Analysis of general quadrilateral orthotropic thick plates with arbitrary boundary conditions by the Rayleigh-Ritz method

This paper presents free vibration analysis of open and closed shells with arbitrary boundary conditions using a spectro-geometric-Ritz method. In this method, regardless of the boundary conditions, each of the displacement components of open and closed shells is represented simultaneously as a standard Fourier cosine series and several auxiliary functions. The auxiliary functions are introduced to accelerate the convergence of the series expansion and eliminate all the relevant discontinuities with the displacement and its derivatives at the boundaries. The boundary conditions are modeled using the spring stiffness technique. All the expansion coefficients are treated equally and independently as the generalized coordinates and determined using Rayleigh-Ritz method. By using this method, a unified vibration analysis model for the open and closed shells with arbitrary boundary conditions can be established without the need of changing either the equations of motion or the expression of the displacement components. The reliability and accuracy of the proposed method are validated with the FEM results and those from the literature.

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Free vibration analysis of three-layer thin cylindrical shell with variable thickness two-dimensional FGM middle layer under arbitrary boundary conditions

Journal of Sandwich Structures & Materials ◽

10.1177/10996362211020429 ◽

2021 ◽

pp. 109963622110204

Author(s):

Xue-Yang Miao ◽

Chao-Feng Li ◽

Yu-Lin Jiang ◽

Zi-Xuan Zhang

Keyword(s):

Cylindrical Shell ◽

Boundary Conditions ◽

Volume Fraction ◽

Ritz Method ◽

Admissible Function ◽

Free Vibration Analysis ◽

Middle Layer ◽

Two Dimensional ◽

Arbitrary Boundary ◽

Arbitrary Boundary Conditions

In this paper, a unified method is developed to analyze free vibrations of the three-layer functionally graded cylindrical shell with non-uniform thickness. The middle layer is composed of two-dimensional functionally gradient materials (2D-FGMs), whose thickness is set as a function of smooth continuity. Four sets of artificial springs are assigned at the ends of the shells to satisfy the arbitrary boundary conditions. The Sanders’ shell theory is used to obtain the strain and curvature-displacement relations. Furthermore, the Chebyshev polynomials are selected as the admissible function to improve computational efficiency, and the equation of motion is derived by the Rayleigh–Ritz method. The effects of spring stiffness, volume fraction indexes, configuration on of shell, and the change in thickness of the middle layer on the modal characteristics of the new structural shell are also analyzed.

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

Shock and Vibration ◽

10.1155/2017/3425298 ◽

2017 ◽

Vol 2017 ◽

pp. 1-10 ◽

Cited By ~ 2

Author(s):

Huimin Liu ◽

Fanming Liu ◽

Xin Jing ◽

Zhenpeng Wang ◽

Linlin Xia

Keyword(s):

Boundary Conditions ◽

Admissible Function ◽

Three Dimensional ◽

Future Research ◽

Pasternak Foundation ◽

Arbitrary Boundary ◽

Thick Plates ◽

Arbitrary Boundary Conditions ◽

Ritz Procedure ◽

Three Dimensional Elasticity

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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A Modified Fourier–Ritz Formulation for Vibration Analysis of Arbitrarily Restrained Rectangular Plate with Cutouts

Shock and Vibration ◽

10.1155/2018/8759074 ◽

2018 ◽

Vol 2018 ◽

pp. 1-22

Author(s):

Yiming Liu ◽

Zhuang Lin ◽

Hu Ding ◽

Guoyong Jin ◽

Sensen Yan

Keyword(s):

Boundary Conditions ◽

Vibration Analysis ◽

Ritz Method ◽

Flexural Vibration ◽

Vibration Characteristics ◽

Geometric Parameters ◽

Arbitrary Boundary ◽

Arbitrary Boundary Conditions ◽

Stiffness Constant ◽

Proper Values

A modified Fourier–Ritz method is developed for the flexural and in-plane vibration analysis of plates with two rectangular cutouts with arbitrary boundary conditions, aiming to provide a unified solving process for cases that the plate has various locations or sizes of cutout, and different kinds of boundary conditions. Under the current framework, modifying the position of the cutout or the boundary conditions of the plate is just as changing the geometric parameters of the plate, and there is no need to change the solution procedures. The arbitrary boundary conditions can be obtained by setting the stiffness constant of the boundary springs which are fixed uniformly along the edges of the plate at proper values. The strain and kinetic energy functions of a plate with rectangular cutout are derived in detail. The convergence and accuracy of the present method are demonstrated by comparing the present results with those obtained from the FEM software. In this paper, free in-plane and flexural vibration characteristics of the plate with rectangular cutout under general boundary conditions are studied. From the results, it can be found that the geometric parameters and positions of the cutout and the boundary conditions of the plate will obviously influence the natural vibration characteristics of the structures.

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Vibration Analysis for Rotating Ring-Stiffened Cylindrical Shells With Arbitrary Boundary Conditions

Journal of Vibration and Acoustics ◽

10.1115/1.4024220 ◽

2013 ◽

Vol 135 (6) ◽

Cited By ~ 17

Author(s):

Lun Liu ◽

Dengqing Cao ◽

Shupeng Sun

Keyword(s):

Free Vibration ◽

Boundary Conditions ◽

Vibration Analysis ◽

Cylindrical Shells ◽

Ritz Method ◽

Free Vibration Analysis ◽

Arbitrary Boundary ◽

Arbitrary Boundary Conditions ◽

Classical Boundary ◽

Stiffened Cylindrical Shells

The free vibration analysis of rotating ring-stiffened cylindrical shells with arbitrary boundary conditions is investigated by employing the Rayleigh–Ritz method. Six sets of characteristic orthogonal polynomials satisfying six classical boundary conditions are constructed directly by employing Gram–Schmidt procedure and then are employed to represent the general formulations for the displacements in any axial mode of free vibrations for shells. Employing those formulations during the Rayleigh–Ritz procedure and based on Sanders' shell theory, the eigenvalue equations related to rotating ring-stiffened cylindrical shells with various classical boundary conditions have been derived. To simulate more general boundaries, the concept of artificial springs is employed and the eigenvalue equations related to free vibration of shells under elastic boundary conditions are derived. By adjusting the stiffness of artificial springs, those equations can be used to investigate the vibrational characteristics of shells with arbitrary boundaries. By comparing with the available analytical results for the ring-stiffened cylindrical shells and the rotating shell without stiffeners, the method proposed in this paper is verified. Strong convergence is also observed from convergence study. Further, the effects of parameters, such as the stiffness of artificial springs, the rotating speed of the ring-stiffened shell, the number of ring stiffeners and the depth to width ratio of ring stiffeners, on the natural frequencies are studied.

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Static and Dynamic Analysis of Annular Sector Plates Subjected to Arbitrary Boundary Conditions

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.624.240 ◽

2014 ◽

Vol 624 ◽

pp. 240-244

Author(s):

Kai Peng Zhang ◽

Cheng Yang ◽

Han Wu

Keyword(s):

Fourier Series ◽

Boundary Conditions ◽

Ritz Method ◽

Current Method ◽

Displacement Function ◽

Current Approach ◽

Arbitrary Boundary ◽

Static And Dynamic Analysis ◽

Arbitrary Boundary Conditions ◽

Annular Sector

In this investigation, an improved Fourier series method (IFSM) is employed to predict the static and dynamic characteristics of annular sector plates with arbitrary boundary conditions. Regardless of boundary supports, the displacement function is invariantly expressed as a modified two-dimensional Fourier series containing sine and cosine function. It is capable of dealing with the possible discontinuities at elastic boundary edges. The unknown Fourier coefficients are treated as generalized coordinates, and determined using Rayleigh-Ritz method. Unlike most of the existing solution techniques, the current approach can be universally applied to a variety of edge restraints including all classical cases and their combinations. The accuracy and reliability of the current method are fully illustrated through all the numerical examples.

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