Study On The Response Of Pre-buckling Beam Under Noise Excitation

Beam is a kind of slim structure which is widely used in many engineering fields.The buckling will occur when the axial force exceed a critical value.With the accelerating development of science and technology,modern engineering structure has became increasingly lightweight and flexible,and the application field of the pre-buckled beam is becoming more and more wide.In view of the pre-buckld beams vibrate easily under external excitation which usually contains many random factors,it is necessary to study the response of the pre-buckled beam under the noise excitation by using the stochastic nonlinear dynamics theory.The paper's main work includes:(1)A modified stochastic averaging method on single-degree-of-freedom oscillators under noise excitation with strongly nonlinearity was proposed.This method applied the energy balance method to overcome the minor defect of the traditional stochastic averaging method.Then,by using this method to the examples,the stationary responses of probability density of amplitudes and energy,together with joint probability density of displacement and velocity are studied to verify the presented approach.The reliability function of the systems were also investigated to offer further support.Digital simulations were carried out and the output of that are coincide with theoretical approximations.(2)A dynamical mode of a pre-buckled simply under random excitation was presented.The random excitation was chosen as the Gaussian white noise and the Gaussian colored noise separately.Applying the Hamilton principle and Galerkin approach the model is expressed as one dimensional vibrating model with parametrical random excitation.(3)When using the Gaussian white noise,the Ito random differential equation governing the beam's vibrating amplitude was obtained by using the mocdified stochastic averaging method.During this procedure,the draft coefficient and the diffusion coefficient were solved out.Subsequently,by applying the Fokker-Planck-Kolmogolov equation,the stationary probability density function(PDF)together with the joint probability density function of the displacement and the velocity were solved.Next,the reliability function and the first passage failure time were researched by the Backward-Kolmogolov equation combined with the boundary conditions.The effect of the system damping coefficient and the noise density were compared and illustrated.The digital simulations were carried out by the Monte Carlo method to verify the analysis.The numerical results were coincide with the theoretical analysis very well.(4)The response of the pre-buckled beam under the colored noise excitation was also studied with similar procedure,and the stationary probability density function together with the joint probability density function of the displacement and the velocity were solved.And the reliability function and the first passage failure time were researched by the Backward-Kolmogolov equation combined with the boundary conditions.The effect of the noise density,system damping coefficient and the noise's correlation time were expounded.Digital simulations were also agreed with the theoretical approximations.