Active Vibration Control of a Fully Clamped Laminated Composite Plate Subjected to Impulsive

Pressure Loadings

Haydar Uyanık Turkish Air Force Academy

Yesilyurt, 34149, Istanbul, Turkey h.uyanik@hho.edu.tr

Abstract— In this numerical study vibrations of a fully clamped composite plate subjected to impulsive pressure loadings are controlled by using piezoelectric patches. Finite element methods are preferred for numerical solutions. For obtaining finite element model of the smart plate structure, semiloof shell finite element including piezoelectric effects is used. Mode summation method is used for reducing the degrees of freedom of the finite element model and state-space equations are obtained from the reduced finite element model for determining appropriate control strategies. Optimal linear quadratic regulator (LQR) approach has been used to determine the feedback gain matrix and vibrations of the plate suppressed successfully.

Keywords-component; blast load, vibration control, semiloof shell element, piezoelectric.

I. INTRODUCTION In aeronautics and astronautics industry, plate and shell

structures are most widely used elements. In addition to this, the use of layered composite materials in the manufacturing of these parts has increased. An aerospace vehicle is effected by time dependent pressure loads while move in the atmosphere and cause structural vibrations. Analysis of structural vibrations of aerospace shell and plate structures subjected to impulsive blast type pressure loadings is one of the most important research subjects in the literature and under blast type impulsive pressure loadings, the dynamic responses of structures have been investigated in many studies [1-4]. Commonly, triangular, step or exponential decay functions can be used for expressing uniform time dependent blast loading which affects object surface through normal direction. Various parameters of the blast loading equation can be obtained from some of the experimental studies.

Control and suppression of structural vibrations are necessary and also important. In recent years, smart materials and structures especially piezoelectric elements are preferred and widely used for vibration control and suppression [5-7]. Applications of smart structures involve the use of piezoelectric parts. The development of piezoelectric composite materials offers great potential for use in advanced aerospace structural applications.

In this numerical study, by the use of piezoelectric patches, vibrations of a fully clamped composite plate subjected to impulsive pressure loadings are controlled and suppressed successfully.

II. ACTIVE SEMILOOF SHELL ELEMENT The semiloof shell element was originally developed by

Irons [8] for linear elastic analysis of thin shell structures. Semiloof shell element is one of the most efficient elements for the solution of shells having arbitrary geometry and it accounts for both membrane and bending actions. It is an isoparametric non-confirming element and Figure 1 shows the unconstrained version of the element configuration.

Figure 1. Semiloof shell element.

Semiloof shell element has three types of nodes as shown in Figure 1. Corner and midside nodes have three components of the global displacement vector ui, vi, wi taken as nodal variables along the global x, y, z directions, respectively. Loof nodes are located at Gaussian quadrature points on the element sides, having two rotations θXZ, θYZ along and perpendicular to the edge, respectively, in the local coordinate system. The central node has three components of the local displacement vector and two rotations as explained. Each element has 17 nodes and 45 degrees of freedom (d.o.f.). The total number of 45 d.o.f. is

978-1-4244-3628-6/09/$25.00 ©2009 IEEE 170

reduced to 32 by applying shear constraints and combining the displacements at the center to produce only a normal deflection. This element was successfully used in various studies [9,10]. In this study an active element is developed based on semiloof shell element for modeling laminated composite shells with piezoelectric layers.

Linear piezoelectric constitutive equations which include electromechanical coupling can be expressed as

T

E P

S P

= −= +

σ c ε e ED eε ε E

(1)

where σ is stress vector, ε is strain vector, EP is electric field vector, D is electric displacement vector, cE is elasticity constant matrix, εS dielectric constant matrix, and e is piezoelectric constant matrix. These equations denote the actuation and the sensing effects of piezoelectric materials, respectively. Seeger et al. [11] have suggested several active semiloof shell elements for vibration control. The Piezoelectric element called S1 based on the finite element approach has no electromechanical coupling and the classical semiloof shell finite element stiffness and mass matrices can be used. Normal forces and moments due to potential differences between upper and lower surfaces of the active layer can be written on the right-hand side of the dynamic equations of motion. In this case, only the first equation from the linear constitutive equations (1) will be considered. In order to calculate the stress value due to the forces and moments acting on a layered structure, Kirchhoff hypothesis was considered. According to the Kirchhoff hypothesis, a fiber, oriented normal to the midplane remains the same after the deformation. It follows that

0p z= +S S κ (2)

where Sp is two dimensional strain vector as Sp={εxx εyy εxy}T. Due to electric field in the z direction of kth piezoelectric layer, Nk and Mk represents in-plane forces and bending moments, respectively. Force and moment resultants can be determined by integrating the stresses over the thickness of the multilayered structure as follows

2

2

k

k

h

k khdz

−= ∫N σ ; 2

2

k

k

h

k khz dz

−= ∫M σ (3)

In the above equation, hk is the thickness of the kth layer. Once again, considering the first equation from the linear constitutive equations (1), stresses in the kth layer may be calculated according to the following equation:

Tk Ek k k Pk= −σ c ε e E ; k Ek=e dc (4)

The d matrix in this expression is the piezoelectric constants matrix and shows piezoelectric coupling between the electric field and the strain. For a piezoelectric material with a

tetragonal or hexagonal structure, which is defined as Class 4 [12], the piezoelectric coupling matrix is:

15

32 3124

24 1531 32 33

0 0 0 0 0; ( )

0 0 0 0 0; ( )

0 0 0

dd d

dd d

d d d

⎡ ⎤=⎢ ⎥= ⎢ ⎥ =

⎢ ⎥⎣ ⎦

d (5)

In an active piezoelectric layer, because of the potential difference value (φk) between the upper and lower surfaces, there will be only an electric field in the z direction and the electric field vector is:

{0 0 }TPk k khφ=E (6)

By substituting equation (4) in equation (3) and integrating over the thickness taking the orientation of the layers into account with respect to the global axis, one gets the global equation which relates the resultant in-plane forces N and bending moments M , to the midplane strain S0, curvature κ and the electrical potential applied to the piezoelectric patches:

0

1

nk

k k=

⎡ ⎤⎧ ⎫ ⎡ ⎤⎧ ⎫= +⎨ ⎬ ⎨ ⎬⎢ ⎥ ⎢ ⎥

⎩ ⎭ ⎣ ⎦⎩ ⎭ ⎣ ⎦∑

NSN A BMκM B D

(7)

The first term on the right hand side of equation (7) is the classical stiffness matrix of a composite laminate, where A is the extensional stiffness matrix, D is the bending stiffness matrix and B is the extension/bending coupling matrix as following,

{ } { }2 2 3 31 11 1 12 3; ; ( ); ( ); ( )k k k k k k kk

z z z z z z− − −= Σ − − −A B C Q (8)

where kQ denotes global stiffness matrix of kth composite layer and depends on the fiber orientation of the layer. These matrices are related to the individual layers according to the classical relationships on composite materials. The second term on the right hand side of the equation (7) expresses the piezoelectric loading.

III. IMPULSIVE PRESSURE LOADINGS The advanced aerospace vehicle is likely to be exposed,

during its operational life, to more severe environmental conditions than in the past. In this sense, blast loads occurring from fuel and nuclear explosions, gust and sonic boom pulses, are likely to act on their structure [13-17].

In many studies, where the time history of the air blast pressure is used as an analytical function, commonly, Friedlander’s decay function is used for modeling uniform time dependent air blast loading which affects object surface through normal direction as following:

171

( )( ) 1 pt tm pp t p t t e α−= − (9)

where pm is peak pressure, t0 is positive phase duration of the blast pulse and α is the pressure waveform parameter [18-21]. These parameters can be obtained from experimental studies. The sonic-boom overpressure can be expressed as follows:

(1 / ) for 0( )

0 for 0 and m p p

p

p t t t rtp t

t t rt− < <⎧

= ⎨ < >⎩ (10)

where r denotes the shock pulse length factor. For r=1, the sonic-boom degenerates into a triangular explosive pulse, while for r=2 and 3, the pulse corresponds to a symmetric and non-symmetric sonic-boom, respectively. The gust can be modeled as a semi-sinusoidal pressure loading as follows:

( )sin for 0( )

0 for 0 and m p p

p

p t rt t rtp t

t t rt

π⎧ < <⎪= ⎨< >⎪⎩

(11)

Rectangular blast pressure loading also can be used for modeling impulsive pressure pulses as following:

for 0( )

0 for 0 and m p

p

p t rtp t

t t rt< <⎧

= ⎨ < >⎩ (12)

IV. VIBRATION CONTROL Including structural damping, stiffness and mass matrices

which express the structural model are obtained from the semiloof shell finite element and used for obtaining the equations of motion expressed as

B P+Mq + Cq + Kq = F F (13)

The force vectors in the right hand side of the equation (13) denote the blast load vector and piezoelectric load vectors respectively can be written in the following form

( )B B p t=F V ; ( )P P t=F V u (14)

where VB represents unit pressure load vector, VP is unit piezoelectric load matrix and

{ }1 2 3( )T

pt u u u u=u (15)

is the time dependent potential differences vector between upper and lower surfaces of each piezoelectric active layer and p is the total number of piezoelectric layer on structure. In order to reduce the d.o.f. of the finite element model, mode summation method is used with weighted modal matrix φ

which is normalized with respect to the mass matrix including dominant modes in the dynamic behavior. After mode summation, the equation of motion becomes

B P+ + = +Mη Cη Kη F F (16)

where M is the weighted mass matrix which equals to unit matrix, C is weighted damping matrix, K is weighted stiffness matrix that includes selected eigenvalues, BF and

PF are the reduced blast and piezoelectric force vectors respectively. State-space equations are obtained from the reduced finite element model and can be written with

1 1− −

⎡ ⎤= ⎢ ⎥⎣ ⎦

0 IA

-M K -M C (17)

1 T

P−

⎡ ⎤= ⎢ ⎥⎣ ⎦

0B

-M Vϕ ;

1 TB

−

⎡ ⎤= ⎢ ⎥⎣ ⎦

0E

-M Vϕ (18)

in the closed form as follows,

( ) ( )t p t

o

x = Ax + Bu + Ey = C x

(19)

where { }T T T=x η η is state vector, A is system matrix, B is control matrix, E is the external disturbance matrix, y is the output vector and Co is output matrix (Fig. 2). Now the problem is suppressing the vibration by applying the input control voltage u(t) by using a feedback control system approach. In this study linear control approach is preferred and control input vector is assumed that

= −u Kx (20)

where K is the control gain matrix. For the determination of the optimum control gain matrix, optimal Linear Quadratic Regulator (LQR) approach can be used. The quadratic performance index is

( )0

ftT TJ dt= +∫ x Qx u Ru (21)

where tf is final time, Q and R are the state penalty matrix and the control penalty matrix, respectively. Q and R are positive definite arbitrary weighting matrices which determine the relative importance of the error. For a more rapid vibration suppression, a large value of Q should be selected. Selecting appropriate weighting matrices for full state feedback (Fig. 2), optimal gain matrix K can be calculated as

172

1 T−=K R B P (22)

where P must satisfy the matrix Riccati equation

1T T−+ − + =A P PA PBR B P Q 0 (23)

After determining the P matrix, the optimum K matrix will be calculated using the equation (22) [22].

Figure 2. Full state feedback control system.

Active vibration suppression using LQR approach has some limitations in spite of numerous successful research activities in this area. For example, active damping may become unstable because of large control gains or non-collocated sensor/actuator configurations [23].

V. NUMERICAL APPLICATION For numerical applications the plate model shown in Figure

3 is used. For this model 7 layers (0.28 mm) composite plate which has 0.25 mm, 5 piezoelectric patches used as actuators on the upper surface of the plate is used. Fiber orientation is [0o/ 90 o]s and structural damping is neglected. The material properties of composite and piezoelectric elements are listed in Table 1.

TABLE I. MATERIAL PROPERTIES

Composite Fiberglass-fabric

Piezoceramics PZT G1195N

Elasticity Modulus (N/mm2) E11=E22=24140 E=63000 Poisson’s Ratio ν12=0.11 ν=0.3 Shear Modulus (N/mm2) G12=3790 G=24200 Density (kg/m3) 1800 7600 Piezoelectric constants (mm/V) − d31=d32=254·10−9 Thickness (mm) 0.28 0.25

In some previous studies by the author, both static and

dynamic analyses of smart plates modeled by using semiloof shell finite elements are performed. The accordance of those results with similar studies were shown in the previous study [23], so there was no need for an additional static and dynamic analyses study of the finite element model used in this research activity.

As shown in Figure 3, a fully clamped layered composite plate is selected as the model. Piezoelectric elements are used on only the upper surface of the plate and it was assumed that

the air blast load affects the lower surface of the plate. Free vibration frequencies of the plate which is modeled as 44×44 elements are calculated by using semiloof shell finite element model and free vibration frequencies are shown in Table 2.

Figure 3. Plate model and piezoelectric elements (dimensions in mm).

The d.o.f of the system obtained by using the semiloof shell finite element is determined as 24467. When the system is defined in terms of the state-space equations for investigating appropriate control strategies, the degrees of freedom will be doubled. In order not to struggle huge matrices, it was decided to reduce the degrees of freedom of the model by using mode summation method by choosing the initial 10 free vibration modes.

TABLE II. FREE VIBRATION FREQUENCIES OF THE PLATE (HZ)

Mode Number 1 2 3 4 5

Frequency (Hz) 229.132 469.053 477.009 665.886 879.949

Mode Number 6 7 8 9 10

Frequency (Hz) 924.665 1042.371 1058.224 1383.884 1425.447

The plate is subjected to the blast load pm=28906 Pa, tp=0.0018 s and α=0.35 values are taken from Reference [24] in equation (9) shown in Figure 4.

Figure 4. Blast pressure curve.

173

Performance index weighting matrices in Eq. (21) are, at first, chosen as Q=103·I20 and R=I5, and later as Q=106·I20 and R=I5 (where I20 is 20×20 and I5 is 5×5 identity matrix) and optimum LQR gain matrices are determined for both cases by using Matlab software tools. Control input voltages applied on the piezoelectric elements are limited to ±500 V for saturation. Time dependent displacement variations of the mid-point of the plate are obtained under blast load as shown in Figure 5. Mid-point displacement of the plate subjected to sonic boom for r=1 triangular pressure pulse shown in Figure 6. Symmetric and non-symmetric sonic pressure pulses, considering r=2 and r=3 are shown in Figure 7 and Figure 8, respectively. Mid-point displacements of the plate for the case of a sinusoidal pressure pulses simulating a gust for r=1 and r=2 are shown in Figure 9 and Figure 10. Under a rectangular pressure pulse for r=3, mid-point displacements are shown in Figure 11.

Figure 5. Mid-point displacements of the plate subjected to blast load.

Figure 6. Mid-point displacements of the plate exposed to a triangular explosive pulse for r=1.

Figure 7. Mid-point displacements of the plate exposed to a sonic-boom pressure pulse characterized by r=2

Figure 8. Mid-point displacements of the plate exposed to a sonic-boom pressure pulse characterized by r=3

Figure 9. Mid-point displacements of the plate for the case of a sinusoidal pressure pulse simulating a gust for r=1

Figure 10. Mid-point displacements of the plate for the case of a sinusoidal pressure pulse simulating a gust for r=2

Figure 11. Mid-point displacements of the plate under a rectangular pressure pulse for r=3

174

VI. CONCLUSIONS In this numerical study, vibrations on a fully clamped

layered composite plate subjected to different types of impulsive pressure loadings such as blast load, sonic boom and gust were actively controlled and successfully suppressed. For the smart plate model, semiloof shell finite element model, which takes the piezoelectric forces and moments into consideration, was used and mode summation method was utilized to reduce the high d.o.f. of the finite element model. State space equations were determined from the motion equations for the reduced system.

Linear control approach and full state feedback control system is used for vibration control. For the determination of the feedback gain matrix, Linear Quadratic Regulator (LQR) approach was used and structural vibrations were controlled. In spite of numerous successful research activities in vibration control of structures using LQR approach, there are some limitations in active vibration suppression. Because of large control gain matrix values, active damping may become unstable. So, its need to be explored for determining the optimum feedback control gain matrix which do not make control system unstable and suppress the structural vibrations effectively.

REFERENCES [1] Librescu, L. and Na, S., “Dynamic Response Control of Thin-Walled

Beams to Blast Pulses Using Structural Tailoring and Piezoelectric Actuation”, Journal of Applied Mechanics, 65(2), 497-504, 1998.

[2] Librescu, L. and Na, S., “Optimal Vibration Control of Thin-Walled Anisotropic CantileversExposed to Blast Loadings”, Journal of Guidance Control and Dynamics, 23(3), 491-500, 2000.

[3] Librescu, L. and Na, S., “Comparative study on vibration control methodologies applied to adaptive thin-walled anisotropic cantilevers”,, European Journal of Mechanics A/Solids, 24, 661–675, 2005.

[4] Kazancı, Z. and Mecitoğlu, Z., “Nonlinear Damped Vibrations of a Laminated Composite Plate Subject to Blast Load”, AIAA Journal, Vol. 44, No. 9, 2002-2008, 2006.

[5] Sunar, M. and Rao, S. S., “Piezoelectricity and its use in disturbance sensing and control of flexible structures: a survey”, Applied Mechanics Review Vol.47, 113-123, 1994.

[6] Benjeddou, A., “Advances in piezoelectric finite element modeling of adaptive structural elements: a survey”, Computers and Structures, 72: 347-363, 2000.

[7] Gaudenzi, P., Carbonaro, R., Barboni, R., “Vibration control of an active laminated beam”, Composite Structures, 38, No. 1-4, 413-420, 1997.

[8] Irons, B., M., “The Semiloof Shell Element, In Finite Elements for Thin Shells and Curved Members”, Willey, New York, 1976.

[9] Mecitoğlu, Z., “Governing Equations of a Stiffened Laminated Inhomogeneous Conical Shell”, AIAA Journal, 35, No. 6, 1997.

[10] Karisiddappa, Viladkar M. N., Godbole P. N., and Krishna P., “Finite Element Analysis of Column Supported Hyperbolic Cooling Towers Using Semi-loof Shell and Beam Elements”, Engineering Structures, Vol. 20, No. 1-2:75-85, 1998.

[11] Seeger, F., Köppe, H., Gabbert, U., “Comparison of Different Shell Elements for the Analysis of Smart Structures”, Journal of Mathematics and Mechanics, Section 1-24 Additional Issue 81, S4, S887-S888, 2001.

[12] Piefort, V., “Finite Element Modelling of Piezoelectric Active Structures”, PhD Thesis, Université Libre de Bruxelles, 2001.

[13] Guruprasad, S., Mukherjee, A., “Layered sacrificial claddings under blast loading Part I analytical studies”, International Journal of Impact Engineering, 24, 957-973, 2000.

[14] Marzocca, P., Librescu, L., Chiocchia, G., “Aeroelastic response of 2-D lifting surfaces to gust and arbitrary explosive loading signatures”, International Journal of Impact Engineering, 25, 41-65, 2001.

[15] Mosalam, K. M., Mosallam, A. S., “Nonlinear transient analyses of reinforced concrete slubs subjected to blast loading and retrofitted with CFRP composites”, Composites, Part B 32, 623-636, 2001.

[16] Marzocca, P., Librescu, L., Chiocchia, G., “Aeroelastic response of a 2-D airfoil in a compressible flow field and exposed to blast loading”, Aerospace Science and Technology, 6, 259-272, 2002.

[17] Hause, T., Librescu, L., “Dynamic response of anisotropic sandwich flat panels to explosive pressure pulses”, International Journal of Impact Engineering, 31, 607-628, 2005.

[18] Gupta, A. D., Gregory, F. H., Bitting, R. L. and Bhattacharya, S, “Dynamic Analysis of an Explosively Loaded Hinged Rectangular Plate”, Computers and Structures, 26, No.1/2, 339-344, 1987.

[19] Türkmen, H. S. and Mecitoğlu, Z., “Nonlinear Structural Response of Laminated Composite Plates Subjected to Blast Loading”, AIAA Journal, 37(12), 1639-1647, 1999.

[20] Kazancı, Z., and Mecitoğlu, Z., “Nonlinear Dynamic Behavior of Simply Supported Laminated Composite Plates Subjected to Blast Load”, Journal of Sound and Vibration, Vol. 317, 3-5, 883-897, 2008.

[21] Ogata,. K., Modern Control Engineering, Second Edition, New Jersey: Prentice-Hall, Englewood Cliffs, 1990.

[22] Balamurugan, V. and Narayanan, S., “Finite Element Formulation and Active Vibration Control Study on Beams Using Smart Constrained Layer Damping (SCLD) Treatment”, Journal of Sound and Vibration, 249(2), 227-250, 2002.

[23] Uyanık H., Mecitoğlu Z., “Vibration Control of a Laminated Composite Plate Subjected to Blast Loading”, III European Conference on Computational Mechanics, 05-09 June, Lisbon, Portugal, 2006.

[24] Türkmen, H. S. and Mecitoğlu, Z., “Dynamic Response of A Stiffened Laminated Composite Plate Subjected to Blast Loading”, Journal of Sound and Vibration, 221(3), 371-389, 1999.

175