【作者】 王昊；

【导师】 陈立群；

【作者基本信息】 上海大学 ， 一般力学与力学基础， 2014， 硕士

【摘要】 梁作为一类细长结构广泛存在于机械、建筑等工程中。当梁受到轴向的压力超过一定值时会发生屈曲。在如今的工程中，屈曲梁应用的范围日趋增大，所以其振动分析有着重要的理论意义。首先，本文建立轴向受力屈曲梁横向受迫振动的数学模型。以Euler梁模型为基础，运用微元法求解得到屈曲梁整体的偏微分控制方程。以屈曲平衡位形为空间坐标进行坐标变换得到新的扰动方程。在简支边界条件下，基于二阶Galerkin方法将包含随空间变系数的偏微分方程描述的连续系统离散化为两自由度非线性振动系统。其次，研究轴向受力屈曲梁在弱受迫和强受迫情形下的非线性振动特性。基于二阶Galerkin方法，得到存在两次非线性项的截断方程，发展多尺度近似解析方法，通过二阶多尺度方法展开，推导出系统的可解性条件。在2:1内共振的条件下，研究轴向受力屈曲梁受迫振动的稳态响应，考察轴向压力、黏弹性系数和外激励幅值对幅频响应曲线的影响，发现饱和现象。再次，研究多尺度方法中的展开阶数对系统稳态幅频响应结果的影响。基于二阶Galerkin方法，得到存在三次非线性项的截断方程。在1:1内共振和弱受迫振动的条件下，分别应用多尺度二阶展开和三阶展开对截断方程进行求解，通过可解性条件得到系统响应曲线。研究发现不同阶数的多尺度展开对系统响应有很大的影响。最后，对本文所做的研究工作以及研究的结论进行了总结，并且展望需要进一步研究的工作。更多还原

【Abstract】 The beam exists widely in machinery, construction and other projects as a kindof slim bar. When the pressure exceeds a critical value, the axial buckling will occuron the beam. Nowadays the applications of the buckled beam have an increasingproportion in engineer. The study of the vibration response of the buckled beam is ofgreat significance.First, the governing equation of a simply-supported viscoelastic buckled beamto a harmonic base excitation and a couple of constant axial force is established.Based on Euler-Bernoulli beam theory, the partial differential governing equations ofthe buckled beam is got by infinitesimal method. In the paper, the disturbanceequation of transverse vibration of the buckled beam is derived from the freegoverning equation via a coordinate transform. The Galerkin method is applied totruncate the systems to a multi degree-of-freedom of nonlinear vibration system.Second, a nonlinear analysis of the response of a simply-supported viscoelasticbuckled beam to a couple of constant axial force and harmonic excitation ispresented. In addition, the harmonic excitation has two types: weak excitation andstrong excitation. The method of multiple scales is developed to present thesolvability condition of approximate solutions. In the presence of the2:1internalresonance and the primary external resonance, various jumping phenomena arerevealed in the amplitude-frequency characteristic curves, and the effects of relatedparameters, such as the coefficient of viscoelasticity, the external excitationamplitude and the axial force, on the phenomena are examined. At the same time, theexistence of the phenomenon of saturation is proved.Third, the effect of the application of the method of multiple scales in different orders to the amplitude-frequency characteristic curves is examined. Base on theexist of the cubic nonlinearities in the a multi degree-of-freedom of nonlinearvibration system, the two-order multi-scale and the three-order multi-scale aredeveloped to present the solvability condition of approximate solutions. In thepresence of the1:1internal resonance and the weak harmonic excitation, differentphenomena are revealed in the amplitude-frequency characteristic curves.Finally, the results of the thesis are summarized and the further work issuggested.更多还原