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Winkler地基模型上功能梯度材料涂层的摩擦接触问题 A frictional contact problem of a functionally graded layer resting on a Winkler foundation
A frictional contact problem of a functionally graded layer resting on a Winkler foundation
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陈耀庚,李星
- , 2016, DOI: 10.7511/jslx201605012
Abstract: 研究Winker地基模型上功能梯度材料涂层在一刚性圆柱形冲头作用下的摩擦接触问题。功能梯度材料涂层表面作用有法线向和切线向集中作用力。假设材料非均匀参数呈指数形式变化,泊松比为常量,利用Fourier积分变换技术将求解模型的接触问题转化为奇异积分方程组,再利用切比雪夫多项式对所得奇异积分方程组进行数值求解。最后,给出了功能梯度材料非均匀参数、摩擦系数、Winker地基模型刚度系数及冲头曲率半径对接触应力分布和接触区宽度的影响情况。
This paper presents the investigation to the frictional contact problem for a functionally graded layer under the action of a rigid circular stamp supported by a Winkler foundation.A segment of the top surface of the graded layer is subject to both normal and tangential traction while rest of the surface is devoid of traction.The graded layer is assumed to be an isotropic nonhomogeneous medium with an exponentially varying shear modulus and a constant Poisson's ratio.The problem is reduced to a Cauchy-type singular integral equations with the use of Fourier integral transform technique and the boundary conditions of the problem.The singular integral equations is solved numerically using Chebychev polynomials.The main objective of this paper is to study the effect of the material non-homogeneity factor,stiffness of the friction coefficient,Winkler foundation and punch radius on the contact pressure distribution and the size of the contact region.
Finite difference analysis of curved deep beams on Winkler foundation
Adel A. Al-Azzawi,Ali S. Shaker
Journal of Engineering and Applied Sciences , 2011,
Abstract: This research deals with the linear elastic behavior of curved deep beams resting on elastic foundations with both compressional and frictional resistances. Timoshenko’s deep beam theory is extended to include the effect of curvature and the externally distributed moments under static conditions. As an application to the distributed moment generations, the problems of deep beams resting on elastic foundations with both compressional and frictional restraints have been investigated in detail. The finite difference method was used to represent curved deep beams and the results were compared with other methods to check the accuracy of the developed analysis. Several important parameters are incorporated in the analysis, namely, the vertical subgrade reaction, horizontal subgrade reaction, beam width, and also the effect of beam thickness to radius ratio on the deflections, bending moments, and shear forces. The computer program (CDBFDA) (Curved Deep Beam Finite Difference Analysis Program) coded in Fortran-77 for the analysis of curved deep beams on elastic foundations was formed. The results from this method are compared with other methods exact and numerical and check the accuracy of the solutions. Good agreements are found, the average percentages of difference for deflections and moments are 5.3% and 7.3%, respectively, which indicate the efficiency of the adopted method for analysis.
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
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.
Dynamic response to moving masses of rectangular plates with general boundary conditions and resting on variable winkler foundation
Awodola, T. O.;Oni, S. T.;
Latin American Journal of Solids and Structures , 2013, DOI: 10.1590/S1679-78252013000200005
Abstract: the dynamic response to moving masses of rectangular plates with general classical boundary conditions and resting on variable winkler elastic foundation is investigated in this work. the governing fourth order partial differential equation is solved using a technique based on separation of variables, the modified method of struble and the integral transformations. numerical results in plotted curves are then presented. the results show that as the value of the rotatory inertia correction factor ro increases, the response amplitudes of the plate decrease and that, for fixed value of ro, the displacements of the plate decrease as the foundation modulus fo increases for the variants of the classical boundary conditions considered. the results also show that for fixed ro and fo, 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. for the rectangular plate, for the same natural frequency, the critical speed for moving mass problem is smaller than that of the moving force problem for all variants of classical boundary conditions, that is, resonance is reached earlier in moving mass problem than in moving force problem. when fo and ro increase, the critical speed increases, hence, risk is reduced.
Vibrations of Three-Layered Cylindrical Shells with FGM Middle Layer Resting on Winkler and Pasternak Foundations  [PDF]
Abdul Ghafar Shah,Aalia Ali,Muhmmad Nawaz Naeem,Shahid Hussain Arshad
Advances in Acoustics and Vibration , 2012, DOI: 10.1155/2012/507031
Abstract: Vibrations of a cylindrical shell composed of three layers of different materials resting on elastic foundations are studied out. This configuration is formed by three layers of material in thickness direction where the inner and outer layers are of isotropic materials and the middle is of functionally graded material. Love shell dynamical equations are considered to describe the vibration problem. The expressions for moduli of the Winkler and Pasternak foundations are combined with the shell dynamical equations. The wave propagation approach is used to solve the present shell problem. A number of comparisons of numerical results are performed to check the validity and accuracy of the present approach. 1. Introduction The circular cylindrical shells have been found in various engineering applications ranging from large civil and mechanical structures to small electrical components for many years. Vibrations of cylindrical shells are the most wide studied area of research because of their simple geometrical shapes. First of all, Love [1] presented the linear thin shell theory established on the Kirchhoff’s hypothesis for plates. All other shell theories have been built on this theory by modifying some physical terms involving therein. Numerical solutions of shell vibration problem started to come out in the thirties of the twentieth century and were presented by Flügge [2]. Sharma [3, 4] gave an analysis of thin circular cylinders. Sharma approximated the axial model dependence by beam functions and they used Rayleigh-Ritz technique to solve the shell problem. Paliwal et al. [5] investigated the vibrations of a thin circular cylindrical shell attached with elastic foundations. Loy et al. [6] investigated the vibrations of functionally graded material cylindrical shells, made up of FG material composed of stainless steel and nickel. The purpose of work was to examine natural frequencies, influence of the constituent volume fractions, and effects of configurations of constituent materials on their frequencies. Pardhan et al. [7] studied the vibrations of functionally graded material (FGM) cylindrical shell structured from stainless steel and zirconia. Zhang et al. [8] studied the vibration frequencies of cylindrical shells with fluid-filled. They used wave propagation approach and they compared the uncoupled frequencies with available results in the literature. Najafizadeh and Isvandzibaei [9] studied the vibrations of thin-walled cylindrical shells with ring supports composed of functionally graded material comprised of stainless steel and nickel. Arshad
Nonlinear Dynamic Analysis of a Timoshenko Beam Resting on a Viscoelastic Foundation and Traveled by a Moving Mass  [PDF]
Ahmad Mamandi,Mohammad H. Kargarnovin
Shock and Vibration , 2014, DOI: 10.1155/2014/242090
Abstract: The dynamic response of a Timoshenko beam with immovable ends resting on a nonlinear viscoelastic foundation and subjected to motion of a traveling mass moving with a constant velocity is studied. Primarily, the beam’s nonlinear governing coupled PDEs of motion for the lateral and longitudinal displacements as well as the beam’s cross-sectional rotation are derived using Hamilton’s principle. On deriving these nonlinear coupled PDEs the stretching effect of the beam’s neutral axis due to the beam’s fixed end conditions in conjunction with the von-Karman strain-displacement relations is considered. To obtain the dynamic responses of the beam under the act of a moving mass, derived nonlinear coupled PDEs of motion are solved by applying Galerkin’s method. Then the beam’s dynamic responses are obtained using mode summation technique. Furthermore, after verification of our results with other sources in the literature a parametric study on the dynamic response of the beam is conducted by changing the velocity of the moving mass, damping coefficient, and stiffnesses of the foundation including linear and cubic nonlinear parts, respectively. It is observed that the inclusion of geometrical and foundation stiffness nonlinearities into the system in presence of the foundation damping will produce significant effect in the beam’s dynamic response. 1. Introduction The topic of vibration study of structural elements such as strings, beams, plates, and shells under the act of a moving mass is of great interest and importance in the field of structural dynamics. It should be noted that the review of numerous reported studies related to the dynamic behavior of mechanical/structural systems discloses that almost linear behavior of such systems is considered. Indeed, in reality such systems inherently and naturally have nonlinear behavior, for example, due to the geometrical nonlinearity or when they are subjected to external loadings comparatively large enough. As we will see later on in the modeling of the problem, the stretching of the beam’s neutral axis due to fixed ends condition adds another nonlinearity to the dynamical behavior of the system. In addition, there are some other external distinct mechanical elements having nonlinear behavior attached to such structures like shock energy absorbers or viscoelastic foundations which will add further other nonlinearities in the model analysis. From mechanical point of view, any beam structure can be modeled as a thin or thick beam for which different theories usually can be implemented. In extending the issue of the
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
 [PDF]

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.
Free-Form Laminated Doubly-Curved Shells and Panels of Revolution Resting on Winkler-Pasternak Elastic Foundations: A 2-D GDQ Solution for Static and Free Vibration Analysis  [PDF]
Francesco Tornabene, Alessandro Ceruti
World Journal of Mechanics (WJM) , 2013, DOI: 10.4236/wjm.2013.31001
Abstract:

This work presents the static and dynamic analyses of laminated doubly-curved shells and panels of revolution resting on Winkler-Pasternak elastic foundations using the Generalized Differential Quadrature (GDQ) method. The analyses are worked out considering the First-order Shear Deformation Theory (FSDT) for the above mentioned moderately thick structural elements. The effect of the shell curvatures is included from the beginning of the theory formulation in the kinematic model. The solutions are given in terms of generalized displacement components of points lying on the middle surface of the shell. Simple Rational Bézier curves are used to define the meridian curve of the revolution structures. The discretization of the system by means of the GDQ technique leads to a standard linear problem for the static analysis and to a standard linear eigenvalue problem for the dynamic analysis. Comparisons between the present formulation and the Reissner-Mindlin theory are presented. Furthermore, GDQ results are compared with those obtained by using commercial programs. Very good agreement is observed. Finally, new results are presented in order to investtigate the effects of the Winkler modulus, the Pasternak modulus and the inertia of the elastic foundation on the behavior of laminated shells of revolution.

A transmission problem for the Timoshenko system
Raposo, C.A.;Bastos, W.D.;Santos, M.L.;
Computational & Applied Mathematics , 2007, DOI: 10.1590/S0101-82052007000200003
Abstract: in this work we study a transmission problem for the model of beams developed by s.p. timoshenko [10]. we consider the case of mixed material, that is, a part of the beam has friction and the other is purely elastic. we show that for this type of material, the dissipation produced by the frictional part is strong enough to produce exponential decay of the solution, no matter how small is its size. we use the method of energy to prove exponential decay for the solution.
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