A Study On Non-linear Random Vibration Responses Of Simply-supported Beams Based On Three-dimensional Path Integration Method

The simply-supported beam is a common member.The analysis of random vibration responses of simply-supported beams under the action of random loads such as earthquake has important scientific significance and engineering value.Especially when the simply-supported beams are subjected to strong random loads,their vibration responses exhibit significant non-linear characteristics,making it difficult to predict and control the random vibration responses.To solve this problem,this paper establishes a three-dimensional path integration method for analyzing the non-linear random vibration responses of simply-supported beams under the action of filtered Gaussian white noise,and studies the probability density function distributions of the non-linear random vibration responses of simply-supported beams.The main work and results of this paper are as follows:(1)The suitable filtered Gaussian white noise model and parameter values for the analysis of random vibration responses are studied.The filtered Gaussian white noise is used to simulate the broadband colored noise,making the random excitation model more in line with the actual engineering load conditions.Gaussian white noise is a commonly used excitation load model in random vibration studies.However,it is too idealized in practical applications and difficult to reflect the actual engineering load characteristics.This paper uses filtered Gaussian white noise to be closer to the actual engineering load and solves the computational difficulties that arise from the filtered Gaussian white noise model.(2)A three-dimensional path integration method based on the Gauss-Legendre integration scheme for filtered Gaussian white noise is established.The effectiveness of the three-dimensional path integration method is verified by four cases,and the influence rules of filter coefficients,non-linear coefficients and excitation strength on the non-linear random vibration responses of simply-supported beams are analyzed.The results show that the calculated values of the three-dimensional path integration method are in good agreement with the Monte Carlo simulation values,even in the tail regions of the probability density functions.As the filter coefficients increase,the spectral densities of the noise excitations decrease,and the dispersion of the probability density functions of the non-linear vibration responses of simply-supported beams decreases.The non-linear responses of simply-supported beams are less stable in the cases of small filter coefficients,weak nonlinearity and strong excitations,should be focused on in the design.(3)The rules of non-linear random vibration responses of the rectangular hollow section simply-supported beams with zero and non-zero mean values of responses in the initial conditions are analyzed.The parameters of the random vibration models are calculated based on the geometric parameters of the sections,and the validity of the threedimensional path integration method in the examples is verified.The probability density function distributions of the non-stationary responses at different times are analyzed.The results show that the distributions of the non-stationary probability density functions of non-linear random vibration responses of simply-supported beams will be significantly concentrated in the initial stage,and will continue to expand with time and eventually reach a steady state.This phenomenon is more pronounced when the mean values of responses in the initial conditions are non-zero.Therefore,in the situations of the random vibrations of simply-supported beams when the mean values of responses in the initial condition are non-zero,we should pay attention to the vibration responses of the stationary states and non-stationary states at the same time.