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Considering the effects of extended tooth contact (ETC), revised fillet-foundation stiffness under double-tooth engagement region, nonlinear contact stiffness and tooth spalling defect, an analytical model for time-varying mesh stiffness (TVMS) calculation of spur gears is established. In addition, the analytical model is also verified by comparing the TVMS under different spalling widths, lengths and locations with that obtained from finite element method. The results show that gear mesh stiffness decreases sharply with the increase of spalling width, especially during the single-tooth engagement; the spalling length only has an effect on the beginning and ending of gear mesh stiffness reduction; the spalling location can affect the range of gear mesh stiffness reduction, and the range will reduce when the spalling location is close to the addendum. This study can provide a theoretical basis for spalling defect diagnosis.

Based on our previous work (Ma et al., 2014, Engineering Failure Analysis , 44, 179–194), this paper presents an improved analytical model (IAM) for the time-varying mesh stiffness (TVMS) calculation of cracked spur gears. In the improved analytical model, the calculation error of TVMS under double-tooth engagement due to repeatedly considering the stiffness of the fillet-foundation is revised, and the effects of reduction of fillet-foundation stiffness of cracked gears and extended tooth contact (ETC) are also considered, which have a great influence on TVMS, especially under the condition of large torques and crack levels. Moreover, the comparisons among the IAM, traditional analytical model (TAM) and finite element (FE) model are also carried out under different torques and crack depths. IAM is also verified by comparing TVMS and vibration responses obtained by FE model, which can be considered as a gauge to evaluate the calculation error. The results show that the maximum error of IAM is about 12.04%, however, that of TAM can be up to 32.73%.

Dynamic characteristics of cracked gear systems, also known as cracked-gear rotor systems, have received increasing interests among industry and academy in the past two decades. This paper reviews published papers on the dynamics of cracked gear systems. These studies mainly focused on three topics: crack propagation prediction, time-varying mesh stiffness (TVMS) calculation and vibration response calculation; Study objects involve the spur gear, helical gear and planetary gear; Different modeling methods including analytical method, finite element (FE) method, combined analytical-FE approach were adopted. More specifically, this review is composed of three related parts according to the above three topics. The first part involves the prediction of the crack propagation path based on two-dimensional (2D) or three-dimensional (3D) gear models, which provides a basis for the hypothesis of crack path in the process of TVMS calculation of cracked gear pairs. The second part summarizes the TVMS calculation methods including analytical methods, FE methods, combined analytical-FE approaches and experimental methods. The final part reviews the dynamic models for vibration analysis of cracked gear systems including lumped mass models and FE models, where the crack effects are characterized by introducing TVMS of cracked gear pairs into the system dynamic models. The well known open problems about cracked gear dynamics are finally stated, and some new research interests are also pointed out. The review will provide valuable references for future studies on dynamics of cracked gears.

Owing to the effect of gear flexibility, the extended tooth contact occurs, which is a phenomenon that the incoming tooth pair enters contact earlier than the theoretical start of contact, and the outgoing tooth pair leaves contact later than the theoretical end of contact. Considering the effects of the extended tooth contact and tooth root crack on the time-varying mesh stiffness (TVMS), a finite element (FE) model of a spur gear pair in mesh is established by ANSYS software. TVMS under different crack depths at constant rated torque (60 Nm) are calculated based on the FE model. Then, a FE model of a geared rotor system is developed by MATLAB software. Frequency-domain features, statistical features (Kurtosis and RMS) and instantaneous energies based on empirical mode decomposition (EMD) under different crack depths are calculated at 1000 rev/min and the corresponding measured results are also performed by model experiment. The results show that considering the effect of extended tooth contact, TVMS of crack gear pair becomes smooth from double-tooth engagement to single-tooth engagement; the gear crack has reduced the gear body rigidity, which leads to the reduction of stiffness of the healthy teeth and has little effect on the vibration response. The instantaneous energy can be chosen as a distinguishing indicator to qualitatively diagnose the gear cracks with different levels.