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弹性约束边界圆环板面内自由振动的二维弹性解 Two-dimensional elasticity solutions for in-plane free vibration of annular plates with elastically restrained edges
Two-dimensional elasticity solutions for in-plane free vibration of annular plates with elastically restrained edges

- , 2016, DOI: 10.7511/jslx201605007
Abstract: 基于二维线弹性理论,应用哈密顿原理导出弹性约束边界圆环板面内自由振动的控制微分方程。采用微分求积法(DQM)数值研究了弹性约束边界圆环板面内自由振动的频率特性。通过设置弹性刚度系数为0或∞,问题退化为四种典型边界圆环板的面内自由振动,与已有文献的计算数值结果进行比较,证实本文的分析求解方法行之有效。最后全面考虑了圆环板边界条件、几何系数及刚度系数对自振频率的影响。
Based on the two-dimension theory of linear elasticity and applying Hamilton's principle,the in-plane free vibration of governing differential equations for annular plates are obtained.Applying differential quadrature method (DQM),the frequencies of in-plane free vibration of annular plates with elastically restrained edges are investigated.All the classical boundary for in-plane displacements can be simulated by setting the stiffnesses of the restraining springs to either zero or infinite.The results presented in this paper has illustrated the analytical method was effective and accurate by comparison of previously reported results with those published in literatures.Finally,The influence of the boundary conditions,geometrical parameter,and stiffness coefficients on the dimensionless frequencies of the annular plates are fully investigated.
Dynamical Analysis of Long Fiber-Reinforced Laminated Plates with Elastically Restrained Edges  [PDF]
Liz G. Nallim,Facundo J. Bellomo,Ricardo D. Quinteros,Sergio Oller
Advances in Acoustics and Vibration , 2012, DOI: 10.1155/2012/189376
Abstract: This paper presents a variational formulation for the free vibration analysis of unsymmetrically laminated composite plates with elastically restrained edges. The study includes a micromechanics approach that allows starting the study considering each layer as constituted by long unidirectional fibers in a continuous matrix. The Mori-Tanaka method is used to predict the mechanical properties of each lamina as a function of the elastic properties of the components and of the fiber volume fraction. The resulting mechanical properties for each lamina are included in a general Ritz formulation developed to analyze the free vibration response of thick laminated anisotropic plates resting on elastic supports. Comprehensive numerical examples are computed to validate the present method, and the effects of the different mechanical and geometrical parameters on the dynamical behavior of different laminated plates are shown. New results for general unsymmetrical laminates with elastically restrained edges are also presented. The analytical approximate solution obtained in this paper can also be useful as a basis to deal with optimization problems under, for instance, frequency constraints. 1. Introduction Fiber-reinforced composite laminated plates are extensively used in many engineering applications. The free vibration analysis of these plates plays a very important role in the design of civil, aerospace, mechanical, and marine structures. In addition to the favorable high specific strength and high specific stiffness, fiber-reinforced composite laminates offer the possibility of optimal design through the variation of stacking pattern, angle of fiber orientation, fiber content, and so forth, known as composite tailoring. All these mechanical and geometrical characteristics, as well as the various coupling effects that take place, must be considered in the prediction of the laminates dynamical response to assure that this is reliable, accurate, and adequate to the design requirements. It is well known that laminated composite plates have relatively low transverse shear stiffness, playing the shear deformation an important role in the global and local behavior of these structures. Among the numerous theories used for laminated plates that include the transverse shear strain, the first-order shear deformation theory (FSDT) [1, 2] is adequate for the computation of global responses (such as natural frequencies) and simultaneously has some advantages due to its simplicity and low computational cost. Many investigations have been reported for free vibration analysis
Free Vibration Analysis of Functionally Graded Beams with General Elastically End Constraints by DTM  [PDF]
Nuttawit Wattanasakulpong, Variddhi Ungbhakorn
World Journal of Mechanics (WJM) , 2012, DOI: 10.4236/wjm.2012.26036
Abstract: The differential transformation method (DTM) is applied to investigate free vibration of functionally graded beams supported by arbitrary boundary conditions, including various types of elastically end constraints. The material properties of functionally graded beams are assumed to obey the power law distribution. The main advantages of this method are known for its excellence in high accuracy with small computational expensiveness. The DTM also provides all natural frequencies and mode shapes without any frequency missing. Fundamental frequencies as well as their higher frequencies and mode shapes are presented. The significant aspects such as boundary conditions, values of translational and rotational spring constants and the material volume fraction index on the natural frequencies and mode shapes are discussed. For elastically end constraints, some available results of special cases for isotropic beams are used to validate the present results. The new frequency results and mode shapes of functionally graded beams resting on elastically end constraints are presented.
Dynamic Bhaviour Under Moving Concentrated Masses of Elastically Supported Finite Bernoulli-Euler Beam on Winkler Foundation
S.T. Oni,T.O. Awodola
Journal of Modern Mathematics and Statistics , 2012,
Abstract: The dynamic response of an elastically supported Bernoulli-Euler beam carrying moving masses and resting on a constant elastic foundation is investigated in this study. This problem, involving non-classical boundary conditions is solved and illustrated with 2 commonest examples often encountered in Engineering practice. Analysis of the closed form solutions shows that, for the same natural frequency the response amplitude for the moving mass problem is greater than that of the moving force problem for fixed axial force and foundation modulli, The critical speed for the moving mass problem is smaller than that for the moving force problem and so resonance is reached earlier in the former. Similarly, an increase in the value of foundation modulli and axial force reduces the critical speed for both illustrative examples. The response amplitudes of both moving force and moving mass problems was also found to decrease when both the foundation moduli K and the axial force N are increased.
Fluid-pipe coupling axis vibration characteristics induced by foundation vibration

- , 2016, DOI: 10.13700/j.bh.1001-5965.2015.0208
Abstract: 摘要 针对轴向基础振动对管道和流体波动的影响,运用轴向基础振动下液压直管道轴向耦合振动数学模型,推导了4种不同管端约束方式下的边界条件,并采用特征线法对不同约束方式下基础振动诱发的管内流体波动进行了研究,分析了管端约束方式、约束刚度、基础振动参数、结构参数对管道出口压力波动幅值的影响。结果表明:与两端固定约束相比,出口轴向自由和入口轴向自由时出口流体压力波动幅值分别增大了很多,且出口处约束刚度越大,压力波动幅值越小;基础振动频率越大,流固耦合作用越剧烈;压力波动幅值随基础振动幅值增大而线性增大;流体流经管道的距离越长,流体波动越剧烈。分析结果能为制定相应的管道振动控制策略提供理论依据。
Abstract:In view of the effect of axial foundation vibration on pipe and fluid fluctuation, an axial coupling vibration mathematical model of direct hydraulic pipeline was used to deduce boundary conditions under four different pipe end constraints, and method of characteristics was adopted to study the fluid fluctuation in pipe induced by foundation vibration under different constraints. The influences of bound manner, restraint stiffness, foundation vibration parameters and structural parameters on pipe outlet pressure fluctuation amplitude were analyzed. The results indicate that the outlet pressure fluctuation amplitudes increase a lot respectively when export axial and entrance axial are free, compared with fixed constraints of both ends. And the greater the exit restraint stiffness is, the smaller the pressure fluctuation amplitude is; the higher vibration frequency is, the stronger fluid-solid interaction is; the pressure fluctuation amplitude increases linearly with the increase of foundation vibration amplitude; the longer the distance of fluid flowing through the pipe is, the severer the fluid fluctuating is. The analysis results could provide a theoretical basis for the formulation of corresponding pipe vibration control strategy.
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.
Finite difference method application in design of foundation girder of variable cross-section loaded on ends  [PDF]
Prolovi? Verka,Boni? Zoran
Facta Universitatis Series : Architecture and Civil Engineering , 2008, DOI: 10.2298/fuace0802179p
Abstract: Deflection of foundation girder supported by the deformable base has been defined by the system of differential equations, where the differential equation of the elastic line of the girder is of the fourth order. The most convenient solution of the problem is application of numerical procedures, in this case it is the finite difference method. In the paper, the mentioned method is applied in the special case of foundation girder of variable cross section loaded by arbitrary load on its ends.

FANG Qin,DU Maolin,

力学与实践 , 2006,
Abstract: Much attention has been paid to the properties of materials and the shapes of cross sections of beams in order to increase the resistance of blast-loaded structures. This paper discusses how to increase the blast resistance of beams effectively by elastic and damping supports at the ends of beams. The equations of the forced vibration of an elastically supported beam with damping are obtained by the Lagrange equations. The responses of the beam subjected to two typical blast loads with long and short durations are analyzed. The numerical results show that the displacements of the ends are decreased with the increase of the damping coefficient and the stiffness of the elastic support. In the case of blast loads with long duration, the resistance of beams can only be increased effectively by using elastic and damping supports simultaneously at the ends of beams, while the resistance of beams can be increased effectively by using elastic and damping supports either jointly or separately in the case of blast loads with short duration.
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.
Parametric Study of Nonlinear Beam Vibration Resting on Linear Elastic Foundation
Journal of Mechanical Engineering and Automation , 2012, DOI: 10.5923/j.jmea.20120206.02
Abstract: Nonlinear beam resting on linear elastic foundation and subjected to harmonic excitation is investigated. The beam is simply supported at both ends. Both linear and nonlinear analyses are carried out. Hamilton’s principle is utilized in deriving the governing equations. Well known forced duffing oscillator equation is obtained. The equation is analyzed numerically using Runk-Kutta technique. Three main parameters are investigated: the damping coefficient, the natural frequency, and the coefficient of the nonlinearity. Stability regions for first mode analyses are unveiled. Comparison between the linear and the nonlinear model is presented. It is shown that first mode shape the natural frequency could be approximated as square root of the sum of squares of both natural frequency of the beam and the foundation. The stretching potential energy is proved to be responsible for generating the cubic nonlinearity in the system.
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