In this paper, an analytical solution on the dynamic mesh forces of planetary gear trains (PGTs) is proposed by investigating a lumped‐parameter model. By using the method of multiple‐scales (MMS), closed‐form expressions of mesh force under the effects of manufacturing and assembly errors are obtained. From these expressions, the effects of several key factors such as the tooth thickness error, pin position error, applied torque, support stiffness of sun gear, and tooth profile modifications (TPM) on dynamic load sharing behaviours are explored. Numerical integration is carried out to verify the validation of the proposed method, and the developed expressions are also validated by comparing the results with previously published predictions. The results for several examined PGT systems show that the key factors abovementioned affect the dynamic load sharing behaviours as both static and dynamic factors. An important new conclusion obtained by this work is that proper tooth profile modifications keep the dynamic load sharing factors almost equal to the results obtained under static conditions. This conclusion provides the possibility to simplify the dynamic analysis to the static analysis on the dynamic load sharing problems.

In this paper, the nonlinear oscillation of planetary gear trains is investigated by the homotopy analysis method. The nonlinearity of planetary gear trains due to the periodically time-varying mesh stiffness and contact loss are included. In contrast to the perturbation analysis, the homotopy analysis method is independent of the contact loss ratio, and then can be applied to both small and large contact loss ratios. In this article, firstly the closed-form approximations for the primary resonance, sub-harmonic resonance, and super-harmonic resonance are obtained by homotopy analysis method. The accuracy of homotopy analysis method solutions is evaluated by numerical integration simulations. Results indicate that with relatively large contact loss ratios, the amplitude–frequency curves obtained by homotopy analysis method agree better with the results obtained by numerical integration than those obtained by the method of multiple scales. This study lays a higher accurate foundation for more complex nonlinear dynamic analysis of planetary gear trains.

In this study, the two-to-one internal resonance between the first two rotational modes of planetary gear trains (PGTs) is investigated. A purely rotational model is applied considering mesh stiffness variations, tooth separations, and tooth profile modifications (TPMs). Semi-analytical solutions for the internal resonance case are obtained using the method of multiple scales (MMS). The solution equations indicate that the mesh stiffness variations and tooth separations are the main factors causing internal resonance. A validation of the MMS was performed by numerical integration (NI). The results from an example analysis indicate that there exists an internal resonance phenomenon in the case of ω_N+2 ≈ ω_2, where ω₂ and ω_N+2 are the natural frequencies associated with the rotational modes, and N is the number of planet gears. Internal resonance in PGTs causes chaos, and part of the energy is transmitted from the ring gear to the sun gear through shocks. Proper TPMs that eliminate the tooth separations could suppress the internal resonance. The internal resonance, in turn, affects the optimal areas of the TPM magnitudes.

A stick-slip dynamic model of a two-gear set is proposed. In this model, time-varying mesh stiffness, tooth separation, friction between the gear teeth surfaces and potential stick-slip are considered. Considering the contributions of the vibration velocity to the gear teeth sliding velocity, transition conditions between the stick state and slip state are revealed. Dynamic analysis shows that, when stick appears, the vibration amplitudes are dramatically increased, and tooth separation occurs. Additionally, stick also induces half-frequency components of the mesh frequency and causes chaos. Parametric studies show that heavy loads and rough tooth surfaces increase the probability of stick and increase the time of the stick state over a single mesh period. However, due to the complex nonlinearity, the existence of a certain relationship between the gear rotation speed and the stick time is not confirmed.