Research Article | Open Access

Volume 2020 |Article ID 7592302 | https://doi.org/10.1155/2020/7592302

Ola Ragb, Mohamed Salah, M. S. Matbuly, R. B. M. Amer, "Vibration Analysis of Piezoelectric Composite Plate Resting on Nonlinear Elastic Foundations Using Sinc and Discrete Singular Convolution Differential Quadrature Techniques", Mathematical Problems in Engineering, vol. 2020, Article ID 7592302, 22 pages, 2020. https://doi.org/10.1155/2020/7592302

# Vibration Analysis of Piezoelectric Composite Plate Resting on Nonlinear Elastic Foundations Using Sinc and Discrete Singular Convolution Differential Quadrature Techniques

Accepted25 May 2020
Published15 Jul 2020

#### Abstract

In this work, free vibration of the piezoelectric composite plate resting on nonlinear elastic foundations is examined. The three-dimensionality of elasticity theory and piezoelectricity is used to derive the governing equation of motion. By implementing two differential quadrature schemes and applying different boundary conditions, the problem is converted to a nonlinear eigenvalue problem. The perturbation method and iterative quadrature formula are used to solve the obtained equation. Numerical analysis of the proposed schemes is introduced to demonstrate the accuracy and efficiency of the obtained results. The obtained results are compared with available results in the literature, showing excellent agreement. Additionally, the proposed schemes have higher efficiency than previous schemes. Furthermore, a parametric study is introduced to investigate the effect of elastic foundation parameters, different materials of sensors and actuators, and elastic and geometric characteristics of the composite plate on the natural frequencies and mode shapes.

#### 1. Introduction

The increase in the use of piezoelectric composite materials resting on elastic foundation structures, especially in aerospace, automotive, and marine environments, leads to the rise of difficulties in nonlinear vibrations in various modern engineering challenges. The elastic foundations play an important role that preserve the structural system under oscillations and avoid mechanical failures. The main role of nonlinear foundations is developing the accuracy of the model to describe the behavior of our system. The cubic nonlinearity is the most common type used in recent research .

Due to the mathematical complexity of such problems, only limited cases can analytically be solved. A number of exact methods of the vibration problems on elastic foundations are issued in . Literature on the numerical solution of the research subject is sparse. Typical useful numerical methods, such as the finite element , the boundary element , differential quadrature , and state space [20, 21] techniques, are used to solve such problems. One of the disadvantages of these methods is consuming too much time and effort to get the solution . Additionally, there is no research in piezoelectric composite plate resting on nonlinear elastic foundations, whether analytically or numerically.

The sinc differential quadrature method (SDQM)  and the discrete singular convolution differential quadrature method (DSCDQM)  are more reliable than the polynomial-based DQM. The nonlinear partial differential equations are reduced to linear equations using perturbation  and iterative quadrature methods . Previous DQ techniques are then employed to reduce the problem to the eigenvalue or bending problem. The same schemes are used for free vibration analysis of piezoelectric nanobeams [45, 46]. Recent studies have investigated the vibration analysis of composite plate resting on elastic foundations, using DSCDQM .

As far as the authors are aware, SDQM and DSCDQM have not been examined for vibration analysis of composite piezoelectric plate materials resting on nonlinear elastic foundations. Based on these versions, numerical schemes are designed for free vibration of piezoelectric composites. The natural frequencies are obtained and compared with previous analytical and numerical frequencies. For each scheme, the convergence and efficiency are verified. Additionally, a parametric study is introduced to investigate the influence of elastic foundation parameters and elastic and geometric characteristics of the composite on the vibrated results.

#### 2. Formulation of the Problem

Consider a three-dimensional piezoelectric composite with , where a, b, and h are the length, width, and total thickness of the composite, respectively. This composite is polarized in the z direction and consists of m layers with different types of materials. The plate rests on a three-parameter foundation model, as shown in Figure 1.

Based on the theory of elasticity and piezoelectricity, the equations of motion and the charge equation of electrostatic can be written as follows [59, 6569]:where , , and are the stresses, displacement, and induction field in the x, y, and z directions, respectively; are the shear stresses; is the density of the material; and (k1, k2, k3) are linear Winkler foundation, linear Pasternak foundation, and nonlinear Winkler foundation parameters, respectively.

The relation between mechanical and electric material properties is constitutive equations, which can be written as follows:where C, e, and are the components of the effective elastic, piezoelectric, and dielectric constants of the same piezoelectric material, respectively, and is the electric potential.

For harmonic behavior, one can assume thatwhere ω is the natural frequency of the plate and . are the amplitudes for , and , respectively.

The elastic material constants can be determined as follows, using the reciprocal theorem :where Ep, Gpq, and (p, q = 1, 2, 3) are Young’s moduli, shear moduli, and Poisson’s ratios.

The isotropic material constants can be expressed as follows :

Substituting equations (5)–(16) into (1)–(4), the problem can be reduced to a quasistatic one, as follows:

The boundary conditions can be described as follows:(1)For a simply supported edge (S):(2)For a clamped edge (C):(3)For a free edge (F):

Mechanical and electrical boundary conditions at the lower and upper surfaces of the composite are as follows:

To ensure the continuity between electric and elastic layers, the following conditions can be considered:

Additionally, the continuity conditions between different elastic materials are as follows:

#### 3. Method of Solution

Two differential quadrature techniques are employed to reduce the governing equations into a nonlinear eigenvalue problem, as follows.

##### 3.1. Sinc Differential Quadrature Method (SDQM)

A cardinal sine function is used as a shape function, such that the unknown ψ and its derivatives can be approximated as a weighted linear sum of nodal values, ψi, (i= −N, N), as follows :where ψ denotes , and ; N is the number of grid points; and hx is the grid size. The weighting coefficients, and , can be determined by differentiating (28) as follows:

##### 3.2. Discrete Singular Convolution Differential Quadrature Method (DSCDQM)

A singular convolution can be defined as follows :where is a singular kernel.

The DSC algorithm can be applied using many types of kernels. These kernels are applied as shape functions, such that the unknown ψ and its derivatives are approximated as a weighted linear sum of ψi, (i = −N, N) over a narrow bandwidth () .

Two kernels of DSC will be employed as follows:(a)A Delta Lagrange kernel (DLK) can be used as a shape function, such that the unknown ψ and its derivatives can be approximated as a weighted linear sum of nodal values, ψi, (i= −N, N), as follows:where 2M + 1 is the effective computational bandwidth. are defined as follows:(b)A regularized Shannon kernel (RSK) can also be used as a shape function, such that the unknown ψ and its derivatives can be approximated as a weighted linear sum of nodal values, ψi, (i= −N, N), as follows:where σ is a regularization parameter and r is a computational parameter. The weighting coefficients and can be defined as follows :

Similarly, one can approximate , and calculate , . Two methods are used to transform the governing equation to linear equations.

###### 3.2.1. Perturbation Method

Assuming the solution of equations (17)–(27) to be a power series that can be solved by a perturbation method of the second order  by substituting into (17)–(27) and equating the terms with the identical powers of , where is a perturbation parameter:

Equate the terms of in (39)–(42):

The previous system is an eigenvalue problem, which is solved to obtain the natural frequencies and the bending deflection W0.

Equate the terms of in (39)–(42):

The previous system is a bending problem, which is solved to obtain W1.

Equate the terms of in (39)–(42):

The previous system is a bending problem, which is solved to obtain W2.

Finally, the series solution can be written as follows:

The convergence condition  due to the perturbation method is set as follows:

We solved the following iterative system:

To get , we solved the following eigenvalue problem:

The boundary conditions (21)–(27) can also be approximated using two DQMs as follows:(1)Simply supported (S):(2)Clamped (C):(3)Free surface (F):

Mechanical and electrical boundary conditions at the lower and upper surfaces of the composite are as follows:

The continuity conditions between the interfaces of layers can be assumed as follows:

We have solved the generalized eigenvalue problem :where K is the coefficient matrix of the previous system, M is the mass matrix and can be diagonal with 0 diagonal elements, and is the free vibration frequencies squared.

For nontrivial solutions for (70), the following determinant should be 0:

Equation (71) gives the natural frequencies for the composite plate.

#### 4. Numerical Results

The present numerical results demonstrate the convergence and efficiency of each one of the proposed schemes for vibration analysis of the piezoelectric composite plate Al/Al2O3 resting on a nonlinear elastic foundation. For all results, the boundary conditions (65)–(69) are augmented in the governing equations (39)–(42).

For practical purposes, the field quantities are normalized as follows:where are the normalized amplitudes of displacements, are the normalized amplitudes of stresses, are the normalized amplitudes of shear stresses, are the thicknesses of the actuator and sensor, and are the flexural rigidity for the bottom layer of composite and the equivalent dimensionless elastic foundation constants.

The computational characteristics of each scheme are adapted to reach accurate results with error of order ≤10−8. The obtained dimensionless frequencies , , and are evaluated as follows: , where and are the density and Young’s modulus, respectively.

For the present results, material parameters for the composite are listed in Table 1.

 Material property Young’s moduli (GPa) Shear moduli (GPa) Poisson’s ratios Density (kg/m3) E2 E1 E3 G12 G13 G23 7 25E2 E2 0.5E2 G12 0.2E2 0.25 0.03 0.4 1600 Metal (aluminum, Al) 70 — — — 0.3 2702 Ceramic (alumina, Al2O3) 380 — — — 0.3 3800 Effective elastic (GPa) C11 C12 C13 C22 C23 C33 C44 C55 C66 Sensor PZT-4 139 78 74 139 74 115 25.6 25.6 30.5 Sensor BaTiO3 166 77 78 166 78 162 43 43 44.5 Actuator Ba2NaNb5O15 239 104 5 247 52 135 65 66 76 Actuator PZT-5A 121 77 77 121 111 21 21 21 23 Material property Piezoelectric constants (C/m2) Dielectric constants (F/m) ∗ 10−9 Density (kg/m3) e1 e2 e3 e4 e5 Sensor PZT-4 −5.2 −5.2 15.1 12.7 12.7 6.5 6.5 5.6 7500 Sensor BaTiO3 −4.4 −4.4 18.6 11.6 11.6 11.2 11.2 12.6 5700 Actuator Ba2NaNb5O15 −0.4 −0.3 4.3 3.4 2.8 19.6 2.01 0.28 5300 Actuator PZT-5A −5.4 −5.4 15.8 3.4 3.4 8.11 8.11 7.34 2330

For the sinc DQ scheme, the problem is solved over a regular grid, ranging from 3 × 5 × 5 to 11 × 5 × 5. Table 2 shows the convergence of the obtained results for the isotropic plate, which is in agreement with the exact results in  over grid size ≥7 × 5 × 5.

 Results Normalized frequencies N Sinc DQM 3 55.9218 103.4553 124.1464 161.3903 177.5293 186.4058 5 55.9721 103.5484 124.2581 161.5355 177.6891 186.5736 7 56.0369 103.6682 124.2982 161.5391 177.6930 186.5777 9 56.0369 103.6683 124.2982 161.5391 177.6930 186.5777 11 56.0369 103.6683 124.2982 161.5391 177.6930 186.5777 Analytical  56.0359 — — — — — RHSDT  56.0310 — — — — — NSDT  56.0311 — — — — — Execution time (sec) 5.615225 over N = 7 × 5 × 5
RHSDT: refined hyperbolic shear deformation theory; NSDT: new shear deformation theory; DQM: differential quadrature method.

For DSCDQ scheme based on Delta Lagrange kernel, the problem is also solved over a uniform grid ranging from 3 × 5 × 5 to 11 × 5 × 5. The bandwidth 2M + 1 ranges from 3 to 11.

Table 3 shows the convergence of the obtained fundamental frequency, which is in agreement with the exact results in  over grid size ≥7 × 5 × 5 and bandwidth ≥5. Table 4 shows that the obtained results are more accurate than those obtained using approximated theories [60, 69]. This table also shows that the execution time of DSCDQM-DLK was less than that of sinc DQM.

 Fundamental frequency DSCDQM-DLK Bandwidth N 3 5 7 9 11 2M + 1 = 3 55.9844 56.0849 56.0685 56.0685 56.0685 2M + 1 = 5 55.9524 56.0349 56.0320 56.0320 56.0320 2M + 1 = 7 56.1798 56.0650 56.0320 56.0320 56.0320 2M + 1 = 9 55.9384 56.2877 56.0320 56.0320 56.0320 2M + 1 = 11 55.9384 56.2877 56.0320 56.0320 56.0320
 Normalized frequencies Results N DSCDQM-DLK 3 55.9524 103.5119 124.2142 161.4660 177.6126 186.4932 5 56.0349 103.6085 124.3302 161.5049 177.6553 186.5380 7 56.0320 103.6031 124.3237 161.4964 177.6444 186.5266 9 56.0320 103.6031 124.3237 161.4964 177.6444 186.5266 Analytical  56.0359 — — — — — RHSDT  56.0310 — — — — — NSDT  56.0311 — — — — — Execution time (sec) 5.611001 over N = 7 × 5 × 5

For the DSCDQ scheme based on the regularized Shannon kernel (RSK), the problem was also solved over a uniform grid ranging from 3 × 5 × 5 to 9 × 5 × 5. The bandwidth 2M + 1 ranged from 3 to 11 and the regularization parameter σ = r hx ranged from 1.0 hx to 2.3 hx, where hx = 1/N − 1. Table 5 shows the convergence of the obtained fundamental frequency to the exact results in  over grid size ≥5 × 5 × 5, bandwidth ≥3, and regularization parameter σ = 1.9 hx.

 Fundamental frequency Regularization parameter DSCDQM-RSK N 2M + 1 σ = 1 × hx σ = 1.5 × hx σ = 1.9 × hx σ = 2.3 × hx 3 3 55.9564 55.9468 55.9853 56.0821 5 55.9561 55.8991 55.9780 56.0706 7 55.9561 55.8991 55.9780 56.0706 5 3 55.8758 55.9715 56.0318 56.0318 5 55.8758 55.9715 56.0318 56.0318 7 55.8758 55.9715 56.0318 56.0318 7 3 55.8507 55.9392 56.0318 56.0318 5 55.9563 55.9790 56.0318 56.0318 7 55.9926 55.9917 56.0318 56.0318 9 3 55.9259 55.9857 56.0318 56.0318 5 55.9823 55.9733 56.0318 56.0318 7 55.9506 56.0193 56.0318 56.0318