Wen L. Li

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 In-Plane Vibration Analysis of Rectangular Plates With Elastically Point-Supported Edges

In comparison with the transverse vibrations of rectangular plates, far less attention has been paid to the in-plane vibrations even though they may play an equally important role in affecting the vibrations and power flows in a built-up structure. In this paper, a generalized Fourier method is presented for the in-plane vibration analysis of rectangular plates with any number of elastic point supports along the edges. Displacement constraints or rigid point supports can be considered as the special case when the stiffnesses of the supporting springs tend to infinity. In the current solution, each of the in-plane displacement components is expressed as a 2D Fourier series plus four auxiliary functions in the form of the product of a polynomial times a Fourier cosine series. These auxiliary functions are introduced to ensure and improve the convergence of the Fourier series solution by eliminating all the discontinuities potentially associated with the original displacements and their partial derivatives along the edges when they are periodically extended onto the entire x-y plane as mathematically implied by the Fourier series representation. This analytical solution is exact in the sense that it explicitly satisfies, to any specified accuracy, both the governing equations and the boundary conditions. Numerical examples are given about the in-plane modes of rectangular plates with different edge supports. It appears that these modal data are presented for the first time in literature, and may be used as a benchmark to evaluate other solution methodologies. Some subtleties are discussed about corner support arrangements.

Structural modal interaction with combination internal resonance under wide-band random excitation

In this paper the non-linear interaction of a three degree of freedom structural model subjected to a wide-band random excitation is examined. The non-linearity of the system results in different critical regions of internal resonance and this has a significant effect on the response statistics. With reference to the combination internal resonance of the summed type the system response is analyzed by using the Fokker-Planck equation approach together with a non-Gaussian closure scheme. The non-Gaussian closure is based on the cumulant properties of order greater than three. As a first order approximation the scheme yields 209 first order differential equations in first through fourth order joint moments of the response co-ordinates. The analysis is carried out with the aid of the computer algebra software MACSYMA. The response statistics are determined, numerically in the time and frequency (internal detuning) domains. Contrary to the Gaussian closure scheme, the non-Gaussian closure solution yields a strictly stationary response in addition to a number of complex response characteristics not previously reported in the literature of the area of non-linear random vibration. These include multiple solutions and jump phenomena (jump and collapse in the response mean squares) at internal detuning slightly shifted from the exact internal resonance condition. At exact internal resonance the system response possesses a unique limit cycle in a stochastic sense. The regions of multiple solutions are defined in terms of system parameters (damping ratios and non-linear coupling parameter) and excitation spectral density level.

Principal internal resonances in 3-DOF systems subjected to wide-band random excitation

Abstract The autoparametric interaction of two normal modes in a three-degree-of-freedom (3-DOF) structural model subjected to a wide-band random excitation is investigated. The analysis deals primarily with the structure response in the neighborhood of three different internal resonance conditions. The Fokker-Planck equation approach, together with a non-Gaussian closure scheme, is used. The analysis is carried out with the aid of the computer algebra software MACSYMA, and leads to 69 differential equations in the first through fourth order moments of the response co-ordinates. Contrary to the Gaussian closure solution, the non-Gaussian closure scheme yields a strictly stationary response. The Gaussian closure scheme fails to predict the system response when two non-adjacent modes are internally tuned. According to the non-Gaussian solution, the autoparametric interaction is found to be sensitive to a relatively high level of excitation spectral density only if the first and second or the first and third normal modes are internally tuned. However, for the case of second and third normal mode interaction, the non-linear response is critical to lower excitation levels. The random responses of the three cases are characterized by energy exchange between the interacted modes. The sensitivity of autoparametric interaction to a certain excitation level is mainly dependent upon the system dynamic properties.

Vibration Analysis of Doubly Curved Shallow Shells With Elastic Edge Restraints

An analytical method is derived for the vibration analysis of doubly curved shallow shells with arbitrary elastic supports alone its edges, a class of problems which are rarely attempted in the literature. Under this framework, all the classical homogeneous boundary conditions for both in-plane and out-of-plane displacements can be universally treated as the special cases when the stiffness for each of restraining springs is equal to either zero or infinity. Regardless of the boundary conditions, the displacement functions are invariably expanded as an improved trigonometric series which converges uniformly and polynomially over the entire solution domain. All the unknown expansion coefficients are treated as the generalized coordinates and solved using the Rayleigh–Ritz technique. Unlike most of the existing solution techniques, the current method offers a unified solution to a wide spectrum of shell problems involving, such as different boundary conditions, varying material and geometric properties with no need of modifying or adapting the solution schemes and implementing procedures. A numerical example is presented to demonstrate the accuracy and reliability of the current method.

Dynamic Analysis of Circular Cylindrical Shells With General Boundary Conditions Using Modified Fourier Series Method

Dynamic behavior of cylindrical shell structures is an important research topic since they have been extensively used in practical engineering applications. However, the dynamic analysis of circular cylindrical shells with general boundary conditions is rarely studied in the literature probably because of a lack of viable analytical or numerical techniques. In addition, the use of existing solution procedures, which are often only customized for a specific set of different boundary conditions, can easily be inundated by the variety of possible boundary conditions encountered in practice. For instance, even only considering the classical (homogeneous) boundary conditions, one will have a total of 136 different combinations. In this investigation, the flexural and in-plane displacements are generally sought, regardless of boundary conditions, as a simple Fourier series supplemented by several closed-form functions. As a result, a unified analytical method is generally developed for the vibration analysis of circular cylindrical shells with arbitrary boundary conditions including all the classical ones. The Rayleigh-Ritz method is employed to find the displacement solutions. Several examples are given to demonstrate the accuracy and convergence of the current solutions. The modal characteristics and vibration responses of elastically supported shells are discussed for various restraining stiffnesses and configurations. Although the stiffness distributions are here considered to be uniform along the circumferences, the current method can be readily extended to cylindrical shells with nonuniform elastic restraints.

Free in-plane vibrations of annular sector plates with elastic boundary supports

In this investigation, a generalized Fourier series method is proposed for the in-plane vibration analysis of annular sector plates with elastic restraints along each of its edges. The in-plane displacement fields are universally expressed as a new form of trigonometric series expansions with a drastically improved convergence as compared with the conventional Fourier series. The expansion coefficients are considered as the generalized coordinates, and determined using the Rayleigh-Ritz technique. Several examples are presented to demonstrate the effectiveness and reliability of the current method for predicting the modal characteristics of annular sector plates with various cutout ratios and sector angles under different boundary conditions. It is also shown that annular and circle plates can be readily included as the special cases of the annular sector plates when the sector angle is set equal to 360°.

Free vibration analysis of doubly curved shallow shells reinforced by any number of beams with arbitrary lengths

This study focuses on the free linear vibrations of doubly curved shallow shells reinforced by any number of beams of arbitrary lengths. Distributed elastic restraints are used to specify generally the boundary conditions along the shell edges and the coupling conditions between the shell and its reinforcing beams. Both the shell and stiffening beams are considered as independent structural components carrying three-dimensional displacement fields. Each of the displacements is invariably expressed as a simple trigonometric series with accelerated and uniform convergence over the solution domains of interest. All the unknown expansion coefficients are treated as the generalized coordinates and solved using the Rayleigh-Ritz technique. As illustrated by examples, the current method provides a unified means for solving a wide range of shell problems involving various practical complications with respect to, for example, the boundary conditions, the coupling conditions, the number of stiffeners, and the lengths and locations of the stiffeners.

Curvature Effects on the Vibration Characteristics of Doubly Curved Shallow Shells with General Elastic Edge Restraints

Effects of curvature upon the vibration characteristics of doubly curved shallow shells are assessed in this paper. Boundary conditions of the shell are generally specified in terms of distributed elastic restraints along the edges. The classical homogeneous boundary supports can be easily simulated by setting the stiffnesses of restraining springs to either zero or infinite. Vibration problems of the shell are solved by a modified Fourier series method that each of the displacements is invariably expressed as a simple trigonometric series which converges uniformly and acceleratedly over the solution domain. All the unknown expansion coefficients are treated equally as a set of independent generalized coordinates and solved using the Rayleigh-Ritz technique. The current method provides a unified solution to the vibration problems of curved shallow shells involving different geometric properties and boundary conditions with no need of modifying the formulations and solution procedures. Extensive tabular and graphical results are presented to show the curvature effects on the natural frequencies of the shell with various 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.