| Non-smooth dynamic system is ubiquitous in real life,and the most representative is all kinds of impact vibration system.Among them,the phenomenon of collision and vibration exists in all aspects of our life,especially for various mechanical devices,which not only affects the performance and life of mechanical devices,but also affects the working efficiency of mechanical devices.Therefore,it is of great practical value to study the dynamics of non-smooth systems.Based on the perturbation theory,the local singularities and grazing bifurcations of two kinds of non-smooth dynamic systems are analyzed from the point of view of optimizing the actual impact vibration phenomena.The parameters needed in the manufacturing process of mechanical devices are optimized,and the service life and working efficiency of mechanical devices are improved.In a two-degree-of-freedom impact vibration system,the chattering phenomenon of the system is theoretically analyzed,and the time it takes to completely chattering is deduced.Then the dynamic behavior around the system when scraping occurs is analyzed.The concrete expressions of zero-time discontinuity mapping and Poincaré cross-section discontinuity mapping of the system are derived.Then the two mappings are compounded with smooth mapping when no collision occurs.The piecewise mapping expression of scratch mapping is given.Then the singularities of the Jocabi matrix of the standard type mapping of the system are studied.Through theoretical derivation,it is found that there is no singularities in the determinant of Jocabi matrix,but the singularities are found in the trace of Jocabi matrix.Next,the grazing bifurcation of the system is deduced in detail,and the conditions to be satisfied when the grazing occurs are deduced.Finally,numerical simulation is carried out with C language programming software,MATLAB and Grapher,and the correctness of the theory is verified.In the bifurcation diagram of the system,there is a fracture phenomenon,that is,the period doubling sequence is discontinuous,which is due to the grazing motion of the system.The Poincaré mapping is discontinuous when the edge movement occurs,which results in singularity of the system.In the first system model based on the promotion,this article to study the second model,which contain nonlinear spring-collision damping vibration system mechanics model,based on the research method of the first model system are analyzed,grazing around the local dynamics behavior,when the system is deduced the zero time of sigmoid get wiped segmentation bifurcation discontinuous expression,for the system to study the standard mapping Jocabi matrix singularity,the singularity of the discovery system did not exist determinant of a matrix,and to meet certain conditions,the singularity of the system will show it.Finally,numerical simulation is carried out.Under the premise that other referenceparameters do not change,the dynamic response of the system when different values are taken is studied,which can be obtained from the numerical simulation,or there are points of fracture or jump on the bifurcation diagram of the system.The analysis shows that these fracture or jump points are the transition points from one motion state to another,and at these points the system will have friction.Moreover,the Poincare mapping of the system is discontinuous when the grazing movement occurs,which is exactly the reason for the singularity of the system.The theoretical derivation and analysis of the two-degree-of-freedom collision vibration model and its extension model are made.The results show that the singularities of the two single-degree-of-freedom systems and the systems with nonlinear springs exist in the trace of the matrix,and the singularities of the systems can only be expressed under certain conditions.It provides a good theoretical basis for its application in practical engineering. |
