The Calculation Method Of Dynamic Response For Non-proportionally Damped Elastic Structures

With the maturity of engineering technology and the increase of seismic demand,it is difficult to meet the requirements of different building functions for the structure which is composed of single damping characteristic material.More and more hybrid structures which are composed of multiple materials with different damping characteristics are applied to actual engineering,such as steel-concrete structure,the structure with energy dissipation dampers,etc.The damping matrix of the hybrid structure no longer satisfies the proportional relationship with the mass matrix and the stiffness matrix.The seismic design method which is commonly used in the design codes is based on the assumption of proportional damping,and it is no longer applied to calculate the structural earthquake effects of non-proportionally damped elastic structures.When the dynamic response of a non-proportionally damped linear system is solved by mathematical dynamic equations,a reasonable damping model is one of the critical factors which are related to the reliability of the calculation and analysis results.In this paper,a more reasonable structure damping model is introduced.The corresponding mode superposition methods are further proposed,which can be directly used to calculate dynamic responses of non-proportionally damped elastic structures.Then the rationality and applicable scope of the proposed methods are verified and analyzed based on the vibration test of the hybrid cantilever plate which is composed of the steel plate and damping film.Finally,the real value form formulas of the corresponding response spectrum method are deduced,which can provide a theoretical basis for the seismic design of non-proportionally elastic structures.The main contents and conclusions are as follows:(1)In order to solve the time-domain divergent problem of the complex damping model,the conjugate relationship of the positive and negative frequency is introduced,and the time-domain motion equation of the hysteretic damping model is obtained based on the frequency-domain motion equation of the complex damping model.The completeness of the complex damping model is retained and the energy dissipation of per cycle is not related to the vibration frequency of the external excitation.Besides,the hysteretic damping model can ensure the stable convergence of the time domain calculation results.However,based on the hysteretic damping model,the damped natural frequency increases with the increase of the damping parameter(the loss factor),and the frequency response function is non-causal.The hysteretic damping model is only suitable for structural dynamic response analysis where the loss factor can be regarded as a constant in a limited frequency range.In order to overcome the two shortcomings,on the basis of the hysteretic damping model,an equivalent viscous damping model and an improved hysteretic damping model based on the frequency-dependent loss factor are further obtained.(2)According to the characteristics of the hysteretic damping model,the equivalent viscous damping model,and the improved damping model which is based on frequency dependent loss factor,the corresponding time-domain calculation methods are proposed based on the structural dynamic response relationship,linear interpolation of external excitation,Fourier series of external excitation and Hilbert-Huang transform of external excitation,respectively.Compared with the time-domain calculation method based on the structural dynamic response relationship,the ones based on the external excitation relationship are semi-analytical algorithms,which are unconditionally stable and convergent.The analyses of the cases show that compared with the time-domain calculation methods based on the structural dynamic response relationship and linear interpolation of external excitation,the time-domain calculation methods based on Fourier series of external excitation and Hilbert-Huang transform of external excitation have higher calculation accuracies,but the calculation efficiency is lower.(3)The complex mode superposition methods based on hysteretic damping model and equivalent viscous damping model can be realized in physical space,respectively.Besides,considering the frequency-dependent loss factor,the real mode superposition method and the complex mode superposition method which are based on the improved hysteretic damping model are proposed,respectively.The damping matrix is easy to construct and the calculation results are unique based on these proposed methods.These proposed methods can be directly used to calculate the time-domain dynamic responses of the non-proportionally damped linear systems.The reasonableness and applicability of each proposed method are verified and analyzed based on the vibration test using a hybrid cantilever plate as a non-proportional damping model specimen.(4)The non-proportionally damped linear system is simplified to the equivalent proportional damping system.On the basis of the real mode superposition methods based on the hysteretic damping model and the equivalent viscous damping model,combined with the virtual excitation method and the stationary random theory,the formulas of response spectrum complete quadratic combination(CQC)are derived based on the equivalent loss factor and the energy method,respectively.On the basis of the complex mode superposition methods based on the hysteretic damping model and the equivalent viscous damping model,the corresponding real value form formulas of response spectrum complex complete quadratic combination(CCQC)are further derived.(5)The modal decomposition response spectrum methods of hysteretic damping model and viscous damping model are compared by aid of the response spectrum and correlation coefficient.The results show that in the case of a small loss factor,the response spectrum value and the correlation coefficient are approximately equal.In the case of a big loss factor,the response spectrum value and the correlation coefficient are quite different and the difference cannot be ignored.Compared with the mode decomposition response spectrum method based on the hysteretic damping model,using the equivalent relationship between the loss factor and the two times of damping ratio,the modal decomposition response spectrum method based on the equivalent viscous damping model can be directly used to calculate the structural earthquake effect with the seismic design spectra.(6)Compared with the response spectrum CQC method based on equivalent loss factor(equivalent damping ratio),the calculation process of the response spectrum CQC method based on the energy method only depends on the material loss factor,and the selection of whole equivalent loss factor is avoided.Besides,non-proportional damping characteristics of the hybrid structures are further considered.Therefore,the response spectrum CQC method based on the energy method is used to calculate the structural earthquake effects of hybrid structures with weak non-proportional damping characteristics in preference.Compared with the response spectrum CCQC method based on the Rayleigh damping model of the substructures,the response spectrum CCQC method based on the equivalent viscous damping model directly depends on the quality,stiffness,and damping parameters,which is not influenced by the mode combination.The corresponding calculation results are unique,and the calculation of the displacement correlation coefficient is only involved.Therefore,the response spectrum CCQC method based on the equivalent viscous damping model can be used to calculate the structural earthquake effects of hybrid structures with significant non-proportional damping characteristics in preference.