Dynamic Response Analysis of Variable Thickness Rectangular Plates on Viscoelastic Foundation with Fractional Derivative
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摘要: 研究了黏弹性地基上变厚度矩形薄板的动力响应。采用分数阶微分的Kelvin-voigt模型描述地基的黏弹性特征,基于弹性板的基本假设,对于小变形问题,建立了黏弹性地基上变厚度矩形薄板的动力控制微分方程。针对该分数阶变系数偏微分方程,采用Galerkin法和Haar小波配点法进行数值求解。分析了四边简支板的动力响应特性,得到了均布载荷作用下线性模型、抛物线模型板中心挠度曲线,讨论了变厚度矩形板的厚度比、长宽比、分数阶导数的阶数、地基弹簧系数、粘滞系数、水平剪切系数等参数变化对板动力特性的影响。Abstract: This paper analyzes the dynamic response of a variable thickness rectangular thin plate on a viscoelastic foundation. The viscoelastic characteristic of the foundation is described with Kelvin-Voigt model with fractional order differential. The governing differential equation of variable thickness rectangular plate on a viscoelastic foundation is established based on the basic hypothesis of elastic plate. The numerical solution of the fractional order differential equation with variable coefficient is obtained with the Galerkin method and the Haar wavelet collocation method. The dynamic response characteristics of the simply supported plate are analyzed, and the central deflection curves of linear model and the parabola model plate under uniform loading are obtained. The influences of length-width ratio and thickness ratio of the rectangular variable thickness plate, and the fractional order, spring coefficient, viscosity coefficient and horizontal shear parameter of the foundation on the dynamic characteristics of the plate are discussed.
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[1] Mansfield E H. On the analysis of elastic plates of variable thickness[J]. The Quarterly Journal of Mechanics & Applied Mathematics, 1962,15(2):167-192 [2] Alipour M M, Shariyat M. Semi-analytical buckling analysis of heterogeneous variable thickness viscoelastic circular plates on elastic foundations[J]. Mechanics Research Communications, 2011,38(8):594-601 [3] Hosseini-Hashemi Sh, Taher H R D, Akhavan H. Vibration analysis of radially FGM sectorial plates of variable thickness on elastic foundations[J]. Composite Structures, 2010,92(7):1734-1743 [4] 滕兆春,丁树声,郑鹏君.弹性地基上变厚度矩形板自由振动的GDQ法求解[J].应用力学学报,2014,31(2):236-242 Teng Z C, Ding S S, Zheng P J. Free vibration analysis of rectangular plates with variable thickness on elastic foundation by using GDQ method[J]. Chinese Journal of Applied Mechanics, 2014,31(2):236-242(in Chinese) [5] 杨柳,彭建设,谢刚,等.Winkler地基上变厚度矩形板弯曲的微分求积解[J].成都大学学报(自然科学版),2011,30(2):131-133 YANG L, PENG J S, XIE G, et al. Differential quadrature solutions for bending problem of rectangular plates with variable thickness on Winkler foundation[J]. Journal of Chengdu University (Natural Science Edition), 2011,30(2):131-133(in Chinese) [6] Gemant A. A method of analyzing experimental results obtained from elasto-viscous bodies[J]. Journal of Applied Physics, 1936,7(8):311-317 [7] Pritz T. Analysis of four-parameter fractional derivative model of real solid materials[J]. Journal of Sound and Vibration, 1996,195(1):103-115 [8] Atanackovic T M. A modified Zener model of a viscoelastic body[J]. Continuum Mechanics and Thermodynamics, 2002,14(2):137-148 [9] Pritz T. Five-parameter fractional derivative model for polymeric damping materials[J]. Journal of Sound and Vibration, 2003,265(5):935-952 [10] Meral F C, Royston T J, Magin R. Fractional calculus in viscoelasticity:an experimental study[J]. Communications in Nonlinear Science and Numerical Simulation, 2010,15(4):939-945 [11] Atanackovic T M, Stankovic B. Stability of an elastic rod on a fractional derivative type of foundation[J]. Journal of Sound and Vibration, 2004,277(1-2):149-161 [12] 寇磊.分数阶微分型双参数黏弹性地基矩形板受荷响应[J].力学季刊,2013,34(1):154-160 Kou L. Response of rectangular plate on fractional derivative two-parameter viscoelastic foundation[J]. Chinese Quarterly of Mechanics, 2013,34(1):154-160(in Chinese) [13] Kerr A D. Elastic and viscoelastic foundation models[J]. Journal of Applied Mechanics, 1964,31(3):491-498 [14] 王苗苗,赵凤群,李娜,等.分数阶微分方程的Haar小波算法研究[J].计算力学学报,2013,30(1):156-160 Wang M M, Zhao F Q, Li N, et al. Study on the Haar wavelet algorithm of fractional differential equations[J]. Chinese Journal of Computational Mechanics, 2013,30(1):156-160(in Chinese) [15] Li Y L, Zhao W W. Haar wavelet operational matrix of fractional order integration and its applications in solving the fractional order differential equations[J]. Applied Mathematics and Computation, 2010,216(8):2276-2285
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