Dynamic And Stability Analysis For The Two-Degree-of-Freedom Vibration Systems Including Dynamic Frictions

Dynamic frictions exist extensively in various structures of machinery and engineering,strong nonlinear behaviors are showed,especially when the relative speed is low between the two objects of the systems.The influence of the systems’ parameters on the dynamic behaviors,the stick-slip characteristic of the systems and the relationship between relative speed and dynamic friction force are analyzed by numerical simulation,and the stability of the systems are judged by the largest Lyapunov exponents for the two-degree-of-freedom vibration systems including Lu Gre and modified LuGre dynamic friction and two-degree-offreedom rigid collision systems including LuGre dynamic friction in this paper.The main research contents as follows:State equations of the systems are presented,Poincaré maps of systems are built by using stroboscopic method for two-degree-of-freedom systems including LuGre and modified LuGre dynamic friction by modeling firstly,the influence of the belt speed on the systems’ dynamic behaviors is analyzed with the systems’ bifurcation diagrams and Poincaré maps by numerical simulation.The results show that the abundant characteristics of bifurcation and chaos are presented on certain parameters,eg: the double periodic bifurcation,Hopf bifurcation,periodic bubble and chaotic bubble and so on.Meamtime,sticking motion is common in both systems,and the mass enters sticking motion at a speed greater than the belt speed.It is found that where the relative velocity is zero,the dynamic friction forces both show the droop characteristic,the maximum positive friction force experienced for a mass is less than the maximum static friction force,under the same set of parameters,the appearance of the negative friction force is affected owing to difference of the belt speeds when analyzing the relationship between the belt speed and dynamic friction force.In the two-degree-offreedom vibration system including LuGre dynamic friction,the phenomena of counterclockwise hysteresis loops are presented.While the phenomena of the clockwise hyateresis loops specifically to modified Lu Gre dynamic friction,and concomitant counterclockwise and clockwise hysteresis loops are presented in the two-degree-of-freedom vibration system including modified LuGre dynamic friction.Then Poincaré maps of rigid collision are built for two-degree-of-freedom rigid collision systems including LuGre dynamic friction,the dynamic changes of the systems are analyzed with the changes of belt speed and mass ratio by the systems’ bifurcation diagrams,Poincaré maps,phase portraits and time histories.The results show that the abundant behaviors of bifurcation are presented on certain parameters,eg: the double periodic bifurcation,periodic bubble and so on.When the belt speed is in a medium speed and the mass ratio is less than 0.5,the collision mass has a sticking motion before the rigid collision,and it collides with a sticking motion.The system flutters when the mass ratio is large.When analyzing the relationship between the belt speed and dynamic friction force,it is found that where the relative velocity is zero,the dynamic friction forces show the droop characteristic,the greater the positive or negative relative speed,the lower friction force.Owing to the influence of rigid collision,the maximum positive and negative friction force experienced for a mass is greater than the maximum static friction force when the change of the belt speed and the mass ratio,and the phenomena of the counterclockwise hyateresis loops are presented.