Jingtao Du

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.

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.

Vibration characteristics and power transmission of coupled rectangular plates with elastic coupling edge and boundary restraints

Coupled-plate structures are widely used in the practical engineering such as aeronautical, civil and naval engineering etc. Limited works can be found on the vibration of the coupled-plate structure due to the increased mathematical complexity compared with the single plate structure. In order to study analytically the vibration characteristics and power transmission of the coupled-plate structure, an analytical model consisting of three coupled plates elastically restrained along boundary edges and elastically coupled with arbitrary angle is considered, in which four groups of springs are distributed consistently along each edge of the model to simulate the transverse shearing forces, bending moments, in-plane longitudinal forces and in-plane shearing forces separately. With elastic coupling condition and general boundary condition of both flexural and in-plane vibrations taken into account by setting the stiffness of corresponding springs, the double Fourier series solution to the dynamic response of the structure was obtained by employing the Rayleigh-Ritz method. In order to validate the model, the natural frequency and velocity response of the model are firstly checked against results published in literatures and the ANSYS data, and good agreement was observed. Then, numerical simulation of the effects of several relevant parameters on the vibration characteristics and power transmission of the coupled structure were performed, including boundary conditions, coupling conditions, coupling angle, and location of the external forces. Vibration and energy transmission behaviors were analyzed numerically. The results show that the power transmission can be significantly influenced by the boundary restraints and the location of excitation. When the excitation is located at the central symmetry point of the model, the energy flow shows a symmetrical distribution. Once the location deviates from the central symmetry point, the power circumfluence occurs and the vortex energy field is formed at high frequency.

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.

Sound Radiation from an Elastically Restrained Plate Covered by an Acoustic Decoupling Layer

The sound radiation from elastically restrained plates covered by a decoupling layer is studied using the Spectrogeometric Method (SGM), which is a meshless and parametric modeling technique. By adopting the Rayleigh-Ritz procedure and the Rayleigh integral, a vibroacoustic coupling system is established. This model studies the situation when the plate is immersed in heavy fluid, such as water, in which the strong coupling between the structure and sound field should be fully considered. The influence of the boundary conditions on the radiated sound power and sound reduction provided by the decoupling layer based on the locally reacting model is studied. The nonuniform distributed decoupling layer is also studied to analyze the sound reduction effect. The sound intensity on the outer surface of the decoupling layer is investigated and tends to be uniform along the plate scale with increasing thickness of the decoupling layer.

Vibration Analysis of Conical Shells by the Improved Fourier Expansion-Based Differential Quadrature Method

An improved Fourier expansion-based differential quadrature (DQ) algorithm is proposed to study the free vibration behavior of truncated conical shells with different boundary conditions. The original function is expressed as the Fourier cosine series combined with close-form auxiliary functions. Those auxiliary functions are introduced to ensure and accelerate the convergence of series expansion. The grid points are uniformly distributed along the space. The weighting coefficients in the DQ method are easily obtained by the inverse of the coefficient matrix. The derivatives in both the governing equations and the boundaries are discretized by the DQ method. Natural frequencies and modal shapes can be easily obtained by solving the numerical eigenvalue equations. The accuracy and stability of this proposed method are validated against the results in the literature and a very good agreement is observed. The centrosymmetric properties of these newly proposed weighting coefficients are also validated. Studies on the effects of semivertex angle and the ratio of length to radius are reported.

Free and forced in-plane vibration of rectangular plates with non-uniform elastic boundary conditions

In this paper, the free and forced in-plane vibration analysis of rectangular plates are performed for the first time using an improved Fourier series method, in which the boundary restraining spring stiffness can vary in any functional pattern along each edge. Two-dimensional improved Fourier series displacement forms are constructed with four supplementary polynomials introduced into the standard 2-D Fourier series to make the field functions sufficiently smooth in the whole solving domain. Energy formulations are employed to describe the in-plane dynamics of plate system, in which the in-plane concentrated point force is taken into account in the form of work term. All the unknown Fourier series coefficients are then solved through the Rayleigh-Ritz procedure. Several numerical examples are given to demonstrate the correctness and effectiveness of the proposed model through the comparison with those calculated via finite element analysis (FEA). The results show that these two results can agree very well with each other for various non-uniform boundary conditions. Based on the established model, the in-plane vibration response is also studied. Some curves and contours are obtained to illustrate how the boundary restraining stiffnesses affect the in-plane point and transfer mobility of rectangular plate structure.

An active vibration isolation system for an air-compressor in marine applications

Similar to a diesel engine, the main vibration sources of an air-compressor are to and fro inertial forces of its piston, rod, and crankshaft. An active vibration isolation system is developed for an air-compressor in a tug boat. This system consists of four inertial actuators and a DSP processor. A four-input and four-output adaptive control strategy is applied and the reference input signal comes from a laser tachometer on the shaft of the motor. The active vibration isolation experiment is conducted in a tug boat when only the air-compressor is working. The experimental results demonstrate that good vibration reductions are obtained at not only error sensors locations but also the hull under the compressor. Some discussion and conclusions are given at last.

Power flows between strongly coupled structural components

In this study, the energy distributions and power flows between some basic structural components such as beams, plates, and shells are studied using a so-called Fourier Spectral Element Method (FSEM). Similar to the SEA modeling, a complex system is here also considered as an assembly of subsystems or components. The FSEM, however, is deterministic in nature in that the solution is obtained by directly and faithfully solving the governing equations for each component under the actual boundary and coupling conditions. What make this model powerful and unique lie in its capability and flexibility of effectively dealing with model uncertainties (due to the probabilistic/stochastic natures of some model parameters) and engineering and manufacturing errors which tend to become critically important at higher frequencies. Since this method does not involve any artificial assumptions or simplifications, it potentially offers a whole frequency solution with adaptive spatial and frequency resolutions.