The Response Characteristics Of The Cantilever Beam Under Fundamental Noise Excitation

A dynamical model of an in-extendable cantilever beam under basal random excitation was presented applying the Kane method and the element method respectively.The random excitation was chosen as Gaussian white noise and Gaussian colored noise separately.Two white noises were imposed to the beam transversely and axially,which yields external and parametrical excitations.When using the Gaussian colored noise,only transverse excitation was considered to avoid the duplicate procedure.Firstly,the Ito random differential equation about the beam's amplitude was obtained by using the standard stochastic averaging method.During this procedure,the draft coefficient and the diffusion coefficient were solved out.Secondly,the system's global stability was studied by the singular boundary theory.Thirdly,the stationary probability density function(PDF)together with the joint probability density function of the displacement and the velocity were solved,applying the Fokker-Planck-Kolmogolov equation(FPK).Subsequently,the reliability function and the first passage failure time were researched by the Backward-Kolmogolov equation(BK)combined with the boundary conditions.Finally,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.The effects of the transverse excitation and axial excitation were discussed and detailed.The response character of the colored noise excited cantilever was also studied with similar procedure,and the stationary probability density function(PDF)together with the joint probability density function of the displacement and the velocity were solved.The effect of the noise's correlation time and the noise density were compared and illustrated.Digital simulations were also agreed with the theoretical approximations.