Shock and Vibration

Volume 2017 (2017), Article ID 5905417, 10 pages

https://doi.org/10.1155/2017/5905417

## A Framework for Extension of Dynamic Finite Element Formulation to Flexural Vibration Analysis of Thin Plates

Department of Aerospace Engineering, Ryerson University, 350 Victoria Street, Toronto, ON, Canada M5B 2 K3

Correspondence should be addressed to Mohammad M. Elahi

Received 6 July 2017; Revised 2 September 2017; Accepted 24 September 2017; Published 19 October 2017

Academic Editor: Yuri S. Karinski

Copyright © 2017 Mohammad M. Elahi and Seyed M. Hashemi. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Dynamic Finite Element formulation is a powerful technique that combines the accuracy of the exact analysis with wide applicability of the finite element method. The infinite dimensionality of the exact solution space of plate equation has been a major challenge for development of such elements for the dynamic analysis of flexible two-dimensional structures. In this research, a framework for such extension based on subset solutions is proposed. An example element is then developed and implemented in MATLAB® software for numerical testing, verification, and validation purposes. Although the presented formulation is not exact, the element exhibits good convergence characteristics and can be further enriched using the proposed framework.

#### 1. Introduction

Dynamic stiffness modeling is a well-established technique in vibrational analysis of structural elements. These methods seek to propose formulations that have the accuracy of the exact solutions and wider applicability of the finite element methods by incorporating some form of the closed form solutions of governing equations instead of polynomials used by classic finite element method (FEM).

Two of the most famous dynamic stiffness formulations, mainly applied to various beam-structures, are dynamic stiffness matrix (DSM) and Dynamic Finite Element (DFE) methods, which produce accurate results with much coarser mesh compared to FEM formulations.

Extensively developed in 1970s, and pioneered again by Banerjee and his coworkers since 1990s [1–8], the beam DSM formulation dates back to much earlier times [9]. The 1941 work by Kolousek [9] is probably the first to derive the dynamic stiffness matrix for the Euler-Bernoulli beam. The DSM formulation is performed by developing a dynamic relation between force and displacement matrix for a problem domain, where the displacement functions are combination of trigonometric and hyperbolic frequency-dependent functions, which satisfy the Euler-Bernoulli or Timoshenko beam equations. In this method, the boundary and loading conditions of the problem in hand are taken into consideration in developing the force-displacement relationship and thus this process yields a one-step formulation to the problem but is very case specific [1–6]. Using truncated Taylor series expansion and assigning penalty for different boundary conditions, the DSM has been more recently generalized by Pagani and his coworkers [7, 8] to vibrational analysis of composite beams and solid and thin-walled structures subjected to various boundary conditions. This approach was also further extended to use higher-order 1D dynamic stiffness elements for the vibration analysis of composite plates [10].

On the other hand, the DFE technique follows the general procedure of FEM element development by using the general displacement solution of beam equation as basis functions. Next, the element dynamic shape functions are developed and supplied to a residual minimization scheme, such as Galerkin’s technique, to develop the frequency-dependent element matrix. The matrix thus developed is not case specific and can be assembled. The boundary and loading conditions are applied in a manner similar to FEM [11–14].

Both DSM and DFE methods generate one matrix per problem which has the effects of mass and stiffness combined. In fact, the mass terms are distributed over the element displacement as a dynamic force using D’Alembert’s principle. The assembly of the element dynamic stiffness matrices and the application of the system boundary conditions then lead to a nonlinear eigenvalue problem, which is then solved using a root finding technique, such as Wittrick-Williams algorithm [15].

Both the DSM and DFE formulations have been very successful for one-dimensional elements such as rods, beams, and beam-structures under various loading and boundary conditions. The expansion of such techniques to two-dimensional (2D) elements such as plates, however, has led to cumbersome mathematics with limited applicability which served as a motive for current research. The particular problems with such extension arise from the fact that the plate governing equations are two-dimensional partial differential equations with infinite number of solutions. This is unlike beam and rod situation where the frequency-dependent solution of the ordinary differential equation can be used directly as the basis functions of approximation space to develop set of dynamic shape functions [16].

The other problem in the case of DFE formulation comes from evaluating the resulting integral equations, written in terms of the dynamic shape functions, to develop element matrices. Since these functions are transcendentally dependent on frequency, the integrations are difficult to handle given the current computational power of computers especially for infinite dimensional domains. This difficulty is particularly important when extension of such elements to irregular shapes is required. In such situations, the area integrals used in mapping of the element to its natural coordinate system will involve the absolute value of Jacobian mapping matrix that is difficult to handle for such complicated displacement functions.

Because the Dynamic Finite Element matrices are a result of these integrations, use of numerical integration schemes such as Gauss quadrature formula will require introduction of numerous evaluation points for an acceptable result. This is unlike FEM extension to plates of arbitrary shapes where the polynomial shape functions provide descent result with small number of evaluation points. Besides, FEM shape functions are much easier to evaluate numerically as they do not have any dependency on vibrational frequency and produce constant matrices of numerical values that can be manipulated with ease. Therefore, much of the available literature for exact solution of plates was focused on solving these equations for simple geometries and loading cases, as presented by Leissa [17–22], or involved variation error minimization techniques, such as finite element method [23–34].

Recently, there have been more researches to extend the dynamic stiffness matrix scheme to plate vibrations. Since this method involves force-displacement relationship development to arrive at the element DSM, it requires developing differential relationships, avoiding the integration problem. Casimir et al. [16] developed the DSM for rectangular thin plates and demonstrated that accurate results can be achieved with limited number of elements. However, their method involved solving infinite dimensional matrices, limited to simple cases, and provided problem specific formulation that must be reformulated for each new configuration. This reformulation for plate elements would therefore involve solving large matrices and is less favourable to be used as a general analysis tool. More recently, Liu and Banerjee [35–37] presented a novel Spectral-DSM (S-DSM) method to generalize the DSM using truncation of Fourier series and the application of spectral analysis. The S-DSM, when applied to the free vibration analysis of complex problems involving isotropic laminated plates under various boundary conditions, demonstrated significantly improved efficiency over the conventional FEM formulations.

In this paper, the focus is to present an alternate dynamic stiffness formulation in the form of DFE for the vibration analysis of Kirchhoff plates. Particularly, the focus is to generate a model that can be easily extended to plates of different shapes without loss of accuracy of the original formulation. The mathematical procedure of beam DFE formulation served as a guideline for development of the current plate DFE design, as presented in the following section. An example 4-node 12-DOF element is then developed and validated against FEM, where its accuracy and efficiency are demonstrated through the free flexural vibration analysis.

#### 2. Mathematical Modeling

To develop Dynamic Finite Element of thin plates, we begin with the partial differential equation governing the free flexural vibration [38]:where is the flexural displacement, is the mass density, is the thickness, and represents bending coefficient. The terms and denote Young’s modulus and Poisson’s ratio of the plate material, respectively.

Applying Galerkin’s weighted residual minimization scheme [39], the integral form of the governing equation (1) is written aswhere is the residual weighting function and is the differential of the area within closed element boundary.

Green’s theorem is a conversion mechanism between area and line integrals performed over a closed boundary. For functions and Green’s theorem stateshere, and are components of the unit normal vector to the element boundary, and is differential segment along the boundary.

Since Galerkin’s method uses the same approximation for both field variable and weighting function , by applying Green’s theorem (4) to the above area integrals twice and taking the conformity of virtual displacements and loads across element boundaries into consideration, one can change Galerkin’s integral equation (3) to its equivalent weak form:where can also be interpreted as the virtual displacement. Equation (5) is the formulation generally used for finite element development of plates. The DFE derivation will deviate from FEM at this point.

Let be the dynamic flexural displacement, where is a row vector containing the dynamic shape function, derived from the frequency-dependent (dynamic) basis functions of approximation space, and is the vector of nodal displacements. The dynamic basis functions, in this case, are defined to be the solutions of the Kirchhoff’s plate governing equation.

*Applying Green’s theorem (4) to the above area integrals twice, (5) can be written as*

Note that the last 4 area integrals can be combined to generate Kirchhoff’s plate equation in terms of virtual displacement and thus can be written as

Assuming, as mentioned earlier, the dynamic shape functions are derived from the dynamic basis function, which are in turn solutions of Kirchhoff’s equation, the terms in the bracket will yield zero. Therefore, the DFE stiffness matrix can be obtained using only evaluation of line integrals performed over element boundaries, as follows:

Although not explicitly visible, the above matrix is symmetric as it was obtained using mathematical manipulation of the symmetric Galerkin’s weak integral form.

##### 2.1. Extension to Arbitrary Shapes

The absence of area integrals in DFE stiffness matrix evaluation enables exact integration of the boundary line integrals when transformed between different coordinate systems using variable substitution. Consider, for example, the quadrilateral case given in Figure 1(b).