Research On The Response Of Cracked Elastic Cantilever Beam Under Foundation Noise Excitation

An elastic cantilever model with surface open crack excited by basal noise was studied.The main purpose of this paper was to explore how the geometrical nonlinearity influence the response of the model excited by the Gaussian white noise as well as the deterministic excitation.Firstly,in order to study the amplitude-fr-equency response of the cracked cantilever beam,by using the knowledge of fracture mechanics we can get the modal equation of the cantilever.The modal functions of the cantilever with different crack depths were obtained by solving the boundary conditions,continuity conditions and compatibility conditions.Secondly,the dynamical equation governing the transverse motion was obtained by the Kane's method after sufficient consideration of the geometrical nonlinearity,and the validity of the equation is verified by the Lagrange method.After the consideration of the linear darmping and introducing the dimensionless transformation,the dimensionless equation can be simplified as an ordinary differential equation with external excitation.Subsequently,in order to compare and analyze the effect of different types of excitation on the system response,the Gaussian white noise excitation was altered to be the detenninistic excitation.Then the amplitude-frequency response of the system under the deterministic excitation was studied by the multi-scale method,and the stability of the system was also analyzed.The author obtains two independent Ito stochastic differential equations about amplitude and phase by applying the stochastic averaging method,then the stationary probability density function(PDF)of the amplitude and the joint stationary PDF of the displacement and velocity were investigated respectively.To explore the reliability and the probability density of the first passage failure time of the cantilever,we can use the initial condition and the boundary conditions of the system to solve the backward Kolmogorov equation(BK equation),and the accuracy of the prediction given by the theoretical analysis was also vindicated through the numerical simulation of the reliability.