<∆∞

/1 .

0 20 40 60 80 1000

0.2

0.4

0.6

0.8

1

1.2x 10-4

Frequency (Hz)

U z (am

plitu

de)

Reduced orderFull-order (ANSYS)

Figure 5. CP1 frequency response, piezo input

IV NUMERICAL RESULTS AND CONCLUSIONS As a first step towards robust aeroservoelastic control design, a simpler model is considered for control design analysis. In addition to the piezoelectric control inputs, a vertical force acting at the wing tip is considered as the exogenous perturbation, together with the noise on the two measurement channels. In the present work, the measured quantities are the z-displacements of two control points, CP1 and CP2, located on the upper wing surface at 4/3bx = and 4/cy = and

4/3cy = . The regulated outputs vector contains the first modal variable and the control input u . The numerical algorithms for modern control synthesis techniques are usually suitable for systems of reduced dimensions. Thus, for control design purposes, a reduced 30 DOF model is extracted from the full ANSYS model using the package mor4ansys (Rudnyi et al. 2004). The system is further reduced to 5 modes using the Modal Hankel Singular Value (MHSV) procedure. We note that introducing small numerical damping is required for the proper functioning of the MHSV algorithm, unless structural damping is considered. In this work we consider a small structural damping of 1% of the critical damping values for each mode. Excellent agreement is observed between the full (ANSYS) and reduced order system frequency responses as shown in Figures 5 and 6. The control synthesis is performed on a reduced system containing the first 4 relevant modes, namely modes 1, 3, 4 and 5. The residual, neglected, modes are accounted for through the weighting functions 1W and 2W , chosen as:

,6400009600

33.13100)( 2221 ×+++

= Iss

ssW

+++

=11

640000960033.13100)( 22 ss

ssW

(13)

0 20 40 60 80 1000

2

4

6

8

x 10-3

Frequency (Hz)

U z (am

plitu

de)

Reduced orderFull-order (ANSYS)

Figure 6. CP1 frequency response, wingtip force input

Parametric uncertainty is introduced in the system parameters, damping and natural frequencies, using the procedure described in Zhou et al., 1996, as follows: 1% for Mode 1 damping and natural frequency, 2% for Modes 3 and 4 and 5% for Mode 5. Constant output multiplicative uncertainty of 2% is considered to account for unmodeled sensor dynamics and measurement errors.

100 101 102-120

-110

-100

-90

-80

-70

-60

-50

-40

-30

Frequency (Hz)

Mod

e 1

(am

plitu

de)

Open LoopClosed LoopPerturbed Closed Loop

Figure 7. Mode 1 response, wingtip force input

A 17 state ∞H optimal controller is synthesized using the numerical optimization approach described in Iorga et al., 2006. The controller is reduced to 8 states using a performance weighted algorithm (Varga, 2002) as implemented in the package SLICOT. A peak reduction of approximately 6 dB can be observed in the dominant Mode 1 frequency response, together with a more modest reduction at lower frequencies, as shown in Figure 7. The corresponding frequency response for Control Point 1 is given in Figure 8. It can be seen from Figures 7 and 8 that the robust controller effectively manages to control the wing in the presence of uncertainties, the closed-loop perturbed

Recent Advances in Aerospace Actuation Systems and Components, June 13-15 2007, Toulouse, France

system responses in the 10 cases shown remaining in a close neighborhood of the nominal system response. To check for the robustness of the closed loop system, and to investigate for a possible robustness degradation following

100 101 102-120

-110

-100

-90

-80

-70

-60

-50

-40

-30

Frequency (Hz)

Mod

e 1

(dB)

Open LoopClosed LoopPerturbed Closed Loop

Figure 8. Control Point 1 response, wingtip force input the controller order reduction we perform µ analysis (Zhou et al., 1996) for the closed loop systems using, first, the full-order controller, and second, the reduced one. The structured singular value µ is shown for the frequency range of interest in Figure 9. No significant differences are noticed between the full and reduced order cases, thus confirming the accuracy of the controller order reduction algorithm. Since the maximum value of µ is significantly lower than 1, it can be concluded that both the full order and reduced order controllers display good performance robustness, with a significant margin available. We note that the controller was synthesized in the mathematical hypothesis of a full-complex uncertainty block, whereas the µ analysis actually treats the parametric uncertainty as strictly real, offering increased accuracy.

10-1 100 101 102 1030

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Frequency (Hz)

µ ∆

Figure 9. Structured Singular Value – Full Order Controller (-) and Reduced Order Controller (--)

Next, the time response of the closed loop system to the exogenous wingtip force is simulated first for the case of a unit step input, and second, that of a unit impulse. The Mode 1 response to the two loading cases is shown in Figures 10

and 11, while that of the Control Point 1 is given in Figures 12 and 13. In both loading cases the dominant characteristic of Mode 1 is clearly displayed, while the contribution of the higher frequency modes to the measured output can only be noticed in the form of rapidly decaying transients at the initiation of motion. It can be noticed that the closed loop control significantly reduces the settling time of the motion.

0 0.5 1 1.5 2 2.5 3 3.5 4-0.06

-0.05

-0.04

-0.03

-0.02

-0.01

0

Time (sec)

Mod

e 1

Open LoopClosed Loop

Figure 10. Mode 1 response, step force input

0 0.5 1 1.5 2 2.5 3 3.5 4-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

Time (sec)

Mod

e 1

Open LoopClosed Loop

Figure 11. Mode 1 response, impulse force input

0 0.5 1 1.5 2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6x 10-3

Time (sec)

Uz (m

)

Open LoopClosed Loop

Figure 12. CP1 response, step force input

Recent Advances in Aerospace Actuation Systems and Components, June 13-15 2007, Toulouse, France

0 0.5 1 1.5 2 2.5 3 3.5 4-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

0.02

0.025

Time (sec)

U z (m)

Open LoopClosed Loop

Figure 13. CP1 response, impulse force input

V CONCLUSIONS An investigation into the use of embedded piezoelectric materials as actuation devices for aeroservoelastic control was performed. The worked focused on system modeling and implementation of a robust control law that accounts for uncertainties and showed that effective robust control is achievable. Further studies should capitalize on these results and refine the control design by investigating distributed multi input control, optimal actuator placement and including aerodynamic states in the open loop system. REFERENCES Abramovich H., Weller T. and Ping S.-Y. (2005), Dynamic

response of a high aspect ratio wing equipped with PZT patches - A theoretical and experimental study, Journal of Intelligent Material Systems and Structures, vol.16, pp. 919-923.

Block J. J. and Strganac T. W. (1998), Applied active control for a nonlinear aeroelastic structure, Journal of Guidance, Control, and Dynamics, vol. 21, pp. 838-845.

Chang W., Gopinathan S.V., Varadan V. V. and Varadan V. K. (2002), Design of robust vibration controller for a smart panel using finite element model, Journal of Vibration and Acoustics, vol. 124, pp. 265–276.

Chopra I. and Sirohi J. (2000), Fundamental behavior of piezoceramic sheet actuators, AIAA Journal, vol. 25, pp. 1373-1385.

Crawley E. F. and de Luis J. (1987), Use of piezoelectric actuators as elements of intelligent structures, AIAA Journal, vol. 25, pp. 1373-1385.

Iorga L., Baruh H. and Ursu I. (2006), A review of ∞H robust of piezoelectric smart structures (submitted to the Applied Mechanics Reviews).

Iorga L., Ursu I. and Baruh H. (2007), ∞H control with µ analysis of a piezoelectric actuated plate (to be published in Journal of Vibration and Control).

Mechbal N. (2005), Simulations and experiments on active vibration control of a composite beam with integrated piezoceramics, Proceedings of 17th IMACS World Congress, France.

Nam C., Chen P.C., Liu D.D., Chattopadhyay A., Kim J., Wereley N. M. (2000), Neural net based controller for flutter suppression using ASTROS with smart structures, Smart structures and integrated systems Conference, Newport Beach, vol. 3985, pp. 98-109.

Raja S., Pashilkar A.A., Sreedeep R. and Kamesh J. V. (2006), Flutter control of a composite plate with piezoelectric multilayered actuators, Aerospace Science and Technology, vol. 10, pg. 435-441.

Rocha J., Moniz P. and Suleman A. (2007), Aeroelastic control of a wing with active skins using piezoelectric patches, Mechanics of Advanced Materials and Structures, vol. 14, pp. 23-32.

Rudnyi E. B., Lienemann J., Greiner A., and J. G. Korvink (2004), mor4ansys: Generating Compact Models Directly from ANSYS Models, The 2004 Nanotechnology Conference and Trade Show, Nanotech 2004, Boston, Massachusetts, vol. 2, p. 279-282.

Tralli A., Rutigliano L., Olivier M., Sciacovelli D. and Gaudenzi P. (2004), Modelling of active space structures for vibration control, Proceedings of Space Conference, Cranfield, GB.

Ursu I. and Ursu F. (2004), New results in control synthesis for electrohydraulic servos, International Journal of Fluid Power, vol. 5, pp. 25-38.

A. Varga (2002), New numerical software for model and controller reduction, Technical Report SLICOT Working Note 2002-5, Institut fur Robotik und Mechatronik, DLR.

Zhou, K., Doyle, J., and Glover, K. (1996), Robust and Optimal control, Prentice Hall.

Wilkie W. K. and Park K. C. (1996), An aeroelastic analysis of helicopter rotor blades incorporating piezoelectric fiber composite twist actuation, Technical Report NASA TM 110252, NASA.

NOTATIONS Generally, all notations are defined in text; however, for convenience, some of more specific are summarised bellow. x nodal displacement (mm)/state vectoru control vector z quality output vector y measured output vector

matrix norm M mass matrix C structural damping matrix K stiffness matrix

AKNOWLEDGEMENTS The work described above was supported (partially) on account of CESAR FP6 Project, Task 4.1, INCAS, and by the CEEX Programme of the Romanian Ministry of Education and Research, Contract No. X2C12/2006.