A Study On The Stochastic Response Of A Nonlinear Cable

As a flexible structure,a cable is sensitive to wind,rain and other random excitations.Vibration with large displacement is prone to occur.Due to the secondary displacement nonlinearity caused by the initial form of the cable,it will have different effects on the vibration response and dynamic characteristics of the cable as the initial form changes.Coupled with the random load,the cable presents a very complex nonlinear random vibration response.In this paper,the stochastic vibration modal response of a cable under Gaussian white noise excitation is studied by using the path integration method,which is based on the Gauss-Legendre integration and the short-time Gaussian transition probability density approximation.The main work is as follows:(1)On the basis of the early scholars' suspension model,the nonlinear vibration mode equation of the cable under the Gaussian white noise excitation are deduced by using the structural dynamics and the random vibration theory and considering the influence of the damping factor about the cable material.(2)The transition probability density is solved by the short-term Gaussian approximation method,and the factors influencing the accuracy and efficiency of the calculation are studied.And the path integration method is used to solve the probability density function of the stationary response.Due to the existence of quadratic nonlinear term of displacement,the displacement modal response has a non-zero mean.The probability density function has an asymmetric distribution,and the failure probability evaluation needs to be further improved.(3)The path integration method is used to study the non-stationary vibration response of the cable at different time events.Compared with the Monte Carlo simulation results,the validity of the path integration method is made.In contrast with the probability density function of stationary response,the adverse effects of the non-stationary response caused by the initial morphology are clarified.(4)Based on the geometrical and physical factors of the cable,the variation law of the nonlinear parameters in the modal vibration equation under the influence of different factors such as span,sag and material properties of the cable is studied.Then,the paper analyzes the influence of the uncertain factors on the probability density of modal vibration response.And the probability distribution function and the probability of reliability analysis of them under different time events are solved by Matlab programming.The results show that the stiffness coefficient and frequency ratio are gradually increased,and the vibration mode is transformed from an antisymmetric sine function to a positive symmetry function with the increase of the sag-to-chord-length ratio.Meanwhile,the mean square of the response decreases correspondingly.Compared with the physical parameters,the geometric parameters of the suspension have a greater effect on the modal vibration response.Therefore,during the initial design of the cable,the structural design should be based on the concept of geometric parameters.The adverse effects of the initial shape in the non-stationary response are comprehensively considered to ensure the safety of the cable structure and to predict the failure probability of the random vibration of the cable.