Nonlinear Motion Stability Analysis Of A Vibrating System With Clearance

Due to the machine processes or the wear error,there are inevitably gaps and constraints between the different parts of the mechanical system.The gaps can cause collisions,and then cause nonlinear behavior in the system.It is a common phenomenon of the nonlinear behavior taking place in the machine with gaps.But in the previous research,the nonlinearity has to be neglected because the theory and tools mastered are not enough.With the development of the science and technology,it is an irresistible trend that the model fits closely with the reality.At present,the nonlinear behavior caused by the non-smooth problem has become a hot point.For example,Floquet theory is used to analyze to the stability of the non-smooth system and the conditions of bifurcation.The Lyapunov exponent can describe quantitatively the nonlinear behavior,such as multi period motion,quasi-periodic motion and chaotic motion.Nowadays,the investigation on the nonlinearity caused by the non-smooth system is also actively involved in the study of the nonlinear fields.In general,the gap-containing systems are basically multi-parameter high-dimensional systems.At present,the theoretical research on nonlinear collisional vibration systems at home and abroad is more limited by one side,and there are fewer sides,especially the analysis of high-dimensional collisions with double-sided constraints.This paper starts with the use of more shoulder blades in engineering applications,and solves the Floquet multiplier and Lyapunov exponent by establishing a two-sided collision system equation.Through numerical simulation,the motion state of the system can be sensed intuitively through the image,and the conclusion can be accurately obtained through numerical values.The numerical results are combined with theoretical analysis to determine the dynamics of the system.It is hoped that the theoretical study of nonlinear collisional vibration and the analysis of the vibration-damping mechanism of the shoulder blade can help theoretical science and engineering practice.The specific research content of this paper is as follows: First of all,the dynamic model of the shoulder-impact vibration system with two shoulders is established.The model is discretized and modeled by finite element method,and its motion equation is deduced and solved.Secondly,Solve the system's Floquet multiplier,judge the value range and system motion through the Floquet multiplier model.Then,by means of numerical simulation,the time history diagram,phase diagram and Poincaré map of the system are drawn under different excitation frequencies.By observing the graph and calculating the Floquet multiplier,the periodic solution stability of the shoulder blade system with double-sided collision is analyzed.In the end,we solve the system's Lyapunov exponent and judge whether the system is chaotic by whether it is positive or not.And by the power of numerical simulation,the time history diagram,phase diagram,Poincaré map and Lyapunov exponent diagram of the system are drawn under different excitation frequencies.By observing the graph and calculating the Floquet multiplier and the maximum Lyapunov exponent,the dynamic characteristics of the shoulder blade system with double-sided collision are analyzed.