- Email: [email protected]
Numerical Investigation on Nonlinear Vibration Behavior of Laminated Cylindrical Panel Embedded with PZT Layers
Available online at www.sciencedirect.com
ScienceDirect Procedia Engineering 144 (2016) 660 – 667
12th International Conference on Vibration Problems, ICOVP 2015
Numerical investigation on nonlinear vibration behavior of laminated cylindrical panel embedded with PZT layers Vijay K. Singha* and Subrata K. Pandab a ,b*
Department of Mechanical Engineering, National Institute of Technology Rourkela, Rourkela -769008, India
Abstract This work investigates the geometrical nonlinear free vibration characteristic of cylindrical composite shell panel embedded with piezoelectric layers. A short circuit configuration (top and bottom layers are grounded i.e. zero potential) has been considered for the present analysis. A general mathematical model has been developed using higher order shear deformation mid-plane kinematics employing Green-Lagrange type of nonlinearity. Numerical solutions are obtained using Hamilton’s principle and discretized isoparametric finite element steps. The validity of present model has been checked by comparing the responses to those available in published literature. In order to examine the efficacy and applicability of the present developed model, few numerical examples are solved for different geometrical parameters (fiber orientation, thickness ratio, aspect ratio, curvature ratio, support conditions and amplitude ratio) with and/or without piezoelectric embedded layers and discussed in details. © 2016 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license © 2016 The Authors. Published by Elsevier Ltd. (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-reviewunder under responsibility of organizing the organizing committee of ICOVP Peer-review responsibility of the committee of ICOVP 2015 2015. Keywords: Laminated composites; Green-Lagrange nonlinearity; PZT; HSDT
1. Introduction Smart structures have got huge attention over the last two decades due to their coupling characteristic between electric, magnetic, thermal and/or mechanical effects for achieving the desired performance in modern structural systems. This lead to wide area of applications in structural health monitoring, vibration isolation and/or control, shape control, medical applications, damage detection and noise attenuation. It is well known that the coupling of mechanical and electrical properties are governed by converse and direct piezoelectric effect which in turn make them well suited
*
Corresponding author. Tel.:0661-2462529 E-mail address: [email protected], [email protected]
1877-7058 © 2016 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of ICOVP 2015
doi:10.1016/j.proeng.2016.05.062
Vijay K. Singh and Subrata K. Panda / Procedia Engineering 144 (2016) 660 – 667
661
for sensors and/or actuators. The piezoelectric materials are used in the form of layers/patches embedded and/or surface bonded to the parent laminated composite structural components. Due to the above increasing applicability, authors are interested in finite element (FE) modelling of smart structures using PZT (lead zirconate titanate) as piezoceramic material in this present analysis. Nomenclature
d H NL RX
RY BI
TI e
D
nodal degree of freedom nonlinear strain matrix radius of curvature parallel to y-axis radius of curvature parallel to x-axis differential operator matrix for potential field thickness coordinate matrix for potential field piezoelectric stress constant piezoelectric strain constant electric displacement
Arefi and Khoshgoftar [1] developed coupled piezo-thermo-elastic generalized model of thick functionally graded hollow spherical shell panel embedded with piezoelectric material under electro-thermo-mechanical loads to improve the relation between mechanical and electrical load. Kishore et al [2] presented nonlinear static analysis of smart composite plate. Benjeddou [3] presented exhaustive review on advances and trends in finite element method for the analysis of adaptive structures. Balamurugan and Narayanan [4,5] investigated active vibration control performance of piezo bonded laminated flat and shell structures by taking the coupled (mechanical and electrical) load effect and solved using finite element method (FEM). Heyliger and Brooks [8] reported free vibration responses of piezoelectric laminated plates with cylindrical bending and also computed the natural frequencies through thickness modal distributions of elastic and electric field variables. Nanda [9] investigated the effects of voltage and different geometrical parameters on the nonlinear free vibration and transient responses of composite shell panels bonded with piezoelectric layers under uniform thermal environment. Kerur and Ghosh [10] reported the active control of nonlinear transient responses of laminated composite plate using the FSDT kinematics and von-Karman nonlinearity under coupled electromechanical loading. Lee et al. [11] investigated the degree of deflection suppression of laminated composite shells bonded with smart layer through nonlinear finite element (FE) steps. Saravanos et al. [12] presented dynamics behaviour composite plate embedded with piezoelectric actuators and sensors using layerwise theory. Singh and Panda [14] developed a geometrical nonlinear mathematical model to analyse large amplitude free vibration behaviour of doubly curved composite shell panel using HSDT kinematics and Green Lagrange type nonlinearity. 2. Mathematical formulations In this present study, a doubly curved laminated composite shallow shell panel (twist radius of curvature Rxy = ∞) of uniform thickness ‘h’ and principal radii of curvatures, Rx and Ry along x and y directions, respectively is shown in Figure1. Its’ outermost surfaces are bonded with the PZT layers whose projection on the xy-plane is a rectangle of dimensions ‘a’ and ‘b’ (Fig.1) [6] 2.1. Element geometry and displacement field The displacement field based on third order shear deformation theory (TSDT) as in [2].
662
Vijay K. Singh and Subrata K. Panda / Procedia Engineering 144 (2016) 660 – 667
u v
k § ¨ ©
w
k ·¸¹ § ¨ ©
x, y , z , t
2 3 u0 x , y zT x x , y z I x x , y z O x x , y
x, y , z , t
2 3 v0 x , y zT y x , y z I y x , y z O y x , y
k ¹¸·
x, y , z , t
w0 x, y zT z
(1)
x, y
Electric potential I can be expressed as [3]
I x, y , z I
0
1 2 x, y zI x, y z 2I x, y
(2)
The electric field vector {E} can be expressed as:
^E`
^ `
where, E 0
T
ª B º ^I` ¬ I¼
^
Ex
0
Ey
0
^ `
ªT º E 0 ¬ I¼ Ex
1
Ey
1
(3) Ex
2
Ey
2
X
1
X
2
differential operator and thickness coordinate matrices correspond to potential field
`
T
and
ª B º and ªT º are the ¬ I¼ ¬« I ¼»
^I` , respectively. The details of
the matrices are presented in Appendix A. where, u(k), v(k) and w(k) denote the displacements of a point along the (x, y, and z) coordinates in the kth layer in function of corresponding mid-plane displacements u0, v0 and w0 at time t. T x and T y are the rotations of normal to the mid-surface i.e. z=0 about the y and x-axes, respectively.. The functions Ix , I y , Ox , Oy and are the higher order terms in the Taylor series expansion and ‘ T z ’ is the transverse extension or thickness stretching term
Fig. 1 Closed-circuit configuration of a cylindrical panel bonded with PZT layers
2.2. Strain Displacement Relations The nonlinear Green–Lagrange strain–displacement relation for the laminated doubly curved shell panel can be
663
Vijay K. Singh and Subrata K. Panda / Procedia Engineering 144 (2016) 660 – 667
^H ` is the summation of the linear ^H L ` and nonlinear ^H NL ` ^H ` ^H L ` ^H NL `
expressed as in [14]. The total strain vector strain vector as follows:
(4)
2.3. Piezoelectric constitutive equations The transformed stress–strain relations for any general kth orthotropic composite lamina embedded with the piezoelectric layer is conceded as follows: (5) ^V ` ªQº ^H ` >e@^E`
¬ ¼
> D@ >e@ ^H` >@^E` T
(6)
2.4. Governing Equation The governing equation of the nonlinear free vibration is derived using Hamilton's principle and expressed as: t2
G ³ T U t1
(7)
0
Eq. (15) is reduced to nonlinear generalized eigenvalue problem and the final form of the nonlinear governing equation of the laminated curved composite panel embedded with PZT can be presented as: T 1 ª§ «¨ ª¬ K q º¼ ª¬ K qI º¼ ª¬ KI º¼ ª¬ K qI º¼ ¬©
^`
º · 2 ¸ Z > M @» I ¹ ¼
^`
0
(8)
where, ‘ω’ is the natural frequency and I corresponding eigenvector, respectively, for any generalized eigenvalue problem and solved using direct iterative method. Table 1. Material properties of graphite/epoxy and PZT-5A used in the present study Elastic properties of graphite/epoxy E11=181.0GPa; E22=E33=10.3GPa; G12=G13=7.17GPa; G23=0.28GPa; ν12=ν13=0.28; ν23=0.33; ρ=1580 kg/m3, Elastic properties of PZT E11=E22=61.0GPa; E33=53.2GPa; G12=22.6GPa; G13=G23=21.1GPa; ν12=0.35; ν13=ν23=0.38; ρ=7750 kg/m3, Piezoelectric stress coefficient of PZT(C/m2) e31=e32=7.209 C/m2; e33=15.118 C/m2; e24=e15=12.322 C/m2 Dielectric constant (F/m): k11=k22=1.53 x 10-8; k33=1.53 x 10-8
3. Results and discussions In order to obtain the desired responses, a general nonlinear FE computer code has been developed in MATLAB environment using the present mathematical model considering the quadratic variation of electric potential through the thickness of laminated panel. The elastic and electrical properties of graphite/epoxy composites and PZT-5A materials are presented in Table1. Table 2 presents the type of support conditions utilized in the present computation to avoid rigid body motion as well as reduce the number of unknowns. The simply supported boundary has been used
664
Vijay K. Singh and Subrata K. Panda / Procedia Engineering 144 (2016) 660 – 667
throughout the analysis unless specified otherwise. The nondimensional form of the fundamental frequencies is obtained using the following formulae for throughout the analysis. 2
Y
§ ©
Zb ®U PZT 5 A / ¨ E22
¯
PZT 5 A
2 ·½ hTOT ¸¾
1 2
(9)
¹¿
Table 2. Different support conditions used in the present analysis CCCC
u0
v0
SSSS
v0
w0
v0
w0 T z
u0
v0
u0
v0
SCSC HHHH
Tx
w0
Tz
Iy
Iy
w0
Ty Oy
Oy
Tz
0 at x = 0, a ;
Tz
Iy
70
Nondimensional fundamental frequency(YL)
Ix
Iy
Ox
O y at x = 0, a and y = 0, b
u0
w0 T z
Ix
Ox
0
at y = 0, b
0 at x = 0 , a;
Tx T y
w0
Tz
Ix
Oy
I y Ox O y 0 at y = 0, b 0 at x = 0, a; u0 v0 w0 T z
Ix
Ox
0 at y = 0,b
P(00/900)2P Nanda[2010]=11.0110, present=10.6536
65
P(00/900)3P Nanda[2010]=11.2326, present =10.4457
60 55 50 45
P(450/-450)P
P(450/-450)2P
P(300/-300)P
P(300/-300)2P
40 35 30 25 20 15 10 5 2x2
3x3
4x4
5x5
6x6
7x7
8x8
Mesh Divison
Fig. 2. Convergence study of fundamental frequency for cylindrical laminated composite shells embedded with PZT layers.
3.1. Convergence and comparison In order to check the accuracy and applicability of present mathematical model, a convergence study of nondimensional fundamental natural frequency is presented in Figure 2 and it can easily be seen that the present values are in good agreement with the reference. Table 3 shows the comparison of present nonlinear frequency values of cross ply piezoelectric shell with those of published literature for different curvature ratio. 3.2. Effect of thickness ratio The effect of thickness ratio on nonlinear frequency is shown in Table 4 for the angle ply laminated piezoelectric composite cylindrical shell. It is observed that the nonlinear frequency values increases instead of decreasing due to the presence of laminate thickness parameter in the denominator part of non-dimensional formula.
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Vijay K. Singh and Subrata K. Panda / Procedia Engineering 144 (2016) 660 – 667
3.3. Effect of support conditions The effect of support constraints on nonlinear frequency parameter of angle ply piezoelectric shell panel is presented in Figure 4. Here, it is observed that the square laminated piezoelectric shell panel with the clamped support condition has the largest value of nonlinear frequency compared to all others due to highest stiffness. Table 3. Comparison study of non-linear frequency of cylindrical laminated composite shells embedded with PZT layers P(0 0/900)2P for different curvature ratio (R/b) Wmax/h
Linear (ϖL)
0.2
0.4
0.6
0.8
R/b
10
20
50
100
Nanda [10]
10.513
10.172
10.100
9.808
Present
10.652
10.577
10.556
10.553
Nanda [10]
11.026
10.528
10.347
10.011
Present
10.767
10.532
10.549
10.546
Nanda [10]
11.775
11.158
10.885
10.501
Present
11.019
10.812
10.619
10.695
Nanda [10]
12.699
11.996
11.651
11.217
Present
11.367
11.106
10.961
10.921
Nanda [10]
13.748
12.981
12.579
12.096
Present
11.771
11.442
11.277
11.215
3.4. Influence of curvature ratio The shell geometries and their types are defined based on the curvature ratio (R/b) i.e., deep to shallow. The shell structures are well known for their higher stretching energy as compared to bending energy as the shell becomes deep which affect the frequency responses greatly. In order to examine the effect of curvature ratio on the nonlinear vibration behaviour of laminated composite cylindrical shell panel embedded with/without PZT layers the present example is solved for four curvature ratios (R/b = 10, 20, 50 and 100) and plotted in Figure 5. It is observed that there is not any appreciable change in responses after R/b=50. Table 4. Effect of thickness ratio (a/h) on nonlinear frequency parameter (ϖNL) of laminated composite cylindrical shell panels P(450/-450)2P (R/a=10, a/b=1) a/h
Linear (ϖL)
0.20
0.40
0.60
10.000
10.504
11.338
12.711
14.413
20.000
20.462
21.229
22.224
23.470
50.000
50.525
51.425
51.037
53.128
100.000
100.573
101.785
103.041
103.068
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Vijay K. Singh and Subrata K. Panda / Procedia Engineering 144 (2016) 660 – 667
Nondimensional nonlinear frequency
30
SSSS CCCC SCSC HHHH
28
26
24
22
20
18 0.0
0.1
0.2
0.3
0.4
0.5
0.6
Amplitude ratio (wmax/h)
Fig. 3. Influence of support conditions on nonlinear frequency parameter (ϖNL) of laminated composite cylindrical shell panels P(450/-450)2P (a/h=20, a/b=1)
Nondimensional nonlinear frequency
19.2
W max/h = 0.2
19.0
W max/h = 0.4
18.8
W max/h = 0.6 W max/h = 0.8
18.6 18.4 18.2 18.0 17.8 17.6 17.4 0
20
40
60
80
100
Curvature ratio(R/a)
Fig.4. Effect of curvature ratio on nonlinear frequency parameter (ϖNL) of laminated composite cylindrical shell panels P(450/-450)2P (a/h=40, a/b=1)
4. Conclusion The following conclusive remarks can be drawn from the above analysis: A nonlinear mathematical model for the free vibration characteristics of laminated cylindrical composite panels embedded with the PZT layers has been investigated and validated with the present available results.
Vijay K. Singh and Subrata K. Panda / Procedia Engineering 144 (2016) 660 – 667
667
It is clear that the thickness ratio, curvature ratio and support conditions have significant effect on nonlinear vibration behavior of piezoelectric cylindrical panel.
Acknowledgements This work is under the project sanctioned by the department of science and technology (DST) through grant SERB/F/1765/2013-2014 Dated: 21/06/2013. Authors are thankful to DST, Govt. of India for its consistent support.
Appendix A. Differential-operator, thickness co-ordinate matrices for the potential field and the nodal degree of freedom are as follows: T ª w º w wy 0 0 0 0 0 0» « wx ª1 0 z 0 z 2 0 0 0º w « » w ªB º « 0 wy wx 0 0 0 1 0 » ; ªT º «0 1 0 z 0 z 2 0 0» I ¬ I¼ « » ¬ ¼ «0 0 0 0 0 0 1 1 » w w «¬ »¼ wy 0 2» « 0 wx 0 0 0
«¬
»¼
{d} {u0 , v0 , w0 ,T x ,T y ,T z , Ix , Iy , Ox , Oy , I0 , I1 , I2 }T References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
M. Arefi, M.J. Khoshgoftar, Comprehensive piezo-thermo-elastic analysis of a thick hollow spherical shell, Smart Structures and Systems. 14 (2014) 225-246. M.D.V. H. Kishore, B.N. Singh, Nonlinear static analysis of smart laminated composite plate, Aerospace Science and Technology.15 (2011), 224–235. A. Benjeddou, Advances in piezoelectric finite element modeling of adaptive structural elements: a survey, Computers and Structures. 76 (2000) 347-363. V. Balamurugan, S. Narayanan, Shell finite element for smart piezoelectric composite plate/shell structures and its application to the study of active vibration control. Finite Elements in Analysis and Design. 37 (2001) 713-738. V. Balamurugan, S. Narayanan, Multilayer higher order piezlaminated smart composite shell finite element and its application to active vibration control. Journal of Intelligent Materials System and Structures. 20(2009) DOI: 10.1177/1045389x08095269. E. Carrera, S. Brischetto, P. Nali, Plates and Shells for Smart Structures: Classical and Advanced Theories for Modeling and Analysis, first ed., John Wiley & Sons Ltd., West Sussex, United Kingdom, 2011, pp. 101-104. R. D. Cook, D.S. Malkus, M.E. Plesha, R.J.Witt, Concepts and applications of finite element analysis, fourth ed., John Wiley & Sons, Singapore, 2009. P.Heyliger, S. Brooks, Free vibration of piezoelectric laminates in cylindrical bending, International Journal of Solids and Structures. 32 (1995), 2945-2960. S.B. Kerur, A. Ghosh: Active Control of Geometrically Non-linear Transient Response of Smart Laminated Composite Plate Integrated With AFC Actuator and PVDF Sensor, Journal of Intelligent Material Systems and Structures. 22 (2011), 1149-1160. N. Nanda,Non-linear free and forced vibrations of piezoelectric laminated shells in thermal environments, The IES Journal Part A: Civil & Structural Engineering. 3 (2010), 147–160. S.J. Lee, J.N.Reddy, F. Rostam-Abadi, Nonlinear finite element analysis of laminated composite shells with actuating layers, Finite Elements in Analysis and Design. 43 (2006) 1 – 21 D.A. Saravanos, P.R. Heyliger, D.A. Hopkins, Layerwise mechanics and finite element for the dynamic analysis of piezoelectric composite plates, International Journal of Solids Structures. 34 (1997) 359-378. J.N.Reddy, Mechanics of laminated composite plates and shells:Theory and Analysis, second ed., CRC Press, Florida, 2004. V.K.Singh, S.K.Panda, Nonlinear free vibration analysis of single/doubly curved composite shallow shell panels, Thin-Walled Structures. 85 2014 341–349.
ScienceDirect Procedia Engineering 144 (2016) 660 – 667
12th International Conference on Vibration Problems, ICOVP 2015
Numerical investigation on nonlinear vibration behavior of laminated cylindrical panel embedded with PZT layers Vijay K. Singha* and Subrata K. Pandab a ,b*
Department of Mechanical Engineering, National Institute of Technology Rourkela, Rourkela -769008, India
Abstract This work investigates the geometrical nonlinear free vibration characteristic of cylindrical composite shell panel embedded with piezoelectric layers. A short circuit configuration (top and bottom layers are grounded i.e. zero potential) has been considered for the present analysis. A general mathematical model has been developed using higher order shear deformation mid-plane kinematics employing Green-Lagrange type of nonlinearity. Numerical solutions are obtained using Hamilton’s principle and discretized isoparametric finite element steps. The validity of present model has been checked by comparing the responses to those available in published literature. In order to examine the efficacy and applicability of the present developed model, few numerical examples are solved for different geometrical parameters (fiber orientation, thickness ratio, aspect ratio, curvature ratio, support conditions and amplitude ratio) with and/or without piezoelectric embedded layers and discussed in details. © 2016 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license © 2016 The Authors. Published by Elsevier Ltd. (http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-reviewunder under responsibility of organizing the organizing committee of ICOVP Peer-review responsibility of the committee of ICOVP 2015 2015. Keywords: Laminated composites; Green-Lagrange nonlinearity; PZT; HSDT
1. Introduction Smart structures have got huge attention over the last two decades due to their coupling characteristic between electric, magnetic, thermal and/or mechanical effects for achieving the desired performance in modern structural systems. This lead to wide area of applications in structural health monitoring, vibration isolation and/or control, shape control, medical applications, damage detection and noise attenuation. It is well known that the coupling of mechanical and electrical properties are governed by converse and direct piezoelectric effect which in turn make them well suited
*
Corresponding author. Tel.:0661-2462529 E-mail address: [email protected], [email protected]
1877-7058 © 2016 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license
(http://creativecommons.org/licenses/by-nc-nd/4.0/). Peer-review under responsibility of the organizing committee of ICOVP 2015
doi:10.1016/j.proeng.2016.05.062
Vijay K. Singh and Subrata K. Panda / Procedia Engineering 144 (2016) 660 – 667
661
for sensors and/or actuators. The piezoelectric materials are used in the form of layers/patches embedded and/or surface bonded to the parent laminated composite structural components. Due to the above increasing applicability, authors are interested in finite element (FE) modelling of smart structures using PZT (lead zirconate titanate) as piezoceramic material in this present analysis. Nomenclature
d H NL RX
RY BI
TI e
D
nodal degree of freedom nonlinear strain matrix radius of curvature parallel to y-axis radius of curvature parallel to x-axis differential operator matrix for potential field thickness coordinate matrix for potential field piezoelectric stress constant piezoelectric strain constant electric displacement
Arefi and Khoshgoftar [1] developed coupled piezo-thermo-elastic generalized model of thick functionally graded hollow spherical shell panel embedded with piezoelectric material under electro-thermo-mechanical loads to improve the relation between mechanical and electrical load. Kishore et al [2] presented nonlinear static analysis of smart composite plate. Benjeddou [3] presented exhaustive review on advances and trends in finite element method for the analysis of adaptive structures. Balamurugan and Narayanan [4,5] investigated active vibration control performance of piezo bonded laminated flat and shell structures by taking the coupled (mechanical and electrical) load effect and solved using finite element method (FEM). Heyliger and Brooks [8] reported free vibration responses of piezoelectric laminated plates with cylindrical bending and also computed the natural frequencies through thickness modal distributions of elastic and electric field variables. Nanda [9] investigated the effects of voltage and different geometrical parameters on the nonlinear free vibration and transient responses of composite shell panels bonded with piezoelectric layers under uniform thermal environment. Kerur and Ghosh [10] reported the active control of nonlinear transient responses of laminated composite plate using the FSDT kinematics and von-Karman nonlinearity under coupled electromechanical loading. Lee et al. [11] investigated the degree of deflection suppression of laminated composite shells bonded with smart layer through nonlinear finite element (FE) steps. Saravanos et al. [12] presented dynamics behaviour composite plate embedded with piezoelectric actuators and sensors using layerwise theory. Singh and Panda [14] developed a geometrical nonlinear mathematical model to analyse large amplitude free vibration behaviour of doubly curved composite shell panel using HSDT kinematics and Green Lagrange type nonlinearity. 2. Mathematical formulations In this present study, a doubly curved laminated composite shallow shell panel (twist radius of curvature Rxy = ∞) of uniform thickness ‘h’ and principal radii of curvatures, Rx and Ry along x and y directions, respectively is shown in Figure1. Its’ outermost surfaces are bonded with the PZT layers whose projection on the xy-plane is a rectangle of dimensions ‘a’ and ‘b’ (Fig.1) [6] 2.1. Element geometry and displacement field The displacement field based on third order shear deformation theory (TSDT) as in [2].
662
Vijay K. Singh and Subrata K. Panda / Procedia Engineering 144 (2016) 660 – 667
u v
k § ¨ ©
w
k ·¸¹ § ¨ ©
x, y , z , t
2 3 u0 x , y zT x x , y z I x x , y z O x x , y
x, y , z , t
2 3 v0 x , y zT y x , y z I y x , y z O y x , y
k ¹¸·
x, y , z , t
w0 x, y zT z
(1)
x, y
Electric potential I can be expressed as [3]
I x, y , z I
0
1 2 x, y zI x, y z 2I x, y
(2)
The electric field vector {E} can be expressed as:
^E`
^ `
where, E 0
T
ª B º ^I` ¬ I¼
^
Ex
0
Ey
0
^ `
ªT º E 0 ¬ I¼ Ex
1
Ey
1
(3) Ex
2
Ey
2
X
1
X
2
differential operator and thickness coordinate matrices correspond to potential field
`
T
and
ª B º and ªT º are the ¬ I¼ ¬« I ¼»
^I` , respectively. The details of
the matrices are presented in Appendix A. where, u(k), v(k) and w(k) denote the displacements of a point along the (x, y, and z) coordinates in the kth layer in function of corresponding mid-plane displacements u0, v0 and w0 at time t. T x and T y are the rotations of normal to the mid-surface i.e. z=0 about the y and x-axes, respectively.. The functions Ix , I y , Ox , Oy and are the higher order terms in the Taylor series expansion and ‘ T z ’ is the transverse extension or thickness stretching term
Fig. 1 Closed-circuit configuration of a cylindrical panel bonded with PZT layers
2.2. Strain Displacement Relations The nonlinear Green–Lagrange strain–displacement relation for the laminated doubly curved shell panel can be
663
Vijay K. Singh and Subrata K. Panda / Procedia Engineering 144 (2016) 660 – 667
^H ` is the summation of the linear ^H L ` and nonlinear ^H NL ` ^H ` ^H L ` ^H NL `
expressed as in [14]. The total strain vector strain vector as follows:
(4)
2.3. Piezoelectric constitutive equations The transformed stress–strain relations for any general kth orthotropic composite lamina embedded with the piezoelectric layer is conceded as follows: (5) ^V ` ªQº ^H ` >e@^E`
¬ ¼
> D@ >e@ ^H` >@^E` T
(6)
2.4. Governing Equation The governing equation of the nonlinear free vibration is derived using Hamilton's principle and expressed as: t2
G ³ T U t1
(7)
0
Eq. (15) is reduced to nonlinear generalized eigenvalue problem and the final form of the nonlinear governing equation of the laminated curved composite panel embedded with PZT can be presented as: T 1 ª§ «¨ ª¬ K q º¼ ª¬ K qI º¼ ª¬ KI º¼ ª¬ K qI º¼ ¬©
^`
º · 2 ¸ Z > M @» I ¹ ¼
^`
0
(8)
where, ‘ω’ is the natural frequency and I corresponding eigenvector, respectively, for any generalized eigenvalue problem and solved using direct iterative method. Table 1. Material properties of graphite/epoxy and PZT-5A used in the present study Elastic properties of graphite/epoxy E11=181.0GPa; E22=E33=10.3GPa; G12=G13=7.17GPa; G23=0.28GPa; ν12=ν13=0.28; ν23=0.33; ρ=1580 kg/m3, Elastic properties of PZT E11=E22=61.0GPa; E33=53.2GPa; G12=22.6GPa; G13=G23=21.1GPa; ν12=0.35; ν13=ν23=0.38; ρ=7750 kg/m3, Piezoelectric stress coefficient of PZT(C/m2) e31=e32=7.209 C/m2; e33=15.118 C/m2; e24=e15=12.322 C/m2 Dielectric constant (F/m): k11=k22=1.53 x 10-8; k33=1.53 x 10-8
3. Results and discussions In order to obtain the desired responses, a general nonlinear FE computer code has been developed in MATLAB environment using the present mathematical model considering the quadratic variation of electric potential through the thickness of laminated panel. The elastic and electrical properties of graphite/epoxy composites and PZT-5A materials are presented in Table1. Table 2 presents the type of support conditions utilized in the present computation to avoid rigid body motion as well as reduce the number of unknowns. The simply supported boundary has been used
664
Vijay K. Singh and Subrata K. Panda / Procedia Engineering 144 (2016) 660 – 667
throughout the analysis unless specified otherwise. The nondimensional form of the fundamental frequencies is obtained using the following formulae for throughout the analysis. 2
Y
§ ©
Zb ®U PZT 5 A / ¨ E22
¯
PZT 5 A
2 ·½ hTOT ¸¾
1 2
(9)
¹¿
Table 2. Different support conditions used in the present analysis CCCC
u0
v0
SSSS
v0
w0
v0
w0 T z
u0
v0
u0
v0
SCSC HHHH
Tx
w0
Tz
Iy
Iy
w0
Ty Oy
Oy
Tz
0 at x = 0, a ;
Tz
Iy
70
Nondimensional fundamental frequency(YL)
Ix
Iy
Ox
O y at x = 0, a and y = 0, b
u0
w0 T z
Ix
Ox
0
at y = 0, b
0 at x = 0 , a;
Tx T y
w0
Tz
Ix
Oy
I y Ox O y 0 at y = 0, b 0 at x = 0, a; u0 v0 w0 T z
Ix
Ox
0 at y = 0,b
P(00/900)2P Nanda[2010]=11.0110, present=10.6536
65
P(00/900)3P Nanda[2010]=11.2326, present =10.4457
60 55 50 45
P(450/-450)P
P(450/-450)2P
P(300/-300)P
P(300/-300)2P
40 35 30 25 20 15 10 5 2x2
3x3
4x4
5x5
6x6
7x7
8x8
Mesh Divison
Fig. 2. Convergence study of fundamental frequency for cylindrical laminated composite shells embedded with PZT layers.
3.1. Convergence and comparison In order to check the accuracy and applicability of present mathematical model, a convergence study of nondimensional fundamental natural frequency is presented in Figure 2 and it can easily be seen that the present values are in good agreement with the reference. Table 3 shows the comparison of present nonlinear frequency values of cross ply piezoelectric shell with those of published literature for different curvature ratio. 3.2. Effect of thickness ratio The effect of thickness ratio on nonlinear frequency is shown in Table 4 for the angle ply laminated piezoelectric composite cylindrical shell. It is observed that the nonlinear frequency values increases instead of decreasing due to the presence of laminate thickness parameter in the denominator part of non-dimensional formula.
665
Vijay K. Singh and Subrata K. Panda / Procedia Engineering 144 (2016) 660 – 667
3.3. Effect of support conditions The effect of support constraints on nonlinear frequency parameter of angle ply piezoelectric shell panel is presented in Figure 4. Here, it is observed that the square laminated piezoelectric shell panel with the clamped support condition has the largest value of nonlinear frequency compared to all others due to highest stiffness. Table 3. Comparison study of non-linear frequency of cylindrical laminated composite shells embedded with PZT layers P(0 0/900)2P for different curvature ratio (R/b) Wmax/h
Linear (ϖL)
0.2
0.4
0.6
0.8
R/b
10
20
50
100
Nanda [10]
10.513
10.172
10.100
9.808
Present
10.652
10.577
10.556
10.553
Nanda [10]
11.026
10.528
10.347
10.011
Present
10.767
10.532
10.549
10.546
Nanda [10]
11.775
11.158
10.885
10.501
Present
11.019
10.812
10.619
10.695
Nanda [10]
12.699
11.996
11.651
11.217
Present
11.367
11.106
10.961
10.921
Nanda [10]
13.748
12.981
12.579
12.096
Present
11.771
11.442
11.277
11.215
3.4. Influence of curvature ratio The shell geometries and their types are defined based on the curvature ratio (R/b) i.e., deep to shallow. The shell structures are well known for their higher stretching energy as compared to bending energy as the shell becomes deep which affect the frequency responses greatly. In order to examine the effect of curvature ratio on the nonlinear vibration behaviour of laminated composite cylindrical shell panel embedded with/without PZT layers the present example is solved for four curvature ratios (R/b = 10, 20, 50 and 100) and plotted in Figure 5. It is observed that there is not any appreciable change in responses after R/b=50. Table 4. Effect of thickness ratio (a/h) on nonlinear frequency parameter (ϖNL) of laminated composite cylindrical shell panels P(450/-450)2P (R/a=10, a/b=1) a/h
Linear (ϖL)
0.20
0.40
0.60
10.000
10.504
11.338
12.711
14.413
20.000
20.462
21.229
22.224
23.470
50.000
50.525
51.425
51.037
53.128
100.000
100.573
101.785
103.041
103.068
666
Vijay K. Singh and Subrata K. Panda / Procedia Engineering 144 (2016) 660 – 667
Nondimensional nonlinear frequency
30
SSSS CCCC SCSC HHHH
28
26
24
22
20
18 0.0
0.1
0.2
0.3
0.4
0.5
0.6
Amplitude ratio (wmax/h)
Fig. 3. Influence of support conditions on nonlinear frequency parameter (ϖNL) of laminated composite cylindrical shell panels P(450/-450)2P (a/h=20, a/b=1)
Nondimensional nonlinear frequency
19.2
W max/h = 0.2
19.0
W max/h = 0.4
18.8
W max/h = 0.6 W max/h = 0.8
18.6 18.4 18.2 18.0 17.8 17.6 17.4 0
20
40
60
80
100
Curvature ratio(R/a)
Fig.4. Effect of curvature ratio on nonlinear frequency parameter (ϖNL) of laminated composite cylindrical shell panels P(450/-450)2P (a/h=40, a/b=1)
4. Conclusion The following conclusive remarks can be drawn from the above analysis: A nonlinear mathematical model for the free vibration characteristics of laminated cylindrical composite panels embedded with the PZT layers has been investigated and validated with the present available results.
Vijay K. Singh and Subrata K. Panda / Procedia Engineering 144 (2016) 660 – 667
667
It is clear that the thickness ratio, curvature ratio and support conditions have significant effect on nonlinear vibration behavior of piezoelectric cylindrical panel.
Acknowledgements This work is under the project sanctioned by the department of science and technology (DST) through grant SERB/F/1765/2013-2014 Dated: 21/06/2013. Authors are thankful to DST, Govt. of India for its consistent support.
Appendix A. Differential-operator, thickness co-ordinate matrices for the potential field and the nodal degree of freedom are as follows: T ª w º w wy 0 0 0 0 0 0» « wx ª1 0 z 0 z 2 0 0 0º w « » w ªB º « 0 wy wx 0 0 0 1 0 » ; ªT º «0 1 0 z 0 z 2 0 0» I ¬ I¼ « » ¬ ¼ «0 0 0 0 0 0 1 1 » w w «¬ »¼ wy 0 2» « 0 wx 0 0 0
«¬
»¼
{d} {u0 , v0 , w0 ,T x ,T y ,T z , Ix , Iy , Ox , Oy , I0 , I1 , I2 }T References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]
M. Arefi, M.J. Khoshgoftar, Comprehensive piezo-thermo-elastic analysis of a thick hollow spherical shell, Smart Structures and Systems. 14 (2014) 225-246. M.D.V. H. Kishore, B.N. Singh, Nonlinear static analysis of smart laminated composite plate, Aerospace Science and Technology.15 (2011), 224–235. A. Benjeddou, Advances in piezoelectric finite element modeling of adaptive structural elements: a survey, Computers and Structures. 76 (2000) 347-363. V. Balamurugan, S. Narayanan, Shell finite element for smart piezoelectric composite plate/shell structures and its application to the study of active vibration control. Finite Elements in Analysis and Design. 37 (2001) 713-738. V. Balamurugan, S. Narayanan, Multilayer higher order piezlaminated smart composite shell finite element and its application to active vibration control. Journal of Intelligent Materials System and Structures. 20(2009) DOI: 10.1177/1045389x08095269. E. Carrera, S. Brischetto, P. Nali, Plates and Shells for Smart Structures: Classical and Advanced Theories for Modeling and Analysis, first ed., John Wiley & Sons Ltd., West Sussex, United Kingdom, 2011, pp. 101-104. R. D. Cook, D.S. Malkus, M.E. Plesha, R.J.Witt, Concepts and applications of finite element analysis, fourth ed., John Wiley & Sons, Singapore, 2009. P.Heyliger, S. Brooks, Free vibration of piezoelectric laminates in cylindrical bending, International Journal of Solids and Structures. 32 (1995), 2945-2960. S.B. Kerur, A. Ghosh: Active Control of Geometrically Non-linear Transient Response of Smart Laminated Composite Plate Integrated With AFC Actuator and PVDF Sensor, Journal of Intelligent Material Systems and Structures. 22 (2011), 1149-1160. N. Nanda,Non-linear free and forced vibrations of piezoelectric laminated shells in thermal environments, The IES Journal Part A: Civil & Structural Engineering. 3 (2010), 147–160. S.J. Lee, J.N.Reddy, F. Rostam-Abadi, Nonlinear finite element analysis of laminated composite shells with actuating layers, Finite Elements in Analysis and Design. 43 (2006) 1 – 21 D.A. Saravanos, P.R. Heyliger, D.A. Hopkins, Layerwise mechanics and finite element for the dynamic analysis of piezoelectric composite plates, International Journal of Solids Structures. 34 (1997) 359-378. J.N.Reddy, Mechanics of laminated composite plates and shells:Theory and Analysis, second ed., CRC Press, Florida, 2004. V.K.Singh, S.K.Panda, Nonlinear free vibration analysis of single/doubly curved composite shallow shell panels, Thin-Walled Structures. 85 2014 341–349.