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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.
 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  has investigated free vibration of hinged beam. Maurizi et al.  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 . Laura et al.  have investigated beams which carry concentrated masses subject to an axial force. Abbas  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 . Free vibration behaviour of an Euler-Bernoulli beam resting on a variable Winkler foundation has been considered by Kacar et al. . Civalek  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  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
 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.
 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.
 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.
 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.
 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.