Considering the curvature nonlinearity and longitudinal inertia nonlinearity caused by geometrical deformations, a slender inextensible cantilever beam model under transverse pedestal motion in the form of Gaussian colored noise excitation was studied. Present stochastic averaging methods cannot solve the equations of random excited oscillators that included both inertia nonlinearity and curvature nonlinearity. In order to solve this kind of equations, a modified stochastic averaging method was proposed. This method can simplify the equation to an Itô differential equation about amplitude and energy. Based on the Itô differential equation, the stationary probability density function (PDF) of the amplitude and energy and the joint PDF of the displacement and velocity were studied. The effectiveness of the proposed method was verified by numerical simulation.

中文翻译：

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基底非线性高斯有色噪声激励下具有惯性非线性的悬臂模型的响应

考虑到几何变形引起的曲率非线性和纵向惯性非线性，研究了在高斯彩色噪声激励形式的横向基座运动下细长的不可伸展悬臂梁模型。当前的随机平均方法无法求解同时包含惯性非线性和曲率非线性的随机激励振荡器方程。为了解决这类方程，提出了一种改进的随机平均方法。这种方法可以将方程简化为关于振幅和能量的Itô微分方程。基于Itô微分方程，研究了振幅和能量的平稳概率密度函数（PDF）以及位移和速度的联合PDF。