| As one of the basic components of modern machinery and equipment,gears are widely used in energy,transportation,equipment manufacturing and other fields.The tooth surface friction,tooth surface wear,pitting,fatigue damage and manufacturing and installation errors existing in the gear pair during the working process will cause the tooth side clearance to increase.Non-linear factors such as time-varying meshing stiffness and tooth backlash will cause complex dynamic response during system motion and reduce equipment reliability.This paper studies the influence of time-varying meshing stiffness and non-linear factors of backlash on the dynamic response of the gear system.Taking the locomotive traction gear as the research basis,the meshing period is calculated according to the basic principle of tooth profile meshing and the involute equation,and the radius of curvature at each meshing point of the tooth profile is calculated.According to the Hertz formula,the meshing point at each meshing point is calculated.The contact surface is wide,and the elastic deformation at each meshing point is obtained by the finite element method.According to the theory of gear meshing stiffness and the principle of series-parallel connection of springs,the single-tooth stiffness and the phase-related comprehensive meshing stiffness are calculated.The polynomial of the comprehensive meshing stiffness function is obtained by curve fitting using the least squares method,and Fourier transform is performed on the polynomial of the comprehensive meshing stiffness.The third-order Fourier series form is retained for the mathematical modeling of the gear system.Based on the single-stage spur gear,the mechanical model of the coupled bending and torsional vibration of the single-degree-of-freedom gear system with backlash was established,and its segmented analytical solution in the single and double-zone meshing state was solved,and the Poincaré map of the symmetrical flexible contact system of the gear pair was established.Considering factors such as time-varying meshing stiffness and tooth side clearance,the numerical solution of the system is obtained by means of numerical integration.The time history response,phase plane diagram,Poincaré cross-section diagram,bifurcation diagram and spectrogram are used to study the dynamic response characteristics of the system under different damping conditions with internal excitation,tooth flank clearance,and initial conditions.It is found that there are multiple solutions in a single degree of freedom gear system,as well as dynamic behaviors such as doubling bifurcation,periodicity,and chaotic sudden changes.According to the actual design and assembly and manufacturing of the gear system,comprehensively consider the time-varying meshing stiffness of the gear pair,tooth backlash,static transmission error,support stiffness and support damping to establish the four degrees of freedom gear system with backlash coupling vibration mechanics and mathematical models,and non-dimensional processing of the model.The fourth-order Runge-Kutta method is used to simulate the system.The results show that under different damping ratio conditions,with the change of meshing frequency and tooth side clearance,there are doubling bifurcation,Hopf bifurcation,Neimark-Saker bifurcation and multiple solutions phenomenon.Under different initial conditions,the overall trend of the dynamic response characteristics of the system motion is basically the same,but there are significant differences in the local interval.The system has both single Hopf bifurcation and doubled bifurcation on the road from periodic motion to chaos.For the one-dimensional bifurcation behavior,there are also three-fold Hopf bifurcation,torus doubling bifurcation,Hopf-flip bifurcation and other co-dimensional two bifurcation behaviors.The system is more stable when the damping ratio and meshing frequency are large.Appropriately increasing the side clearance can improve the stability of the system. |
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