中文摘要：

基于经典Winkler地基模型及Euler—Bernoulli梁理论，考虑梁的几何非线性效应，运用Newton第二定律建立了弹性地基上有限长梁的非线性运动方程。采用Galerkin方法对运动方程进行一阶模态截断，进而利用多尺度法求得了该系统自由振动的一阶近似解。揭示了两端简支梁的非线性自由振动特性，分析了弹性模量、长细比及地基刚度系数等参数对系统固有频率的影响。并通过该系统的位移时程曲线，分析了阻尼对弹性地基上梁运动特性的影响。

英文摘要：

The non-linear free vibration of a finite-length beam on the elastic foundation is investigated. Based on the Winkler foundation model and Euler-Bernoulli beam theory, the nonlinear motion equation of the finite-length beam on an elastic foundation with geometric nonlinearity is deduced based on the Newton＇s Second Law. The first-order mode truncation of the vibration function is obtained using the Galerkin method. The approximate solution of the free vibration of the finite-length beam is derived utilizing the multi-scale method to illustrate the behaviour of the non-linear free vibration. The effects of the slenderness ratio of beam, the modulus of elastic system and the stiffness of foundation on the natural frequency of the hinged-hinged beam on the Winkler foundation are analyzed. The influence of damping of the soil-beam system on the motion of the beam is also discussed.