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非均匀Winkler弹性地基上变厚度矩形板自由振动的DTM求解 Free vibration analysis for rectangular plates with variable thickness resting on a non-uniform Winkler elastic foundation by DTM
Free vibration analysis for rectangular plates with variable thickness resting on a non-uniform Winkler elastic foundation by DTM
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滕兆春,衡亚洲,刘露
- , 2018, DOI: 10.7511/jslx20170217002
Abstract: 针对非均匀Winkler弹性地基上变厚度矩形板的自由振动问题,通过一种有效的数值求解方法——微分变换法(DTM),研究其量纲固有频率特性。已知变厚度矩形板对边为简支边界条件,其他两边的边界条件为简支、固定或自由任意组合。采用DTM将非均匀Winkler弹性地基上变厚度矩形板量纲化的自由振动控制微分方程及其边界条件变换为等价的代数方程,得到含有量纲固有频率的特征方程。数值结果退化为均匀Winker弹性地基上矩形板以及变厚度矩形板的情形,并与已有文献采用的不同求解方法进行比较,结果表明,DTM具有非常高的精度和很强的适用性。最后,在不同边界条件下分析地基变化参数、厚度变化参数和长宽比对矩形板量纲固有频率的影响,并给出了非均匀Winkler弹性地基上对边简支对边固定变厚度矩形板的前六阶振型。
For free vibration problem of rectangular plates with variable thickness resting on a non-uniform foundation and by an effective solving numerical method called differential transformation method (DTM),the dimensionless natural frequency characteristics are investigated.Two opposite edges of plates are assumed to be simply supported and other two edges can be changed into simply supported,camped or free boundary conditions arbitrarily.By using DTM,dimensionless normalized governing differential equation of rectangular plates with variable thickness resting on a non-uniform Winkler elastic foundation and boundary conditions are transformed to the equivalent algebraic equations,which can derive equations of dimensionless natural frequency.The example results are back to cases for uniform Winkler rectangular plates and rectangular plates with variable thickness,which are compared with different methods in present literature.The result shows that DTM have very higher accuracy and stronger applicability.Finally,the influence of the varied foundation parameter,the varied thickness parameter and the aspect ratio on dimensionless natural frequencies are analyzed for different boundary conditions and deriving the first six mode shapes for CSCS plate with variable thickness resting on a non-uniform Winkler elastic foundations.
An Efficient Analytical Method for Vibration Analysis of a Beam on Elastic Foundation with Elastically Restrained Ends  [PDF]
Mustafa ?zgür Yayli,Murat Aras,Süleyman Aksoy
Shock and Vibration , 2014, DOI: 10.1155/2014/159213
Abstract: An efficient analytical method for vibration analysis of a Euler-Bernoulli beam on elastic foundation with elastically restrained ends has been reported. A Fourier sine series with Stoke’s transformation is used to obtain the vibration response. The general frequency determinant is developed on the basis of the analytical solution of the governing differential equation for all potential solution cases with rigid or restrained boundary conditions. Numerical analyses are performed to investigate the effects of various parameters, such as the springs at the boundaries to examine how the elastic foundation parameters affect the vibration frequencies. 1. Introduction Beams resting on elastic foundations have wide application in engineering practice. The vibration analysis of beams is investigated using various elastic foundation models, such as, Vlasov, Pasternak, and Winkler models. A number of studies have been performed to predict the dynamic response of beams on elastic foundations with different boundary conditions. Numerous works have been performed to explore the static deflection and vibration response of the beams resting on various elastic foundations. Chun [1] has investigated free vibration of hinged beam. Maurizi et al. [2] have considered the vibration frequencies for a beam with different boundary conditions. Vibration of beams on partial elastic foundations has been studied by Doyle and Pavlovic [3]. Laura et al. [4] have investigated beams which carry concentrated masses subject to an axial force. Abbas [5] has investigated vibration of Timoshenko beams with elastically restrained ends. Free vibration and stability behavior of uniform beams and columns with nonlinear elastic end rotational restraints has been considered by Rao and Naidu [6]. Free vibration behaviour of an Euler-Bernoulli beam resting on a variable Winkler foundation has been considered by Kacar et al. [7]. Civalek [8] has implemented differential quadrature and harmonic differential quadrature methods for buckling analysis of thin isotropic plates and elastic columns. H. K. Kim and M. S. Kim [9] have considered vibration of beams with generally restrained boundary conditions. A number of studies have been reported investigating the free vibration of beams on elastic foundation [10–25]. Although vibration analysis of beams on elastic foundation is a widely studied topic, there are only few papers that exist in the literature pertaining to the analysis of beams with elastically restrained ends. In this study, an efficient method is introduced for the analysis of the free
Effect of Vertical Vibration on Block Foundation Resting on Homogeneous and Layered Medium
Ankesh Kumar, Dr. Bappaditya Manna, Prof. K. S. Rao
International Journal of Engineering Research , 2014,
Abstract: In the present study, an investigation is carried out to determine the effect of soil–rock and rock–rock foundation systems on dynamic response of block foundations under vertical mode of vibration. The half-space theory is used for the analysis of foundation resting on homogeneous soil and rocks. The finite element program having transmitting boundaries is considered for layered system considering soil–rock and rock–rock combinations. The analysis is carried out in details for soil– rock and weathered rock–rock systems and the different equations are presented for above combinations. The effect of top layer thicknesses, shear wave velocity and eccentric moments are also simulated. The rock–rock systems considered are sandstone, shale and limestone underlain by basalt rock. It is interpreted that as the shear wave velocity ratio increase the natural frequency increases and the peak displacement amplitude decreases.
Nonlinear Forced Vibration Analysis for Thin Rectangular Plate on Nonlinear Elastic Foundation  [cached]
Zhong Zhengqiang,Xiao Yonggang,Yang Cuiping
Research Journal of Applied Sciences, Engineering and Technology , 2013,
Abstract: Nonlinear forced vibration is analyzed for thin rectangular plate with four free edges on nonlinear elastic foundation. Based on Hamilton variation principle, equations of nonlinear vibration motion for thin rectangular plate under harmonic loads on nonlinear elastic foundation are established. In the case of four free edges, viable expressions of trial functions for this specification are proposed, satisfying all boundary conditions. Then, equations are transformed to a system of nonlinear algebraic equations by using Galerkin method and are solved by using harmonic balance method. In the analysis of numerical computations, the effect on the amplitude-frequency characteristic curve due to change of the structural parameters of plate, parameters of foundation and parameters of excitation force are discussed.
Parametric Study of Dynamic Wrinkling in a Thin Sheet on Elastic Foundation  [PDF]
M. N. Hamdan, A. A. Al-Qaisia, S. Abdallah
International Journal of Modern Nonlinear Theory and Application (IJMNTA) , 2012, DOI: 10.4236/ijmnta.2012.13008
Abstract: This work presents an approximate analytical study of the problem of dynamic wrinkling of a thin metal sheet under a specified time varying tension. The problem is investigated in the framework of the dynamic stability of a nonlinear plate model on elastic foundation which namely takes into account the nonlinear mechanics of mid-plane stretching and the dependence of the membrane force on this mechanics. The plate is assumed to be a wide rectangular slab, hinged at two opposite ends and free at the long ends, which can be deformed in a cylindrical shape so that the governing in-plane bending equation of motion takes the same form as that of a beam (e.g. lateral strip) element. An approximate analytical analysis of the beam wrinkling behavior under sinusoidal parametric excitation is carried out by using the assumed single mode wrinkling motion to reduce the beam field nonlinear partial differential equation to that of a single degree of freedom non-linear oscillator. A first order stability analysis of an approximate analytical solution obtained using the Multi-Time-Scales (MMS) method is used to derive a criterion defining critical driving frequency in terms of system parameters for the initiation of wrinkling motion in the thin metal sheet. Results obtained using this criterion is presented for selected values of system parameters.
Dynamic behaviour under moving concentrated masses of simply supported rectangular plates resting on variable Winkler elastic foundation
Oni, S. T.;Awodola, T. O.;
Latin American Journal of Solids and Structures , 2011, DOI: 10.1590/S1679-78252011000400001
Abstract: the response of simply supported rectangular plates carrying moving masses and resting on variable winkler elastic foundations is investigated in this work. the governing equation of the problem is a fourth order partial differential equation. in order to solve this problem, a technique based on separation of variables is used to reduce the governing fourth order partial differential equations with variable and singular coefficients to a sequence of second order ordinary differential equations. for the solutions of these equations, a modification of the struble's technique and method of integral transformations are employed. numerical results in plotted curves are then presented. the results show that response amplitudes of the plate decrease as the value of the rotatory inertia correction factor r0 increases. furthermore, for fixed value of r0, the displacements of the simply supported rectangular plates resting on variable elastic foundations decrease as the foundation modulus f0 increases. the results further show that, for fixed r0 and f0, the transverse deflections of the rectangular plates under the actions of moving masses are higher than those when only the force effects of the moving load are considered. therefore, the moving force solution is not a safe approximation to the moving mass problem. hence, safety is not guaranteed for a design based on the moving force solution. also, the analyses show that the response amplitudes of both moving force and moving mass problems decrease both with increasing foundation modulus and with increasing rotatory inertia correction factor. the results again show that the critical speed for the moving mass problem is reached prior to that of the moving force for the simply supported rectangular plates on variable winkler elastic foundation.
A Finite Element Study of the Bending Behavior of Beams Resting on Two-Parameter Elastic Foundation  [cached]
Iancu-Bogdan Teodoru,Vasile Mu?at,M. Vrabie
Bulletin of the Polytechnic Institute of Jassy, Constructions, Architechture Section , 2006,
Abstract: Although the Winkler’s model is a poor representation of the many practical subgrade or subbase materials, it is widely used in soil-structure problems for almost one and a half century. The foundations represented by Winkler model can not sustain shear stresses, and hence discontinuity of adjacent spring displacements can occur. This is the prime shortcoming of this foundation model which in practical applications may result in significant inaccuracies in the evaluated structural response. In order to overcome these problem many researchers have been proposed various mechanical foundation models considering interaction with the surroundings. Among them we shall mention the class of two-parameter foundations -- named like this because they have the second parameter which introduces interactions between adjacent springs, in addition to the first parameter from the ordinary Winkler’s model. This class of models includes Filonenko-Borodich, Pasternak, generalized, and Vlasov foundations. Mathematically, the equations to describe the reaction of the two-parameter foundations arc equilibrium ones, and the only difference is the definition of the parameters. For the convenience of discussion, the Pasternak foundation is adopted in present paper. In order to analyse the bending behavior of a Euler-Bernoulli beam resting on two-parameter elastic foundation a (displacement) Finite Element (FE) formulation, based on the cubic displacement function of the governing differential equation, is introduced. The resulting effects of shear stiffness of the Pasternak model on the mechanical quantities are discussed in comparison with those of the Winkler’s model. Some numerical case studies illustrate the accuracy of the formulation and the importance of the soil shearing effect in the vertical direction, associated with continuous elastic foundation.
Analysis of Critical Velocities for an Infinite Timoshenko Beam Resting on an Elastic Foundation Subjected to a Harmonic Moving Load  [PDF]
Bin Zhen,Wei Luo,Jian Xu
Shock and Vibration , 2014, DOI: 10.1155/2014/848536
Abstract: Critical velocities are investigated for an infinite Timoshenko beam resting on a Winkler-type elastic foundation subjected to a harmonic moving load. The determination of critical velocities ultimately comes down to discrimination of the existence of multiple real roots of an algebraic equation with real coefficients of the 4th degree, which can be solved by employing Descartes sign method and complete discrimination system for polynomials. Numerical calculations for the European high-speed rail show that there are at most four critical velocities for an infinite Timoshenko beam, which is very different from those gained by others. Furthermore, the shear wave velocity must be the critical velocity, but the longitudinal wave velocity is not possible under certain conditions. Further numerical simulations indicate that all critical velocities are limited to be less than the longitudinal wave velocity no matter how large the foundation stiffness is or how high the loading frequency is. Additionally, our study suggests that the maximum value of one group velocity of waves in Timoshenko beam should be one “dangerous” velocity for the moving load in launching process, which has never been referred to in previous work. 1. Introduction Moving-load problems have received a great deal of attention worldwide in the past several decades. The earliest moving-load problems are about railway bridges excited by traveling trains. Then the application areas gradually have been extended to high-speed commuter trains, missile sled test tracks, high-speed projectile launchers, and so on. “Critical velocity” is a phenomenon that has been found in beams subjected to moving loads. A resonant wave in a beam can be induced when a load moves at the critical velocity, which results in an unbounded increase of the displacements, rotation, and bending moments of the beam for an undamped case. In reality, damping always exists, but even so, a very large deflection of the beam may occur if the moving load approaches the critical velocity [1–3]. Researches show that the “critical velocity” phenomenon may shorten life of launchers due to high stress, fatigue, premature wear, and gouging of rails [4]. Therefore, it is very necessary to determine the critical velocity for a beam subjected to a moving load. Timoshenko [3] firstly derived an expression for the critical velocity of a concentrated load moving along the Bernoulli-Euler beam resting on a continuous elastic foundation. It was proposed that the critical velocity is analogous to an additional longitudinal compressive force
Finite Element Formulation of Forced Vibration Problem of a Prestretched Plate Resting on a Rigid Foundation  [PDF]
M. Er z,A. Yildiz
Journal of Applied Mathematics , 2007, DOI: 10.1155/2007/56360
Abstract: The three-dimensional linearized theory of elastodynamics mathematical formulation of the forced vibration of a prestretched plate resting on a rigid half-plane is given. The variational formulation of corresponding boundary-value problem is constructed. The first variational of the functional in the variational statement is equated to zero. In the framework of the virtual work principle, it is proved that appropriate equations and boundary conditions are derived. Using these conditions, finite element formulation of the prestretched plate is done. The numerical results obtained coincide with the ones given by Ufly and in 1963 for the static loading case.
弹性地基上转动FGM梁自由振动的DTM分析 DTM analysis for free vibration of rotating FGM beams resting on elastic foundations
DTM analysis for free vibration of rotating FGM beams resting on elastic foundations
 [PDF]

滕兆春,衡亚洲,张会凯,马永斌
- , 2017, DOI: 10.7511/jslx201706006
Abstract: 基于Euler-Bernoulli梁理论,利用广义Hamilton原理推导得到弹性地基上转动功能梯度材料(FGM)梁横向自由振动的运动控制微分方程并进行量纲化,采用微分变换法(DTM)对量纲控制微分方程及其边界条件进行变换,计算了弹性地基上转动FGM梁在夹紧-夹紧、夹紧-简支和夹紧-自由三种边界条件下横向自由振动的量纲固有频率,再将控制微分方程退化到转动和地基时的FGM梁,计算其不同梯度指数时第一阶量纲固有频率值,并和已有文献的FEM和Lagrange乘子法计算结果进行比较,数值完全吻合。计算结果表明,三种边界条件下FGM梁的量纲固有频率随量纲转速和量纲弹性地基模量的增大而增大;在一定量纲转速和量纲弹性地基模量下,FGM梁的量纲固有频率随着FGM梯度指数的增大而减小;但在夹紧-简支和夹紧-自由边界条件下,一阶量纲固有频率几乎不变。
Based on Euler-Bernoulli beam theory,the governing differential equation of motion of the lateral free vibration a rotating functionally graded material (FGM) beam on elastic foundation is derived by using generalized Hamilton principle,and differential transform method (DTM) is used to transform the dimensionless governing differential equation and the boundary conditions.At the same time,the dimensionless natural frequencies of transverse free vibration of rotating FGM beam on elastic foundation at the clamped-clamped,clamped-simply supported and clamped-free three boundary conditions are determined,then the governing differential equation is degenerated to the FGM without rotation and elastic foundation.The values of first non-dimensional natural frequency with different FGM gradient index are calculated and they are completely consistent with the results by either the FEM or the Lagrange multipliers method in the literature.The results show:at the above three kinds of the boundary conditions,the dimensionless natural frequencies increase with the growth of the dimensionless rotating speed and the dimensionless elastic foundation modulus.Under a certain dimensionless rotating speed and dimensionless elastic foundation modulus,the dimensionless natural frequencies decrease along with the growth of the FGM gradient index.However,when at clamped-simply supported and clamped-free boundary conditions,the first dimensionless natural frequency is almost constant.
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