Qingshan Wang

A unified method for free vibration analysis of circular, annular and sector plates with arbitrary boundary conditions

The vibrations of circular, annular, and sector plates are traditionally considered as different boundary value problems and often treated using different solution algorithms and procedures. This problem is further compounded by the fact that the solution for each type of plate typically needs to be adapted to different boundary conditions. In this paper, a simple solution method is proposed for a unified vibration analysis of annular, circular and sector plates with arbitrary boundary conditions. Regardless of the shapes of the plates and the types of boundary conditions, the displacement solutions are invariably expressed as a new and simple form of trigonometric series expansion with an accelerated convergence rate. The unification of seemingly different boundary value problems for the circular and annular plates and their sector counterparts is physically accomplished by applying a set of coupling springs to ensure appropriate continuity conditions along the radial edges. The accuracy, reliability and versatility of the current method are fully demonstrated and verified through numerical examples involving plates with various shapes and boundary conditions. It should be noted that the current method can be easily applied to sector plates with an arbitrary inclusion angle up to 2π.

Free vibration analysis of axially loaded laminated composite beams with general boundary conditions by using a modified Fourier–Ritz approach

In this paper, a modified Fourier–Ritz approach has been adopted to analyze the free vibration of axially loaded laminated composite beams with arbitrary layup and general boundary conditions, which include classical boundaries, elastic boundaries, and their combination. The influences of Poisson effect, axial deformation, couplings among extensional, bending and torsional deformations, shear deformation, and rotary inertia are incorporated in the formulation. In this present method, regardless of boundary conditions, the displacements and rotation components of the beam are invariantly expressed as a standard Fourier cosine series and several auxiliary closed-form functions. These auxiliary functions are introduced to eliminate any potential discontinuities of the original displacement function and its derivatives, throughout the whole beam including its ends, and to effectively enhance the convergence of the results. Since the displacement field is constructed to be adequately smooth in the whole solution domain, an accurate solution can be obtained by using Ritz procedure based on the energy functions of the beam. Numerical examples are presented for several different boundary conditions, geometric properties, and material parameters. The results show that the present method enables rapid convergence, high reliability, and accuracy. Numerous new free vibration results for axially loaded laminated composite beams with different lamination schemes and elastic restraints are presented.

An accurate solution method for the vibration analysis of Timoshenko beams with general elastic supports

In this investigation, an accurate solution method is presented for the free vibrations of Timoshenko beams with general elastic restraints at the end points, a class of problems which are rarely attempted in the literatures. Unlike in most existing studies where solutions are often developed for a particular type of boundary conditions, the current method can be generally applied to a wide range of boundary conditions with no need of modifying solution algorithms and procedures. Under the current framework, the displacement and rotation functions are generally sought, regardless of boundary conditions, as an improved trigonometric series in which several supplementary functions introduced to remove the potential discontinuities with the displacement components and its derivatives at the end points and accelerated the series expansion. Mathematically, the current Fourier series expansion is an exact solution for a class of problems with the Timoshenko beam such that both the governing equations and the boundary conditions simultaneously satisfy any specified degree of accuracy. The effectiveness and reliability of the presented solution are demonstrated by comparing the present results with those results published in literatures and finite element method data, and numerous new results for beams with elastic boundary restraints is presented, which may serve as benchmark solution for future researches.

An improved Fourier series solution for the dynamic analysis of laminated composite annular, circular, and sector plate with general boundary conditions

In this article, the authors presented a unified solution for the dynamic analysis of laminated composite annular, circular, and sector plate with general boundary conditions. The first-order shear deformation theory is employed to formulate the theoretical model. Regardless of the shapes of the plates and the types of boundary conditions, each displacement and rotation component of the elements is expanded as an improved Fourier series expansion which is composed of a double Fourier cosine series and several auxiliary functions introduced to eliminate all the relevant discontinuities with the displacement and its derivatives at the boundaries and to accelerate the convergence of series representations. Since the displacement fields are constructed adequately smooth throughout the entire solution domain, an exact solution is obtained based on the Rayleigh–Ritz procedure by the energy functions of the plates. The accuracy, reliability, and versatility of the current solution is fully demonstrated and verified through numerical examples involving plates with various shapes and boundary conditions. Some new results of free vibration analysis for composite laminated annular sector plate, circular sector plate, annular plate, and circular plate are presented, which may be served as benchmark solution for future computational methods. The effects of the sector angles, layer numbers, and boundary spring stiffness on vibration characteristics of the plates are reported. In addition, the force vibration analysis of the plates is also studied. The influence of the boundary spring stiffness, layer number, orthotropic stiffness ration, and fiber orientation angle on dynamic characteristics of the plates is investigated.

A Unified Spectro-Geometric-Ritz Method for Vibration Analysis of Open and Closed Shells with Arbitrary Boundary Conditions

This paper presents free vibration analysis of open and closed shells with arbitrary boundary conditions using a spectro-geometric-Ritz method. In this method, regardless of the boundary conditions, each of the displacement components of open and closed shells is represented simultaneously as a standard Fourier cosine series and several auxiliary functions. The auxiliary functions are introduced to accelerate the convergence of the series expansion and eliminate all the relevant discontinuities with the displacement and its derivatives at the boundaries. The boundary conditions are modeled using the spring stiffness technique. All the expansion coefficients are treated equally and independently as the generalized coordinates and determined using Rayleigh-Ritz method. By using this method, a unified vibration analysis model for the open and closed shells with arbitrary boundary conditions can be established without the need of changing either the equations of motion or the expression of the displacement components. The reliability and accuracy of the proposed method are validated with the FEM results and those from the literature.

A unified solution for vibration analysis of moderately thick, functionally graded rectangular plates with general boundary restraints and internal line supports

ABSTRACT The objective of this article is to present a Fourier-Ritz method-based solution approach for the free vibration analysis of moderately thick, functionally graded (FG) rectangular plates with general boundary restraints and internal line supports. Under the current framework, regardless of boundary conditions, each of the displacements and rotations of the FG plates is invariantly expressed as a modified Fourier series in both directions. Then, the accurate solutions are obtained using the Ritz procedure based on the energy function of the plates. The convergence and accuracy of the present method are verified by a considerable number of convergence tests and comparisons.

Vibrations of Composite Laminated Circular Panels and Shells of Revolution with General Elastic Boundary Conditions via Fourier-Ritz Method

Abstract A Fourier-Ritz method for predicting the free vibration of composite laminated circular panels and shells of revolution subjected to various combinations of classical and non-classical boundary conditions is presented in this paper. A modified Fourier series approach in conjunction with a Ritz technique is employed to derive the formulation based on the first-order shear deformation theory. The general boundary condition can be achieved by the boundary spring technique in which three types of liner and two types of rotation springs along the edges of the composite laminated circular panels and shells of revolution are set to imitate the boundary force. Besides, the complete shells of revolution can be achieved by using the coupling spring technique to imitate the kinematic compatibility and physical compatibility conditions of composite laminated circular panels at the common meridian with θ = 0 and 2π. The comparisons established in a sufficiently conclusive manner show that the present formulation is capable of yielding highly accurate solutions with little computational effort. The influence of boundary and coupling restraint parameters, circumference angles, stiffness ratios, numbers of layer and fiber orientations on the vibration behavior of the composite laminated circular panels and shells of revolution are also discussed.

Free Vibration Analysis of Moderately Thick Rectangular Plates with Variable Thickness and Arbitrary Boundary Conditions

A generalized Fourier series solution based on the first-order shear deformation theory is presented for the free vibrations of moderately thick rectangular plates with variable thickness and arbitrary boundary conditions, a class of problem which is of practical interest and fundamental importance but rarely attempted in the literatures. Unlike in most existing studies where solutions are often developed for a particular type of boundary conditions, the current method can be generally applied to a wide range of boundary conditions with no need of modifying solution algorithms and procedures. Under the current framework, the one displacement and two rotation functions are generally sought, regardless of boundary conditions, as an improved trigonometric series in which several supplementary functions are introduced to remove the potential discontinuities with the displacement components and its derivatives at the edges and to accelerate the convergence of series representations. All the series expansion coefficients are treated as the generalized coordinates and solved using the Rayleigh-Ritz technique. The effectiveness and reliability of the presented solution are demonstrated by comparing the present results with those results published in the literatures and finite element method (FEM) data, and numerous new results for moderately thick rectangular plates with nonuniform thickness and elastic restraints are presented, which may serve as benchmark solution for future researches.

In-Plane Vibration Analysis of Annular Plates with Arbitrary Boundary Conditions

In comparison with the out-of-plane vibrations of annular plates, far less attention has been paid to the in-plane vibrations which may also play a vital important role in affecting the sound radiation from and power flows in a built-up structure. In this investigation, a generalized Fourier series method is proposed for the in-plane vibration analysis of annular plates with arbitrary boundary conditions along each of its edges. Regardless of the boundary conditions, the in-plane displacement fields are invariantly expressed as a new form of trigonometric series expansions with a drastically improved convergence as compared with the conventional Fourier series. All the unknown expansion coefficients are treated as the generalized coordinates and determined using the Rayleigh-Ritz technique. Unlike most of the existing studies, the presented method can be readily and universally applied to a wide spectrum of in-plane vibration problems involving different boundary conditions, varying material, and geometric properties with no need of modifying the basic functions or adapting solution procedures. Several numerical examples are presented to demonstrate the effectiveness and reliability of the current solution for predicting the in-plane vibration characteristics of annular plates subjected to different boundary conditions.

Vibration Analysis of Annular Sector Plates under Different Boundary Conditions

An analytical framework is developed for the vibration analysis of annular sector plates with general elastic restraints along each edge of plates. Regardless of boundary conditions, the displacement solution is invariably expressed as a new form of trigonometric expansion with accelerated convergence. The expansion coefficients are treated as the generalized coordinates and determined using the Rayleigh-Ritz technique. This work allows a capability of modeling annular sector plates under a variety of boundary conditions and changing the boundary conditions as easily as modifying the material properties or dimensions of the plates. Of equal importance, the proposed approach is universally applicable to annular sector plates of any inclusion angles up to 2π. The reliability and accuracy of the current method are adequately validated through numerical examples.