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Considering the effects of the tip-fillet and friction between teeth surfaces, an improved analytical model for time-varying mesh stiffness (TVMS) calculation is further developed based on our previous works (Ma et al. in Mech Mach Theory, 98:64–80, 2016) in which the effects of extended tooth contact, revised fillet-foundation stiffness and nonlinear contact stiffness are taken into account. The proposed model is also verified by comparing the TVMS with that obtained from finite element method. Combined effects of the tip-fillet and friction on the TVMS, loaded transmission error and loaded sharing factor are discussed. The results show that TVMS between double- and single-tooth engagements becomes smooth under the small radii of the tip-fillet, however, an abrupt change of the TVMS from double-tooth contact to single-tooth contact can be observed under the large radii of the tip-fillet. The range of the double-tooth engagement decreases and the single-tooth engagement increases with the increase of the radius of the tip-fillet. In addition, the tooth-surface friction mainly affects the mesh stiffness in single-tooth engagement region due to the change of friction force direction near pitch point, and the extent of variation increases with the increase of friction coefficient. The friction has a larger effect on the TVMS than the tip-fillet for the selected parameters of tip-fillet and friction. This study can provide a basis for the structural design of the spur gears.

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.

Due to the effects of gear flexibility, the extended tooth contact (ETC) can appear, which is the phenomenon that the incoming tooth pair gets into contact ahead of the theoretical start of contact and the outgoing tooth pair is out of contact later than the theoretical end of contact. A large calculation error for the time-varying mesh stiffness (TVMS) calculation can be caused if the effects of ETC are ignored, especially under the larger torques. In this paper, an improved analytical method (IAM) suitable for gear pairs with tip relief is established to determine time-varying mesh stiffness (TVMS), where the effects of ETC, nonlinear contact stiffness, revised fillet-foundation stiffness, and tooth profile modification are considered. Based on the improved analytical model, TVMS under different torques, lengths, and amounts of profile modification is compared with that obtained from analytical finite element approach [29] and from FE method. The results show that TVMS obtained from the IAM agrees well with that from FE method and from analytical FE approach [29], and the computational efficiency of the IAM is also much higher than that of FE method.

In this paper, a mesh stiffness model is developed for profile shifted gears with addendum modifications and tooth profile modifications (TPMs). The time-varying mesh stiffness (TVMS), load sharing factor (LSF), loaded static transmission error (LSTE) and non-loaded static transmission error (NLSTE) of a profile shifted spur gear pair with TPMs are obtained by the analytical model. The optimum profile modification curve under different amounts of TPM is determined by analyzing the LSTE first. Then, considering the effect of NLSTE, finite element (FE) model of a geared rotor system is established. The system vibration responses under different TPM curves are analyzed and the optimum modification curve is further evaluated by amplitude frequency responses. The results show that the optimum modification curve is related to the amount of TPM and modification coefficients. The comparison of the optimum profile modification curves is evaluated by LSTE and vibration responses, which shows that the optimum modification curve should be determined by evaluating the vibration response of the geared rotor system in the low mesh frequency range.

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.

This paper studies the mesh characteristics and vibration responses of a gear system with crack and spalling faults using different modeling methods. These modeling methods for the time-varying mesh stiffness (TVMS) include Yang's method, Tian's method and Liang's method which can not consider the effect of the fillet foundation stiffness, Zhou's method, Wan's method, two methods proposed by Ma and finite element method which include the effect of the fillet foundation stiffness. The effects of the crack and spalling are introduced into these models to calculate the TVMS for gear pairs with faults, and TVMS are compared based on these methods. By combining the TVMS with a two-dimensional lumped mass model of a gear system, the system vibration responses are compared using these modeling methods. The comparison results show that the fillet foundation stiffness has an important influence on the TVMS and vibration responses of the gear systems. Moreover, the error of repeated calculation of the fillet foundation stiffness must be revised.