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

Zhou, Ding; Cheung, Y.K.; Au, Ftk; Lo, S.H.

doi:10.1016/s0020-7683(02)00460-2

Cited by 150 publications

(57 citation statements)

References 24 publications

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“…Wei et al [16] explored the natural frequencies of partially-supported plates using the discrete singular convolution method. Zhou et al [17,18] analyzed the natural frequency of moderately thick rectangular plates using the Chebyshev polynomial as the admissible function in the Ritz method. Recently, Gharaibeh et al [19,20] used a combination between series solutions and the Ritz method to solve for the first natural frequency and mode shape of a squared elastic plate.…”

Section: Introductionmentioning

confidence: 99%

Numerical Investigation of the Free Vibration of Partially Clamped Rectangular Plates

Gharaibeh

Obeidat

Obaidat

2018

International Journal of Applied Mechanics and Engineering

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This paper studies the free vibration characteristics of rectangular plates with partially clamped edges around the corners using the finite element method. ANSYS Parametric Design Language (APDL) was utilized to produce the finite element (FE) models and to run the analysis. The FE models were used to obtain the plate first natural frequency and mode shape. A comprehensive investigation of the effect of the plate geometric parameters and different boundary condition properties on the natural frequency and mode shapes is presented. The results showed that the vibration characteristics of the structure are greatly dependent on the plate size and the constraint properties.

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Section: Introductionmentioning

confidence: 99%

Numerical Investigation of the Free Vibration of Partially Clamped Rectangular Plates

Gharaibeh

Obeidat

Obaidat

2018

International Journal of Applied Mechanics and Engineering

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“…Lim et al [21,22] presented a concise formulation and an efficient method of solution to study the free vibration of thick, shear deformable plates with classical boundary conditions. Zhou et al [23] used the Chebyshev polynomials as admissible functions and applied Ritz method to study the three-dimensional vibration of rectangular plates with classical boundary conditions. By means of the review of the above references, most of the studies for the vibration problem of the thick rectangular plates do not consider the elastic foundations.…”

Section: Introductionmentioning

confidence: 99%

Three-Dimensional Vibration Analysis of Rectangular Thick Plates on Pasternak Foundation with Arbitrary Boundary Conditions

Liu

Jing

et al. 2017

Shock and Vibration

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This paper presents the first known vibration characteristic of rectangular thick plates on Pasternak foundation with arbitrary boundary conditions on the basis of the three-dimensional elasticity theory. The arbitrary boundary conditions are obtained by laying out three types of linear springs on all edges. The modified Fourier series are chosen as the basis functions of the admissible function of the thick plates to eliminate all the relevant discontinuities of the displacements and their derivatives at the edges. The exact solution is obtained based on the Rayleigh-Ritz procedure by the energy functions of the thick plate. The excellent accuracy and reliability of current solutions are demonstrated by numerical examples and comparisons with the results available in the literature. In addition, the influence of the foundation coefficients as well as the boundary restraint parameters is also analyzed, which can serve as the benchmark data for the future research technique.

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“…In particular, Zhou and coworkers employed the Chebyshev-Ritz procedure to study three dimensional free vibration characteristics of arbitrary thick rectangular plates with various uniform boundary conditions [38], a torus with a circular cross section [39], circular and annular plates with any boundary conditions [40], solid and hollow circular cylinders [18], rectangular thick plates resting on elastic Pasternak foundations [41], generalized super-elliptical plates [42], cantilevered thick skew plates [43], and just recently, circular toroidal sectors with circular cross-section [5].…”

Section: Introductionmentioning

confidence: 99%

“…1). It is noteworthy that due to the excellent properties of Chebyshev polynomials in numerical operations, the adopted method is well known to predict more frequencies and modes with higher convergence rate, better numerical stability, and improved accuracy in comparison with other types of admissible functions such as simple algebraic polynomials, particularly in the 3-D vibration analysis of an elastic bodies where numerical instability may occur with a great number of terms of admissible functions [38,41,53]. This fact was underlined by Zhou et al [40] where they demonstrated that by using Chebyshev polynomials instead of simple polynomials as the admissible functions, the immunity against ill-conditioned behavior in computing eigenfrequencies of completely free solid and annular thick circular plates can be greatly enhanced.…”

Section: Introductionmentioning

confidence: 99%

Three Dimensional Vibration Analysis of a Class of Traction-Free Solid Elastic Bodies with an Eccentric Cavity

Hasheminejad

Mirzaei

2012

Shock and Vibration

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Abstract.A three-dimensional elasticity-based continuum model is developed for describing the free vibrational characteristics of an important class of isotropic, homogeneous, and completely free structural bodies (i.e., finite cylinders, solid spheres, and rectangular parallelepipeds) containing an arbitrarily located simple inhomogeneity in form of a spherical or cylindrical defect. The solution method uses Ritz minimization procedure with triplicate series of orthogonal Chebyshev polynomials as the trial functions to approximate the displacement components in the associated elastic domains, and eventually arrive at the governing eigenvalue equations. An extensive review of the literature spanning over the past three decades is also given herein regarding the free vibration analysis of elastic structures using Ritz approach. Accuracy of the implemented approach is established through proper convergence studies, while the validity of results is demonstrated with the aid of a commercial FEM software, and whenever possible, by comparison with other published data. Numerical results are provided and discussed for the first few clusters of eigen-frequencies corresponding to various mode categories in a wide range of cavity eccentricities. Also, the corresponding 3D mode shapes are graphically illustrated for selected eccentricities. The numerical results disclose the vital influence of inner cavity eccentricity on the vibrational characteristics of the voided elastic structures. In particular, the activation of degenerate frequency splitting and incidence of internal/external mode crossings are confirmed and discussed. Most of the results reported herein are believed to be new to the existing literature and may serve as benchmark data for future developments in computational techniques.

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

Cited by 150 publications

References 24 publications

Numerical Investigation of the Free Vibration of Partially Clamped Rectangular Plates

Numerical Investigation of the Free Vibration of Partially Clamped Rectangular Plates

Three-Dimensional Vibration Analysis of Rectangular Thick Plates on Pasternak Foundation with Arbitrary Boundary Conditions

Three Dimensional Vibration Analysis of a Class of Traction-Free Solid Elastic Bodies with an Eccentric Cavity

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