轴向运动梁的横向随机响应

Transverse random response of an axially moving beam

轴向运动梁是许多飞行器结构的简化模型,随着长细比增加和质量减小,梁的弹性特征愈加明显,同时运动速度对运动梁的振动特性也有显著影响。根据汉密尔顿原理(Hamilton’s principle),推导出轴向运动欧拉-伯努利(Euler-Bernoulli)梁模型受横向激励作用时的动力学控制方程。首先,在有轴向力和无轴力情况下分别对方程进行无量纲化、复模态分析,得到统一形式的频率方程和模态函数,可以用数值方法求解其固有频率和模态函数。然后,将动力学方程解耦为一个微分方程组,求解方程组,得到轴向运动梁在横向激励下位移的响应。最后,用数理统计的方法,计算随机响应的相关函数,再做傅里叶变换(Fourier transform)后得到复数形式的随机响应谱。数值算例的结果表明,轴向运动速度对自由梁的振动特性和随机响应有显著影响。

An axially moving beam is a simplified model for many aircraft structures.Its elasticity is patently more obvious with its slenderness ratio increased and mass reduced,and its velocity has a significant effect on its vibration characteristics at the same time.Here,the dynamic equation of transverse vibration of an axially moving beam subjected to a transverse excitation was derived with Hamilton＆#39;s principle.At first,the dimensionless method and complex modal analysis method were applied to simplify the equation with an axial force or without an axial one.The frequency equation and modal functions were obtained,they were solved using the numerical method.Then,the decoupling method was used to simplify the control equation into a set of differential equations,the displacement responses of the beam were gained after solving those equations.Finally,the random response＆#39;s correlation function was calculated by using the method of mathematical statistics,and the random response spectrum of the beam was achieved via Fourier transformation.The numerical example illustrated that the beam＆#39;s moving velocity can affect its vibration characteristics and random responses significantly.