Simple analytical solution for active vibration control of ...nopr.niscair.res.in/bitstream/123456789/8434/1... · Simple analytical solution for active vibration control of intelligent

Indian Journal of Engineering & Materials Sciences Vol. 12, June 2005, pp. 196-206

Simple analytical solution for active vibration control of intelligent composite and sandwich plates

R Suresh & G Venkateswara Rao

Structural Engineering Group, Vikram Sarabhai Space Center, Trivandrum 695 022, India Received 19 July 2004; accepted 31 January 2005

A simple analytical formulation of laminated composite plates with integrated piezoelectric layer, used as sensor and actuators for the vibration suppression of intelligent composite and sandwich plates is presented. The first order shear deformation theory is employed to account for the transverse shear deformation. The equations are transformed into the state space form and solved by the linear quadratic regulator (LQR) based control algorithm. The effect of the state and control parameters on the system response is studied. The effect of the thickness ratio of the intelligent plate is also investigated.

IPC Code: H01H35/14; G05D19/00

The technological developments in aerospace engineering have created new avenues of research particularly in the areas of active vibration control and shape control of flexible structures and structural health monitoring using the concept of smart structures. The aerospace structures should be designed as light weight structures due to the high cost of transportation. Under external loading conventional structures deform in a passive manner with no control over the final deformed configuration. These structures are also lightly damped because of the low internal damping of the material used in the construction and the increased flexibility may cause dynamic instability. This will affect the precision and performance of the system. Since these structures are distributed parameter systems having an infinite set of vibration modes, distributed measurement and control are required. However, conventional control systems use discrete sensors and actuators to control the vibration of distributed elastic systems. If the sensors and actuators, which are discrete, are placed at the nodal points or on the nodal lines of a vibration mode, that mode will be missed. Thus, it is essential to use an active control system with distributed sensors and actuators in these structures to control and stabilize the structure during the operation. Piezoelectric material is a class of smart material that utilizes the converse piezoelectric effect to actuate a structure, in addition to the direct piezoelectric effect for sensing structural deformation. Various finite element models have been developed to predict the active vibration control behaviour due to a piezoelectric actuator. Tzou and Tseng1 developed the

coupled finite element equations for piezoelectric continuum and solved it simultaneously. The active vibration control of a smart beam with direct velocity feedback and modified independent modal space control was demonstrated by Ghate and Seshu2, though the finite element method (FEM) models which can be applied widely However, these are computationally intensive. Hence, analytical methods offer a better alternative for the purpose of active vibration control. Analytical model of the beam by assuming linear variation of strain in the substrate and constant of strain in the actuators are studied by Crawley and de Luis3. Crawley and Anderson4 proposed that the entire cross-section, both beam and actuator, undergoes consistent Bernoulli-Euler strain. Yoon and Wahington5 applied the consistent Bernoulli-Euler strain model to the curved beams. Lee and Moon6 used the classical plate theory to derive a simple formulation for piezoelectric plates. Analytical solution for the static analysis of simply supported intelligent plate, subjected to electromechanical load, was presented by Haylinger7 and Ray et al.8. Two strategies for solving structural and control problem have been presented by Onoda and Haftka9. One is a nested optimization strategy by taking advantage of optimal linear quadratic control theory from which the optimal gain matrix can be found using the well known Riccati equation, the other is direct optimization that all structural parameters and entries of the gain matrix are equally treated as design variables. Compared to the direct method, nested method is more advantageous and is therefore used by many others, Sunar and Rao10 and Tsujioka et al11.

SURESH & RAO: SIMPLE ANALYTICAL SOLUTION FOR VIBRATION CONTROL OF COMPOSITE PLATES

197

Active vibration control of the structure has received wide attention for many years with the advent of smart structures, on account of its well-known benefits. The application of active control system is quite involved in terms of the fabrication of the structure, design of control law and implementation of controller. The design of such a system requires the multi-input and multi-output feedback control system. Several methods are available in the literature for the design of such feedback control systems both in the frequency and the time domain. As the subject is still in the developmental stage, a simple solution to provide the understanding of their response characteristics is of utmost importance. In the present study, a simple analytical formulation of laminated composite plates with integrated piezoelectric layer, used as the sensor and the actuator for the vibration suppression of the intelligent composite and sandwich plates are presented. The first order shear deformation theory is employed to account for the transverse shear deformation. The equations are transformed into state space form and solved by the linear quadratic regulator (LQR) based control algorithm. The effect of the state and control parameters on the system response and the effect of these parameters on the different thickness ratios of the intelligent plate are studied. Kinetic and Electric Potential Distribution The piezoelectric layer can be used as an actuator or a sensor. The piezoelectric actuator and sensor layers are also considered as load bearing components in the plate. The length and width of the plate are denoted by ‘a’ and ‘b’ respectively. The material considered for the distributed sensor and actuator layers is, piezoelectric PVDF. A laminated composite plate is considered with co-ordinates x, y along the inplane direction and z along the thickness direction (Fig. 1). Using the Mindlin’s formulation, the displacements u, v, w in x, y, z directions respectively at any generic point in the laminate are expressed as functions of the mid-plane (z = 0) displacements uo, vo, wo and rotation of plate normals θy and θx with respect to the initial configuration is as follows: ( ) ( ) ( ), , ; , ; , ;o yu x y z t u x y t z x y t= + θ ,

( ) ( ) ( ), , ; , ; , ;o xv x y z t v x y t z x y t= + θ and

( ) ( ), , ; , ;ow x y z t w x y t= …(1)

where yθ and xθ are rotations of the plate normal to the XZ and YZ planes respectively. The plate normals though straight, with respect to the deformed configuration, are not necessarily normal to the mid-plane after deformation and hence permit a constant shear strain through the thickness. Strain-displacement relations for the intelligent laminated structure (laminated composite substrate + piezo layer) are given by

, ,x o x y xu zε = + θ ; , ,y o y x yv zε = + θ ;

( ), , , ,xy o y o x x x y yu v zγ = + + θ +θ ; …(1a)

xz o ywγ = +θ ; yz o xwγ = +θ

{ }{ }T

x y xy xz yzε = ε ε γ γ γ

The electric potential distribution is chosen in such a way that the surfaces of the piezoelectric layers which are in contact with the substrate are suitably grounded. Also, since the thickness of the piezoelectric layer is very small it is reasonable to assume that the electric potential function which yields zero potential at the interface between actuator/sensor and substrate and provides a linear variation across the thickness of the sensor and actuator layer is:

1

1

( , , ; ) ( , ; )i

i i

z zx y z t x y tz z

+

+

⎛ ⎞−φ = φ⎜ ⎟−⎝ ⎠

…(2)

where z is the thickness location (Fig. 1).

Fig. 1— Geometry of an intelligent plate

INDIAN J. ENG. MATER. SCI., JUNE 2005

198

The simply supported spatial boundary conditions considered are: At x = 0, a; w = v = Nx = Mx= 0 At y = 0, b; w = u = Ny = My= 0 …(3) The admissible displacement functions and the electric potential function8, which exactly satisfy the geometric boundary condition of a plate with the simply supported edges, are:

( )1 1

cos sink l

om n

m x n yu U ta b= =

π π= ∑∑

( )1 1

sin cosk l

om n

m x n yv V ta b= =

π π= ∑∑

( )1 1

sin sink l

om n

m x n yw W ta b= =

π π= ∑∑

( ) 1

1 1 1

sin sinak l

a a ia a

m n i i

z z m x n ytz z a b

+

= = +

⎛ ⎞− π πφ = φ ⎜ ⎟−⎝ ⎠

∑∑

( ) 1

1 1 1

sin sinsk l

s s is s

m n i i

z z m x n ytz z a b

+

= = +

⎛ ⎞− π πφ = φ ⎜ ⎟−⎝ ⎠

∑∑

( )1 1

sin cosk l

x xm n

m x n yta b= =

π πθ = θ∑∑ ;

( )1 1

cos sink l

y ym n

m x n yta b= =

π πθ = θ∑∑ …(4)

where, the superscript ‘s’ and ‘a’ in the above expressions represent sensor and actuator respectively. The k and l are the limit for the modes to be considered in the a and b directions, which depends on the lower most frequencies for an intelligent composite plate. In the present analysis subspace iteration is used to extract the frequencies. The m and n are the modes correspond to the length and width directions (x and y) of the intelligent plate. The surface charge density on the surface of the actuator and the surface traction force can be expressed in the form of the Fourier series. In the present study, surface charge density and the surface traction force are considered as

3

1 1 1

sin sinak l

o a am n i i

m x n yz z a b= = +

−β π πΦ = Φ

−∑∑ and

1 1

sin sink l

om n

m x n ya b= =

π πΘ = Θ∑∑ …(5)

where β3 is the dielectric constant in thickness direction. Constitutive Equation The constitutive equations which relate the strain vector {ε} and the electric field vector {Ef} to the stress vector {σ} and the electric displacement vector {D} are as follows12: { } [ ]{ } [ ]{ }fD d E= ε + β and

{ } [ ]{ } [ ] { }TfQ d Eσ = ε − …(6)

where [d], [β] and [Q] are the piezoelectric constant, the dielectric constant at a constant mechanical strain and the constitutive matrices respectively. The coefficient of the piezoelectric constant and the dielectric constant matrices for the sensor and actuator are given by Cady13. Solution procedure For an intelligent laminate which is vibrating harmonically with angular frequency ω, the strain energy (Π) can be expressed as

{ } [ ]{ } { } [ ]{ }1/ 2

{ } [ ]{ } { } [ ]{ }

T T

T Tv

Q d Edv

E d E E

⎡ ⎤ε ε − εΠ = ⎢ ⎥

− ε − β⎢ ⎥⎣ ⎦∫ …(7)

The kinetic energy (T) of the intelligent laminate with volume ‘v’ is given by

2 2 2

2v

T u v w dvρ ⎡ ⎤= + +⎣ ⎦∫ & & & …(8)

where ρ is the mass density, Using the Lagrangian equation and taking the indicated partial derivatives, from the strain energy expression we obtain

7

10ui i

i

KU =

∂Π= δ

∂ ∑ , 7

10vi i

i

KV =

∂Π= δ

∂ ∑ ,

7

10wi i

iK

W =

∂Π= δ

∂ ∑ , 7

1xi i

ix

K=

∂Π= δ

∂θ ∑ ,


199

7

1yi i

iy

K=

∂Π= δ

∂θ ∑ , 7

1si i

is

K=

∂Π= δ

∂φ ∑ and

7

1ai i

ia

K=

∂Π= δ

∂φ ∑ …(9)

where Kui, Kvi, Kwi, Kvi, Kyi, Ksi, Kai are the coefficients of stiffness matrix, the non-zero terms of the stiffness matrix is given in Appendix A. In general, the terms for the rth equation of the sets can be expressed as follows:

7

1ri i

ir

K=

∂Π= δ

∂δ ∑ …(10)

The displacement and the electric potential vector as well as mechanical force vector and electrical force vectors are defined as:

{ } { }0 0, , , , , ,Ts a

o x yU V Wδ = θ θ φ φ

{ } { }00,0, , 0,0,0,0 Tef q= ,

{ } { }3

10,0,0, 0,0,0,

T

ai i

f z z +

−β φ= − …(11)

From the kinetic energy expression we obtain,

7

10ui i

io

T uu dv MU U =

∂ ∂= ρ = δ

∂ ∂ ∑∫& &&

& &,

7

10vi i

i

T MV =

∂= δ

∂ ∑ &&

, 7

10wi i

i

T MW =

∂= δ

∂ ∑ &&

,

7

1xi i

ix

T M=

∂= δ

∂θ ∑ &&

and 7

1yi i

iy

T M=

∂= δ

∂θ ∑ &&

…(12)

In general, terms for the rth equation of the sets for the mass matrix are obtained as

7

1ri i

ir

T M=

∂= δ

∂δ ∑ &&

…(13)

The mass matrix is of diagonal form. The rth equation of the set for stiffness and mass matrices is as follows:

2ri i rr rK Mδ = ω δ …(14)

The non-zero terms of the mass matrix is given in Appendix-A. In the present study subspace iteration is

employed to get the eigen values and eigen vectors. The initial three frequencies are obtained, the values of m and n depend on the lower three modes of an intelligent composite plate. The system of equations is reduced to the modal form by using these three eigen vectors. The system of equations after modal condensation can be written as [ ]{ } [ ]{ } { } [ ]{ }E AM δ K δ F F φ+ = +&& …(15)

where [M], [K] and [FA] are generalized mass, structural stiffness and actuator stiffness matrices. Feedback control strategy In this section, the full state-space approach is used to design the control law. The state space representation of Eq. (15) can be rewritten after reducing it to the modal form as { } [ ]{ } [ ]{ }X A X B= + φ& …(16)

where

{ } { }T T TX = δ δ& , [ ] 1 1

0 IA

M K M C− −

⎡ ⎤= ⎢ ⎥− −⎣ ⎦

and [ ] 10TAB M F−⎡ ⎤= ⎣ ⎦ …(17)

The objective function (J) is defined as

{ } [ ]{ } { } [ ]{ }( )f

o

tT T

t

J X Q X R dt= + φ φ∫ …(18)

where to and tf are the initial and final times respectively, [Q] and [R] are the weighing matrices for the state and control respectively. The control law is designed based on this objective function ‘J’. The steady state solution ( )P t⎡ ⎤⎣ ⎦ satisfies the following Algebraic Riccati Equation (ARE) given by [ ][ ] [ ] [ ] [ ][ ][ ] [ ] [ ] [ ]1 0T TP A A P P B R B P Q−+ − + = …(19) There are various algorithms in the literature to obtain the solution of the ARE. These methods are used to obtain the steady state solution, by integrating up to the time when a steady state solution is reached, that is ( )P t⎡ ⎤⎣ ⎦ does not change with the special ARE norm value. The optimum gain in the steady state is


200

then expressed as [ ] [ ] [ ] [ ]1 TG R B P−= . In the present study MATLAB14 is used to solve the above ARE equation. Results and Discussion To validate the present formulation, the transient response of an isotropic plate was carried out, the numerical results are compared with those available in the open literature. Detailed numerical studies are carried out on intelligent composite and sandwich plates and the results for the same are presented in the subsequent sections. To study the effect of the state and control parameters on the response of the system the weighing matrix for state [Q], in Eq. (18), is normalized by ‘q’ and the weighing matrix for control [R] is also normalized by ‘r’. These two parameters q, r influence the response of the controlled structure, hence the effect of this parameter on the frequency response of the system is studied in detail. The effect of the state and control parameters on the different thickness ratios of the plate is also presented in the

following sections. The free vibration analysis of the structure is carried out first and then the initial three modes are used to transform the system to the modal coordinates. These modal equations are used for the control strategy. Response of a square plate To validate the above formulation transient response of a simply supported isotropic square plate with thickness ratio (side/thickness) of five is investigated for a suddenly applied uniform pulse loading. The geometry and loading details are shown in Fig. 2a. The material properties used in the present analysis, with the standard notation, are: E = 2.1 × 106 N/cm2, ν = 0.25, ρ = 8 × 10-6 N s2/cm4. The intensity of the pulse load applied on the top surface as an uniformly distributed load of 10 N/cm2. The open loop response of the square plate under the uniform pulse load is compared with that obtained by Reddy15, and shown in Fig. 2b. The time period of the system under pulse loading is 3.1×10-4 s. It can be inferred from Fig. 2 that the open loop response of the system is also the same. The results are in good agreement with those obtained by Reddy15 . Effect of the state and control parameters on the response of the system The response of a square intelligent composite plate with a thickness ratio of 100 is investigated. The PVDF layer of 40 μm thickness was mounted on the top and bottom surface and act as a distributed sensor

Fig. 2a— Geometry and uniform pulse loading for a square plate

Fig. 2b— Center deflection versus time for a square plate under uniform pulse load.


201

and actuator. The substrate material properties used in the present analysis are: EL = 172.369×109 N/m2; ET = 6.89 × 109 N/m2; νLT = 0.25; ρ = 1.7 × 103 N s2/m4. The lay up sequence of the substrate is 0/90/0. The piezoelectric layer properties are E = 0.2×109 N/m2; ν= 0.29; ρ = 1.8 × 103 N s2/m4; piezoelectric constant d11 = d12 = d13 = 0.046 c/m2; dielectric constant β11 = β22 = β33 = 0.1062×109 F/m. The peak amplitude of the square plate under sinusoidal forcing function is given in Fig. 3 for the excitation frequency varied from 10 Hz to 100Hz. The state and control parameter ratios (q/r) of 2, 4, 6, 8, 10 and 100 are considered as shown in Fig. 3. Though LQR control is familiar to researchers in control system area the influence of q/r on structural

Fig. 3—Effect of state and control parameters on the response of

the intelligent plate (beginning of the excitation)

parameter is not demonstrated for researchers on q/r ratio of 2, the amplitude of response is high at structural area. It can be seen from the figure that for resonance frequency, (It should be noted that, the fundamental frequency of the intelligent composite plate is 53.79 Hz and the next higher frequency is 80.85 Hz). This shows the controlling effect is poor for the system with smaller q/r ratio as expected. However, as a penalty the control effort (energy) required will be greater for higher q/r. It can also be seen that the response amplitude at the second resonance frequency is lesser than the first resonance frequency as expected.

The transient response of the intelligent composite plate under sinusoidal forcing function with excitation frequency of 52.5 Hz is plotted for the selected few q/r ratios in Figs 4-7. The excitation frequency is nearer to the fundamental frequency of the intelligent plate. It can be seen from Fig. 4 that the initial peak amplitude occurs at 0.6 s for q/r ratio of 2.0. Similarly from Figs 5 and 6 it can be noted that the initial peak amplitude for q/r ratios of 6 and 10 occurs at 0.55 s. and 0.5 s respectively. It is also to be noted that the subsequent occurrence of the maximum peak amplitude is lesser than the initial one. Fig. 7 shows that the initial peak amplitude and time at initial peak amplitude is reduced for q/r ratio of 100. Similarly, the transient response of the intelligent plate with sinusoidal forcing function with the excitation frequency of 82.5 Hz is plotted in Figs 8-11 for the q/r ratios of 2, 6, 10 and 100 respectively. It can be noted that the excitation frequency is nearer to the second resonance frequency. Unlike for the first resonance frequency the reduction in the peak amplitude is not very significant at the second resonance frequency.

Fig. 4— Transient response of an intelligent plate at first resonance frequency for q/r = 2

Indian Journal of Engineering & Materials Sciences Vol. 12, June 2005, pp. 196-206




203


Fig. 8— Transient response of an intelligent plate at second resonance frequency for q/r = 2



204

Fig. 10—Transient response of an intelligent plate at second resonance frequency for q/r = 10


Fig. 12—Effect of state and control parameters on the response of the intelligent plate (at 5, s)

The frequency response of the plate at 5 s after the feedback control invoked is given in Fig. 12 for various values of q/r ratio. It can be noted from this figure that, the maximum amplitude corresponding to the first resonance frequency is lower than the peak amplitude corresponding to the beginning of the excitation (Fig. 3) as expected. Effect of the thickness ratio on the response of the system The response of a square intelligent composite plate with different thickness ratios of 10, 20, and 50 is investigated. The substrate and the piezoelectric material properties used in the present analysis are same as given in the previous sub-section. The composite substrate lay up sequence is 0/90/0. The


205

Fig. 13—Transient response of a plate under pulse load with thickness ratio of 10

response of the plate under mechanical pulse load (as shown in Fig. 2a) with the intensity of unity is studied. The state space equation is solved separately for each thickness ratio. The equation is solved for q/r ratio of 6. The transient response of the plate with the thickness ratio of 10 is plotted in Fig. 13. It can be seen from the figure that the amplitude of the system under open loop condition is 1.9×10-9. After invoking the feedback control the amplitude is reduced to 1.038×10-9 and the time taken to reach this constant amplitude is 2380 μs. Similar transient response is plotted for the thickness ratio of 20 and 50 in Figs 14 and 15 respectively. It can be noted from these figures that in the uncontrolled response the peak amplitude for the thickness ratio of 20 and 50 are 2.15×10-8 and

7.2×10-7 respectively. This amplitude is reduced by nearly half of the corresponding values, once the feedback control is invoked, but the time taken to reach this half amplitude is increasing for the higher thickness ratio of the plates. This shows that time required to control the higher thickness ratio plate is more. Intelligent sandwich plate The intelligent square sandwich plate subjected to electrical pulse loading is considered for this example. Piezoelectric layer is bonded on the top and bottom of the sandwich plate, the mechanical and electrical properties of the face sheet and piezoelectric layer used in the present analysis are the same as given in

Fig. 14— Transient response of a plate under pulse load with thickness ratio of 20

Fig. 16—Transient response of the smart sandwich plate under electrical load


206

the previous section. The mechanical properties for the core material considered, using the standard notation, are: EL = ET = 0.2758 GN/m2, EZ = 3.4474 GN/m2, GLT = 0.1103 GN/m2, GLz = GTz = 0.4137 GN/m2 and νLT = νTT = νLZ = 0.25. The thickness ratio of the plate is 100. The thickness of each face sheet is H/10, where H is the total thickness of the plate. The thickness of PVDF layer is 40 μm. The first three frequencies of the sandwich plate are 53.7, 72.0 and 200.1 Hz respectively. The transient response of the uncontrolled vibration of the sandwich plate under unit electrical pulse load is plotted in Fig. 16. It can be seen from this figure that this plate is vibrating with 0.018 s time period. Response of the plate under feed-back control is also plotted in Fig 16. It can be

Fig. 15—Transient response of a plate under pulse load with thickness ratio of 50

noted from the figure that the plate amplitude is controlled in 0.5 s. Conclusions In this paper a simple analytical approach is used to study the active vibration control capability of the intelligent plate using LQR algorithm. To validate the formulation the transient response of the square isotropic plate is compared with the results available in the literature. The effect of state and control parameter ratios (q/r) on the active vibration control behaviour of the plate is also presented. The response amplitude at the first resonance is more than that of the second resonance at the initial instant of time and after the initial settling time the resonance amplitudes for both the first and second resonances are almost the same. The effect of a particular state and control

parameter on the different thickness ratio of the plate is also studied. The amplitude of the excited plate is reduced by nearly half, once the feedback control is invoked. Time required to control the vibration level in the higher thickness ratio plates is more. References

1 Tzou H S & Tseng C I, J Sound Vib, 138 (1990) 17-34. 2 Ghate A A & Seshu P, Int. Conf Smart Materials Structures,

ISSS-SPIE 2002, (2002), 558-565. 3 Crawley E F & De Luis J, AIAA J 25 (1987) 1371-1385. 4 Crawley E F & Anderson E H, J Int Mat Syst Struct, 1 (1990)

4-25, 5 Yoon H & Wahington G, J Smart Mater Struct, 7 (1998)

537-542. 6 Lee C K & Moon F C, J Acoust Soc Amer, 85 (1989) 2432-

2439. 7 Heylinger P, ASME J Appl Mech, 64 (1997) 299-306. 8 Ray M C, Bhattacharya R & Samanta B, AIAA J, 31 (1993)

1684-1691. 9 Onoda J & Haftka R T, AIAA J, 28 (1987) 1133 - 1138.

10 Sunar M & Rao S S, AIAA J, 31 (1993) 714-720. 11 Tsujioka K, Kajwara I & Nagamatsu A, Proc 36th Struct Dyn

Mat Conf, (1995) 2983- 2991. 12 Tiersten H F, Linear piezoelectric plate vibration, (Plenum

Press: New York), 1969. 13 Cady W G, Piezoelectricity (McGrawHill: New York), 1946. 14 Anon, User’s Guide for MATLAB, The Math Works Inc.,

Natick, 1993. 15 Reddy J N, Int J Numer Methods Eng, 19 (1987) 237-255. Appendix A The nonzero terms of the symmetric stiffness matrix [K] in Eq. (9) is as follows:

2 11 331 2 2u

A AKa b

⎛ ⎞= π +⎜ ⎟⎝ ⎠

, ( )2

2 21 33uK A Aabπ

= + ,

2

5 11uK Baπ⎛ ⎞= ⎜ ⎟

⎝ ⎠ 6 31

suK d

aπ⎛ ⎞= ⎜ ⎟

⎝ ⎠,

7 31a

uK daπ⎛ ⎞= ⎜ ⎟

⎝ ⎠, 2 33 22

2 2 2vA AKa b

⎛ ⎞= π +⎜ ⎟⎝ ⎠

,

2

4 22vK Baπ⎛ ⎞= ⎜ ⎟

⎝ ⎠, 5 31

svK d

bπ⎛ ⎞= ⎜ ⎟

⎝ ⎠,

6 31a

vK dbπ⎛ ⎞= ⎜ ⎟

⎝ ⎠ 2 44 55

3 2 2wA AKa b

⎛ ⎞= π +⎜ ⎟⎝ ⎠

,

4 55wK Abπ⎛ ⎞= ⎜ ⎟

⎝ ⎠, 5 44wK A

aπ⎛ ⎞= ⎜ ⎟

⎝ ⎠,

2 22 334 552 2x

D DK Ab a

⎛ ⎞= π + +⎜ ⎟⎝ ⎠

, ( )2

5 21 33xK D Dab

⎛ ⎞π= +⎜ ⎟⎝ ⎠

,

( )16 312

i i sx

z zK d

b−+ π⎛ ⎞= ⎜ ⎟

⎝ ⎠, ( )1

7 312i i a

x

z zK d

b−+ π⎛ ⎞= ⎜ ⎟

⎝ ⎠,

2 11 335 442 2y

D DK Aa b

⎛ ⎞= π + +⎜ ⎟⎝ ⎠

, ( )16 312

i i sy

z zK d

a−+ π⎛ ⎞= ⎜ ⎟

⎝ ⎠,


207

( )17 312

i i ay

z zK d

a−+ π⎛ ⎞= ⎜ ⎟

⎝ ⎠,

( )( ) 2 2

116 1

1 3

si is

si i

z zK

z z a b−

−

⎛ ⎞−β π π⎛ ⎞ ⎛ ⎞= + β +⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟− ⎝ ⎠ ⎝ ⎠⎝ ⎠,

( )( ) 2 2

117 1

1 3

ai ia

ai i

z zK

z z a b−

−

⎛ ⎞−β π π⎛ ⎞ ⎛ ⎞= + β +⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟− ⎝ ⎠ ⎝ ⎠⎝ ⎠

In the above equations Aij, Bij and Dij are the well-known in-plane, bending-extension coupling and bending stiffness coefficients for the composite plate respectively.

The nonzero terms of the mass matrix [M] in Eq. (14) is as follows:

( )1 2 3 11

n

u u u i i ii

M M M z z +=

= = = ρ −∑

( )3 34 5 1

1 3

ni

u u i ii

M M z z +=

ρ= = −∑

In the above equation ρ is the density and n is the total number of layers.

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Simple analytical solution for active vibration control of ...nopr.niscair.res.in/bitstream/123456789/8434/1... · Simple analytical solution for active vibration control of intelligent