Time-varying Gear Mesh Stiffness Research Articles

New deterministic model for calculating mesh stiffness and damping of rough-surface gears considering elastic–plastic contact and energy-dissipation mechanism

For a long time, meshing parameter calculations and dynamic modeling of gear systems have assumed smooth tooth surfaces, disregarding the impact of actual three-dimensional (3D) microscopic topography. Based on elastic–plastic contact and energy-dissipation mechanisms, this work developed a comprehensive deterministic model that integrates 3D rough surface modeling, reconstruction, and contact analysis, for calculating the time-varying mesh stiffness (TVMS) and mesh damping (TVMD) of deterministic rough-surface gears. Distinct from fractal and statistical models, the deterministic model analyzes deterministic, real 3D rough surfaces rather than simple surfaces defined by a few parameters. Utilizing stochastic process theory, different-topography surfaces are obtained using height and spatial features and reconstructed employing watershed algorithm and ellipsoidal asperity fitting. Subsequently, a deterministic elastic–plastic contact model based on equivalent rough curved-surface contact is constructed, emphasizing local contact states and energy dissipation for calculating contact stiffness (CS) and damping (CD) of gears. Accordingly, a meshing characteristics analysis model is established for calculating TVMS, TVMD, and load sharing ratio (LSR). Experimental validation and comparisons with several statistical and gear contact models demonstrate the effectiveness, reliability, and superiority of the proposed deterministic model. Multi-parameter impact analysis reveals small Sq (meaning low roughness Sa) and Ssk and high load correlate with increased CS and TVMS, while CD and TVMD depend on their interaction and hardness. Overall, by bridging the micro-surface topography and macro-meshing parameters, our model provides essential insights for the design and manufacturing of high-performance gear to reduce vibration and noise.

Calculation of Time-Varying Mesh Stiffness of Internal Gears based on Precise Tooth Profile and Dynamic Analysis of Planetary Systems with Root Cracks

The internal meshing spur gear pair is the research object, and the potential energy method is applied to calculate the time-varying meshing stiffness of the internal gear. The internal gear tooth profile is divided into two parts: involute and transition curves, and the gear tooth stiffness is calculated by numerical integration based on accurate tooth profile, which improves the calculation accuracy. Analyzed the influence of different crack size parameters on the stiffness of internal meshing gears. A coupled dynamic model of a planetary system with internal gear crack faults was established, and the influence of cracks on the dynamic response of the planetary system was studied using Zoom-FFT spectrum and cepstrum analysis methods. The simulation results show that as the crack parameter size increases, the mesh stiffness of the internal gear pair gradually weakens, and the periodic vibration impact of the planetary-internal gear pair is also more severe. Cepstrum analysis can easily capture weak fault characteristic frequency information in the system and discover their variation patterns. The research results can provide a theoretical basis for the state monitoring and fault diagnosis of gear systems with crack defects.

Dynamic modeling of spur gear system considering the coupling effect between bearing deflection and gear mesh stiffness

Bearing deflection has a direct impact on the mesh condition of gear systems via the center distance of the gear pair and corner contact. These factors need to be considered to accurately model the time-varying mesh stiffness and dynamic responses of gears. To address this issue, a spatial coordinate conversion is used to obtain the instantaneous spatial position of the gear pair under the bearing deflection. The tooth pair separation distance is derived through the geometric relationship analysis to model the corner contact effect. A dynamic model is established which employs a step-to-step time integration algorithm that considers the coupling effect between bearing deflection and time-varying mesh stiffness to obtain the dynamic response of the spur gear system. The simulation results of the proposed model show good agreement with the experimental results which demonstrate that the proposed model enables a proper analysis of the influence of bearing deflection on gear mesh behavior. This study provides an original dynamic model for analyzing the dynamic performance of the gear system considering the influence of bearing deflection.

Crack detection of a modified high contact ratio gear

For the purpose of vibration-based fault detection and to avoid gear failure consequences it is necessary to investigate the possibility of early fault detection. A reduction in the gear time-varying mesh stiffness (TVMS) due to the existence of a crack can be used to detect and evaluate tooth damage. The reduction in TVMS has an impact on the system’s dynamic response signal, and then the used fault detection indicators are affected. The current article sheds light on the fault detection possibility of high contact ratio (HCR) gears as compared to those standard ones of low contact ratio (LCR). The influence of modifying the gear tooth addendum dimension for obtaining a high contact ratio on the possibility of early fault detection is studied. Twenty different crack cases are examined by using both the LCR and HCR design cases. For every crack case, the mesh stiffness is evaluated and introduced to the used gear dynamic model for obtaining the system’s dynamic response. The TVMS evaluation is presented for both the LCR and HCR gear cases. Four fault detection indicators, namely the RMS, kurtosis, peak value and crest factor are applied as different measures to enhance the research outcome. The results show that the studied indicators will have less increase when the crack size increases in the HCR design than in the LCR one. Therefore, the HCR design will be less sensitive or more difficult to detect from the fault detection perspective. Lowering the amount of signal noise is applied to the HCR case to examine the impact on enhancing the obtained indicators. The four applied indicators show different behaviours with reducing noise.

A novel modeling method for evaluating the time-varying mesh stiffness of gears with pitting based on point cloud processing algorithm

A novel modeling method for evaluating the time-varying mesh stiffness of gears with pitting based on point cloud processing algorithm

An analytical method to calculate the time-varying mesh stiffness of spiral bevel gears with cracks

Time-varying mesh stiffness (TVMS) is one of the main dynamic excitation sources of gear transmission systems. The accurate calculation of TVMS is the foundation to comprehensively investigate the vibration of powertrain. Compared with the spur gear, the meshing process of spiral bevel gear is spatially time-varying, which makes the analytical calculation of TVMS more difficult. In this paper, the meshing relationship of spiral bevel gears is first established by extending the slice method of spur and helical gears to spiral bevel gear. Secondly, the analytical models of perfect and cracked spiral bevel gears are presented to evaluate TVMS. Meanwhile, the effects of crack length and depth on mesh stiffness are investigated. Thirdly, the influence of crack length and depth on the dynamic response are simulated based on the 8-DOF dynamic model of spiral bevel gear. Finally, the experiments are conducted to verify the effectiveness of the analytical model. Experimental results demonstrate that the existence of tooth crack fault makes the response characterized by periodic impact with the rotating frequency of cracked pinion.

Analysis of Time-Varying Mesh Stiffness and Dynamic Response of Gear Transmission System with Pitting and Cracking Coupling Faults

The gear transmission system is an important part of the mechanical system, so it is essential to judge its running state accurately. To solve the difficult problem of identifying the components of coupling faults, this paper derives the calculation method of gear time-varying mesh stiffness for coupling faults of pitting and cracking based on the energy method and considering the coupling between teeth, establishes the dynamics model of two-stage gear transmission system with coupling faults and studies the influence of coupling faults on gear time-varying mesh stiffness and dynamic characteristics. The accuracy of the proposed method is verified by experiments. The results show that both pitting and cracking can lead to a reduction in mesh stiffness. The stiffness of pitting will fluctuate irregularly due to the influence of pitting on the tooth surface, while the stiffness of cracked teeth is relatively smooth. The coupling fault stiffness is dominated by more serious faults. By analyzing the periodic impact components in time domain and the sideband components around the harmonics in frequency domain the faulty gears in the transmission system can be distinguished. It provides an effective reference for the diagnosis of faulty gears.

Electromechanical Coupling Dynamic Characteristics of the Dual-Motor Electric Drive System of Hybrid Electric Vehicles

The electric mode is the main operational mode of dual-motor hybrid electric vehicles (HEVs), so the reliability of the dual-motor electric drive system (DEDS) is particularly important. To research the electromechanical coupling mechanism of the DEDS of HEVs, firstly, considering the time-varying mesh stiffness of gears and the nonlinear characteristics of inverters, an electromechanical coupling dynamics model of the DEDS was established, including the permanent magnet synchronous motor (PMSM) and the gear transmission system. Then, the electromechanical coupled dynamic characteristics of the DEDS in the single-motor and dual-motor drive modes were analyzed under steady-state and impact load conditions, respectively. The results show that the motor stator current frequency is modulated by the complicated gear meshing frequency, and the operation state of the gear transmission system can thus be monitored in the stator current. Impact load causes the instantaneous torsional vibration of the transmission system dominated by the first-order natural frequency, and the vibration characteristic frequency appears in the form of a side frequency in the stator current signal; moreover, compared with the single-motor drive mode, the speed synchronization error in the dual-motor drive mode will aggravate torsional vibration in the gear system. The impact energy of the gear system caused by external impact load can be suppressed by reducing the speed synchronization error.

An accurate and numerically efficient time-varying mesh stiffness model for modified straight bevel gear under load

Gear time-varying mesh stiffness (TVMS) plays an important role in the analysis accuracy of a gear transmission mechanism. In previous research, the TVMS of a modified straight bevel gear (SBG) was mainly obtained by the finite element method (FEM), whose computation time is usually long to ensure computational accuracy. By contrast, this work proposes a fast calculation method with high accuracy for the TVMS of a modified SBG. First, B-spline surfaces are introduced to reconstruct a modified SBG using discrete tooth surface points. Next, based on the tooth contact analysis and loaded tooth contact analysis, combined with energy conservation principle and infinitesimal method, a semi-analytical instantaneous single mesh stiffness (SMS) model for the modified SBG is derived. Then, using the additional deformation compatibility condition and principle of force balance, the calculation method of the TVMS of the modified SBG is obtained. Finally, a comparison of the mesh stiffness calculation results between the method of this study and that of FEM is completed. The results show that the semi-analytical TVMS model used in this study can be accurately and quickly solved.

Dynamic modelling of two-stage gear systems with localized wear defects

Abstract Considering the profile deviation induced by localized wear defects, a dynamic model of two-stage gear systems is established. In the proposed dynamic model, the time-varying mesh stiffness of worn gears is calculated by the loaded tooth contact analysis method. The mesh stiffness and the non-loaded transmission error are regarded as the time-varying stiffness excitation and the displacement excitation, respectively. Based on the proposed dynamic model, the dynamic responses of the two-stage gear system with localized wear defects are acquired. The simulated vibration responses are compared to the measured results to test the effectiveness of the proposed dynamic model. It is found that the proposed model can yield similar fault feature with the experiment, which shows the validity of the proposed dynamic model. This paper can afford theoretical basis for the fault mechanism and dynamic characterization of localized wear defects.

A new analytical model for evaluating the time-varying mesh stiffness of helical gears in healthy and spalling cases

Accurately evaluating the time-varying mesh stiffness (TVMS) of a mating gear pair is one of the most important aspects in gear vibration dynamic simulation. Existing work on estimating TVMS of helical gears with tooth spalls generally assumes the defects are in flat bottom shape which deviates from the spalling features as observed in curved bottom shape in practice. The cliff like spalling feature results in inaccurate calculation of TVMS of helical gears with such spalls. This paper proposes an advanced method to evaluate the TVMS of helical gears with tooth spalls with curved-bottom features. In the proposed method, the gear tooth is sliced along the center line of tooth spall in order to easily evaluate the equivalent cross-section area and area moments of inertia of the gear tooth which enables the proposed method adapts to tooth spall(s) with curved-bottom features in arbitrary position, angle and size. The proposed method corrects the foundation stiffness errors of existing methods and takes the influence of axial mesh forces into consideration. Its effectiveness on modelling tooth spalls with curved-bottom features for helical gears is validated by FEM (finite element method).

Probability distribution model of gear time-varying mesh stiffness with random pitting of tooth surface

Pitting is one of the fault forms of gears, which directly causes a reduction in gear time-varying meshing stiffness (TVMS). In this paper, the probability distribution model of gear TVMS is established with random pitting. Firstly, a distribution model of tooth surface pitting is proposed based on the Monte Carlo method, which considers the relationship of tooth surface fault degree and pits number. Then, the TVMS with different characteristics of pitting is calculated by combining image processing. Pits number and distribution parameters of pitting are taken as random variables. The probability distribution models of TVMS with random pitting of tooth surfaces are established. Furthermore, the influence of pits number and distribution parameters on gear meshing stiffness are investigated. The results show that the probability distribution models of TVMS of gears with random pitting agree well with the normal distribution with angular displacement.

Numerical and experimental study of gear rattle based on a refined dynamic model

A refined single pair gear dynamic model is established to accurately predict the gear rattle. The model considers various key parameters, such as backlash, time-varying mesh stiffness, nonlinear oil film force between teeth and drag torque. The time-varying mesh stiffness of helical gears is calculated by using the potential energy method. The nonlinear oil film force is introduced into the model by considering the entrainment and squeeze effects of the lubricating oil between the teeth. In addition, the friction resistance caused by the squeeze of lubricating oil between the teeth is considered in the proposed model. The rotation speed and angular acceleration of the loose gear were measured on the bench, and compared with the simulation results, which proved the accuracy of the model. The results of the parametric analysis show that excessive torque fluctuation and backlash may lead to severe impact of the gear pair. Increasing the drag torque has a positive effect on the control of the gear rattle. Larger viscosity will make faster attenuation of high frequency impact and reduce the impact strength.

Dynamic modelling of traction motor bearings in locomotive-track spatially coupled dynamics system

For railway locomotive, its motor bearing is one of the key components, which is likely to be damaged due to the intensified vibration conditions caused by the wheel-rail interaction and gear mesh. In this paper, a locomotive-track spatially coupled dynamics model considering the dynamic effect of the traction power transmission is established to investigate the dynamic responses of traction motor bearings. The detailed structure of the traction motor, complex internal interactions of the rolling bearing itself, time-varying gear mesh stiffness, and wheel-rail interaction are considered. The simulated results indicate that the gear mesh excitation and internal interactions of the motor bearings significantly affect the dynamic characteristics of the rotor. The motor bearing at the driving end surfers more dynamic loads, and the internal interactions between the components are more intensified than that of the bearing at the non-driving end. Moreover, the skidding phenomenon of the motor bearing at the driving end is alleviated by the greater radial load. Additionally, the wheel-rail interaction with track random irregularity will deteriorate the working conditions of the motor bearings. The modelling method and results can provide helpful theoretical guidance for the motor bearing design and structure optimization of the locomotive traction motor.

Dynamic performance of locomotive electric drive system under excitation from gear transmission and wheel-rail interaction

Development of railway transportation capabilities towards high-power, high-speed and heavy axle-load makes the dynamic interactions between the electric drive subsystem and the mechanical subsystem of a locomotive more intensive and complicated, which may threaten the locomotive operational safety. However, the included dynamics mechanism and influencing law have not been revealed completely. Based on the co-simulation method and vehicle-track coupling dynamics, a model which enables considering dynamic interactions between the electric drive subsystem and the mechanical subsystem of a locomotive through an electromagnetic coupling interface and motor control subsystem is developed in this paper. By using this co-simulation model, the dynamic responses of the mechanical subsystem and the electric drive subsystem are analysed under complicated dynamic excitation, such as the time-varying gear mesh stiffness and the track geometric irregularity. The results indicate an obvious interaction between the electric drive subsystem and the mechanical subsystem, where the vibration status of the major mechanical components can be reflected by the electric current signals. Besides, the characteristic frequencies of the gear transmission can be extracted from the frequency spectrum of the traction motor current signals, which provides the possibility for condition monitoring and fault diagnosis of the locomotive mechanical components through the electrical signals.

Numerical Modelling of Vibration Responses of Helical Gears under Progressive Tooth Wear for Condition Monitoring

Gear wear is a common fault that occurs in a gear transmission system that degrades the operating efficiency and may cause other catastrophic failures such as tooth breakage and fatigue. The progressive wear of a helical gear and its influences on vibration responses are rarely investigated due to the combined effects of the complicated lubrication state and the time-varying characteristic. To fill this gap, a numerical study was put forward to investigate the interactions between gear wear and dynamic response. In this study, an Archard’s wear model with elastohydradynamic lubrication (EHL) effect is adopted to simulate the helical gear wear, which is incorporated with an eight-degree of freedom dynamic model for understanding the gear dynamic at different wear degrees. The wear model shows that the gear wear mainly happens at the gear root due to the relative high slide-to-roll ratio. The dynamic modelling results demonstrate that the wear causes a reduction in time-varying gear mesh stiffness further leads to more vibration. Besides, the simulated vibration responses and experimental validation show that the wear cause increases in the amplitudes of the gear mesh frequency and its harmonics, which can reflect the evolution of progressive gear wear and can be used as monitoring features of gear wear.

A Model for Analysis of Time-Varying Mesh Stiffness of Helical Gears with Misalignment Errors

A Model for Analysis of Time-Varying Mesh Stiffness of Helical Gears with Misalignment Errors

Influences of gear design parameters on dynamic tooth loads and time-varying mesh stiffness of involute spur gears

Gears are one of the most significant machine elements in power transmission due to the many advantages such as high load capacity, long life, and reliability. Due to the increasing power and speed values, the characteristics expected from the transmission elements are also increasing. Significant changes occur in the dynamic behavior of the gears at high speed due to the resonance. For this reason, determining the resonance frequencies is becoming an important issue for designers. This paper presents a method for determining the resonance regions of the gear system under different design parameters. The main purpose of this study is to examine the effects of different gear design parameters on spur gear dynamics. For this aim, the effects of these parameters on the mesh stiffness and contact ratio are examined, and the interaction of mesh stiffness, contact ratio, and dynamic response is presented. Different mesh stiffness calculation methods used to calculate time-varying mesh stiffness and a parametric gear dynamic model are proposed. To solve the equations of motions, a computer program is developed in MATLAB software. Five different design parameters, which are teeth number, pressure angle, reduction ratio, profile shifting factor, addendum factor, and damping ratio, are taken into consideration. The dynamic factor variation is calculated for 1600 rpm a constant pinion speed for each parameter for a single mesh period. Furthermore, the dynamic factor is calculated for the pinion speed between 400–30000 rpm and the frequency response and the resonance regions of the gear system are defined. As a result of the study, the profile shifting and the addendum factor are the most effective two parameters on the gear dynamics. Also, the contact ratio and mesh stiffness have a great effect on the dynamic response of the system. The methods decreasing dynamic load factors are also discussed at the end of the study.

Frictional dynamic model predictions of FZG-A10 spur gear pairs considering profile errors

In this work, a nonlinear dynamic model of an FZG-A10 spur gear was investigated by taking into account for the actual time-varying gear mesh stiffness and the frictional effects between meshing gear teeth to evaluate the influence of the dynamic effects on frictional gear power loss predictions. The equations of motion of the generalized translational-torsional coupled dynamic system derived from Lagrange principle was extended compared to authors’ previous work in order to account for time dependent coefficient of friction and profile errors. The dynamic response of spur gears, computed by an iterative implicit scheme of Newmark, is changed due to the presence of coefficient of friction and profile errors. A dynamic analysis was performed and the influence of frictional effect including tooth shape deviations, in particular, was scrutinized since a time-dependent coefficient of friction is deeply related to the gear surface roughness and all parameters dependent on gears error profiles are introduced in the proposed model. The predicted meshing gear power losses with constant and local friction coefficient were compared. The influence of constant and variable profile errors considered in the local coefficient of friction formulation was also studied and their corresponding root mean square (RMS) power loss was compared to the experimental results. The results using FZG A10 spur gear pairs running under several operating conditions (different loads and speeds) validate the superiority of the proposed model against previous similar models.

Research on Time-Varying Mesh Stiffness of Double Involute Gears

The tooth of double involute gear (DIG) is considered as a cantilever beam on the tooth root circle, considering the real tooth profile of DIG, an improved energy method is used to solve the time-varying mesh stiffness. The influence of tooth waist order parameters and the difference of time-varying mesh stiffness between DIG and common involute gear(CIG) are studied. The results show that the time-varying mesh stiffness of the single tooth pair is less than that of CIG with the same parameters when the contact line passing through the tooth waist grading position, and the comprehensive time-varying mesh stiffness of DIG is less than that of CIG. The tooth waist order parameters will affect the time-varying mesh stiffness of DIG, and compared with the tooth waist altitude coefficient (l*), the tooth waist tangential modification coefficient (y*) has little effect on the time-varying mesh stiffness.