Vibration of a Rectangular Plate Carrying a Massive Machine with Elastic Supports

2016 ◽  
Vol 16 (10) ◽  
pp. 1550069 ◽  
Author(s):  
Lingzhi Wang ◽  
Zhitao Yan ◽  
Zhengliang Li ◽  
Zhimiao Yan
Keyword(s):  
Boundary Conditions ◽  
Rectangular Plate ◽  
Engineering Practice ◽  
Element Analysis ◽  
Optimal Position ◽  
Elastic Supports ◽  
Fourier Cosine ◽  
Cosine Series ◽  
Fourier Cosine Series ◽  
Plate Size

In engineering practice, a massive machine may be placed on a plate supported by beams considered as elastic boundary conditions. The vibration of the plate due to the periodic excitation of the massive machine will cause noises or damages to the building in which the machine is housed. An analytical approach for the vibration analysis of a rectangular plate carrying a massive machine with uniform elastic supports is presented. The machine is simplified as a distributed mass. The transverse plate displacement is determined by the superposition of a two-dimensional (2D) Fourier cosine series and several supplementary functions. All the unknown Fourier coefficients are calculated directly from the Rayleigh–Ritz formulation. To validate the present approach, several numerical examples with classical boundary conditions are presented. The results reveal good agreement between the analytical results and those based on the finite element analysis (ANSYS). The effects of the plate size, location of the machine, and support stiffness on the modal, and transient response of the plate are investigated. From the results it is found that the transient displacement amplitude of the plate decreases almost linearly as the thickness increases, it increases nonlinearly along with the increase in the support stiffness, and that the optimal position for deploying the transformer is the center of the plate.

scholarly journals Analytical Solutions Based on Fourier Cosine Series for the Free Vibrations of Functionally Graded Material Rectangular Mindlin Plates

Materials ◽  
2020 ◽  
Vol 13 (17) ◽  
pp. 3820
Author(s):  
Chiung-Shiann Huang ◽  
S. H. Huang
Keyword(s):  
Boundary Conditions ◽  
Material Properties ◽  
Free Vibrations ◽  
Functionally Graded ◽  
Rectangular Plates ◽  
Series Solutions ◽  
Fourier Cosine ◽  
Cosine Series ◽  
Fourier Cosine Series ◽  
Graded Material

This study aimed to develop series analytical solutions based on the Mindlin plate theory for the free vibrations of functionally graded material (FGM) rectangular plates. The material properties of FGM rectangular plates are assumed to vary along their thickness, and the volume fractions of the plate constituents are defined by a simple power-law function. The series solutions consist of the Fourier cosine series and auxiliary functions of polynomials. The series solutions were established by satisfying governing equations and boundary conditions in the expanded space of the Fourier cosine series. The proposed solutions were validated through comprehensive convergence studies on the first six vibration frequencies of square plates under four combinations of boundary conditions and through comparison of the obtained convergent results with those in the literature. The convergence studies indicated that the solutions obtained for different modes could converge from the upper or lower bounds to the exact values or in an oscillatory manner. The present solutions were further employed to determine the first six vibration frequencies of FGM rectangular plates with various aspect ratios, thickness-to-width ratios, distributions of material properties and combinations of boundary conditions.

scholarly journals Dynamic Analysis of Rectangular Plate Stiffened by Any Number of Beams with Different Lengths and Orientations

2019 ◽  
Vol 2019 ◽  
pp. 1-22
Author(s):  
Yuan Cao ◽  
Rui Zhong ◽  
Dong Shao ◽  
Qingshan Wang ◽  
Xianlei Guan
Keyword(s):  
Rectangular Plate ◽  
Transient Response ◽  
Dynamic Characteristics ◽  
Ritz Method ◽  
Current Method ◽  
Fourier Cosine ◽  
Cosine Series ◽  
Structural Intensity ◽  
Fourier Cosine Series ◽  
Fourier Series Method

The present work is concerned with dynamic characteristics of beam-stiffened rectangular plate by an improved Fourier series method (IFSM), including mobility characteristics, structural intensity, and transient response. The artificial coupling spring technology is introduced to establish the clamped or elastic connections at the interface between the plate and beams. According to IFSM, the displacement field of the plate and the stiffening beams are expressed as a combination of the Fourier cosine series and its auxiliary functions. Then, the Rayleigh–Ritz method is applied to solve the unknown Fourier coefficients, which determines the dynamic characteristics of the coupled structure. The Newmark method is adopted to obtain the transient response of the coupled structure, where the Rayleigh damping is taken into consideration. The rapid convergence of the current method is shown, and good agreement between the predicted results and FEM results is also revealed. On this basis, the effects of the factors related to the stiffening beam (including the length, orientations, and arrangement spacing of beams) and elastic parameters, as well as damping coefficients on the dynamic characteristics of the stiffened plate are investigated.

scholarly journals Free and Forced Vibration of the Moderately Thick Laminated Composite Rectangular Plate on Various Elastic Winkler and Pasternak Foundations

2017 ◽  
Vol 2017 ◽  
pp. 1-23 ◽  
Author(s):  
Dongyan Shi ◽  
Hong Zhang ◽  
Qingshan Wang ◽  
Shuai Zha
Keyword(s):  
Boundary Conditions ◽  
Forced Vibration ◽  
Shear Deformation Theory ◽  
Laminated Composite ◽  
Future Research ◽  
Rectangular Plates ◽  
Fourier Cosine ◽  
Cosine Series ◽  
Fourier Cosine Series ◽  
Foundation Model

An improved Fourier series method (IFSM) is applied to study the free and forced vibration characteristics of the moderately thick laminated composite rectangular plates on the elastic Winkler or Pasternak foundations which have elastic uniform supports and multipoints supports. The formulation is based on the first-order shear deformation theory (FSDT) and combined with artificial virtual spring technology and the plate-foundation interaction by establishing the two-parameter foundation model. Under the framework of this paper, the displacement and rotation functions are expressed as a double Fourier cosine series and two supplementary functions which have no relations to boundary conditions. The Rayleigh-Ritz technique is applied to solve all the series expansion coefficients. The accuracy of the results obtained by the present method is validated by being compared with the results of literatures and Finite Element Method (FEM). In this paper, some results are obtained by analyzing the varying parameters, such as different boundary conditions, the number of layers and points, the spring stiffness parameters, and foundation parameters, which can provide a benchmark for the future research.

scholarly journals A Mode-Matching Solution for TE-Backscattering from an Arbitrary 2D Rectangular Groove in a PEC

2020 ◽  
Vol 20 (3) ◽  
pp. 159-163 ◽  
Author(s):  
Mehdi Bozorgi
Keyword(s):  
Singular Integrals ◽  
Logarithmic Singularity ◽  
Linear Equations ◽  
Mode Matching ◽  
Electric Conductor ◽  
Fourier Cosine ◽  
Cosine Series ◽  
Fourier Cosine Series ◽  
Highly Oscillatory Integrals ◽  
Rectangular Groove

In this paper, the simple yet effective mode-matching technique is utilized to compute TE-backscattering from a 2D filled rectangular groove in an infinite perfect electric conductor (PEC). The tangential magnetic fields inside and outside of the groove are represented as the sums of infinite series of cosine harmonics (half-range Fourier cosine series). By applying the continuity of the tangential magnetic field, these modes are matched on the groove to obtain the series coefficients by solving a system of linear equations. For this purpose, some oscillatory logarithmic singular integrals involving Hankel and trigonometric functions are solved numerically, starting by removing the logarithmic singularity via integration by parts. In the following, the new well-behaved highly oscillatory integrals are computed using efficient methods, and several comparisons are made to demonstrate the validity and ability of the presented procedure.

scholarly journals An Exact Series Solution for the Vibration of Mindlin Rectangular Plates with Elastically Restrained Edges

2014 ◽  
Vol 2014 ◽  
pp. 1-11
Author(s):  
Xue Kai ◽  
Wang Jiufa ◽  
Li Qiuhong ◽  
Wang Weiyuan ◽  
Wang Ping
Keyword(s):  
Boundary Conditions ◽  
Plate Theory ◽  
Mindlin Plate ◽  
Rectangular Plates ◽  
Cross Sectional ◽  
Fourier Cosine ◽  
Cosine Series ◽  
The Matrix ◽  
Elastically Restrained Edges ◽  
Elastically Restrained

An analysis method is proposed for the vibration analysis of the Mindlin rectangular plates with general elastically restrained edges, in which the vibration displacements and the cross-sectional rotations of the mid-plane are expressed as the linear combination of a double Fourier cosine series and four one-dimensional Fourier series. The use of these supplementary functions is to solve the possible discontinuities with first derivatives at each edge. So this method can be applied to get the exact solution for vibration of plates with general elastic boundary conditions. The matrix eigenvalue equation which is equivalent to governing differential equations of the plate can be derived through using the boundary conditions and the governing equations based on Mindlin plate theory. The natural frequencies can be got through solving the matrix equation. Finally the numerical results are presented to validate the accuracy of the method.

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