轴向窄带随机激励悬臂梁的单模态参激共振

Single modal parametric resonances in axially narrow-band random oscillating slender cantilever beams

以轴向基础窄带随机激励悬臂梁非线性动力学方程组为分析对象,采用多尺度法,获得了系统主参激共振的非线性调谐方程组。在假设带宽较小的前提下,利用摄动法,获得了系统非平凡幅-频响应的1,2阶稳态矩近似理论表达式,并通过直接的数值积分获得了相应的曲线形式并进行了比较,取得了较好的一致性。分析结果表明:对于第1阶模态的主参激共振,其1,2阶稳态矩-频率特性呈现硬特性,而对于2阶及以上模态的主参激共振,系统1,2阶稳态矩-频率特性呈现软特性;带宽的小范围变化对1,2阶稳态矩产生的效应甚微。通过对概率密度进一步的数值计算,首次发现了系统的响应在非平凡平稳响应与平凡平稳响应间的随机跳跃现象,计算结果显示,随着带宽的增加,非平凡平稳响应处的概率密度逐渐减小,而平凡平稳响应处的概率密度随之增加。

According to the nonlinear dynamic equations of motion for slender cantilever beams axially excited by narrow-band random processes of the base, a set of nonlinear modulation equations for the principal parametric resonances is developed based on the method of multiple scales. The first and second order non-trivial steady state responses of the system are obtained by perturbation method and the corresponding amplitude-frequency curves are calculated by both approximate theory and direct numerical integration. Results show that for the first mode, the amplitude expectations of both the first and the second order steady state responses are of the hardening type, whereas for the second mode, the amplitude expectations for both the first and the second order steady state responses are of the softening type. Based on the results of both theory and numerical integration, the small changes of the narrow-band widths have almost no influences on the amplitude expectations of both the first and second order steady state responses. Furthermore, via a direct numerical integration method, the probability density of the displacement and the amplitude and the joint probability density function of the displacement and velocity of the stationary response of the system are obtained, and through which stochastic jumps between the non-trivial stationary response and the trivial stationary response are examined for the first time. The digital simulation shows that with the increase of the narrow-band width, the probability density near the non-trivlal stationary solution gradually decreases whereas the probability density near the trivial one gradually increases.