Free vibration characteristics of compound planetary gear train sets

In this study, a plane model of a compound planetary gear train (CPGT) is proposed in order to study its free vibration characteristics. Two translations and one rotation are considered for each element (sun, carrier, ring-1, ring-2, and planets) of the CPGT. The contact between the teeth in mesh is modelled by linear springs with stiffness called gearmesh stiffness. The phasing between these stiffness is taken into account. Using Lagrange formulation, the equation of motion is derived. Starting from the eigenvalue problem of the system, the influence of the planets' number and position, and the effect of the gyroscopic phenomena on the free vibration of the CPGT are studied. The natural modes are classified into three groups: translational (the carrier, ring 1, and sun have pure modal translational deflection with no rotation), rotational (the carrier, ring 1, and sun have pure modal rotational deflection with no translation), and planets' modes (only the planets have a modal deflection). It was found that the change of the planets' angular position does not affect this classification. On the other hand, the study of the influence of the carrier rotation speed show that the gyroscopic effect separates the repeated translational modes into distinct ones. The obtained results are much important in the design process in order to avoid critical operating conditions which lead to undesirable vibrations.