Fast-slow Dynamical Analysis Of The Nonlinear Torsional Model Of A Harmonic Gear Reducer

Harmonic gear reducer is a new type of mechanical transmission device,it has been widely used by its simple structure,lightweight,small size,high precision,large transmission ratio,high transmission efficiency,and many other advantages.A variety of nonlinear factors in the system will induce complex fast-slow dynamical characteristics,which will aggravate the fatigue damage of system components,shorten the service life of gears,and seriously affect the normal operation of the system.Due to the complexity of the problem,previous scholars’ research on the complex dynamic of harmonic gear reducer system mainly focused on conventional dynamics,while the research on fast-slow dynamics was rarely reported.In this paper,considering the nonlinear factor of torsional stiffness,a mathematical model of a harmonic gear reducer system involving nonlinear torsional stiffness is established.To discuss the fast-slow dynamics of the harmonic gear reducer system by adopting research methods such as the fast-slow analysis method or attraction field analysis and combining with numerical simulation,a variety of new relaxation oscillation modes are discovered,and the dynamical mechanism of their generation is revealed.The specific research content of this paper is as follows:(1)Firstly,the coexistence of multiple attractors in the nonlinear torsional system of the harmonic gear reducer was discovered,and then the two-parameter bifurcation diagram was drawn to explore the possible change behaviors of the relaxation oscillations.It is found that the number and stability of the attractors near fold bifurcation points may change when the system parameters vary.This is manifested in two ways: first,when a fold bifurcation occurs in the upper equilibrium branch,the system is bi-stable;while a fold bifurcation occurs in the lower equilibrium branch,the system is stable.Second,the system is bi-stable when fold bifurcations occur in the upper and lower equilibrium branches.For the above two cases,we use fast-slow analysis and attraction domain analysis to explore the dynamical mechanisms of the relaxation oscillation behaviors.As a result,several different relaxation oscillation patterns are obtained.In particular,relaxation oscillations with asymmetric structure,i.e.,compound relaxation oscillations,characterized by the system trajectory transitions passing through the middle branch at one end and crossing it at the other,are discovered and researched.(2)Based on the research on the dynamical behaviors of the nonlinear torsional system of the harmonic gear reducer,another special relaxation oscillation mode in the system is found by changing the torsional stiffness condition.Its manifestation is mainly as follows: when the upper equilibrium branch or the lower equilibrium branch has a fold bifurcation,the system trajectory will reciprocate between the two equilibrium point attractors,and finally be attracted by a certain equilibrium point attractor.Therefore,the special dynamical behavior of “circuitous transition” is induced.In order to further explore the dynamical mechanism related to the “circuitous transition” the relaxation oscillation behaviors of the system under asymmetric stability conditions and symmetric stability conditions are studied respectively.It is worth noting that under the symmetric stability conditions,one special compound relaxation oscillation pattern is found,i.e.,the system trajectories with “circuitous transitions” at one end and normal relaxation transitions at the other end.