The Influence Of Parameters On The Dynamics Of Mechanical Vibration Systems With Gaps

In recent years,non-smooth dynamical systems have attracted the attention of many scholars and engineering technicians due to its universality in daily life and importance in practical engineering fields.As a typical representative of non smooth dynamic systems,collision systems widely exist in the field of mechanical engineering.The nonlinear and singular problems caused by factors such as collision,stickiness,and edge scraping make the collision system exhibit complex dynamic behaviors,which have a significant impact on the efficiency and performance of the system.Therefore,it is of important value for doing the dynamic research of collision systems in engineering application.This article firstly studies a simplified mechanical model of mechanical collision systems with pre-tensioned springs in engineering practice,constructs a Poincarémapping composed of smooth flow mapping and discontinuous mapping,and provides a numerical method for calculating Floquet multipliers.The periodic attractor mode and its parameter domain in the two parameter plane of the system are numerically simulated.The stability and evolution laws of the periodic 1 attractor,as well as the discontinuous grazing bifurcation,grazing and period-doubling induced bifurcation,saddle-node type and subcritical period-doubling bifurcation,crisis and other discontinuous bifurcation behaviors are studied by combining Runge kutta method,shooting method,continuation method and cell mapping method,and revealed the formation mechanism of hysteresis domain and subharmonic inclusion domain.When the pre-compression is not equal to 0,the discontinuous grazing bifurcation of the 1–m attractor can produce a Uk–(km+1)attractor sequence,thus hysteresis domain and subharmonic inclusion domain are generated during the transition of adjacent 1–m and1–(m+1)attractor.The main mode of the periodic attractor in the subharmonic inclusion domain is k–(km+1)(k=2,3,4,...).When the pre-compression is equal to 0,the grazing bifurcation of the 1–m attractor is continuous,but the grazing induced saddle-node bifurcation or period-doubling bifurcation form hysteresis domain or subharmonic inclusion domain during the transition of adjacent 1–m attractor.The main modes of periodic attractors in the subharmonic inclusion domain SIR_m are 2–(2m+1)and 2–2(m+1).In the small neighborhood of period-doubling bifurcation,saddle-node bifurcation enables the final state of the system to jump,and period-doubling bifurcation exhibits subcritical characteristics.This bifurcation process is called SN-type period-doubling bifurcation.Discontinuous grazing bifurcation,subcritical period-doubling bifurcation and SN-type period-doubling bifurcation cause coexistence of multiple attractor in the transition of adjacent attractor.When the periodic attractor and chaos coexist,the chaotic attractor disappears through the boundary crisis.Then,taking a single-degree-of-freedom elastic mechanical vibration system with gaps as an example.In the numerical simulation system,the periodic attractor mode and its parameter domain in the two parameter plane are combined with the phase diagram,Poincarémap and Floquet eigenvalue multiplier to analyze the bifurcation characteristics of adjacent periodic 1 attractor transiting through the hysteresis domain and subharmonic inclusion domain.The results show that the attractor of adjacent period 1 can be directly transferred through the grazing bifurcation,and can also be transferred through the hysteresis domain or subharmonic inclusion domain through the saddle-node bifurcation or period-doubling bifurcation induced by the grazing.The vibro-impact system has very rich dynamics.This article has discovered many complex new phenomena from a new perspective,and the research results can provide reference for the parameter design and selection of mechanical collision systems with pre-tensioned springs in engineering practice.Through analyzing the two parameter dynamics and the global dynamics analysis of the coexistence attractor,the dynamic optimization design of the system can be realized,which consequently enables the system presents the desired attractor mode in a larger parameter space.