M.K. Lim

A continuum three-dimensional vibration analysis of thick rectangular plates

Abstract This paper presents a continuum three-dimensional Ritz formulation for the vibration analysis of homogeneous, thick, rectangular plates with arbitrary combinations of boundary constraints. This model is formulated on the basis of the linear, three-dimensional, small deformation elasticity theory to predict the vibratory responses of these thick rectangular plates. The displacement fields in the transverse and in-plane directions are expressed by sets of orthogonally generated polynomial functions. These shape functions are intrinsically a product of a class of orthogonal polynomial functions and a basic function which are chosen to satisfy the essential geometric boundary conditions at the outset. Sets of frequency data for plates with various aspect ratios and thickness ratios have been presented. These data are used to examine the merits and limitations of the classical plate theory and Mindlin plate theory by direct comparisons. Finally, using the three-dimensional continuum approach, sets of first known deformed mode shapes have been generated thus helping to understand the vibratory motion. Furthermore, these results may also serve as the benchmark to further research into the refined plate theories.

A solution method for analysis of cracked plates under vibration

Abstract An investigation on the vibrational behaviour of cracked rectangular plates is reported. Vibration analysis is carried out for plates with a crack (i) emanating from an edge or (ii) centrally located. A highly computationally efficient and accurate domain decomposition method is presented. To establish this discrete model, the cracked plate domain is assumed to be an assemblage of small subdomains with the appropriate shape functions formed according to the boundary conditions. The complete coupling process leads to a governing eigenvalue equation that can be solved to obtain the vibration frequencies. These results, where possible, are compared with the work of other investigators. With these results, the effects of various geometric parameters on the vibration response of cracked plates can be examined. Some concrete conclusions can be deduced from a careful examination of these results.

Improved Inverse Eigensensitivity Method for Structural Analytical Model Updating

In order to update analytical models of practical engineering structures, inverse eigensensitivity method (IEM) has been developed. Though it has nowadays been widely accepted, the classical inverse eigensensitivity method does have some drawbacks such as the assumption of small error magnitudes and slow speed of convergence due to the fact that the sensitivity coefficients are calculated purely based on modal data of analytical model. In the present paper, an improved inverse eigensensitivity method, which avoids the existing problems of classical inverse eigensensitivity method, has been developed. The improved method employs both analytical and experimental modal data to calculate the required eigensensitivity coefficients which are very close to their true values. The method has been further extended to the case where measured coordinates are incomplete. Practical applicability of the method has been assessed by its application to the updating of the finite element model of a plane truss structure

Vibration of Stress-Free Hollow Cylinders of Arbitrary Cross Section

A three-dimensional elasticity solution to the vibrations of stress-free hollow cylinders of arbitrary cross section is presented. The natural frequencies and deformed mode shapes of these cylinders are obtained via a three-dimensional displacement-based energy formulation. The technique is applied specifically to the parametric investigation of hollow cylinders of different cross sections and sizes. It is found that the cross-sectional property of the cylinder has significant effects on the normal mode responses, particularly, on the transverse bending modes. By varying the length-to-width ratio of these elastic cylinders, interesting results demonstrating the dependence of frequencies on the length of the cylinder have been concluded.

Three-dimensional vibration of rectangular plates : Variance of simple support conditions and influence of in-plane inertia

Plate models developed on the basis of the three-dimensional elasticity theory are crucial to the correct solutions in dynamics problems. In a three-dimensional setting, there exist numerous possible definitions of boundary conditions. This paper attempts to identify the variance of three-dimensional simple support conditions existing in practice. The hard simple support condition assumes vanishing normal stresses at the edges and zero transverse and tangential displacement at the peripheries. The soft simple support, on the other hand, imposes zero transverse displacement with vanishing normal and shear stresses at the boundaries. To quantify the relative effects of each boundary condition on the vibratory responses of thick plates, a global three-dimensional Ritz formulation is employed for analysis. This technique uses sets of finite polynomial series in Cartesian co-ordinates to approximate the global normal mode variations of the plate. These separable one-dimensional polynomials are orthogonally generated within the plate domain such that all the essential edge conditions are identically satisfied. The accuracy of the proposed three-dimensional Ritz method is assured from the convergence and comparison studies. Numerical experiments have been conducted to study firstly, the effects of geometric parameters on the overall normal mode characteristics of simply supported plates ; and secondly, the effects of in-plane inertia on the vibration frequencies of plates with different thicknesses. The three-dimensional deformed mode shapes for selected cases have also been computed. These have served to describe more vividly the normal mode characteristics of different types of simply supported rectangular plates.

Vibration Characteristics of Simply Supported Thick Skew Plates in Three-Dimensional Setting

A procedure is presented for determining the three-dimensional elasticity solutions for free vibration analysis of simply supported thick skew plates. The exact expressions of strain and kinetic energies are derived from linear, small-strain, three-dimensional elasticity theory. To allow the treatment of soft and hard simple support conditions, sets of three-dimensional spatial displacement functions are expressed in terms of unit normals to the edges. By virtue of the three-dimensional elasticity theory, the present method does not require a special treatment for stress ingularity at the obtuse corners. This method is also demonstrated to be free from shear locking phenomena. The significant difference in the vibration response of skew plates with soft and hard simple support conditions is highlighted. The influence of skew angle on the eigenvalues of thick skew plate is discussed in the context of the three-dimensional elasticity solutions.

Roles of domain decomposition method in plate vibrations: Treatment of mixed discontinuous periphery boundaries

Abstract The detailed development of a domain decomposition method (DDM) for the vibrational modelling of rectangular plates with mixed-edge boundary conditions is presented. In the DDM, the complex plate domain is decomposed into small simple subdomains and the appropriate shape function of each subdomain is represented by sets of admissible orthogonal polynomials generated using the Gram-Schmidt recurrence process. The continuity matrices that couple the eigenvectors of adjacent subdomains are derived based on the satisfaction of continuity conditions along the interconnecting boundaries. The stiffness and mass matrices of each subdomain after pre- and post-multiplication by the respective continuity matrices are assembled to form the global stiffness and mass matrices. To demonstrate the effectiveness and accuracy of the DDM, a vibration study of several partially mixed edge plates has been carried out. Convergence tests for example problems are presented in which the accuracy of the results is established. The frequency parameters and mode shapes obtained, where possible, are verified by comparison with data published in the open literature.

Method of domain decomposition in vibrations of mixed edge anisotropic plates

Abstract This paper presents the first known study on the flexural vibration of anisotropic plates with mixed discontinuous periphery boundaries. A newly developed domain decomposition method is used in the analysis to derive the governing eigenvalue equation. In the solution process, the complex plate domain is decomposed into appropriate subdomains. The displacement functions of each subdomain are represented by sets of orthogonally generated polynomials that satisfy the essential geometric boundary conditions. From the compatibility requirements at the interconnecting boundaries, sets of continuity matrices are computed. These matrices are used to couple the respective eigenvectors of the adjacent subdomains. The stiffness and mass matrices of each subdomain after being pre- and post-multiplied by the corresponding continuity matrix, are assembled to form the global stiffness and mass matrices of the anisotropic plate. Convergence and comparison studies have been carried out on selected cases to establish the rate of convergence and degree of accuracy of the present formulation. A comprehensive range of frequency parameters and deflection mode shapes of anisotropic plates with different mixed edge configurations are obtained using the proposed method. The effects of fiber orientation and the partial mixed ratio on the vibrational response of these plates have been investigated in detail.

Vibration of Rectangular Mindlin Plates with Intermediate Stiffeners

A first known investigation into the vibratory characteristics of rectangular Mindlin plates with intermediate stiffeners is presented. The Rayleigh-Ritz method is used, with displacement and rotational functions assumed in the form of mathematically complete algebraic polynomials. Sets of numerical frequency parameters for rectangular plates of various boundary conditions, thicknesses and plate dimensions are presented. In the study, the effects of shear deformation and rotary inertia on the vibrational response of the plate structures are investigated. The influence of torsional rigidity and geometric properties of stiffeners on the natural frequency parameters are included. To validate the proposed formulation, numerical results for some simplified problems have been determined where existing literature for these problems can be found. Finally sets of new vibration frequencies for plates with one or more stiffeners in various directions are presented.

A global approach for vibration of thick trapezoidal plates

A global pb-2 Rayleigh-Ritz method is used to study the free vibration behaviour of trapezoidal Mindlin plates with different combinations of boundary conditions. The Mindlin theory is used to account for the effects of shear deformation and rotary inertia. The governing energy functional for the plate is derived and minimized to arrive at the eigenvalue equation. In the process, the shape functions are assumed in a set of two-dimensional mathematically complete polynomials which satisfy the geometric boundary conditions of the plate. This procedure has been implemented numerically to compute the vibration frequencies for trapezoidal Mindlin plates. Convergence and comparison studies have been carried out to verify the accuracy of this method. Sets of first known results have been presented for trapezoidal Mindlin plates of varying thickness, aspect ratio and boundary constraint. These new results enhance the database of vibration solutions.