Non-random Vibration Analysis For General Viscous And Viscoelastic Damping Systems

In order to deal with dynamic uncertain vibrations with limited samples,the interval process model is introduced to model the dynamic uncertain excitation,and then a kind of non-probabilistic vibration analysis method is developed in recent years,named as “non-random vibration analysis method” to distinguish it from the traditional random vibration theory.In non-random vibration analysis,both the excitation and the response are given in the form of boundary functions over the entire time history,which can reduce the dependence on large experimental samples.In this paper,the non-random vibration analysis is further extended into general viscous damping systems and viscoelastic damping systems to calculate dynamic response bounds of vibration systems under uncertain dynamic loads,which will be helpful for the safety evaluation and reliability design of real vibration systems.The main research contents of this paper are as follows:(1)A non-random vibration analysis method for general viscous damping vibration systems is developed to calculate the dynamic response bounds of a viscous damping vibration system under uncertain excitations.Firstly,the unit impulse response matrix of the system is obtained by using a complex mode superposition method.Secondly,the dynamic response middle point and radius functions of the system under uncertain excitations are derived based on the Duhamel’s integral,and then the upper and lower response bounds of the system can be obtained.(2)A non-random vibration analysis method for single-degree-of-freedom viscoelastic damping vibration systems is proposed based on the fractional damping vibration model.Firstly,the fractional derivative model is introduced to describe the viscoelastic damping characteristic and then the fractional damping vibration model is formed;Secondly,the forced vibration response expression of a linear single-degree-of-freedom system with a fractional derivative damping is obtained based on Laplace transform;Finally,the dynamic response bounds of the system under dynamic uncertain excitations can be obtained by calculating the dynamic response middle point and radius functions.(3)A non-random vibration analysis method for multi-degree-of-freedom viscoelastic damping vibration systems is developed based on the mode superposition theory,which can calculate the dynamic response bounds of the system under uncertain excitations.Firstly,the fractional derivative damping vibration model of a linear multi-degree-of-freedom system with a viscoelastic damping is established,and then the multi-degree-of-freedom problem can be transformed into the linear superposition of multiple single-degree-of-freedom systems based on the modal superposition method.Secondly,the response function expression of the system under uncertain excitations can be obtained by calculating all the response functions corresponding to each mode based on Laplace transformation.Finally,for the fractional derivative damping multi-degree-of-freedom system under uncertain excitations,the uncertain excitations can be treated as interval processes,and then the dynamic response bounds of the system can be obtained by calculating the response middle point and radius functions.