Dynamic excitation caused by time-varying meshing stiffness is one of the most important excitation forms in gear meshing process. The mesh phase relations between each gear pair are an important factor affecting the meshing stiffness. In this paper, the mesh phase relations between gear pairs in an encased differential gear train widely used in coaxial twin-rotor helicopters are discussed. Taking the meshing starting point where the gear tooth enters contact as the reference point, the mesh phase difference between adjacent gear pairs is analyzed and calculated, the system reference gear pair is selected, and the mesh phase difference of each gear pair relative to the system reference gear pair is obtained. The derivation process takes into account the modification of the teeth, the processing, and assembly of the duplicate gears, which makes the calculation method and conclusion more versatile. This work lays a foundation for considering the time-varying meshing stiffness in the study of system dynamics, load distribution, and fault diagnosis of compound planetary gears.
Planetary gears are widely used in mechanical transmission systems, but the vibration and noise affect their reliability and life. In this paper, the torsional dynamic model of an encased differential planetary gear with coaxial contrarotating outputs considering the time-varying meshing stiffness, damping, and phase difference of all gear pairs is established. By solving the equations of the derived system, three types of natural frequencies with different multiplicities of the system are obtained. The multiscale method is used to study the parametric vibration stability caused by the time-varying meshing stiffness, and the results are verified by numerical simulation. The dynamic characteristics of elastic meshing force are analyzed from time domain and frequency domain. The variation of the dynamic load factor of each gear pair with input speed and the relationship between its peak position and the natural frequency of the derived system are discussed. The results show that there is an unequal coupling phenomenon of meshing frequency between the meshing forces of different planetary sets. In the absence of external excitation, the meshing stiffness parameters not only excite the main resonance response of the system but also cause superharmonic resonance, subharmonic resonance, and combined resonance.
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