Single-stage Gear System With Coexisting Attractors And Its Stability Study

Gear systems are widely used in the power transmission of mechanical equipment because of their compact structure and high transmission efficiency.With the rapid development of the manufacturing industry,the gear system has higher requirements for transmission accuracy,reliability and smooth operation.In most machinery and equipment,gears are the main source of noise and vibration.Therefore,the study of the stability and bifurcation of coexisting attractors in gear systems can provide a parametric basis for optimal design of the dynamics of the system,structural reliability and improved equipment life.In this paper,the single stage spur gear transimission system is taken as the research object,and the non-linear dynamics models of the three-degree-of-freedom gear system and the single-degree-of-freedom gear system with friction are established respectively,taking into account the tooth side clearance,time-varying stiffness,comprehensive transmission error and tooth surface friction,etc.The stability and bifurcation of the coexisting attractors of the system are analysed to study their dynamics characteristics.The main studies are as follows:(1)A mechanical model of the three-degree-of-freedom gear system and the single-degree-of-freedom gear system with friction is established,the differential equations of motion are solved and the differential equations of motion of the system are dimensionlessly simplified.Based on the characteristics of the segmented smooth vibration system,the local Poincaré mapping of the system is established and the Jacobi matrix of the composite mapping of the system is derived.Theoretical derivations of the simple and continuation shooting methods are carried out to obtain methods for solving the coexisting attractors of the system and their Jacobi matrix eigenvalues.(2)A three-degree-of-freedom gear system is used as the object of study,combining numerical simulation,the continuation shooting method and Floquet eigenmultipliers to extend the tracking of the stability and bifurcation of the coexisting attractors.The cell mapping method is applied to calculate the domain of attraction of the coexisting attractor and to analyse the erosion evolution of the domain of attraction of the coexisting attractor with the change of system parameters.Combining the bifurcation diagram with the Maximum Lyapunov Exponent(TLE)diagram,the effects of the engagement frequency,damping coefficient and integrated transfer error fluctuation coefficient on the system dynamics are analysed.(3)The stability and bifurcation of the coexisting attractors are traced in a single-degree-of-freedom gear system with friction,and the erosion evolution of the attraction domain of the coexisting attractor with the change of system parameters is analysed,and the effects of meshing frequency,tooth side clearance,torque and damping coefficient on the system dynamics are investigated.It is found that Hopf bifurcation,grazing bifurcation and period-doubling bifurcation have no effect on the topology of the attractor domain and do not affect the global characteristics of the system.In contrast,the occurrence of saddle-node bifurcation generates new periodic attractors and changes the topology of the attraction domain of the original attractors,so saddle-node bifurcation is the main reason for the coexistence of periodic attractors.In part of the parameter interval,the saddle-node bifurcation causes the final state of the system to jump between two different periodic motions,and the period-doubling bifurcation exhibits subcritical properties.Therefore,the essence of subcritical period-doubling bifurcation is that the system undergoes a period-doubling-induced saddle-node bifurcation,resulting in a hysteresis zone.The presence of grazing bifurcations of unstable periodic motion in the partial hysteresis region causes the steady-state response of the system to jump to 2n-2 p-(2q-1)or 2n-(2p-1)-2q motion,rather than 2n-2p-2q motion,via SN-type period-doubling bifurcation of n-p-q motion,in contrast to the behaviour of conventional subcritical period-doubling bifurcations.Boundary crisis are an important factor in the termination of chaotic attractors.