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

In this work, free vibration of the piezoelectric composite plate resting on nonlinear elastic foundations is examined. 'e threedimensionality 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. 'e 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. 'e 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.


Introduction
e 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. e elastic foundations play an important role that preserve the structural system under oscillations and avoid mechanical failures. e main role of nonlinear foundations is developing the accuracy of the model to describe the behavior of our system. e cubic nonlinearity is the most common type used in recent research [1][2][3][4][5][6][7][8].
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. e 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.

Formulation of the Problem
Consider a three-dimensional piezoelectric composite with (0 ≤ x ≤ a, 0 ≤ y ≤ b, 0 ≤ z ≤ h), where a, b, and h are the length, width, and total thickness of the composite, respectively. is composite is polarized in the z direction and consists of m layers with different types of materials. e 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,[65][66][67][68][69]: τ xy,x + σ y,y + τ yz,z � ρv ,tt , τ xz,x + τ yz,y + σ z,z + k 1 w + k 3 w 3 − k 2 w ,xx + w ,yy � ρw ,tt , where (σ x , σ y , σ z ), (u, v, w), and (D x , D y , D z ) are the stresses, displacement, and induction field in the x, y, and z directions, respectively; (τ yz , τ xz , τ xy ) are the shear stresses; ρ is the density of the material; and (k 1 , k 2 , k 3 ) are linear Winkler foundation, linear Pasternak foundation, and nonlinear Winkler foundation parameters, respectively. e relation between mechanical and electric material properties is constitutive equations, which can be written as follows: where C, e, and η are the components of the effective elastic, piezoelectric, and dielectric constants of the same piezoelectric material, respectively, and ϕ is the electric potential.
For harmonic behavior, one can assume that where E p , G pq , and v pq (p, q � 1, 2, 3) are Young's moduli, shear moduli, and Poisson's ratios.
e isotropic material constants can be expressed as follows [56]: , Substituting equations (5)-(16) into (1)-(4), the problem can be reduced to a quasistatic one, as follows: e 1 + e 5 U ,xz + e 2 + e 4 V ,yz + e 5 W ,xx + e 4 W ,yy + e 3 W , Mathematical Problems in Engineering 3 e 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:

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

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, ψ i , (i � −N, N), as follows [27][28][29][30][31][32]: where ψ denotes U, V, W, and Φ; N is the number of grid points; and h x is the grid size. e weighting coefficients, A x ij and B x ij , can be determined by differentiating (28) as follows:
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, ψ i , (i � −N, N), as follows: where 2M + 1 is the effective computational bandwidth. A x ij and B x ij 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, ψ i , (i � −N, N), as follows: where σ is a regularization parameter and r is a computational parameter. e weighting coefficients A x ij and B x ij can be defined as follows [33][34][35][36][37][38][39][40][41][42]: Similarly, one can approximate ψ y , ψ z , ψ yy , and ψ zz and calculate A y ij , A z ij , B y ij , and B z ij . Two methods are used to transform the governing equation to linear equations.

Mathematical Problems in Engineering
Equate the terms of ε 0 in (39)- (42): e previous system is an eigenvalue problem, which is solved to obtain the natural frequencies and the bending deflection W 0 .
Equate the terms of ε 1 in (39)- (42): e previous system is a bending problem, which is solved to obtain W 1 .
Equate the terms of ε 2 in (39)-(42): e previous system is a bending problem, which is solved to obtain W 2 .
Finally, the series solution can be written as follows: e convergence condition [43] due to the perturbation method is set as follows: 3.2.2. Iterative Quadrature. We solved the following iterative system: 6 Mathematical Problems in Engineering To get W o , we solved the following eigenvalue problem: e boundary conditions (21)- (27) can also be approximated using two DQMs as follows: (1) Simply supported (S): Mathematical Problems in Engineering 7 (2) Clamped (C): (3) Free surface (F): Mechanical and electrical boundary conditions at the lower and upper surfaces of the composite are as follows: e continuity conditions between the interfaces of layers can be assumed as follows: We have solved the generalized eigenvalue problem [65]: where K is the coefficient matrix of the previous system, M is the mass matrix and can be diagonal with 0 diagonal elements, and ω 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. For practical purposes, the field quantities are normalized as follows:

Numerical Results
where U * , V * , and W * are the normalized amplitudes of displacements, σ * x , σ * y , and σ * z are the normalized amplitudes of stresses, τ * xy , τ * xz , and τ * yz are the normalized amplitudes of shear stresses, h a and h s are the thicknesses of the actuator and sensor, and D, K 1 , K 2 , and K 3 are the flexural rigidity for the bottom layer of composite and the equivalent dimensionless elastic foundation constants. e computational characteristics of each scheme are adapted to reach accurate results with error of order ≤10 −8 . e obtained dimensionless frequencies Ω, ω, and β are evaluated as follows: Ω � ωh ��� ρ/E, ω � ωb/π 2 ���� � ρh/D, and β � ωa 2 ���� � ρh/D, where ρ and E are the density and Young's modulus, respectively.
For the present results, material parameters for the composite are listed in Table 1.
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 [68] over grid size ≥7 × 5 × 5.
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. e 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 [68] 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].
is table also shows that the execution time of DSCDQM-DLK was less than that of sinc DQM.
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. e bandwidth 2M + 1 ranged from 3 to 11 and the regularization parameter σ � r h x ranged from 1.0 h x to 2.3 h x , where h x � 1/N − 1. Table 5 shows the convergence of the obtained fundamental frequency to the exact results in [68] over grid     Table 5: Comparison between the normalized fundamental frequency β using the discrete singular convolution differential quadrature method on regularized Shannon kernel (DSCDQM-RSK), bandwidth (2M + 1), regularization parameter σ, and grid size N for the simply supported square isotropic plate (h c /h p � 200, a/h � 5, K 1 � 1000, K 2 � 100, and K 3 � 0).     Table 6 shows that the obtained results from DQ schemes were more accurate than other models in the literature [60,69]. Further, the execution time of this scheme is the least. erefore, the DSCDQM-RSK scheme is the best choice among the examined quadrature schemes for vibration analysis of piezoelectric composite materials resting on nonlinear elastic foundations. Furthermore, a parametric study is introduced to investigate the influence of foundation   parameters, elastic and geometric characteristics of the composite, and types of material on the values of natural frequencies and mode shapes. Tables 7 and 8 show the natural frequency for isotropic plates. ese tables show the effect of elastic foundation parameters on the frequency for different boundary conditions. e natural frequency increased with the increase in foundation parameters K 1 and K 2 . Additionally, the SCSC plate had the highest frequency and the SSSF had the lowest frequency. Moreover, the efficiency of the nonlinear treatment methods was investigated. e execution time for the iterative quadrature method was less than the perturbation method, with a factor close to 1.5 until 1.9 for all boundary conditions used in Table 8. Tables 8-10 and Figures 2-5 show that the natural frequencies increase with increasing side-to-thickness ratio (a/h), foundation parameters K 1 , K 2 , K 3 , total thickness, Young's modulus gradation ratio (E 1 /E 2 ), shear modulus gradation ratio (G 13 /G 12 ), and number of layers. e natural frequencies converge at a total thickness (h � 0.075) in the   CCCC plate and a total thickness (h � 0.035) in the SSSSS plate, as shown in Tables 10 and 11. Figure 6 shows the natural frequencies decrease with the decrease of the piezoelectric layer thickness (h c /h p ). Furthermore, Figure 7 shows that the natural frequencies decreased with an increase in the aspect ratio (a/b) at different values (a/h). Moreover, the natural frequencies remained constant when h c /h p > 20 in all different edge conditions. Figures 2, 3, and 8 show the effect of the material type on the natural frequency. It can be seen that the sensor was more significant than the actuators. e natural frequency was almost unchanged when a/h > 40. Additionally, the influence of the BaTiO 3 material was more affected by the natural frequencies than the PZT-4 material. Furthermore, Figures 9-12 show the first three mode shapes of normalized transverse displacement (w) and electric potential (φ). From the previous figures, it can be seen that the normalized transverse displacement was at a maximum for the CCCF plate and was at a minimum for the CCCC plate. Additionally, the electric potential was at a maximum for the CCCF plate and at a minimum for the CSCS plate. Furthermore, the previous figures show that the amplitude of displacement and electric potential increased with the increase of the nonlinear Winkler parameter K 3 .

Conclusions
Accurate quadrature schemes have been applied for free vibration analysis of piezoelectric composite plates with and without nonlinear elastic foundations for different types of boundary conditions. Numerical results were given to demonstrate the convergence, accuracy, and efficiency of the present methods. A MATLAB program was designed for each one, such that the maximum error (comparing with the previous exact results) was ≤10 −8 . Additionally, the execution time for each scheme was determined. It is concluded that the convergence speed for the discrete singular convolution differential quadrature method based on regularized Shannon kernel (DSCDQM-RSK) with grid size ≥5 × 5 × 5, bandwidth 2M + 1 ≥ 3, and regularization parameter σ � 1.9 h x is very fast and has the best accurate efficient results for the concerned problem (three-layered piezoelectric composite with total thickness � 0.075 in the CCCC plate and h � 0.035 in the SSSS plate). e efficiency of the iterative quadrature method is better than the perturbation technique. e influences of linear and nonlinear elastic foundation on the natural frequency and higher modes of frequency are investigated. e effect of the Pasternak parameter is larger than the effect of the linear and nonlinear Winkler coefficients on the natural frequencies.
e natural frequency does not change when varying the nonlinear Winkler parameter. Additionally, the natural frequency converges at a larger value of the linear Winkler parameter. A parametric study is also introduced to investigate the influence of elastic and geometric characteristics of the composite and types of materials of the vibrated plate on the results. e nondimensional frequency for the material type BaTiO 3 is larger than that for the PZT-4 material. Further, the natural frequency decreases with the decrease in the thickness of the piezo material or with the increase in the composite thickness until the value of ratio h c / h p closes to 10, so the natural frequency remains constant. e thinner composite has frequencies larger than thick composite. e composite plate with CCCC conditions has the minimum normalized transverse displacement, whereas it has the maximum value for CCCF conditions. Additionally, the electric potential is the maximum for the CCCF plate and has a minimum value for the CSCS plate. Moreover, the vertical displacement increases slightly when enlarging the nonlinear parameter. Finally, the aim is for these results to be useful for modern engineering smart structures.
Data Availability e data in this paper are all open, and the data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
e authors declare that they have no conflicts of interest.