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.

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.

Dynamic Analysis of Circular Cylindrical Shells With General Boundary Conditions Using Modified Fourier Series Method

2012 ◽  
Vol 134 (4) ◽  
Author(s):  
Lu Dai ◽  
Tiejun Yang ◽  
W. L. Li ◽  
Jingtao Du ◽  
Guoyong Jin
Keyword(s):  
Fourier Series ◽  
Boundary Conditions ◽  
Dynamic Analysis ◽  
Cylindrical Shells ◽  
Ritz Method ◽  
Current Method ◽  
General Boundary ◽  
General Boundary Conditions ◽  
Arbitrary Boundary ◽  
Circular Cylindrical Shells

Dynamic behavior of cylindrical shell structures is an important research topic since they have been extensively used in practical engineering applications. However, the dynamic analysis of circular cylindrical shells with general boundary conditions is rarely studied in the literature probably because of a lack of viable analytical or numerical techniques. In addition, the use of existing solution procedures, which are often only customized for a specific set of different boundary conditions, can easily be inundated by the variety of possible boundary conditions encountered in practice. For instance, even only considering the classical (homogeneous) boundary conditions, one will have a total of 136 different combinations. In this investigation, the flexural and in-plane displacements are generally sought, regardless of boundary conditions, as a simple Fourier series supplemented by several closed-form functions. As a result, a unified analytical method is generally developed for the vibration analysis of circular cylindrical shells with arbitrary boundary conditions including all the classical ones. The Rayleigh-Ritz method is employed to find the displacement solutions. Several examples are given to demonstrate the accuracy and convergence of the current solutions. The modal characteristics and vibration responses of elastically supported shells are discussed for various restraining stiffnesses and configurations. Although the stiffness distributions are here considered to be uniform along the circumferences, the current method can be readily extended to cylindrical shells with nonuniform elastic restraints.

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 Response of Fractionally Damped Beams with General Boundary Conditions Subjected to Moving Loads

2012 ◽  
Vol 19 (3) ◽  
pp. 333-347 ◽  
Author(s):  
R. Abu-Mallouh ◽  
I. Abu-Alshaikh ◽  
H.S. Zibdeh ◽  
Khaled Ramadan
Keyword(s):  
Dynamic Response ◽  
Boundary Conditions ◽  
Fractional Derivative ◽  
Fractional Derivatives ◽  
Initial Conditions ◽  
General Boundary ◽  
General Boundary Conditions ◽  
Damping Characteristics ◽  
The Laplace Transform ◽  
The Right

This paper presents the transverse vibration of Bernoulli-Euler homogeneous isotropic damped beams with general boundary conditions. The beams are assumed to be subjected to a load moving at a uniform velocity. The damping characteristics of the beams are described in terms of fractional derivatives of arbitrary orders. In the analysis where initial conditions are assumed to be homogeneous, the Laplace transform cooperates with the decomposition method to obtain the analytical solution of the investigated problems. Subsequently, curves are plotted to show the dynamic response of different beams under different sets of parameters including different orders of fractional derivatives. The curves reveal that the dynamic response increases as the order of fractional derivative increases. Furthermore, as the order of the fractional derivative increases the peak of the dynamic deflection shifts to the right, this yields that the smaller the order of the fractional derivative, the more oscillations the beam suffers. The results obtained in this paper closely match the results of papers in the literature review.

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.

Dynamic Response of a Spinning Timoshenko Beam With General Boundary Conditions and Subjected to a Moving Load

1994 ◽  
Vol 61 (1) ◽  
pp. 152-160 ◽  
Author(s):  
J. W.-Z. Zu ◽  
R. P. S. Han
Keyword(s):  
Dynamic Response ◽  
Boundary Conditions ◽  
Timoshenko Beam ◽  
Moving Load ◽  
Original System ◽  
Adjoint System ◽  
General Boundary ◽  
General Boundary Conditions ◽  
Adjoint Systems ◽  
Analysis Technique

The dynamic response of a spinning Timoshenko beam with general boundary conditions and subjected to a moving load is solved analytically for the first time. Solution of the problem is achieved by formulating the spinning Timoshenko beams as a non-self-adjoint system. To compute the system dynamic response using the modal analysis technique, it is necessary to determine the eigenquantities of both the original and adjoint systems. In order to fix the adjoint eigenvectors relative to the eigenvectors of the original system, the biorthonormality conditions are invoked. Responses for the four classical boundary conditions which do not involve rigidbody motions are illustrated. To ensure the validity of the method, these results are compared with those from Euler-Bernoulli and Rayieigh beam theories. Numerical simulations are performed to study the influence of the four boundary conditions on selected system parameters.

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.

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