Free vibration analysis of symmetrically laminated thin composite plates by using discrete singular convolution (DSC) approach: Algorithm and verification

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Abstract

This study presents a detailed procedure for the implementation of a discrete singular convolution (DSC) approach to the free vibration analysis of composite plates based on classical laminated plate theory (CLPT). The approach performs a numerical solution of differential equation of motion by using a grid discretization based on distribution theory and wavelets. In the paper, firstly, computational algorithm of the DSC method is presented. Then, the accuracy of the computer code developed is verified by comparing DSC solutions with the exact results of simply supported isotropic thin beams, fully simply supported one-layer isotropic and specially orthotropic plates, and also some symmetrically laminated thin composite plates orientated to become specially orthotropic. Besides, DSC predictions for laminated composite plates with different boundary conditions and ply numbers, for which there is no analytical solution, are compared with those of several distinguished works available in the literature. It is noteworthy that DSC results completely match with the exact solutions and are in perfect agreement with those of compared studies.

Introduction

Laminated composites are increasingly used in various mechanical structures and industrial applications such as aircrafts, automobiles, marines, buildings and several house-hold appliances due to their, in particular, higher stiffness and higher strength-to-weight ratio compared to isotropic or wooden materials. In vibration engineering, modal parameters of a structure are primary design information, because they directly affect the forced response characteristics. Conventional methods for vibration analysis are generally based on either theoretical solutions or experimental studies. However, in general, practical problems are either too difficult or impossible to deal with by analytical methods and experiments are rather expensive. Therefore, numerical simulations and algorithms are of significant role in modern vibration analysis. The finite element method (FEM) has been commonly used in the vibration analysis of composite plates. Significant studies up to the 1980s on the vibration analysis of laminated composite plates by the finite element method were reviewed by Reddy [1]. Reddy and Averill [2] also presented refined two-dimensional theories and computational models of laminated composite plates and reviewed the computational aspects of finite element models of these refined theories. Ritz, p-Ritz and Rayleigh–Ritz approaches are successfully employed in the vibration analysis of laminated plates [3], [4], [5], [6], [7], [8], [9], [10]. The differential quadrature technique introduced by Bellman et al. [11] has been applied in the vibration analysis of both isotropic and composite plates [12], [13], [14], [15], [16]. Besides, several alternative techniques have been increasingly used [17], [18], [19], [20], [21].

In the last decade, a novel approach originally introduced by Wei [22], [23] and called “discrete singular convolution (DSC)” analysis has presented a powerful technique for the numerical solution of differential equations. The solution technique of DSC is based on the theory of distribution and wavelets. The technique includes both the flexibility of local methods and the accuracy of global methods. The DSC method has been reliably used in various vibration analyses: Wei [24], [25], [26] and Wei et al. [27], [28], [29] showed that the DSC method can be effectively used in the vibration analysis of isotropic beams and plates with several uniform and non-uniform boundary conditions. Wei et al. [30] and Zhao et al. [31] proved the accuracy of the DSC method in the prediction of high natural frequencies of beams and plates. At present, these high-frequency predictions are unique results numerically obtained. Furthermore, Ng et al. [32] clearly indicated that the DSC yields more accurate predictions compared to the differential quadrature method for higher-order eigenfrequencies.

In the literature, the implementation procedure of DSC is presented rather implicitly. In this paper, the basic algorithm of the DSC approach and boundary condition implementation are clearly introduced. A computer code has been developed on the basis of the DSC for the free vibration analysis of composite plates based on classical laminated plate theory (CLPT). The accuracy of the code is verified by comparing the DSC free vibration results with the exact ones for simply supported isotropic thin beams, fully simply supported one-layer isotropic and specially orthotropic plates, and some symmetrically laminated thin composite plates orientated to become specially orthotropic. In addition, free vibrations of several laminated thin composite plates, which have no analytical solutions, are predicted by DSC for different boundary conditions and ply numbers. The results are compared with those of various published studies utilizing different methods.

Section snippets

Bending vibrations of symmetrically laminated plates based on CLPT

Time-independent differential equation of harmonic bending vibration for a symmetrically laminated thin composite plate with natural frequency ω having side lengths a and b, total thickness h, average mass density ρ0 and Poisson rate υ can be written in Cartesian co-ordinates (x, y) in terms of flexural displacement w as follows [33]: D114w(x,y)x4+4D164w(x,y)x3y+2(D12+2D66)4w(x,y)x2y2+4D264w(x,y)xy3+D224w(x,y)y4-ρ0hω2w(x,y)=0.Here, D11, D12, D22 and D66 are the bending rigidities

Theory of the DSC

Singular convolution is defined by the theory of distributions. Let T be a distribution and η(t) be an element of the space of test functions. Then, a singular convolution can be given by [22]F(t)=(T*η)(t)=-T(t-x)η(x)dx.Here, the sign * is the convolution operator, F(t) is the convolution of η and T, T(tx) is the singular kernel of the convolution integral. Depending on the form of the kernel T, singular convolution can be applied to different science and engineering problems. Delta kernel

Implementation of the DSC to symmetrically laminated plates

Applying linear DSC operator L, which performs the DSC approach in Eq. (23), to Eq. (4), one can obtain a discretized governing equation of symmetrically laminated composite plates in a non-dimensional form:Dγk=-MMδπ/Δ,σ(4)(kΔ)W(Xi+k,Y)+2λ2Dφ(k=-MMδπ/Δ,σ(2)(kΔ)W(Xi+k,Y)k=-MMδπ/Δ,σ(2)(kΔ)W(X,Yi+k))+λ4k=-MMδπ/Δ,σ(4)(kΔ)W(X,Yi+k)+4λDα(k=-MMδπ/Δ,σ(3)(kΔ)W(Xi+k,Y)k=-MMδπ/Δ,σ(1)(kΔ)W(X,Yi+k))+4λ3Dβ(k=-MMδπ/Δ,σ(1)(kΔ)W(Xi+k,Y)k=-MMδπ/Δ,σ(3)(kΔ)W(X,Yi+k))=Ω2W(X,Y).

The DSC full matrix: DSC

Conclusions

Thin plates made of composite materials present many advantages in the use of several industrial applications. Although a number of commercial codes based on conventional methods are used in the vibration analysis of composite structural elements, researchers have been working to develop more accurate, more effective, easy to use, operational frequency-independent new approaches. In this regard, this study proves the applicability of the DSC approach to the free vibration analysis of composite

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