Abstract
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.
Highlights
One essential and famous conclusion is that the geometrical nonlinearity will induce hardening effect, while the longitudinal inertia nonlinearity will lead to a softening effect
During the enthusiasm of researching the dynamics of the cantilever excited by deterministic signals, some researchers explored the responses of randomly excited cantilever models
A new stochastic averaging method for solving the cantilever beam model with both inertia nonlinearity and geometric nonlinearity which is excited by colored noise is proposed in this paper. e old method is extended to a new filed. e key issue of this method is balancing the input energy and diffusion energy of the Hamiltonian function
Summary
Feng et al [11,12,13,14] studied responses of cantilever model with curvature nonlinearity and longitudinal inertia nonlinearity under narrow bounded noise excitation by the method of multiple scales. Ge and Yan [15] studied a cantilever model with longitudinal inertia nonlinearity and the geometrical nonlinearity under basal white noise excitation. We will investigate an improved stochastic averaging method to deal with the randomly excited differential equations with both inertia nonlinearity and the geometric nonlinearity. In order to study the cantilever model excited by Gaussian colored noise, a stochastic averaging method which can handle the equations with inertia nonlinearity and the geometrical nonlinearity is presented. Numerical simulations on the randomly excited equations are employed to verify the theoretical deductions
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