Statistical Linearization Of Energy Dissipation Devices

It is very important to predict the structural vibration response in the engineering design.Because of the uncertainty of seismic activity and the randomness of seismic wave propagation in the crust,ground motion parameters such as ground displacement,velocity and acceleration are random.The motion process is extremely irregular,then the non-linearity of the structure with additional energy dissipation and shock absorber is represented by the combination of non-linear restoring force,non-linear inertia,non-linear damping or the combination of the three.At the same time,under the reciprocating action of seismic loads,the relationship curve between restoring force and deformation of components does not change along the same path during loading and unloading,showing significant hysteretic characteristics.Therefore,the linearization of the nonlinear model is an effective method to solve the nonlinear random vibration.How to obtain the equivalent linear form of different nonlinear models and ensure that the equivalent linear system is similar to the original nonlinear model,reduce the error as much as possible and improve the accuracy is of great engineering significance.This paper carries out the following four research tasks:(1)The piecewise non-linear model and Bouc-Wen hysteretic model under Clough-Penzien non-stationary excitation are solved by Monte Carlo simulation method.The statistical eigenvalues and probability distribution of the non-linear model are further analyzed based on the structural response data obtained.The Monte Carlo simulation results are retained and compared with the response results of the subsequent equivalent linear model to measure the approximation degree of the linearization method.(2)The piecewise damping nonlinear model,piecewise stiffness nonlinear model and piecewise stiffness-damping nonlinear model are linearized respectively.According to Monte Carlo simulation results,the assumption of Gauss distribution for response variables is put forward,and the principle of minimum mean square deviation of non-linear model and linearized model is adopted.Then the equivalent linearization form is derived.Finally,the statistical eigenvalues of response variables obtained by statistical linearization and Monte Carlo method are compared,and the excitation input of non-linear model and equivalent linearization form is exactly the same.Time history analysis shows that the equivalent linear form can replace the original nonlinear system well,but there are still some errors to be improved.(3)For the equivalent linearization form of Bouc-Wen hysteretic model which has been deduced when n=1,for the intermediate parameter z_u(representing the maximum response Z),this paper proposes to replace the theoretical value calculated according to the formula with statistical value.It is found that the linearization effect of statistical value is closer to the Monte Carlo simulation result than the theoretical value,which indicates that the maximum value of the hysteretic variable Z needs further study.(4)For Bouc-Wen hysteretic model,the probability density function diagrams of response variables under Clough-Penzien non-stationary model excitation with model parameters N of 1,2 and 3 are obtained by Monte Carlo simulation.It is proposed that when n=2 and n=3,the mixed distribution of Gauss and uniform distribution is used to describe the probability distribution of hysteretic variable Z,which is not the principle of minimum mean square deviation between linear model and linearized model.The parameters of equivalent linear model are derived by statistical linearization method.Compared with the results of previous assumptions that hysteretic variable Z obeys pure Gauss distribution,the linearization effect of the proposed method is better.The pure Gauss hypothesis and Monte Carlo simulation results show that the method has better accuracy.