Thesis for Dynamic stability of laminated cylindrical panel with piezoelectric layer

8/19/2019 Thesis for Dynamic stability of laminated cylindrical panel with piezoelectric layer

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CONTENTS

1. Abstract 5

2. Introduction 6-7

3. Literature review 8-11

4. Objective of present work 12

5. Formulation of problem 13-18

6. Finite element formulation 19-24

7. Dynamic stability problem 25-29

8. Results and discussion 30-41

9. Conclusions 42

10. Scope of future work 43

11. References 44-46


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LIST OF SYMBOLS:

u,v,w displacements

, Rotational displacements , εi stress and strain componentsEi electric field components

Di electric displacement components

Qij plane-stress reduced elastic constants

eij piezoelectric constants

{ } N in-plane force resultant vector

D laminate elastic stiffness matrix

T,U kinetic energy and potential energy

Wc work of the conservative part of thrust P

M mass matrix

Ni Shape function of node i

K stiffness matrix

K G geometric stiffness matrixGc gain of the current amplifier

Gi gain to provide feedback control

G control gain

ω natural frequency

Ω non-dimensional frequency

α static load factor

β pulsating load factor

ρ density


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LIST OF FIGURES:

1. Geometry of a laminated panel.

2. Panel configuration.

3. Piezolaminated composite cylindrical panel.

4. Scheme for partial edge loading.

5. Regions of resonances of anti-symmetric angle-ply cylindrical panel

under partial edge loading at both ends (a/b=1; b/h=100; c/b=1)

6, Regions of resonances of anti-symmetric angle-ply cylindrical panel

under partial edge loading at both ends (a/b=1; b/h=100; c/b=0.75)

7. Regions resonance of anti-symmetric angle-ply cylindrical panel

subjected to partial edge loading at both ends (a/b=1; b/h=100;

c/b=0.5)

8. Regions simple resonance (2ω3) of anti-symmetric angle-plycylindrical panel with different b/R ratios (a/b=1; b/h=100; c/b=1)

9. Regions combination resonance (ω1+ω2) of anti-symmetric angle-ply

cylindrical panel with different no. of layers (a/b=1; b/h=100; c/b=1)

10. Regions simple resonance (2ω2) of anti-symmetric angle-ply

cylindrical panel with different ply-orientation (a/b=1; b/h=100; c/b=1)




cylindrical panel for different static load factors (a/b=1; b/h=100;

c/b=1)

13. Regions simple resonance (2ω2) of anti-symmetric angle-ply

cylindrical panel for different static load factors (a/b=1; b/h=100;c/b=1)


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14. Unstable zones at simple resonances (2 ω1) with piezoelectric layer

with different feedback gains.(clamped clamped boundary condition)

15. Unstable zones at combination resonances with piezoelectric layer with

different feedback gains.(clamped clamped boundary condition)

16. Mode Shapes for clamped-clamped boundary condition.

17. Unstable zones at simple resonances with piezoelectric layer with

different feedback gains.(Simply Supported SS-1 boundary condition)

18. Unstable zones at combination resonance (ω1+ω6) with piezoelectric

layer with different feedback gains.(Simply Supported SS-1 boundary

condition)

19. Mode Shapes for simply supported SS-1 boundary condition.


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1. ABSTRACT:

This work is mainly focused on dynamic stability analysis of laminated

cylindrical panel with and without piezoelectric layers and the control of instability

regions by incorporating the negative velocity feedback into the system using the

piezoelectric layers as sensor and actuator. Equation of motion is derived based on

first order shear deformation theory by using finite element method. The top and

bottom piezoelectric layers are utilized as sensor and actuator for the active control of

the structure. The effect of feedback control gain on angle-ply laminates is also

studied. Method of multiple scales is used to study the behaviour of parametrically

excited system. . Numerical results are obtained for various combinations of the

system parameters such as the radius of curvature, number of layers, ply orientation of

the laminas and the effects of these on the zones of parametric instability are studied.

The results show that occurrence of resonances under dynamic axial loading. The

effect of feedback shows that there is a critical value of load below which the system

will not become dynamically unstable.


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2. INTRODUCTION:

The fibre-reinforced laminate composite plates play an important role in

modern industry for its high strength-to-weight ratio. The idea of applying smart

materials to mechanical and structural systems has been used in various disciplines

because of their coupled mechanical and electrical properties. These properties make

them well suited for use as sensors and actuators in active control of the structures.

These materials include piezoelectric polymers and ceramics; shape memory alloys,

electro rheological fluids. Among these, piezoelectric materials have been most

preferred because of its advantages like it is inexpensive; light weighted, and can be

easily shaped and bonded to surfaces or embedded into structures. Another main

advantage of these materials is the direct and converse piezoelectric effects.

Aircraft structures and spacecraft structures consist of a large number of shell

type elements which are subjected to different types of in-plane as well as out-of-

plane loads. Many other structures like bridges, ships, vehicles etc. also uses shell

type elements. These elements being thin are prone to buckling and dynamic

instabilities. The dynamic instability may result in large deflection or resonance

resulting complete failure of the structure.

The plates subjected to in-plane dynamic (periodic) forces experience resonant

transverse vibrations under certain combination of the natural frequency of transverse

vibration, the frequency of the in-plane forcing function and the magnitude of the in-

plane load. This phenomenon is called dynamic instability or parametric instability or

parametric resonance. The spectrum of the values of parameters causing unstable

motion is called the region of dynamic instability or parametric resonance. If the

frequency of in-plane forcing function at parametric resonance has relation with only

one natural frequency of transverse vibration of the plate, the resulting resonance is

called simple resonance; otherwise it is called combination resonance.


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Basics of composite Materials:

The word composite in the term composite material signifies that two or more

materials are combined on a microscopic scale to form a useful third material. Thekey is the microscopic examination of a material where in the components can be

identified by the naked eye. Different materials can be combined on a microscopic

scale, such as in alloying of metals, but the resulting materials is, for all practical

purposes, macroscopically homogeneous, i.e. the components cannot be distinguished

by the naked eye and essentially act together. The advantage of composite materials is

that, if well designed, they usually exhibit the best qualities of their components or

constituents and often qualities that neither constituent possesses.

Laminated composite materials consist of layers of at least two different

materials that are bonded together. Lamination is used to combine the best aspects of

the constituent layers and bonding material in order to achieve a more useful material.

The properties that can be emphasized by lamination are stiffness, strength, low

weight, corrosion resistance, wear resistance, beauty or attractiveness, thermal

insulation, acoustical insulation etc.


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3. LITERATURE REVIEW:

The parametric instability of elastic structures like plates, shells, columns has

been investigated by Bolotin [1], where instability zones were constructed by Fourier

analysis. In this study the method used is to first obtain the Mathieu-Hill equations

after which the dynamic stability is investigated.

P K Datta et al.[2] studied the dynamic stability of doubly curved laminated

panels. He studied the problem of the occurrence of combination resonances in

contrast to simple resonances in parametrically excited anti-symmetric angle-ply and

symmetric cross-ply laminated composite doubly curved panels with central circular

cutout. The method of multiple scales is used to obtain analytical expressions for the

simple and combination resonance instability regions. The method of multiple scales

is used to obtain analytical expressions for the simple and combination resonance

instability regions.

K.Chandrasekhara and N.Agarwal [3] studied the active vibration control of

laminated composite plates using piezoelectric sensors and actuators.

Reddy [4] has done various formulations of laminated composite plates with

integrated sensors and actuators. He has studied finite element model for the active

control of laminated composite plate containing piezoelectric sensor and actuator at

its top and bottom surfaces. Static and dynamic analysis of cantilever beam is carried

out.

Liu et al. [5] and Song Cen et al. [6] studied the finite element modeling of

piezoelectric laminates. Cen et al. developed 4-node quadrilateral finite element for

formulating their problem. The element, denoted as CTMQE, is free of shear locking

and exhibits excellent capability in the analysis of thin piezoelectric laminated

composite plates.

K M Liew et al. [7] studied active control of laminated composite plates with

piezoelectric patches used as sensor and actuator. They formulated using mesh-free


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method i.e., the element-free Galerkin (EFG) method. The study was based on first-

order shear deformation Theory (FSDT) and principle of virtual displacement.

Arcangelo Messina et al. [8] studied the free vibration of open, laminated

composite, circular cylindrical panels having a rectangular plan-form and all their

edges free of external tractions. The analysis is based on the application of the Ritz

approach on the energy functional of the Love-type version of a uniÞed shear

deformable shell theory.

T. Y. Ng et al. [9] studied dynamic stability of simply supported isotropic

cylindrical panels under combined static and periodic axial forces. An extension of

Donnell’s shell theory to a first order shear deformation theory is used and a system

of Mathieu-Hill equations are obtained via a normal mode expansion and the

parametric resonance response was analyzed using Bolotin’s method.

Hiroyuki Matsunaga [10] analysed the buckling stresses of cross-ply

laminated composite shallow shells. The natural frequencies are calculated by taking

into account the effects of transverse shear and normal deformations and rotary

inertia. Three types of simply supported shallow shells with positive, zero and

negative Gaussian curvature are considered. Numerical results are compared with

those of the published three-dimensional models.

The free vibration of laminated composite cylindrical panels is solved by the

meshfree approach by X. Zhao [11]. This study examines in detail the effects of

different boundary conditions on the frequency characteristics ofthe cylindrical

panels. The effects of the curvature of the cylindrical panels as well as the lamination

scheme, on the frequencies of the panels, are also investigated.

J H Kim et al. [12] studied the effect of piezoelectric damping layers on

dynamic instability of plate. The structure is damped with piezoelectric layers and

adopted a control mechanism for suppressing the vibration. The piezoelectric layers

are embedded on the top and bottom surfaces. The structure model is based on the

first-order shear deformation plate theory, and the finite element method is applied in

the numerical analysis.


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M.S. Qatu, A. W. Leissa [13] studied vibration frequencies of completely free

laminated plates and shallow shells. The Ritz method with algebraic polynomial

displacement functions is used. Convergence studies are made and accurate results are

obtained by using 192 displacement terms for spherical, circular cylindrical,

hyperbolic paraboloidal shallow shells and 64 terms for plates. Results are compared

with those obtained experimentally and by finite element methods.

A. Alibeigloo et al. [14] studied the solution for static analysis of cross-ply

rectangular plate imbedded in piezoelectric layers using differential quadrature

method (DQM) and Fourier series approach. Applying the DQM to the governing

differential equations new state equations about state variables at discrete points are

derived. The stress, displacement and electric potential distributions are obtained by

solving these state equations. Both the direct and the inverse piezoelectric effects are

investigated and the influence of piezoelectric layers on the mechanical behaviour of

plate is studied.

S. Wanga, D.J. Dawe [15] studied the dynamic instability analysis of

composite laminated rectangular plates and prismatic plate structures, based on the

use of first-order shear deformation plate theory (SDPT). The equations of motion of

a structure are established by using Lagrange’s formulation and they are a set of

coupled Mathieu equations. The boundary parametric resonance frequencies of the

motion are determined by using the method suggested by Bolotin.

Nayfeh [16] used the method of multiple scales to analyze the response of

two-degree-of-freedom systems to multi-frequency parametric excitations. Ayech

Benjeddou and Deu [17] present a two-dimensional (2D) closed-form solution for the

free-vibrations analysis of simply-supported piezoelectric sandwich plates. The

formulation considers full layerwise first-order shear deformation theory and through-

thickness quadratic electric potential. Its independent mechanical and electric

variables are decomposed using Fourier series expansions, then substituted in the

derived mechanical and electric 2D equations of motion.


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Guanghui Qing, Jiajun Qiu [18] established a modified mixed variational

principle for piezoelectric materials and the state-vector equation of piezoelectric

plates is deduced directly from the principle. the exact solution of the state-vector

equation is simply given, and based on the semi-analytical solution of the state-vector

equation, a realistic mathematical model is proposed for static analysis of a hybrid

laminate and dynamic analysis of a clamped aluminum plate with piezoelectric

patches. Both the plate and patches are considered as two three-dimensional

piezoelectric bodies, but the same linear quadrilateral element is used to discretize the

plate and patches.

K.Y. Lam, Wu Qian [19] studied analytical solutions for the vibrations of

thick symmetric angle-ply laminated composite cylindrical shells using the first-order

shear deformation theory. The frequency characteristics for thick symmetric angle-ply

laminated composite cylindrical shells with different H/R and L/R ratios are studied in

comparison with those of symmetric cross-ply laminates.

V. Balamurugan et al. [20] studied the mechanics for the coupled analysis of

piezolaminated plate and Piezolaminated curvilinear shell structures and their

vibration control performance are considered. A plate/shell structure with thin PZT

piezoceramic layers embedded on top and bottom surfaces to act as distributed sensor

and actuator is considered. Active vibration control performance of plates and shells

with distributed piezoelectric sensors and actuators have been studied.

Chien-Chang Lin et al. [21] studied vibration control of beam-plates with

bonded piezoelectric sensors and actuators. Basic equations for piezoelectric sensors

and actuators are presented. The equation of motion for a beam-plate structure bonded

with pairs of piezoelectric sensors or actuators is derived by using the Hamilton's

principle, and a fnite element method is used for the analysis.


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4. OBJECTIVE OF PRESENT WORK:

1. To study the zones of dynamic stability analysis of laminated cylindrical panel

with and without piezoelectric layers.

2. To study the effect of feedback control gain on angle-ply laminates.

3. To study the effect of using viscoelastic layer as the core layer on instability

zones.

4. The effects of material properties, radius to thickness ratio etc are also studied.


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5. FORMULATION OF THE PROBLEM:

A Cylindrical panel of dimensions a and b and of total thickness h composed

of N orthotropic layers with the principal material coordinates (1 , 2 , 3 ) of thek th lamina oriented at an angle to the laminate coordinate x, is considered (seefig.1). The geometry of the composite laminated panel is as shown in fig.2 for N-

layered laminated case.

Figure 1. Geometry of a laminated panel

Figure 2. Panel configuration

A piezoelectric layer, with thickness t p is, now bonded at the top and bottom

surfaces of the panel. The other dimensions of the piezoelectric layers are considered

as same as those of the panel.


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Considering xy-plane of the problem to be in the underformed mid-plane of

the laminate and z-axis is taken positive downward from the mid-plane. The k th layer

of panel is located between the points z=zk and z=z k+1 in the thickness direction as

shown in figure 1.

Before formulation, we make certain assumptions as stated below:

• The layers are assumed to be perfectly bonded together.

• The material of each layer is assumed to be linearly elastic and has

two planes of material symmetry (i.e., orthotropic)

• Each layer is assumed to be of uniform thickness.

• The strains and displacements are small.

Strain-Displacement Relationship:

Based on first order shear deformation theory the displacement field u, v, and

w at a point ( x, y and z) is [Reddy text book]

( ) ( )

( ) ( )

( ) ( )

0

0

0

, , , , , ( , , )

, , , , , ( , , )

, , , , ,

x

y

u x y z t u x y t z x y t

v x y z t v x y t z x y t

w x y z t w x y t

φ

φ

= +

= +

=

(1)

u0, v

0, and w0 are the in-plane and transverse displacements of a point ( x, y) on the mid

plane and x

φ , y

φ are the rotations of the normal to the midplane about the y and x axes

respectively, t is time.

The linear strains associated with displacements based on the shear

deformable version of the Sanders shell theory are as follows

( )( )

0

0

0 12

x

y

y x

u xx xx xx x x

v w yy yy yy y R y

u v v u xy xy xy y x y x R x y

w v yz y y R

w xz x x

z z

z z

z z

φ

φ

φ φ

ε ε κ

ε ε κ

γ γ κ

γ φ

γ φ

∂∂∂ ∂

∂∂∂ ∂

∂∂∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂

∂∂

∂∂

= + = +

= + = + +

= + = + + + + −

= + −

= +

(2)


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where 0 0 0( , , )

xx yy xyε ε γ , ( , , )

xx yy xyκ κ κ are membrane strains and flexural strains, ( , ) yz xzγ γ

are transverse strains.

The non linear strain-displacement relations based on sanders non-linear

theory of panel are expressed as

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( )

222 2 2 21 1 1 12 2 2 2

222 2 221 1 1 1

2 2 2 2

2

y x

y x

y y x x

u v w xxn x x x x x

u v w yyn y y y y y

u u v v w w v xyn x y x y x x R x y x y

z

z

z

φ φ

φ φ

φ φ φ φ

ε

ε

γ

∂∂∂ ∂ ∂∂ ∂ ∂ ∂ ∂

∂∂∂ ∂ ∂∂ ∂ ∂ ∂ ∂

∂ ∂∂ ∂∂ ∂ ∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

= + + + +

= + + + +

= + + − + +

(3)

In vector form

{ } { }T

xx yy xy yz zxε ε ε γ γ γ = (4)

{ }0 0 0{ }T

xx yy xy xx yy xy yz xzε ε ε γ κ κ κ γ γ = (5)

{ }T

nl xxn yyn xynε ε ε γ = (6)

where { }ε is generalized strain vector corresponding to midplane.

Constitutive Relations:

The stress-strain relation for a lamina about any axes is given by:

=

⎣

�11 �12 �16 0 0�12 �22 �26 0 0

�

16

�

26

�

66 0 0

0 0 0

�44 �450 0 0 �45 �55⎦

⎩

∈∈∈∈∈⎭

where,

�11 = 114 + 2 (12 + 2 66)22 + 114 �12 = (11 + 22 − 466)22 + 12(4 + 4) �22 = 114 + 2 (12 + 2 66)22 + 224

�16 = (11 − 12 − 266)3 + (12 − 22 + 2 66)

3


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�26 = (11 − 12 − 266)3 + (12 − 22 + 2 66)3 �66 = (11 + 22 − 212 − 266)22 + 66(4 + 4)

The elastic stiffness matrix corresponding to transverse shear deformation is

derived as follows:

�44 = 132 + 232 �55 = 232 + 132 �45 = (13 − 23)

Where, m=cos

and n=sin

11 = 11(1−1221) , 12 = 1121(1−1221) , 21 = 2212(1−1221) ,22 = 22(1−1221) , 16 = 12 , 44 = 13 and 55 = 23

Considering the piezoelectric composite cylindrical panel made up of N L

number of layers as shown in Figure 1. The linear constitutive relations for coupled

electro elastic behaviour of the k th

lamina are expressed with respect to the laminate

coordinate system ( x, y and z) by the direct and converse piezoelectric equations

respectively as [4]:

14 15 11 12 1

24 25 12 22 2

31 32 36 33 3

0 0 0 0

0 0 0 0

0 0 0 0

x xx

yz

y yy

zx k z xyk k k k k k

D e e E

D e e E

D e e e E

ε γ

ε γ

γ

∈ ∈ = + + ∈ ∈ ∈

(7)

3111 12 16

3212 22 26

3616 26 66

14 2444 45

15 2545 55

0 00 00 00 0

0 00 0

00 0 0

00 0 0

xx xx

yy yy x

xy xy y

z yz yz

xz xz k k k k

eQ Q Qe E Q Q Q

e E Q Q Q

e e E Q Q

e eQ Q

σ ε σ ε

σ γ

σ γ

σ γ

= −

k

(8)

where, ( ), ,ij ij ijQ e ∈ are the plane-stress reduced elastic constants, the piezoelectric

constants and the permittivity coefficients, respectively, of the k th lamina in its


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laminate coordinate system and ( ), , , ,ij ij ij i i E Dσ ε γ are the stresses, normal strains,

shear strains, electric field, and electric displacement components, respectively,

referred to the laminate coordinate system. In lamina coordinates piezoelectric

constant matrix has been expressed as

[ ] [ ]e d Q= (9)

The piezoelectric strain coefficient matrix

15

24

31 32 36

0 0 0 0

0 0 0 0

0 0

d

d d

d d d

=

(10)

Integrating equation (8) through the panel thickness leads to following

laminate constitutive relations

{ } { } { } p N D N ε = − (11)

where { } N is the in-plane force resultant and total moment resultant vector, D is

the laminate elastic stiffness matrix

{ } { }, , , , , , ,T

x y xy x y xy y x N N N N M M M Q Q= (12)

[ ] [ ] 0

[ ] [ ] 0

0 0 [ ]s

A B

D B D

A

=

(13)

The laminate elastic stiffness coefficients in the above equation are defines as follows

[ ] [ ] [ ]( ) ( ) ( )/2 2

/2, , 1, , , 1, 2,6

h

ijh

A B D Q z z dz i j−

= =∫ (14)

[ ] /2

/2( , 4, 5)

h

s i j ijh

A k k Q dz i j−

= =∫ (15)

with ik and jk being the shear correction factors.


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Actuator Equation:

The force and moment resultant vector { } p N due to piezoelectric actuator

{ }PA

p PA

PA

N

N M

Q

=

(16)

{ } ( ) ( )11

[ ] , 1,2,6 L N T

PA ij k x y z k k k k

N e E E E h h i j−=

= − =∑

(17)

{ } ( ) ( )2 2 11

1[ ] , 1,2,6

2

L N T

PA ij k x y z k k k k

M e E E E h h i j−=

= − =∑ (18)

{ } ( )24 11 15

0

0

L N T

PA x y k k k k k

eQ E E h h

e −

=

= −

∑ (19)

Electric field intensity across each lamina is

{ } { }0 0 / x y z k Ak E E E V h= (20)

V k is the constant voltage applied across the k th layer and h A is the thickness of the

layer.


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6. FINITE ELEMENT FORMULATION:

In the present analysis an isoparametric finite element model with five degrees

of freedom at each node is used for studying the dynamic behaviour of the cylindrical

panel. The same kind of interpolation is assumed for all the variables over each

element. Accordingly,

( ){ } ( ) ( ){ }1

, , ,n N

e e e

i i

i

u x y t N x y u t =

= ∑ (21)

{ } { }0 0 0 T

e

i i i i xi yiu u v w φ φ = (22)

[ ]e ei i N N I = (23)

where { } [ ]( ), , ,e en i i N u N I are number of nodes, displacement vector at node i,

element shape functions and fifth order unit matrix respectively. The superscript e

denotes the parameter at element level.

Using equation (21), the generalised strain vector shown in equation (4) is expressed

as

{ } { } { } { }{ }1 2 1 2... ...n nT

T T T e e e e e e e

N N B B B u u uε = (24)

where

( ) ( )

1

1 12 2

1

0 0 0 0

0 0 0

0 0 0

0 0 0 0

0 0 0 0

0

0 0

0 0 0

i

i

i i

i

i

i i i i

i

i

N

x

N

y R

N N

y x

N

xe

N i

y

N N N N

R y R x y x

N

R y

N

x

N i

N i

B

∂

∂

∂

∂

∂ ∂

∂ ∂

∂

∂

∂

∂

∂ ∂ ∂ ∂−∂ ∂ ∂ ∂

∂−∂

∂

∂

=

(25)


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The kinetic energy at the element level is defined as

{ } { }12e

T e

vT u u dv ρ = ∫ (26)

The strain energy due to linear and nonlinear strains is defined as

{ } { } { } { }01 12 2e e

T T e

nlv v

U dv dvε σ ε σ = +∫ ∫ (27)

where { } { }0 0 0 0 x y xyσ σ σ τ = are the in plane stresses due to external loading.

The dynamic equations of a laminated composite cylindrical panel is derived by using

Hamilton’s principle

( )2

1

0t

ct

T U W dt δ δ δ − + =∫ (28)

where T is the kinetic energy, U is the strain energy and W c is the work done by the

external forces.

Parametric Loading:

Substituting equations (26), (27) into equation (28) and using (6), (21), and (24) the

dynamic equation of motion for parametric loading is as follows

[ ]{ } [ ]{ } [ ]{ } { }e e e e e e eG p M u K u P K u F + − = (29)

Here, edge load

( ) coss d P t P P t = + Ω

(30)

Substituting equations (26), (27) into equation (28) and using (6), (21), and (24) the

dynamic equation of motion for follower loading is derived as follows

[ ]{ } [ ]{ } [ ]{ } { }e e e e e e eG p

M u K u P K u F + − =

(31)


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where,

1 1

1 1[ ] [ ] [ ][ ]d et[ ]e e T e M N M N J d d ξ η

+ +

− −= ∫ ∫ (32)

1 1

1 1[ ] [ ] [ ][ ]d et[ ]e e T e

GK G S G J d d ξ η + +

− −= ∫ ∫ (33)

1 2[ ] [ ] [ ]...[ ]ee e e e

N G G G G = (34)

1 1

1 1[ ] [ ] [ ][ ]d et[ ]e e T eK B D B J d d ξ η

+ +

− −= ∫ ∫ (35)

{ } { }1 1

1 1[ ] det[ ]

T e e e

p A pF K V B N J d d ξ η + +

− − = = ∫ ∫ (36)

1 2

1 2

1

2 3

32

0 0 0

0 0 0

0 0 0 0[ ]

0 0 0

0 0 0

I I

I I

I M

I I

I I

=

(37)

( ) ( )1

2 311 2 3 1 1 2 2

1

2, , 1, , , ,

Lk

k

N z

k

zk

I I I I I z z dz I I I I

R R ρ

−=

= = + = +∑∫ (38)

2

1

0 01

1 2 1120 01

2

2

0 0 0 0[ ]

0 0 0 0[ ]

0 0 0 0[ ] , [ ] an d[ ] [ ][ ]

0 0 0 0[ ]

0 0 0 0 [ ]

x xy h

xy y

S

S h h

S S S S S h h

S

S

σ τ

τ σ

= = =

(39)


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0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0[ ]

0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

i

i

i

i

i

i i

i

i

i

i

N

x

N

y

N

x

N

y

N

xe

i N N

R y

N

x

N

y

N

x

N

y

G

∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

− ∂

∂

∂

∂

∂

∂

∂

∂

∂

∂

= (40)

Sensor Equation:

As the electrodes are placed on the transverse surfaces of panel with the poling

direction z and no charge is externally applied to the sensor layer then electric field

displacement of k th layer in thickness direction derived from equation (7) is

{ }{ } { }{ }31 32 36{ } 0 0T T

z D e e e eε ε = = (41)

Figure 3. Piezolaminated composite cylindrical panel


23/46

23

Now, the closed circuit charge measured through the surface electrodes of a k th

layer is ,

1

12

( ) ( )

( )k k

z z

Se z z Se z z

q t D ds D ds−

= =

= +

∫ ∫ (42)

wheree

S is the effective surface electrode of piezoelectric patch, in the present

analysis it is considered that the whole piezoelectric lamina serves as an effective

electrode.

Therefore the total charge developed by all elements on the sensor layer is

1 1

1 21 1

1

( ) { }[[ ] [ ].....[ ]]det[ ] { }s

n

N

e e e e

N

e

q t e H H H J d d uξ η + +

− −=

= ∑∫ ∫ (43)

1

( ) [ ]{ }s N

e e

S

e

q t K u=

= ∑ (44)

where N s denote number of elements and

( ) ( )

0

0

0 0

1

2 2

1

0 0 0

0 0

[ ] 1 1 0

0 0

0 0 0

i i

i i

i i i i

i

i

N z N

x x

N z N

y R y

z N z N N N e

i R y R x y x

N

i R y

N

i x

H

N

N

∂ ∂

∂ ∂

∂ ∂

∂ ∂

∂ ∂ ∂ ∂

∂ ∂ ∂ ∂

∂−∂

∂

∂

= − +

(45)

10 12 ( )k k z z z −= + (46)

The voltage applied to the actuator is expressed as

( )[ ]{ }

S

dq t V G K u

dt = = (47)


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24

where G is the gain provided by velocity feedback control.

G=Gi Gc

[

] =

1

2

∫[

3][

]

(=)+

∫[

3][

]

(=+1) (48)

Assembling the element equations and using equations (37) and (47), global equations

can be written as

[ ]{ } [ ]{ } [ ]{ } [ ]{ } 0G M u C u K u P K u+ + − = (49)

for parametric and follower loading respectively.

where [ ] [ ][ ][ ] A S

C K G K = and {u} is the global displacement vector.


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25

7. DYNAMIC STABILITY PROBLEM:

The application of the finite element method to a plate subjected to in-plan

loading yields the following equilibrium equation (49):

[]{̈} + []{̇} + []{} − []{} = 0 where, [M], [C], [K] and [K G ] are mass, damping , elastic and stress stiffness

matrices respectively. All are the symmetric square matrices of order N, the number

of degree of freedom of the system.{u} is the nodal displacement vector of the N and

P is the magnitude of the total edge load on each of the two opposite edges.

For buckling problem, equation (17) reduces to:

[]{} − []{} = 0 where, is the buckling load and {u} gives the mode shapes of buckling.

For free vibration problem without damping equation (17) can be expressed

as:

[] − []{} − 2[]{} = 0

where, is Natural frequency of vibration and {q} gives the normal modes ofvibrations.

Above both equations are the eigenvalue problems. Solutions of these

equations give 2 respectively, and corresponding eigenvectors {d}.The edge load is periodic P(t) and is expressed in the form :

() = + Ω where, is the static portion of P , is the amplitude of the dynamic componentof P and Ω is the angular frequency of dynamic loading.


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26

P(t) can also be expressed as:

() = + Ω (50)where =/ and = / are termed as static and dynamic load factors .Substituting equation (50) in to (49) lead to

[ ]{ } [ ][ ][ ]{ } [ ]{ } [ ]{ } [ ]{ }cos 0a s cr G cr G M u K G K u K u P K u P t K uα β − + − − Ω = (51)

Applying the modal transformation, (51) can be converted into a set of

coupled Mathieu-Hill equations

̈ + ̂ ̇ + [Λ]{} + cos Ω ̂{} = 0 (52)where,

[ ] [ ] [ ] [ ] [ ]

[ ] [ ][ ] [ ] [ ][ ]

, / 2T

cr G

T T

Gcr

K P K and

S K and C C P

α φ ε β

φ φ

φ

φ φ ∧ ∧

Λ = − =

= − =

In component form,

2

1 1

ˆ2 cos 0 M M

m mn n m m mn n

n n

C t S ζ µ ξ ω ζ ε ζ = =

+ + + Ω =∑ ∑ (53)

Using method of multiple scales by first order expansion of µ and ε ,

0 0 1 2 1 0 1 2 2 0 1 2( , , ) ( , , ) ( , , ) .......m m m mT T T T T T T T T ζ ζ µζ εζ = + + + (54)

where, 0T =t, 1 2andT t T t µ ε = = are the so-called fast-scale and slow scales,

respectively.

0 1 2 D .... D D Dµ ε = + + + (55)2 2

0 0 1 0 2=D 2 D 2 .... D D D Dµ ε + + + (56)


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27

Substituting equations (54),(55) and (56) into (53) and equating according power of ε

we get,

2 2

0 0 0 0m m m D ζ ω ζ + = (57)

2 2

0 1 1 0 1 0 0 0

1

2D D M

m m m m mn n

n

D D C ζ ω ζ ζ ζ =

+ = − + ∑ (58)

2 2

0 2 2 0 2 0 0 0 0ˆ2D [exp( ) exp( )]m m m m ms s

s

D D S i T i T ζ ω ζ ζ ζ + = − − Ω + − Ω∑

(59)

By solving equation (57)

0 1 2 0 1 2 0( , ) exp( ) ( , ) exp( )m m m m m A T T i T A T T i T ζ ω ω = + − (60)

m0Substitute ζ expression in equation (58)

2 2

0 1 1 0 1 0 0 0

1

2 exp( ) D exp( ) . M

m m m m m mn n n

n

D D D A i T C A i T c cζ ω ζ ω ω =

+ = − + +∑ (61)

m0Substitute ζ expression in equation (59)

2 2

0 1 1 0 1 0 0 0

1

2 exp( ) D exp( ) . M

m m m m m mn n n

n

D D D A i T C A i T c cζ ω ζ ω ω =

+ = − + +∑ (62)

Considering the case Ω near m nω ω +

The nearness of Ω to m nω ω + can be expressed by introducing the detuning

parameter σ ,

m nω ω εσ Ω = + + (63)

Removing secular terms from the equation (60)

2 2ˆ2 exp( ) 0

m m mn ni D A S A i T ω σ − + = (64)

In the same way considering 1nζ in place of 1mζ we get,

2 2ˆ2 exp( ) 0

n n nm mi D A S A i T ω σ − + = (65)

1 2exp( )exp( )m m m m A i T ib T α β = (66)

where, 2 2( ) exp( )m m ma T ib T α = and mα is constant.

Similarly, n 2 1A ( )exp( )n na T ib T = (67)


28/46

28

mA , nA expressions substitute in the equations (65) and solving we get ,

1 1ˆ{ 2 exp( )} { exp[ ( ) ]}

m m m m mn n nb ib T a S i b T aω σ − + − =0 (68)

1 1ˆ{ exp[ ( ) ]} { 2 exp( )}nm m m n n n nS i b T a b ib T aσ ω − + − =0 (69)

From the above two equations for non-trivial solution we get,

21 ( )2

mb Lσ σ = + −

21 ( )

2nb Lσ σ = − −

where,

ˆ ˆmn nm

m n

S S L

ω ω =

But,

2

1 2 1 2

1exp( )exp( ) exp{ [ ( ) ]}

2 2m m m m m mm

i A i T ib T i C T L T α β α σ σ = == − + + − (70)

If

2

0 Lσ − ≤

,20

0

1 1exp( ) exp{[ ( )] }

2 2 2m m mm

i T A C L T

σε α µ ε σ = − −

m A is bounded if,

21 1 ( ) 02 2

mmC Lµ ε σ − − ≤ (71)

After solving equation (71),

2

mmC

L µ

σ ε

= ± −

(72)


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29

σ value substituted in equation (63)

2

mmm n

C L

µ ω ω ε

ε

Ω = + ± −

if 2 0 Lσ − ≤ (73)

2

mmm n

C L

µ ω ω ε

ε

Ω = + ± − −

if 2 0 Lσ − ≥ (74)

In the above equation, m=n corresponds to simple resonance

m≠n corresponds to combination resonance


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30

8. RESULTS AND DISCUSSION:

The basic configuration of the problem considered here is laminated

composite cylindrical panel with piezoelectric layers subjected to compressive in-

plane edge loading. The following dimensions are considered in the present analysis

[a/b=1; b/h=100; b/R=0.2 as shown in figure1.].

The following material properties as per the composite terminology are used,

Composite material :

E11=140 GPa, E11/E22=40, G12=2.1 GPa,

G12 =G13= 0.6E2, G23= 0.5E2, 12=0.23,=1600 kg/m3 Piezoelectric material PZT G1195:

E11=E22=63GPa, G11=G13=G23= 24.2 GPa, =0.3,=7000 kg/m3 ;

Piezoelectric constants (m V−1): d31 = d32=2.54× 10−12 ;

Electrical permittivity (F m-1): 1.53 × 10-8

Clamped clamped (CCCC) boundary conditions:

Along x = 0 & a; u = w = v = ==0Along y = 0 & b; u = w = v = ==0

Non dimensional natural frequencies:

Ω=

2

/

1

ℎ2

Lamination scheme: (450/-450/450/-450)


31/46

31

The Scheme for partial edge loading is shown in figure 4.

Figure 4. Scheme for partial edge loading

Computer program:

A computer program has been developed to perform all the necessary

computations. Element elastic stiffness matrices are obtained with 2 x 2 Gauss

sampling points to avoid possible shear locking. Element mass matrices are obtained

with 2 x 2 Gauss sampling points, as higher order integration is often unnecessary.

The geometric stiffness matrix is essentially a function of the in-plane stress

distribution in the element due to applied edge loading. Since, the stress field is non-

uniform, plane stress analysis is carried out using finite element technique to

determine the stresses at 3 x 3 Gauss sampling points.

For validation of the program non-dimensional natural frequencies is

compared with standard results as shown in table1.

Table 1.Fundamental frequencies (in Hz) of clamped and cylindrical Cross-ply

[0/90] shells a=b=2.54 mm, h = 0.254mm.

R/a Chandrasekhara [22] Present

10 458.21 459.43

20 331.54 332.67


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32

In the following results the natural frequencies are taken as ω1


33/46

33

Figure7. Regions resonance of anti-symmetric angle-ply cylindrical panel

subjected to partial edge loading at both ends (a/b=1; b/h=100; c/b=0.5)

It is observed from figures 5-7 that the widths of combination resonance zones

are comparable with the widths of simple resonance zones for anti-symmetric angle-

ply laminates. It is shown that the width of combination resonance zone is large in

case of partial loading. This signifies that the combination resonance zones are

important instability effects similar to simple resonance zones.

Figure 8. Regions simple resonance (2ω3) of anti-symmetric angle-ply cylindrical

panel with different b/R ratios (a/b=1; b/h=100; c/b=1)

1.5 2 2.5 3 3.5 4 4.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency Ratio

P /

P c r

( 2 ω1)

( 2 ω2)

( ω1 + ω

3)

1.9 2 2.1 2.2 2.3 2.4 2.50

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency Ratio

P /

P c r

b/R = 0.3

b/R = 0.4


34/46

34

Figure 9. Regions combination resonance (ω1+ω2) of anti-symmetric angle-ply

cylindrical panel with different no. of layers (a/b=1; b/h=100; c/b=1)

Figure 8 shows the effect of edge length to the radius of curvature of the panel

on simple resonance. It is observed that as b/R ratio increases the instability region

moves outwards along the frequency ratio axis and the width decreases.

Figure 9 shows the effect of number of layers on combination resonance. As

the number of layers increases, the instability regions move outward on the frequency

ratio axis and their width increases.

2 2.2 2.4 2.6 2.8 3 3.2 3.4 3.60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency Ratio

P

/ P c r

layer =6

layer =4

layer =2


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35

Figure 10. Regions simple resonance (2ω2) of anti-symmetric angle-ply




The effect of ply orientation on simple and combination resonance is shown in

figures 10 and 11 respectively. It is observed that the greater the ply orientation the

smaller the instability region for anti-symmetric angle-ply laminates. It is observed

that with the increase of ply-orientation the instability region moves outwards along

the frequency ratio axis.

1.6 1.8 2 2.2 2.4 2.6 2.8 3 3.20

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency Ratio

P /

P c r

theta =0

theta = 15

theta = 30

2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.80

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency Ratio

P /

P c r

theta = 0

theta = 15

theta = 30


36/46

36


cylindrical panel for different static load factors (a/b=1; b/h=100; c/b=1)

Figure 13. Regions simple resonance (2ω2) of anti-symmetric angle-ply

cylindrical panel for different static load factors (a/b=1; b/h=100; c/b=1)

From figure 12-13 it is observed that as the static load factor increases, the

instability regions shift inward in the frequency ratio axis and their widths increase. It

means laminated composite panels are more susceptible to dynamic instability due to

higher static load.

3 3.1 3.2 3.3 3.4 3.5 3.6 3.70

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency Ratio

P

/ P c r

α= 0.0

α= 0.2

α= 0.4

1.5 2 2.5 3 3.5 40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency Ratio

P /

P c r

α = 0.0

α = 0.2

α = 0.4


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37

Figure 14. Unstable zones at simple resonances (2 ω1) with piezoelectric layer


Figure 15. Unstable zones at combination resonances with piezoelectric layer


1.7 1.8 1.9 2 2.1 2.2 2.3 2.40

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency Ratio

P /

P c r

Gain = 0

Gain = 1000

Gain = 2000

3.3 3.35 3.4 3.45 3.5 3.55 3.6 3.650

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency Ratio

P /

P c r

(ω1+ω

5) Gain=0

(ω1+ω

6) Gain=0

(ω1+ω

5) Gain=1000

(ω1+ω

6) Gain=1000

(ω1+ω

5) Gain=2000

(ω1+ω

6) Gain=2000


38/46

38

1st mode 2nd mode

3rd mode 4th mode

5th mode 6th mode

Figure 16. Mode Shapes for clamped-clamped boundary condition

It is observed from figure 14 and 15 that with an increase in the control gain,

the zones of instability moves up and hence dynamic instability improves.

In case of clamped-clamped boundary condition (figure 16), 1st, 5th, 6th modes

are not symmetric. Thus these modes are controlled by providing negative velocity

feedback. It is shown from the mode shapes that 2nd , 3rd and 4th modes are symmetric.

So, these modes are not controlled. The combination resonance zones for

(ω1+ω5),(ω1+ω6) are also controlled.

0 1

2 3

4 5

6 7

8 9

10

0

2

4

6

8

10

0

0.2

0.4

0.6

0.8

1

02

46

810

0

5

10-1

-0.5

0

0.5

1

01

23

45

67

89

10

01

23

45

67

89

10

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

01

23

45

67

89

10

01

23

45

67

89

10

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

01

23

45

67

89

10

01

23

45

67

89

10

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0

2

4

6

8

10

0

5

10

-1

-0.5

0

0.5

1


39/46

39

Figure 17. Unstable zones at simple resonances with piezoelectric layer with

different feedback gains.(Simply Supported SS-1 boundary condition)

Figure 18. Unstable zones at combination resonance (ω1+ω6) with piezoelectric

layer with different feedback gains.(Simply Supported SS-1 boundary condition)

1.5 2 2.5 3 3.5 4 4.5 5 5.5 60

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency Ratio

P /

P c r

(2ω1) Gain=0

(2ω6) Gain=0

(2ω1) Gain=250

(2ω6) Gain=250

(2ω1) Gain=500

(2ω6) Gain=500

3.308 3.309 3.31 3.311 3.312 3.313 3.314 3.315 3.316 3.317 3.3180

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Frequency Ratio

P

/ P c r

Gain=0Gain=250

Gain=500


40/46

40

1st mode 2nd mode

3rd mode4th mode

5th mode 6th mode

Figure 19. Mode Shapes for simply supported SS-1 boundary condition

02

46

81

0

5

100

0.2

0.4

0.6

0.8

1

02

46

810

0

5

10-1

-0.5

0

0.5

1

02

4 6

81

0

5

10-1

-0.5

0

0.5

1

0

2 4

6

8

1

0

5

10

-1

-0.5

0

0.5

1

02

46

810

0

5

10-1

-0.5

0

0.5

1

02

46

81

0

5

10-1

-0.5

0

0.5

1


41/46

41

In case of simply supported boundary condition (figure 19), the 1st, 5th and 6th

modes are not symmetric. Thus from figure 17 we observe that we can control these

modes by increasing the gain value.

It is shown from the mode shapes that 2nd , 3rd and 4th modes are symmetric.

So, these modes are not controlled. The combination resonance zones for

(ω1+ω5),(ω1+ω6) are also controlled. For these modes with an increase in the control

gain, the zones of instability move up. In comparison to clamped clamped condition,

here control is done with less value of gain.


42/46

42

9. CONCLUSIONS:

• In the study of dynamic instability it has been observed that the width of the

primary resonance is more and the width of the combination resonance zone is

almost negligible.

• The above figures (fig. 5-7) confirm the fact that as c/b increases, the system

becomes more unstable since the unstable regions grow.

• It is observed that with the increase of b/r ratio the instability region moves

outwards along the frequency ratio axis and the width decreases. (fig.8)

• It is observed that as the no of layers increases, the instability regions move

outward on the frequency ratio axis. (fig. 9)

• The greater the ply-orientation the smaller the instability region for simple

resonances and the case is reverse for combination resonances. (fig. 10-11)

• It is observed that as the static load factor increases, the instability regions

move inward and their widths decreases. (fig. 12-13)

• The above figure shows that with an increase in the control gain, the zones of

instability moves up and hence dynamic instability improves. (fig.14-19) The

control gain affects only when the mode shape is not symmetric.


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43

10. SCOPE OF FUTURE WORK:

1. To predict the stability characteristic of laminated composite cylindrical

panel with piezoelectric layer for partial and point loading.

2. To study the effect of different boundary conditions on simple and

combination resonance characteristics of a laminated composite panel.

3. To study the dynamic instability of cylindrical composite panels with

viscoelastic layers.


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44

11. REFERENCES:

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shells, Computers & Structures Vol. 33. No.2., 435-440, 1989.

Date post:	07-Jul-2018
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Thesis for Dynamic stability of laminated cylindrical panel with piezoelectric layer

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