scholarly journals Three-dimensional free vibration of thick plates with general end conditions and resting on elastic foundations

2018 ◽  
Vol 38 (1) ◽  
pp. 110-121
Author(s):  
Zhuang Lin ◽  
Shuangxia Shi
Keyword(s):  
Boundary Conditions ◽  
Ritz Method ◽  
Three Dimensional ◽  
Series Representation ◽  
General Boundary ◽  
Geometrical Parameters ◽  
Penalty Function Method ◽  
Rectangular Plates ◽  
General Boundary Conditions ◽  
Elastic Foundations

This paper presents a three-dimensional formulation for the free vibrations of thick rectangular plates with general boundary conditions and resting on elastic foundations. The general boundary conditions are imposed by means of penalty function method. The displacements of the plates are expressed by a three-dimensional cosine series and some simple polynomial functions which introduced to ensure and accelerate the convergence of the series representation. All the unknown coefficients can be obtained by using the Rayleigh–Ritz method. Comparisons of the present results with those in available literature demonstrate the accuracy and reliability of the present formulation. Furthermore, parametric investigations are presented including the effects of boundary conditions, geometrical parameters, and elastic foundation.

Vibration Analysis of Laminated Composite Rectangular Plates With General Boundary Conditions

2018 ◽  
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.

scholarly journals Static analysis of four-parameter functionally graded plates with general boundary conditions

2019 ◽  
Vol 13 (2) ◽  
pp. 12-23
Author(s):  
Hoang Thu Phuong ◽  
Tran Huu Quoc ◽  
Ho Thi Hien
Keyword(s):  
Boundary Conditions ◽  
Power Law ◽  
Material Properties ◽  
Load Distribution ◽  
Plate Theory ◽  
Ritz Method ◽  
Functionally Graded ◽  
General Boundary ◽  
Geometrical Parameters ◽  
General Boundary Conditions

In this study, the Ritz variational method is used to analyze and solve the bending problem of rectangular functionally graded material plate with general boundary conditions and subject to some types of load distribution over the entire plate domain. Based on the Kirchoff plate theory, the equilibrium equations are obtained by minimizing the total potential energy. The material properties are assumed to be graded through the thickness of the plates according to a power law with four parameters. The accuracy of the solution has been checked and validated through different comparisons to that available literature. A wide variety of examples have been carried out to reveal the influences of different geometrical parameters, FGM power law index, type of load distribution and boundary conditions on the bending responses of the plates. The results show that the gradients in material properties play an important role in determining the response of the FGM plates.Keywords: FGM; Kirchhoff plate; Ritz method; boundary conditions.

scholarly journals Three-Dimensional Solution for the Vibration Analysis of Functionally Graded Rectangular Plate with/without Cutouts Subject to General Boundary Conditions

Materials ◽  
2021 ◽  
Vol 14 (22) ◽  
pp. 7088
Author(s):  
Wenhao Huang ◽  
Kai Xue ◽  
Qiuhong Li
Keyword(s):  
Boundary Conditions ◽  
Rectangular Plate ◽  
Vibration Analysis ◽  
Ritz Method ◽  
Three Dimensional ◽  
Functionally Graded ◽  
General Boundary ◽  
Global Coordinate System ◽  
General Boundary Conditions ◽  
Characteristic Analysis

Functionally graded materials (FGMs) structures are increasingly used in engineering due to their superior mechanical and material properties, and the FGMs plate with cutouts is a common structural form, but research on the vibration characteristics of FGMs plate with cutouts is relatively limited. In this paper, the three-dimensional exact solution for the vibration analysis of FGMs rectangular plate with circular cutouts subjected to general boundary conditions is presented based on the three-dimensional elasticity theory. The displacement field functions are expressed as standard cosine Fourier series plus auxiliary cosine series terms satisfying the boundary conditions in the global coordinate system. The plate with circular cutout is discretized into four curve quadrilateral sub-domains using the p-version method, and then the blending function method is applied to map the closed quadrilateral region to the computational space. The characteristic equation is obtained based on the Lagrangian energy principle and Rayleigh–Ritz method. The efficiency and reliability of proposed method are verified by comparing the present results with those available in the literature and FEM methods. Finally, a parametric study is investigated including the cutout sizes, the cutout positions, and the cutout numbers from the free vibration characteristic analysis and the harmonic analysis. The results can serve as benchmark data for other research on the vibration of FGMs plates with cutouts.

scholarly journals Dynamic and Sound Radiation Characteristics of Rectangular Thin Plates with General Boundary Conditions

2019 ◽  
Vol 6 (1) ◽  
pp. 117-131
Author(s):  
Yuan Du ◽  
Haichao Li ◽  
Qingtao Gong ◽  
Fuzhen Pang ◽  
Liping Sun
Keyword(s):  
Dynamic Response ◽  
Boundary Conditions ◽  
Ritz Method ◽  
Sound Radiation ◽  
Current Method ◽  
Thin Plates ◽  
General Boundary ◽  
General Boundary Conditions ◽  
Radiation Characteristics ◽  
Aspect Ratios

AbstractBased on the classical Kirchhoff hypothesis, the dynamic response and sound radiation of rectangular thin plates with general boundary conditions are studied. The transverse displacements of plate are represented by a double Fourier cosine series and three supplementary functions. The potential discontinuity associated with the original governing equation can be transferred to auxiliary series functions. All kinds of boundary conditions can be easily achieved by varying stiffness value of springs on each edge. The natural frequencies and vibration response of the plates are obtained by means of the Rayleigh–Ritz method. Sound radiation characteristics of the plate are derived using Rayleigh integral formula. Current method works well when handling dynamic response and sound radiation of plates with general boundary conditions. The accuracy and reliability of current method are confirmed by comparing with related literature and FEM. The non-dimensional frequency parameters of the rectangular plates with different boundary conditions and aspect ratios are presented in the paper, which may be useful for future researchers.Meanwhile, some interesting points are foundwhen analyzing acoustic radiation characteristics of plates.

Effect of the multiple projectile on the low-velocity impact response of CNTs reinforced beam

2020 ◽  
Vol 17 (1) ◽  
pp. 1-17
Author(s):  
Ali Sadik Gafer Qanber ◽  
Raed Salman Saeed Alhusseini ◽  
Bashar Dheyaa Hussein Al-Kasob ◽  
Manar Hamid Jasim ◽  
Mehdi Ranjbar
Keyword(s):  
Boundary Conditions ◽  
Volume Fraction ◽  
Ritz Method ◽  
Low Velocity Impact ◽  
General Boundary ◽  
General Boundary Conditions ◽  
Lagrange Equations ◽  
Low Velocity ◽  
Content Type ◽  
Velocity Impact

PurposeThe main objective of this article is to develop a theoretical formulation for predicting the response of CNTs reinforced beam under multiple impactors with general boundary conditions, using first-order shear deformation beam theory.Design/methodology/approachThe rule of mixtures is implemented to derive the material properties of the beam. The nonlinear Hertz contact law is applied for simulation between impactors and the surface of the beam. A combination of approaches includes energy method, Ritz method and generalized Lagrange equations are used to extract the matrix form of equations of motion. The time-domain solution is obtained using implementing the well-known Runge Kutta 4th order method.FindingsAfter examining the accuracy of the present method, the effects of the number of impactors include one impactor, and three impactors in various CNTs volume fraction are studied for CNTs reinforced beam with clamped-clamped, clamped-free and simply supported boundary conditions under the low-velocity impact. The most important finding of this article is that contact force and beam indentation at the middle of the beam in the case of one impactor are greater than those reported in the case of three impactors.Originality/valueThis article fulfills an identified need to study how CNTs reinforced beam behaviour with general boundary conditions under multiple low-velocity impacts can be enabled.

Free In-Plane Vibration Analysis of Annular Plates with General Boundary Conditions

2013 ◽  
Vol 572 ◽  
pp. 189-192
Author(s):  
Dong Yan Shi ◽  
Xian Jie Shi ◽  
Wen L. Li ◽  
Zheng Rong Qin
Keyword(s):  
Boundary Conditions ◽  
Vibration Analysis ◽  
Ritz Method ◽  
Current Method ◽  
General Boundary ◽  
General Boundary Conditions ◽  
Series Solutions ◽  
Plane Vibration ◽  
Annular Plates ◽  
Special Cases

An analytical method is derived for the free in-plane vibration analysis of annular plates with general boundary conditions. Under this framework, all the classical homogeneous boundary conditions can be treated as the special cases when the stiffness for each restraining springs is equal to either zero or infinity. The improved Fourier series solutions for the in-plane vibrations are obtained by employing the Rayleigh-Ritz method. A numerical example is presented to demonstrate the accuracy and reliability of the current method.

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