| Nonlinear factors such as clearance and dry friction are widely exist in wheel-rail systems,brake systems and some other vehicle systems,which may affect the operational performance of various types of vehicle systems.In order to make the vehicle systems run safely and stably and provide a reference for the parametric design and a comprehensive optimization control law of the system,a theoretical mechanical model with dry friction and clearance is established and its dynamic characteristics are analyzed in view of the non-smooth characteristics of the various vehicle systems.The research on this kind of model can partly reflect the kinematic law of many vehicle systems under the influence of non-smooth factors,and provide a reference for the actual application of non-smooth dynamics theories in vehicle systems.In this paper,the mechanical models of two kinds of systems with clearance and dry friction are established.The dynamic characteristics of these systems in each excitation frequency range are analyzed by the Poincarémapping method and the numerical simulation.The changes of the diversity,existing intervals and impact velocity of the various periodic motions in the system with the variation of system parameters are considered,and the non-smooth characteristics of these systems are revealed.Firstly,the mechanical model of a single-degree-of-freedom vibro-impact system with dry friction is established,and the numerical simulation results of its dynamic characteristics are analyzed.As the result shows,the sticking-sliding motions exist because of the dry friction.The frequency ranges of motions with the sticking phenomena are scattered.Most periodic or chaotic windows have frequency intervals containing the sticking phenomena,but the frequency ranges of all kinds of motions with the sticking phenomena are narrow.In the low excitation frequency range,the system almost has only two kinds of motions,they are the chattering-impact motions and the the p/1 motions.The Grazing bifurcations of the p/1motions occur successively with the decrease of excitation frequency so that the number of impacts increases gradually and finally the p/1 motions turn into chattering-impact vibrotion.When the excitation frequency is small enough,the Sliding bifurcation occurs thus make the chattering-impact vibrotion change into complete chattering-impact motion with the characteristic of sticking.The sequences of 1/n subharmonic periodic motions exist in the middle or high frequency range.When the excitation frequency is decreased,the Grazing bifurcations and the inverse period doubling bifurcations of the 1/n motions occur thus make the 1/n motion turn into various subharmonic impact motions or chaotic motions which are relatively complicated.Finally,the number of the excitation periods of 1/n motion decreases and the 1/n motion evolves into the stable 1/(n-1)motion.The Grazing bifurcations in the system are divided into two categories:the Real-grazing bifurcations and the Bare-grazing bifurcations.When the Real-grazing bifurcation occurs,the number of impacts of a kind of periodic motion decreses by once in a motion period and this motion directly turn into another stable periodic motion.When the Bare-grazing bifurcation occurs,the instability caused by the grazing appears which may induces various types of complex subharmonic impact motions or chaotic motions.Secondly,the influences on the dynamic characteristics of the system with the change of the parameters is considered.The existence,diversity and the evolution rules of periodic motions in all the ranges of the excitation frequency are analyzed by changing the parameters of the system.According to the numerical simulation results,the variation of system parameters will affect the distribution of periodic motions,and make several kinds of periodic motions appear or disappear.Among them,the variation of the friction coefficient f_u has a great influence on the sticking phenomenon of the system.In the middle or high frequency range,the increase of f_u will make the frequency intervals of different periodic motions which contain the sticking phenomena grow in number,some kinds of 1/n motions gradually disappear,and the frequency ranges of some kinds of subharmonic impact motions or chaotic motions may appear and expand as a result of that.This will make the dynamic behaviors of the system more complicated.However,in general,the change of system parameters has little effect on the rules of transition of different periodic motions,and those slight changes only take place in very narrow ranges of the frequency.Finally,the Stribeck friction model is used to describe the friction in the system.The dynamic model of the two-degree-of-freedom system with gap and dry friction is established.The dynamic characteristics of the system and the influence of the system parameters on its dynamic characteristics are analyzed.As the result shows,The kinds of periodic motions and the change rules between different periodic motions of the two-degree-of-freedom system are very similar to those in single-degree-of-freedom system.The two-degree-of-freedom system also has p/1 motions and chattering impact motions in the low excitation frequency range.The generation mechanism of the complete chattering impact motions is the same as that of the single-degree-of-freedom system,and the change rules between different 1/n motions in the middle or high frequency range is also similar to that of the single-degree-of-freedom system.However,the distribution of the various kinds of periodic motions in two-degree-of-freedom system is slightly different from that of the single-degree-of-freedom system and the phenomenon of frictional sticking in it is significantly reduced compared to the single-degree-of-freedom system.In addition,the system parameters of the two-degree-of-freedom system are more so that the influencing factors of the dynamics of the system are more complicated.As a result,the complex dynamic phenomena such as chaotic motions or long-period motions with multiple impacts are more likely to occur in some range of different parameters. |
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