【摘要】 针对轴向基础窄带随机激励柔性梁而建立的非线性动力学方程,采用多尺度法并结合笛卡尔坐标变换,导出了系统两阶模态间受组合参数共振时的非线性调制方程组。随后,在假设谐和激励下,获得了系统平凡响应稳定性边界条件的解析表达式,相应的Hop f分岔类型及产生的极限环在中心流型定理与数值计算间相互得到了验证。最后,计算了窄带随机激励下系统平凡响应的最大Lyapunov指数,获得了几乎肯定稳定性边界的数值形式,发现带宽的增大导致系统不稳定区域的增大,数值观察了类似的极限环的存在,同时,也发现相应的极限环的厚度随带宽的增大而变厚且存在相应环面趋于杂乱现象。更多还原

【Abstract】 According to the nonlinear dynamic equations of motions for flexible beams excited axially by narrow-band random processes,a set of nonlinear modulation equations for the combination parametric resonance between two natural modes is established based on the method of multiple scales and Cartesian transformation.Then the case of harmonic parametric excitation is taken into consideration.The equations of approximate transition curves that separate stable solutions from the unstable trivial ones,which belong to unstable foci,are derived.The type of Hopf bifurcation is determined in the vicinity of the bifurcation point via the center manifold theorem and the corresponding limit cycle is numerically found.Finally,the case of narrow-band random parametric excitation is focused on.To determine the almost sure stability of the trivial response of the system,the numerical results for the largest Lyapunov exponent are obtained,which show that with the increase of narrow-band width γ,the unstable region will be widened.Furthermore,the increase of γ results in a fact that the limit cycle gradually becomes a diffused one,i.e.the width of the limit cycle is increased to some extent.更多还原