Three-dimensional vibration analysis of thick rectangular plates using Chebyshev polynomial and Ritz method

Vibration analysis of the classical elastic structures is not only essential for the study of vibration reduction by predicting the dynamic behavior, but also important to ensure a reliable, safe, and lasting structural performance through the proper design procedure. In this paper, the influence of boundary conditions on the free and forced three-dimensional vibration analysis of thick rectangular plates has been performed using the improved Fourier series method. For the elastically restrained thick rectangular plate, the three-dimensional improved Fourier series displacement forms are used to model the vibration field. The energy formula is employed to describe the three-dimensional dynamics of the plate. All the unknown Fourier series coefficients are then solved by the Rayleigh-Ritz method. In order to validate the proposed model, several numerical examples are provided and compared against the results from the literature and Finite Element Analysis (FEA). In addition, the effects of the boundary restraining spring stiffness and the thickness ratios of thick rectangular plates are analyzed under elastically restrained boundary conditions to develop an in-depth understanding of the three-dimensional vibration characteristics of thick rectangular plates.

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Three-Dimensional Vibration Analysis of Twisted Cylinder With Sectorial Cross Section

Journal of Applied Mechanics ◽

10.1115/1.4007475 ◽

2013 ◽

Vol 80 (2) ◽

Cited By ~ 1

Author(s):

D. Zhou ◽

S. H. Lo

Keyword(s):

Cross Section ◽

Chebyshev Polynomial ◽

Numerical Solutions ◽

Twist Angle ◽

Ritz Method ◽

Three Dimensional ◽

Cylindrical Domain ◽

Linear Elasticity Theory ◽

Boundary Functions ◽

Polynomial Series

The three-dimensional (3D) free vibration of twisted cylinders with sectorial cross section or a radial crack through the height of the cylinder is studied by means of the Chebyshev–Ritz method. The analysis is based on the three-dimensional small strain linear elasticity theory. A simple coordinate transformation is applied to map the twisted cylindrical domain into a normal cylindrical domain. The product of a triplicate Chebyshev polynomial series along with properly defined boundary functions is selected as the admissible functions. An eigenvalue matrix equation can be conveniently derived through a minimization process by the Rayleigh–Ritz method. The boundary functions are devised in such a way that the geometric boundary conditions of the cylinder are automatically satisfied. The excellent property of Chebyshev polynomial series ensures robustness and rapid convergence of the numerical computations. The present study provides a full vibration spectrum for thick twisted cylinders with sectorial cross section, which could not be determined by 1D or 2D models. Highly accurate results presented for the first time are systematically produced, which can serve as a benchmark to calibrate other numerical solutions for twisted cylinders with sectorial cross section. The effects of height-to-radius ratio and twist angle on frequency parameters of cylinders with different subtended angles in the sectorial cross section are discussed in detail.

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Vibration Analysis of Laminated Composite Rectangular Plates With General Boundary Conditions

ASME 2018 Noise Control and Acoustics Division Session presented at INTERNOISE 2018 ◽

10.1115/ncad2018-6106 ◽

2018 ◽

Cited By ~ 1

Author(s):

Yu Fu ◽

Jianjun Yao ◽

Zhenshuai Wan ◽

Gang Zhao

Keyword(s):

Boundary Conditions ◽

Vibration Analysis ◽

Ritz Method ◽

Laminated Composite ◽

Free Vibration Analysis ◽

Geometric Parameters ◽

General Boundary ◽

Rectangular Plates ◽

General Boundary Conditions ◽

Benchmark Solutions

In this investigation, the free vibration analysis of laminated composite rectangular plates with general boundary conditions is performed with a modified Fourier series method. Vibration characteristics of the plates have been obtained via an energy function represented in the general coordinates, in which the displacement and rotation in each direction is described as an improved form of double Fourier cosine series and several closed-form auxiliary functions to eliminate any possible jumps and boundary discontinuities. All the expansion coefficients are then treated as the generalized coordinates and determined by Rayleigh-Ritz method. The convergence and reliability of the current method are verified by comparing with the results in the literature and those of Finite Element Analysis. The effects of boundary conditions and geometric parameters on the frequencies are discussed as well. Finally, numerous new results for laminated composite rectangular plates with different geometric parameters are presented for various boundary conditions, which may serve as benchmark solutions for future research.

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