Based on the single degree of freedom torsion model, Zajkek et al. used parabolic functions to describe the engagement stiffness of single contact gear pairs, and used the truncated Fourier series form of Green’s periodic function (PGF) method to find the stable periodic analytical solution. By using the real eigenvalues of the system, the unstable boundary of the system was found. The periodic analytical solution obtained by Green’s periodic function is compared with the Runge-Kutta numerical solution, and the results are consistent. Wang set up typical locomotivetransmission system such as three degrees of freedom torsional vibration model of rice by Ishikawa method in the process of meshing gear pair of time-varying mesh stiffness analysis and calculation, with the aid of the bifurcation diagram, phase map, poincare map, time response and amplitude frequency response output data statistical characterization methods, such as, drive pinion speed is studied and the influence of the stiffness of the gear transmission system dynamic performance, reveals the multicycle response bifurcation and chaos response mechanism of the dynamic evolution of the process.
Yangd.c.h. Considering the transmission error, contact deformation and damping factors under the involute tooth profile, the torsional vibration characteristics under three kinds of external load excitation (free vibration, load constant, sinusoidal load distribution), such as angular velocity, displacement along the engagement line, damping attenuation and other response characteristics, were studied. Chen et al. established a microscopic fractal tooth profile mathematical boundary model and adopted a pure torsion mechanics model to study the influence of meshing surface roughness, fractal dimension and initial meshing clearance, stiffness coefficient and dimensional damping on the system response. The results show that the roughness can cause the response instability of the system, and the stability of each response index can be improved by increasing the surface roughness level, and the fractal dimension can approximate simulate the surface texture morphology, and predict the possible response of the system under different processing processes. The initial clearance and stiffness coefficient have little effect on the system response under the stable load input state.
YousfiNourhaine using integral method and continuous wavelet transform methods, such as damping estimation for the spur gear pair is calculated, in the form of integral evolution equation of motion for second formulation of the model, and guide the most appropriate damping with constant section model calculation model, based on the wavelet envelope extraction process of demodulation methods to verify the model, related to the structure model of damping ratio. Yangd.c.h. Based on the established model and research results, the anti-deformation stiffness of involute gear teeth and the dynamic characteristics of the system driven by prime movers (internal combustion engine and motor) and the response output are further considered. The results show that the energy loss caused by Hertz damping is greater than that caused by Coulomb friction, and it is helpful to understand the impact behavior of high-speed gear system under frequent start/stop or intermittent motion. Based on this torsion model, Xiong et al., Xiao et al., took oil film stiffness and backlash into consideration and studied the response of these two factors to the system stability.