Research and Application of Data-Driven Modeling Method for Gear System Based on Time-Varying Characteristics

The current research on gear system dynamics mainly utilizes linear spring damping model to calculate the contact force between gears. However, this linear model cannot correctly describe the energy transfer process of collision that often occurs in gear system. Focus on the contact-impact events, this paper proposes an improved gear contact force model for dynamic analysis in helical gear transmission system. In this model, a new factor associated with hysteresis damping is developed for contact-impact state, whereas the traditional linear damping factor is utilized for normal meshing state. For determining the selection strategy of these two damping factors, the fundamental contact mechanics of contact-impact event affected by supporting forces are analyzed. During this analysis, an effect factor is proposed for evaluating the influence of supporting forces on collision. Meanwhile, a new restitution of coefficient is deduced for calculating hysteresis damping factor, which suitable for both separation and non-separation states at the end of collision. In addition, the time-varying meshing stiffness (TVMS) is obtained based on the potential energy approach and the slice theory. Finally, a dynamic analysis of a helical gear system is carried out to better understand the contact force model proposed in this paper. The analysis results show that the contribution of supporting forces to the dynamic response of contact-impact event within gear pair is important. The supporting forces and dissipative energy are the main reasons for gear system to enter a steady contact state from repeated contact-impact state. This research proposes an improved contact force model which distinguishes meshing and collision states in gear system.

Electric vehicle has been gaining popularity in the worldwide market. The drive system for an electric vehicle usually consists of a permanent magnet synchronous motor (PMSM) and a two-stage gear system. The vibration characteristics of electric drive system is of great importance to the noise and durability performance. Recently, rotor-step skewing has drawn attention as a means to reduce the vibration response for the PMSM, since it can effectively reduce the torque ripple as shown in the previous studies. However, the coupling relationship between the torque ripple and the gear transmission system has not been studied thoroughly. To explore the influence of torque ripple on the motor-gear system when considering rotor-step skewing, this paper proposes a coupled electro-mechanical system model to study the vibration characteristics of electric drive system. The effect of rotor-step skewing on cogging torque and torque ripple of PMSM, dynamic transmission error (DTE) of gear transmission system, and vibration characteristics of electric drive system is studied. The simulation results show that the torque ripple has a significant effect on the DTE and the vibration responses of the electric drive system. The rotor-step skewing not only effectively suppress the torque ripple, but also optimize the DTE harmonics induced by the torque ripple.

Tooth surface wear is an essential factor that affects gear transmission system dynamic characteristics and must be considered in monitoring the health status of gear systems. In this study, a coupled bending and torsion dynamic model is first established. The internal excitation changes under the influence of tooth surface wear and geometric errors, including the time‐varying meshing stiffness and comprehensive meshing error, are calculated using the method proposed by the authors in a previously published paper. Subsequently, a quasi‐static wear model of a tooth surface is established based on the Archard wear formula. The two proposed models are combined to simulate the changes in the wear area and depth of involute gear pairs and to investigate the influence of tooth surface wear on gear system dynamic characteristics. Several sets of simulation results are used to demonstrate the relationship between the tooth surface wear and gear system dynamic characteristics. The coupled analysis method and findings presented in this study are of great significance for the fault diagnosis and health monitoring of gear transmission systems.

The influence of involute contact ratio on the torsional vibration behavior of a spur gear pair is investigated experimentally by measuring the dynamic transmission error of several gear pairs using a specially designed gear test rig. Measured forced response curves are presented, and harmonic amplitudes of dynamic transmission error are compared above and below gear mesh resonances for both unmodified and modified gears having various involute contact ratio values. The influence of involute contact ratio on dynamic transmission error is quantified and a set of generalized, experimentally validated design guidelines for the proper selection of involute contact ratio to achieve quite gear systems is presented. A simplified analytical model is also proposed which accurately describes the effects of involute contact ratio on dynamic transmission error.

The effect of gear contact state change due to shaft misalignment on meshing stiffness is usually neglected in the traditional stiffness calculation model with misalignment error, the further influence mechanism of shaft misalignment on gear dynamic characteristics is also unclear. To address these shortcomings, a new mesh stiffness calculation model with misaligned gear considering the effects of tooth contact state is proposed by combining the improved loaded tooth contact analysis (LTCA) model. Then the effects of tooth contact state changes aroused by shaft misalignment on the meshing stiffness excitation are investigated. Moreover, a dynamic model of the misaligned gear system with 8 degrees of freedom (DOF) is established, and based on which the dynamic characteristics of the gear system are investigated and verified by experiment. The study results indicate that the proposed model can be used to evaluate the stiffness excitation and dynamic characteristics of the misaligned gear system with the tooth contact state taken into consideration. This study can provide a theoretical method for evaluating and identifying shaft misalignment errors.

To investigate the mechanisms of unexpected failures in a multi-stage gear transmission system under a relatively low load, a rigid–flexible coupled multi-body dynamics model with 10 spur gears and 12 helical gears is established. The dynamic condensation theory is applied to improve computational efficiency. The construction of this model incorporates critical nonlinear factors, ensuring high precision and reliability. Based on the proposed model, four critical dynamic parameters, including acceleration, mesh stiffness, dynamic transmission error, and vibration displacement, are analyzed. This research systematically reveals the nonlinear dynamic mechanism under the multi-gear coupling effect. The spectrum of the gears exhibits prominent low-frequency peaks at 320 Hz and 750 Hz. Notably, alternate load-dominated gears show a shift in prominent low-frequency peaks. The phenomenon of marked oscillations in mesh stiffness suggests a potential risk of localized weakening in the system’s load-carrying capacity. Critically, alternating torques induce periodic double-tooth contact regions in the gear at specific time points (0.115 s and 0.137 s), which are identified as critical factors leading to gear transmission system failures. The variation characteristics of the dynamic transmission error (DTE) demonstrate that the DTE is strongly correlated with the meshing state. The analysis of vibration displacement further indicates that the alternating external loads are the dominant excitation source of vibrations, noise, and failures in the gear transmission system.

Nonlinear characteristics of a multi-degree-of-freedom spur gear system with bending-torsional coupling vibration

Study on nonlinear dynamic characteristics of gear system with 3D anisotropic rough tooth surface based on fractal theory

The meshing process includes high-speed impact between the teeth. The vibration and noise in the gear transmission become prominent owing to the backlash between gears. Currently, the concentrated mass model is primarily used to analyse the vibration and noise mechanism of gear transmission, and it is difficult to effectively explain high-speed gear collision. In this study, an improved impact vibration model was applied to the meshing state of a high-speed gear. The gear transmission was simplified as a collision vibration system with absolute and traversable boundaries, and the piecewise-smooth dynamics model of the gear system is established. A mapping dynamic analysis method was then used to describe the impact and meshing boundary of gears. The impact-meshing situation of the gear system was observed, and the vibration response of the system under different speeds and loads was analysed using numerical simulation. The results revealed that the main source of the gear vibration was the meshing frequency of a gear pair. The resonance generated by the modulation of the rotation and meshing is the main reason for the long-term impact-meshing phenomenon of the gear transmission system. The results of this study are of great significance as they reveal the vibration characteristics of a high-speed gear transmission system.

In practice, gear backlash and bearing clearance usually exist together in a gear system. They may induce complicated dynamic responses and degrade transmission performance. Up to now, although each of them has been researched, little attention has been paid to the coupling dynamic characteristics of gear backlash and bearing clearance. In a limited number of relevant studies, since the linear collision models they adopted are difficult to realistically depict actual collision behaviors caused by bearing clearance, these studies cannot accurately reveal the coupling dynamic characteristics of gear backlash and bearing clearance. Furthermore, system stability of the gear system considering gear backlash and bearing clearance has not been thoroughly investigated. In view of this, this paper contributes to the research on dynamic modeling and stability analysis for the spur gear system considering gear backlash and bearing clearance. A nonlinear collision model with time-varying contact stiffness/damping is suggested for describing the bearing collision behaviors. Based on the geometrical relationship of dynamic center distance, dynamic working pressure angle, and dynamic backlash, the coupling motion model of gear backlash and bearing clearance is developed. On this basis, the dynamic model of the spur gear system considering gear backlash and bearing clearance is established and verified by numerical simulations, virtual prototyping simulations and experiments. Afterwards, to thoroughly explore the complicated dynamic characteristics of the gear system considering gear backlash and bearing clearance, several important parameters, i.e., rotational speed, gear backlash and bearing clearance, are chosen as bifurcation parameters to study their influences on system stability via bifurcation diagrams, time-domain waveforms, FFT spectra, Poincaré maps, and phase diagrams. Various complicated nonlinear behaviors, such as hopping, multiple periodic motion, quasi-periodic motion, and chaotic motion, are revealed. This study can provide useful reference for the multi-clearance coupling research of the gear system in complicated working environments.

PurposeThe paper is devoted to presenting a systematic investigation on the mechanical model and nonlinear dynamic characteristics of spur gear system with and without input shaft crack.Design/methodology/approachConsidering the backlash, load-distribution, time-varying meshing stiffness and sliding friction, the modelling of a 5DOF gear system is proposed. Likewise, stiffness and damping models under elastohydrodynamic lubrication are developed, and sliding friction between gear pair is also outlined. In particular, a cracked input shaft which affects the support stiffness is presented, and breathing crack in keyway is adopted. On this basis, the dynamic responses of a gear system with and without input shaft crack are examined using numerical method, and some classical response diagrams are given, illustrating the effect of the important parameters on the gear system.FindingsDynamic simulation demonstrates that there exist periodic, quasi-periodic and chaotic motions in the gear system, and rational speed of the gear pair has noteworthy effects on vibration characteristic. Besides, comparison between healthy and cracked condition of input shaft indicates that occurring of crack convert periodic motion to quasi-periodic or chaotic motion.Originality/valueThe results give an understanding of the operating conditions under which undesirable dynamic behavior occurs, and provide some useful information to design and diagnose such gear system with crack fault.

Analysis of effect of random perturbation on dynamic response of gear transmission system

As the most widely used power transmission device in mechanical equipment, the vibration characteristics of gears have a very important influence on the working performance. It is of great theoretical and practical significance to study the vibration characteristics of gear system. In this paper, a gear transmission system model is set up in a forcefully nonlinear form; the continuity mapping and discontinuity mapping are utilized to analyze the nonsmooth vibration. Then, the sliding dynamics of separation boundaries is studied by using the perturbation method and the differential inclusion theory. In addition, the periodic response of gear pair system is illustrated and Floquet’s theory is presented to confirm the stability and bifurcation of periodic response. Concurrently, the maximal Lyapunov exponent is obtained to accurately determine the chaotic state in gear pair system, which is consistent with the bifurcation diagram and Poincare section. Finally, a reasonable explanation is given for the jump phenomenon in bifurcation diagram.

In order to extend the service life of the long-chain gear transmission system of a drum shearer, an electro-mechanical coupling model of a drum shearer cutting unit is established. The model considers the dynamic characteristics of the motor, time-varying meshing stiffness, as well as the drum load characteristics. Additionally, the dynamic characteristics and control strategy for suppressing the dynamic load of the gear transmission system under impact load are investigated based on this model. Firstly, the influence of the gear transmission system of the drum shearer cutting unit under impact load is analyzed. Then, on that basis, the active control strategy based on motor torque compensation is proposed to suppress the dynamic load of the gear transmission system caused by mutational external load. Finally, the suppression effect on the dynamic load of the gear transmission system is analyzed. Research results indicate that this control strategy has good control effects to suppress the dynamic load caused by a mutational external load, which confirms the effectiveness of the proposed control strategy.

Scaling law of gear transmission system obtained from dynamic equation and finite element method