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Teeth disengagement or back-side teeth engagement induced by backlash reduces the transmission quality and dynamic performance of gear systems, and the accurate interpretation of multi-state meshing behavior can provide guidance for structural optimization and performance evaluation. Therefore, the multi-state meshing behavior of the gear system is...
Citations
... The error excitation is affected by many complex factors, including manufacturing errors [37], thermal deformation [38], oil film thickness deformation [39] and elastic deformation. In this paper, the error excitation refers to the geometrical transmission error. ...
... Therefore, appropriate analytical lubrication models should be adapted in TVMS and TVMD models to improve the precision of gear dynamic responses [31]. Shi et al. [51] attempted to combine numerical model of lubricant in [26] with analytical potential energy models to better predict elastic properties of the contact. The contact stiffness due to gear temperature rise was also included in the model. ...
An integrated gear tribodynamics model is proposed for the study of EV powertrains’ performance. The model considers the transient effects of lubrication regimes, non-Newtonian shear thinning, inlet shear heating, deformation states of asperities in mixed regime of lubrication and contact temperature using a set of analytical routines, which are computationally efficient. The proposed gear tribodynamics model provides a breakdown of the interdependency of these attributes and studies their impact on the performance of gear contacts. The results indicate that up to 30% of the contact load can be carried by asperities, of which 80% undergo elastoplastic deformation. In addition, the contribution of lubricant to contact stiffness can be greater than that of surface asperities by an order of magnitude.
... Planetary gear trains (PGTs) are widely and extensively used in various mechanical transmission systems, such as robots, aircrafts and vehicles thanks to its small size, light weight, high load capacity and large transmission ratio. However, the vibration and noise problems of PGTs are also prominent due to their complex structure and operating conditions, such as including internal and external meshing gear pairs, single-and double-tooth engagements caused by the non-integer contact ratio, gear separation and back-side tooth contact induced by backlash [1][2][3]. In particular, the long-term accumulation of unhealthy -meshing status or dynamic instability caused by changes in the number of meshing tooth pairs and gear disengagement restricts gear life and aggravates failure behaviors, such as fatigue pitting, root cracking and even fracture of gears, thereby affecting equipment operating accuracy, transmission efficiency and service life [4][5][6]. ...
Gear disengaging, back-side tooth contact or poor dynamic behavior during operating leads to dynamic instability in planetary gear trains (PGTs). A novel nonlinear dynamic model of PGTs with internal and external gear pairs considering multi-state engagement induced by backlash and contact ratio is established. An improved time-varying meshing stiffness model including temperature stiffness is analytically derived. The time-varying meshing stiffness with temperature effect, friction, backlash, time-varying pressure angle, and time-varying friction arm are integrated into the dynamic model of PGTs. Multi-state engaging behavior is efficiently identified by constructing different Poincaré mappings. A method to calculate dynamic instability is proposed in the time-domain trace. The intrinsic relationship between multi-state engaging and dynamic instability is investigated via multi-section bifurcation plots and phase trajectory topology. The global dynamic instability is revealed based on the bifurcation and evolution of coexistence behavior under the parameter-state synergy. The results show that the multi-state engagement is heavily depending on bifurcation and phase trajectory topology, which whereby affects the dynamic instability. Two special phenomena, complete and incomplete bifurcations, are discovered under parameter-state synergy. Complete bifurcation causes global instability and incomplete bifurcation results in local instability and yields coexistence responses. Incomplete bifurcation brings about new bifurcation branches.
... Keywords Gear fault diagnosis Á Dynamic model Á Mesh stiffness Á Body crack Á Tip relief Abbreviations subscript i Subscript i = 1, 2, 3 represents Zone I (i = 1), II (i = 2) or III (i = 3), respectively subscript k Subscript k = p, g represents driving gear (k = p) or driven gear (k = g), respectively a 1 The angle of the meshing force a 2 Half of the tooth angle corresponding to the base circle a p ...
... The horizontal distance between the meshing position and the intersection of tooth profile and base circle d 2 The horizontal distance between the intersections of tooth profile with dedendum circle and base circle d 3 The horizontal distance between the crack tip and the intersection of tooth profile and dedendum circle e pg ...
... The gear system is an important part of mechanical equipment, which is widely used as the key component of power and motion transmission in aerospace, energy development, industrial production and other modern mechanical equipment [1][2][3]. As one of the important basic components of the gear system, gear often produces crack failures in the operation of the system due to excessive load, improper assembly or fatigue caused by long-term operation [4,5]. ...
Due to ignoring the effects of the change of the tooth attachment position caused by the cracks, traditional time vary mesh stiffness (TVMS) calculation models and dynamic simulations for cracked gears will lose their precision in the body crack case. To address this shortcoming, a new analytical TVMS calculation model of cracked gear considering tip relief (TR) is developed based on a proposed variable-angle deformation energy integration method. On this basis, a dynamic model of the gear system for the analysis of fault vibration characteristics is established. The effectiveness and accuracy of the proposed TVMS calculation model are verified by the finite element method. A comprehensive investigation is finally carried out to reveal the effects of the parameters of TR, load and crack on the TVMS and dynamic characteristics of the cracked gears. The study results indicate that the proposed models can meet the accurate TVMS calculation and dynamic simulation for both the tooth- and body-cracked gears, and the influences of the tooth attachment position change caused by the crack cannot be ignored. This study could provide a systematic methodology and meaningful reference for the dynamic modelling, simulation and fault diagnosis of gear systems with crack failures.
... Recently, Liu et al. [8] established the multi-time scale Poincaré mapping sections to identify the transition of the neighboring periodic motion due to the changing system parameters. Shi et al. [23,24] disclosed the bifurcation process of the system and the tooth engagement based on the four particular Poincaré mapping sections. These four particular Poincaré mapping sections were also used to study the influence of the system-inherent phase on the back-side impact frequency of the spur gear pair by Huang et al. [25]. ...
... where R dbij and R kbij (i = p, g; j = a, 1, 2, r) represent the equivalent base circle radii of the jth pair mesh tooth in gear i. S dij and S kij (i = p, g; j = a, 1, 2, r) represent the friction arms of the jth pair mesh tooth in gear i. The gear dynamics system with two rotational degrees of freedom (DOFs) can be reduced to a single DOF model by subtracting the equation of gear from the equation of pinion in Eq. (1) [17][18][19][21][22][23][24][25], which is given as, ...
The bifurcation of the gear transmission system is very important for the stability and safety characteristics of the system from the perspective of nonlinear dynamics. Meanwhile, the extended tooth contact and the coupling effect between gear neighboring teeth are focused on the establishment of the dynamic models of the gear transmission system. However, there has been nearly no literature revealing the effect of the extended tooth contact and the coupling effect between gear neighboring teeth on the bifurcation characteristics of the gear system. In this paper, a dynamic model of spur gear pair is established by considering the extended tooth contact, the coupling effect between gear neighboring teeth, and some time-varying parameters. The extended tooth contact and coupling effect between gear neighboring teeth are directly involved in the calculation of the dynamic meshing force. Then, a solving method with two loop statements is proposed to numerically solve the gear dynamic model. Besides, two new Poincaré mapping sections are established to unify the characterization of system motion and extended tooth contact. Finally, the bifurcation characteristics of the system are studied by using the established Poincaré mapping sections, bifurcation diagrams with multi-mapping sections, phase portraits, Poincaré mapping points, and contact forces. The results show that the system bifurcation and chaotic motion become more severe due to the coupling effect between neighboring teeth. The contact of the extended teeth is accentuated by the system bifurcation. But more extended tooth contact decreases the parameter range of system bifurcation. The connection between system bifurcation, extended tooth contact, and coupling effect between gear neighboring teeth is revealed. This paper can also provide a reference for the stability of the gear transmission system.
... Shi et al. [17] formulated the timevarying geometric parameters due to the moving meshing position under two meshing states. And they [18,19] further established the dynamic model with multi-state meshing behavior considering the number of meshing teeth. Huang et al. [20] analyzed the timevarying friction torque affected by two-side impact. ...
When the helical gear contains mixed modification, the backlash will vary along the tooth width and tooth profile, which will lead to uneven contact of the meshing tooth pairs and meshing impact on both sides of the gear teeth, and deeply affect the dynamic characteristics of the system. In this paper, each meshing excitation with time variation is analyzed based on the function of the meshing surface, and the quantitative calculation model of the meshing force and friction torque considering the meshing state and the contact state is developed. The nonlinear dynamic model of helical gear pair with time-varying backlash caused by mixed modification is established, and the influence mechanism of each modification amount on the bifurcation characteristics of the system is analyzed. Through optimizing the multi-modification parameters, the vibration amplitude of the chaotic motion is significantly weakened, and the obtained modification parameters are also generally applicable to the vibration suppression of other high-speed motions. Furthermore, a strong advantage of this work is that although the method is proposed for modified tooth surfaces, it is also suitable for other unconventional tooth surfaces that can be described by functions.
... Sainsot et al. [31] evaluated the tooth fillet-foundation deflection using a simple empirical formula to account for the tooth deflections induced by the gear body compliance. Other researchers adapted this model to determine the fillet-foundation stiffness [25,27,[32][33][34]. A few studies combined the bending, axial, shear, and tooth fillet compliances in the prediction of the total TVMS [25,27,33]. ...
... Lubricant damping varies with contact load, speed, lubricant's rheological properties, film thickness and squeeze film effect. Analytical models use formulations based on either constant damping ratio or simplified solutions of Reynolds equation [6,11,[32][33][34][35][41][42][43][44][45]. Chen et al. [21] combined the analytical TVMS model with a damping coefficient which varied with the compressive deformations of the contacting teeth. ...
... This study focuses on the effects of lubricant-surface interactions in predicting TVMS and the overall nonlinear dynamics response of the gear system. Therefore, the analytical formulation for the fillet-foundation stiffness in [31] is used based on the research carried out in [25,27,[32][33][34]. The more comprehensive approach in [38] can be adapted in future to account for the gear body compliance coupling with its neighbouring teeth. ...
Noise, vibration, and harshness (NVH) issues pose considerable challenges for electric vehicle powertrain engineers. Gear vibrations generate an intrusive gear whine noise, with significant impact on the sound quality of electric powertrains. Dynamic transmission error (DTE) is the most quantitative indicator for gear NVH. Backlash, time variable meshing stiffness and damping contribute to DTE. Hence, a better understanding of these excitation sources is essential. A gear tribodynamics model is developed using potential energy method to estimate time variable meshing stiffness (TVMS). A fully analytical time-efficient model is proposed for lubricated contact stiffness based on transitions in the regimes of lubrication. The model accounts for the combined effects of surface elasticity and lubricant stiffness. Film thickness and damping coefficients are transiently updated at each instant during meshing cycle. The predictions from this model are compared with measured results from the literature and predicted results from Hertz contact model. The lubricated contact model successfully shows the contribution of the lubricant stiffness to TVMS and its variations with elasticity and viscosity parameters during meshing cycle. Gear harmonic and super-harmonic resonances are accurately estimated in terms of amplitude, frequencies and stiffness softening nonlinearities. Time history responses and phase-displacement diagrams show good agreement with the gear dynamics response at the main harmonic and second super-harmonic frequencies. The proposed model has a reasonable accuracy, significantly better than those from Hertzian contact models, and is considerably time efficient in comparison to numerical EHL solvers.
... Shi et al. [4] calculated time-varying backlash including elastic deformations, thermal deformations and film thickness. They also improved further a dynamics model of spur gear system considered time-varying backlash [5] based on their previous works [4,6], and found that time-varying backlash changes the transitions of meshing states and dynamics motions in the gear transmission system. Based on the above, the value of backlash is time-varying rather than constant, which is determined by the center distance, rotational speed, load torque, gear tooth temperature, oil film thickness, meshing point position, single-tooth engagement and double-tooth engagement in reality. ...
... The switching conditions of the different meshing states of the system are changed by the time-varying backlash. The periodic motion may become chaotic motion when the backlash changes from constant to time-varying, and the meshing state of the system is changed accordingly [5,6]. It is necessary and practical that the time-varying backlash is applied to the dynamics model of the gear transmission system. ...
... They also established three meshing safety domains to identify the tooth disengagement and back-side teeth contacting, and studied the global integrity of the safe basin in the meshing safety domain [26]. Backlash in these dynamics models and established safety domains were constant and mismatched with actual values [1][2][3][4][5], as well as the influence of the time-varying characteristics of backlash on the motions and safety characteristics of the system cannot be revealed. ...
Multi-state meshing characteristics induced by backlash and contact ratio in the gear pair are an unsolved puzzle about its relationship to safety characteristics of the gear transmission system. A nonlinear dynamics model of spur gear pair considering time-varying backlash, multi-state meshing, contact ratio and other time-varying parameters is improved. The time-varying backlash comprehensively determined by static backlash, tooth elastic deformation, thermal deformation and oil film thickness is calculated to identify the multi-state meshing of the system. The safety characteristics of the system are classified into nine safety levels according to the influence of meshing states on operation stability. Nine safety domains corresponding to the safety levels are established, respectively. An algorithm is proposed to calculate the safe basin of the gear transmission system in established safety domains. The evolution of safety characteristics in established safety domains is investigated from a local and global perspective by safe basin, nonlinear analysis methods, bifurcation and safety dendrograms and statistical methods. The results show that the value of backlash is affected by meshing states and working conditions of the system. Time-varying backlash is important in identifying multi-state meshing and establishing multi-level safety domains. The safety levels of system responses can be judged by established multi-level safety domains, and the safety characteristics of the system are affected by system parameters, motions and bifurcations. It provides a theoretical basis for the prediction of meshing states and the design of system parameters.
... established the coupled nonlinear dynamic model of a gear-shaft-bearing transmission system considering the tooth contact temperature, friction and load fluctuation to analyze the temperature effect on the dynamic characteristics. Shi et al [20]. analyzed the effect of the gear temperature rise on the time-varying meshing stiffness, and studied the nonlinear dynamics of a spur gear system. ...
... Based on Ref. [20], the comprehensive time-varying meshing stiffness, k com,j (t), is calculated by: ...
This paper aims to reveal the relationship between the high-speed gear wear and nonlinear dynamics for a wind turbine gearbox considering the tooth contact temperature. The temperature and deformation of the tooth surface are calculated respectively using the Blok's flash temperature theory and thermal deformation formula, and the meshing stiffness caused by the tooth contact temperature is obtained using the Hertz theory. The wear depth and time-varying meshing stiffness of high-speed gears are calculated using the Archard's equation and potential energy method, respectively. The nonlinear dynamic model of the gearbox is established to study the characteristics, and the frequency domain values of gear wear are compared with experimental data. The results show that the tooth contact temperature makes the time-varying meshing stiffness decrease, then affecting the dynamics of the system, the stability decreases with the increase of comprehensive transmission error, and for the gearbox considering the contact temperature, the gear wear makes the chaotic motion occur in advance and its region increase obviously. The research provides a basis for the wear fault diagnosis and detection of gear transmission systems.
A tooth fracture is one of the most common failures in the gear systems used in industrial sectors and manufacturing. In order to prevent fracture defects, great research interest has emerged in fault detection in gear systems. Statistical process control charts have been recently used for fault detection based on experimental data. Despite the advantages of simulation data for dynamic models recognized in gear systems, this approach has yet to be used to train SPC charts. This paper proposes to apply an exponentially weighted moving average chart (EWMA) to monitor and detect fracture deterioration generated by the 5% reduction in gear mesh stiffness in the dynamic model of the spur gear system. Next, white Gaussian noise is added at different levels to raw signals, and the appropriate noise level is chosen based on the highest signal-to-noise ratio (SNR). Then, time-domain features of both root mean square (RMS) and kurtosis are extracted as statistical control indicators. After that, EWMA charts are constructed using the statistical features of the gear signal under healthy conditions. Finally, EWMA charts are being tested based on faulty status features. Results show that the EWMA by RMS is more effective than the EWMA by kurtosis in detecting fracture faults.
