第一作者蒲育男，硕士,讲师，1984年5月生

通信作者滕兆春男，副教授，1969年8月生

Winkler-Pasternak弹性地基FGM梁自由振动二维弹性解

蒲育1，滕兆春2

(1. 兰州工业学院土木工程学院，兰州730050；2. 兰州理工大学理学院，兰州730050)

摘要：基于二维线弹性理论，建立Winkler-Pasternak弹性地基上功能梯度(Functionally Graded Material,FGM)梁自由振动控制微分方程。假设材料物性沿梁厚度方向按幂律分布，采用微分求积法(Differential Quadrature Method,DQM)数值求解4种不同边界FGM梁自由振动无量纲频率特性。将计算结果与Winkler-Pasternak弹性地基梁对比表明，该分析方法对弹性地基梁自由振动研究行之有效,并考虑边界条件、梯度指数、跨厚比、地基系数对FGM梁自振频率影响。

关键词：Winkler-Pasternak弹性地基；FGM梁；自由振动；无量纲频率；微分求积法

基金项目：国家自然科学基金(11372123)

收稿日期：2014-07-16修改稿收到日期：2014-10-11

中图分类号:O343

文献标志码：A

DOI:10.13465/j.cnki.jvs.2015.20.013

Abstract:Based on the two-dimension theory of linear elasticity, the free vibration differential equations for FGM beams resting on Winkler-Pasternak elastic foundations were derived. The material properties were supposed to change continuously along the thickness of the beam according to the power law distribution. Using the differential quadrature method (DQM), the dimensionless natural frequencies of FGM beams under four different boundary conditions were investigated. The formulations were validated by comparing the results obtained with those available in the literature for homogeneous beams on Winkler-Pasternak elastic foundations. The influences of the boundary conditions, material graded index, length-to-thickness ratio and elastic coefficients of foundations on the non-dimensional frequency parameters of FGM beams were discussed.

Two-dimensional elasticity solutions for free vibration of FGM beams resting on Winkler-Pasternak elastic foundations

PUYu1,TENGZhao-chun2(1. College of Civil Engineering，Lanzhou Institute of Technology，Lanzhou 730050，China；2. School of Science, Lanzhou University of Technology, Lanzhou 730050, China)

Key words:Winkler-Pasternak elastic foundation; functionally graded material beam; free vibration; dimensionless frequency; differential quadrature method

弹性地基梁的动力学研究在工程领域有十分重要意义及广泛应用背景，因此已有诸多对其动态响应的研究[1-3]。Rosa等[4]研究弹性地基Euler梁在集中质量块作用下的自由振动。Chen等[5]结合状态空间法(SSM)及微分求积法(DQM)数值研究双参数弹性地基梁的弯曲与自由振动。Ding等[6]用Airy应力函数多项式形式求解弹性地基功能梯度梁在不同边界条件下的动态响应。Ying等[7]用SSM分析简支-简支边界条件的二维弹性地基FGM梁弯曲、自由振动。Alshorbagy等[8]采用FEM分析FGM梁自由振动。Thai等[9]基于高阶剪切梁理论，研究FGM梁弯曲、自由振动。然而对梁自由振动研究大多基于不同的梁理论。

经典梁理论忽略横向剪切变形影响，故仅适用细长梁。……