Response Study Of A Cantilever Beam With End Masses Under Base Noise Excitation

As a simplified model in many engineering structures,the dynamical problems of cantilever beam have always been the focus of many scholars.A lot of practical engineering structures can be simplified modeled as a cantilever beam with a tip mass,such as towers,water tower,tower crane and so on,they are disturbed by high winds,waves,storms,etc.,most of these disturbances are random,so the studies on the responses of the cantilever beam under random noise excitation,are of great theoretical and practical value.A model of an in-extensible cantilever beam with a tip mass subjected to basal noise excitation is studied.The main purpose of the study is to explore how the cantilever model with a tip mass is affected by the noise intensity and geometric nonlinearity when the cantilever is excited by the Gaussian white noise excitation.Firstly,a cantilever dynamical model with a tip mass excited by the pedestal movement with the form of the Gaussian white noise was established in this dissertation.The dynamical equation with the geometrical nonlinearity and the axial inertia nonlinearity which governs the transverse motion was gained with the Kane's method.The damping term of the system is introduced and dimensionless transformation is done,thus the differential equation can be simplified.The key procedures are as follows.1.Based on the Hamiltonian function,the oscillator was transformed to be a pair of ordinary stochastic differential equations about transient equivalent amplitude and transient phase.2.Both the strong nonlinear random average method and the improved random average method which is effective on strong nonlinearity were applied to analyze the responses of the cantilever model theoretically.The random differential equations were simplified to be an Ito equation using the stochastic averaging method.On this basis,the stationary probability density functions(PDFs)of the amplitude as well as the joint PDF of displacement and velocity are obtained.3.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).The numerical simulations verified the theoretical analysis.