非齐次边界条件下轴向运动梁的非线性振动

轴向运动系统的横向非线性振动一直是国内外研究的热点课题之一.目前相关研究大都是针对齐次边界条件的.但是在工程实际中,非齐次边界条件更为常见,而针对非齐次边界条件的研究相对较少.为深入研究非齐次边界条件对轴向运动系统横向非线性振动的影响,本文以轴向变速运动黏弹性Euler梁为例,引入由黏弹性引起的非齐次边界条件,同时还引入由轴向加速度引起的径向变化张力,建立梁横向振动的积分-偏微分型运动方程,并导出了相应的非齐次边界条件.采用直接多尺度法分析了梁的次谐波参数共振.由可解性条件得到了梁的稳态响应,并根据Routh-Hurvitz判据确定了系统稳态响应的稳定性.通过数值例子讨论了黏弹性系数,轴向运动速度,轴向速度脉动幅值和非线性系数对幅频响应的影响,并详细对比分析了非齐次边界条件和齐次边界条件对幅频响应的影响.结果表明:随着黏弹性系数的增大,非齐次边界条件下的零解失稳区域和稳态响应幅值比齐次边界条件下的失稳区域和幅值大,非齐次边界条件对高阶次谐波参数共振的影响更加显著.最后,引入微分求积法来验证直接多尺度法的近似解结果. 

NONLINEAR VIBRATIONS OF AXIALLY MOVING BEAMS WITH NONHOMOGENEOUS BOUNDARY CONDITIONS

The transverse nonlinear vibration of the axial moving system has become one of the hot subjects at home and abroad. At present, most of the related studies are on the homogeneous boundary conditions. However, the nonhomogeneous boundary conditions are more common in the engineering practice. There are relatively few publications on the axial moving system with homogeneous boundary conditions. In order to study the effect of nonhomogeneous boundary conditions on the transverse nonlinear vibration of an axial moving beam, the nonlinear parametric vibrations of an axially accelerating viscoelastic Euler beam under nonhomogeneous boundary conditions are studied in this paper. The variable tension caused by the axial acceleration is introduced. A nonlinear integro-partial-differential equation and corresponding nonhomogeneous boundary conditions of an axially accelerating viscoelastic beam are presented. The effects of nonhomogeneous boundary conditions are highlighted. The method of multiple scales is used to establish the solvability conditions. The steady-state response of the beam was obtained from the solvability condition. According to the Routh-Hurvitz criterion, the stability of the response was determined. Some numerical examples are introduced to demonstrate the effect of the viscoelastic coefficient, mean speed, axial speed fluctuation amplitude, nonlinear coefficient, and the nonhomogeneous boundary conditions on the steady-state response. The larger viscoelastic coefficient leads to the smaller instability interval of the trivial solutions and the smaller stable steady-state response amplitude especially for the second mode. The instability interval and the stable steady-state response amplitude with the nonhomogeneous boundary conditions are larger than that with the homogeneous boundary conditions. The differential quadrature scheme is introduced to confirm the approximate analytical results. The numerical results show reasonable agreement with the approximate analytical results.