Energy Method Study On Random Vibration Of Structure Under Modulated Ground Motion

At present,the energy spectrum of total input energy has been a popular research project in energy method in the field of seismic engineering in domestic and overseas.By selecting the actual recorded acceleration time history and calculating the seismic response of structures with different periods,the quantitative relationship between the total input energy of the earthquake and the damping ratio,the duration of the earthquake response and the type of the site can be obtained,and the energy spectrum of the total input energy can be obtained finally.For the actual engineering structure,it is difficult to get the explicit formula of the total input energy time history because of the uncertainty of the ground motion and the nonlinearity of the structure.By using the same power spectral density function of ground motion,numerous time history samples with the same spectral characteristics can be synthesized,and the corresponding input energy time history curves of these samples are quite different,which makes the traditional deterministic seismic design method based on energy method have certain limitations.Based on the theory of random vibration,the statistical characteristics of input energy of linear structure system with SDOF and MDOF under the action of modulated non-stationary ground motion are studied.The specific research contents are as follows:1.According to the theory of structural random vibration,the selected ground motion in this paper is the non-stationary ground motion with uniform modulation,and its power spectral density function is the product of the power spectral density function of steady ground motion and the modulation function with only time variable.In this paper,the trigonometric series superposition method of synthetic ground motion is introduced,which can be used to calculate the input energy of SDOF and MDOF structure system under the action of modulated non-stationary ground motion.2.Taking the SDOF structural system as the research object,using the energy balance equation and the random vibration theory,the theoretical formulas of the time history of mean and variance of the transient input energy and the total input energy of the structure under the non-stationary seismic action are derived.The theoretical formula shows that the total input energy is related to the mass of the structure,the damping ratio,the natural frequency,the modulation function of the non-stationary ground motion and the power spectral density function of the steady ground motion.Numerical examples show that the mean and mean square deviation of transient input energy based on the time history sample statistical method are fluctuant,and the curves fluctuate up and down around the curves of theoretical formula in this paper.For the statistical value of total input energy,the results of the two methods are in high degree of coincidence.3.The theoretical formulas of the mean and variance of the transient input energy and the total input energy time history of the MDOF structural system under the nonstationary seismic action are studied.The theoretical formula shows that the total input energy of earthquake is related to the generalized mass of structural system,mode participation coefficient,damping ratio of each mode,natural frequency,modulation function of non-stationary ground motion and power spectral density function of steady ground motion.The results of numerical examples show that the time history sample statistics are in high degree of coincidence with the mean value of the total input energy time history given by the theoretical formula,but the mean square deviation of the time history sample statistics is slightly less than the theoretical formula value.4.Combining SDOF and MDOF structural system,the accuracy of theoretical formula of the mean value and the variance of the non-stationary transient input energy and total input energy derived in this paper is verified by comparing the time history of the total input energy obtained by the time history sample statistical method.This study lays a theoretical foundation for the further study of the statistical characteristics of the total input energy spectrum of linear structures under random ground motions,and provides a reference for the future study of the random energy spectrum of nonlinear structures.