K.C. Hung

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.

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.

Three-dimensional vibratory characteristics of solid cylinders and some remarks on simplified beam theories

Abstract A set of linear three-dimensional frequency equations that describes the vibratory characteristics of elastic solid cylinders of different end supports is derived. Starting from the linear, small-strain, three-dimensional elasticity theory, integral expressions for strain and kinetic energies are formulated. The three-dimensional mode shapes are expressed in terms of sets of one- and two-dimensional orthogonal polynomial functions which are to approximate the longitudinal and lateral surface variations of solid cylinders. From the resulting displacement-based energy expressions, the variational form of the three-dimensional energy functional is minimized to yield the linear eigen-value equation. Frequency solutions for elastic solid cylinders of different lengths and end support conditions are determined. The accuracy of solutions is validated through convergence tests and comparisons with the existing three-dimensional analytical solutions and empirical data. From the present three-dimensional elasticity solutions, some remarks on the existing simplified beam theories are made. Particular attention is drawn to Timoshenkos shear deformable beam theory for transverse bending modes. The validity of the one-dimensional wave equation for longitudinal and torsional modes is also discussed.

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.

Boundary beam characteristics orthonormal polynomials in energy approach for vibration of symmetric laminates — II: Elastically restrained boundaries

Abstract This paper presents an eigenvalue formulation for the vibration analysis of symmetrically laminated rectangular plates subjected to translational and rotational restraints at the edges. The Rayleigh-Ritz method, along with the deflection functions assumed in sets of orthogonally generated polynomials, is used to perform the analysis. The total strain energy of the elastically restrained rectangular plate is the sum of the bending strain energy and elastic strain energy of translational and rotational restraints. This resulting strain energy combined with the kinetic energy of the plate formed the total energy functional which is minimized to obtain the governing eigenvalue equation of the elastically restrained symmetrically laminated rectangular plate. In this paper, several examples of elastically restrained laminated plates with different fiber orientation angles and stacking sequences have been solved to demonstrate the accuracy and efficiency of the present method. The combined effects of laminate stacking sequences, fiber orientation angle and translational and rotational stiffnesses of the elastic edges on the vibrational response have been carefully examined.

Vibration of Thick Prismatic Structures With Three-Dimensional Flexibilities

This paper presents an investigation on free vibration of thick prismatic structures (thick-walled open sections of L, T, C, and I shapes). The derivation of a linear frequency equation based on an exact three-dimensional small-strain linearly elastic principle is presented. This formulation uses one and two-dimensional polynomial series to approximate the spatial displacements of the thick-walled open sections in three dimension. The proposed technique is applicable to vibration of thick-walled open sections of different cross-sectional geometries and end support conditions. In this study, however, we focus primarily on the cantilevered case which has high value in practical applications. The perturbation of frequency responses due to the variations of cross-sectional geometries and wall thicknesses is investigated. First-known frequency parameters and three-dimensional deformed mode shapes of these thick-walled open sections are presented in vivid graphical forms. The new results may serve as a benchmark reference to future research into the refined beam and plate theories and also for checking the accuracy of new numerical techniques.