Gear Dynamic Model Research Articles

Dynamic features of planetary gear set with tooth plastic inclination deformation due to tooth root crack

Gear tooth root crack, as one of the popular gear tooth failures, is always caused by the dynamic load or excessive load applied to the tooth. It will devastate the working performance of the gear system, by problems such as vibration and noise, or even lead to a broken tooth, which will stop the normal working process of the gear system. It has attracted wide attention from researchers. However, the previous studies focused their concentration only on the mesh stiffness reduction due to tooth root crack, while the tooth plastic inclination due to tooth bending damages like gear tooth root crack is seldom considered. In this paper, a tooth plastic inclination model for spur gear with tooth root crack is developed by regarding the cracked tooth as a cantilever beam. It influences not only the displacement excitation but also the mesh stiffness and load-sharing factor among tooth pairs in mesh. The simulation results obtained by incorporating the tooth plastic inclination deformation model together with the tooth root crack model into a 21-Degree-of-Freedom planetary gear dynamic model indicate that the tooth plastic inclination has a significant effect on the performance of the gear system rather than the mesh stiffness reduction due to tooth root crack.

Three Kinds of Gear Transmission Nonlinear Dynamics Models

Based on the Lagranges equations, a new nonlinear dynamic gear model is established by introducing two variables of relative rotation angleand mean rotation angle. The motion equations derived with Lagranges equation exhibit nonlinear terms which are absent in the equations derived on Newtons equations. Combining with the numerical simulation, the dynamic responses in time domain and frequency domain are deduced, and it can be concluded that the responses at low speed of three different models are different. However, they are similar at the designed speed without the consideration of dissipation energy. On the contrary, the dynamic responses are similar at low speed and the simplified Newtons equation differs at the designed speed including dissipation energy.

Gear tooth pitting modelling and detection based on transmission error measurements

In this study, an experimental validation of a 3D gear dynamic model in the presence of localised faults such as pitting on tooth flanks is proposed. The corresponding numerical model accounts for spur and helical gear systems including gear errors and deviations along with the supporting shafts and bearings. Simulation results are compared with the evidence from a back-to-back test rig and the model validation relies on loaded transmission error (TE) measurements. Many numerical and experimental results on dynamic behaviours due to the presence of tooth pitting in geared systems are presented. Based on TE measurements, it is demonstrated that the actual vibrations generated by gear tooth pitting validate the gear model and its extension to consider such tooth surface failures.

Dynamic Features of a Planetary Gear System With Tooth Crack Under Different Sizes and Inclination Angles

Planetary gears are widely used in the industry due to their advantages of compactness, high power-to-weight ratios, high efficiency, and so on. However, planetary gears such as that in wind turbine transmissions always operate under dynamic conditions with internal and external load fluctuations, which accelerate the occurrence of gear failures, such as tooth crack, pitting, spalling, wear, scoring, scuffing, etc. As one of these failure modes, gear tooth crack at the tooth root due to tooth bending fatigue or excessive load is investigated; how it influences the dynamic features of planetary gear system is studied. The applied tooth root crack model can simulate the propagation process of the crack along tooth width and crack depth. With this approach, the mesh stiffness of gear pairs in mesh is obtained and incorporated into a planetary gear dynamic model to investigate the effects of the tooth root crack on the planetary gear dynamic responses. Tooth root cracks on the sun gear and on the planet gear are considered, respectively, with different crack sizes and inclination angles. Finally, analysis regarding the influence of tooth root crack on the dynamic responses of the planetary gear system is performed in time and frequency domains, respectively. Moreover, the differences in the dynamic features of the planetary gear between the cases that tooth root crack on the sun gear and on the planet gear are found.

Uncertainty Quantification in Gear Remaining Useful Life Prediction Through an Integrated Prognostics Method

Accurate health prognosis is critical for ensuring equipment reliability and reducing the overall life-cycle costs. The existing gear prognosis methods are primarily either model-based or data-driven. In this paper, an integrated prognostics method is developed for gear remaining life prediction, which utilizes both gear physical models and real-time condition monitoring data. The general prognosis framework for gears is proposed. The developed physical models include a gear finite element model for gear stress analysis, a gear dynamics model for dynamic load calculation, and a damage propagation model described using Paris' law. A gear mesh stiffness computation method is developed based on the gear system potential energy, which results in more realistic curved crack propagation paths. Material uncertainty and model uncertainty are considered to account for the differences among different specific units that affect the damage propagation path. A Bayesian method is used to fuse the collected condition monitoring data to update the distributions of the uncertainty factors for the current specific unit being monitored, and to achieve the updated remaining useful life prediction. An example is used to demonstrate the effectiveness of the proposed method.

Simulation of crack propagation in spur gear tooth for different gear parameter and its influence on mesh stiffness

Most of the gear dynamic model relies on the analytical measurement of time varying gear mesh stiffness in the presence of a tooth fault. The variation in gear mesh stiffness reflects the severity of tooth damage. This paper proposes a cumulative reduction index (CRI) which uses a variable crack intersection angle to study the effect of different gear parameters on total time varying mesh stiffness. A linear elastic fracture mechanics based two dimensional FRANC (FRacture ANalysis Code) finite element computer program is used to simulate the crack propagation in a single tooth of spur gear at root level. A total potential energy model and variable crack intersection angle approach is adopted to calculate the percentage change in total mesh stiffness using simulated straight line and predicted crack trajectory information. A low contact ratio spur gear pair has been simulated and the effect of crack path on mesh stiffness has been studied under different gear parameters like pressure angle, fillet radius and backup ratio. The percentage reduction of total mesh stiffness for the simulated straight line and predicted crack path is quantified by CRI. The CRI helps in comparing the percentage variation in mesh stiffness for consecutive crack. From the result obtained, it is observed that the proposed method is able to reflect the effect of different gear parameters with increased deterioration level on total gear mesh stiffness values.

Modeling of Multiteeth Contact Meshing Gear Dynamics with Backlash and Coincidence Degree

In the present paper, a multiteeth meshing gear contact dynamics model is proposed by introducing a modified robotic contact model. The inertial property, backlash of gear teeth and coincidence degree of gear meshing are considered into the model. In addition, the proposed model is used to simulate discontinuous meshing gear contact. Simultaneously, the gear meshing contact dynamical finite element model is also simulated using the Ansys/LS-DYNA software to demonstrate the rationality of the proposed model.

Parametric Analysis of Gear Mesh and Dynamic Response of Loaded Helical Beveloid Transmission With Small Shaft Angle

A synthesized gear mesh and dynamic model assuming line contact that is derived from a set of manufacturing parameters is formulated for analyzing the beveloid gear mesh-coupling mechanism. Using the proposed model, the effect of the dominant geometry design parameter that is the crossed angle between the first principal directions of the tooth surface curvatures (FPD-angle) on gear mesh characteristic and dynamic response is investigated. Also, the analysis of the gear mesh characteristic and dynamic response subject to torque load variation is performed. It is shown that the dynamic transmission error and dynamic mesh force worsen as the geometry FPD-angle increases for a specific torque load level. Furthermore, even though higher torque load can produce larger contact area, which is desirable, it also increases the gear mesh stiffness and transmission error that tend to aggravate dynamic response.

Enhanced friction model for high-speed right-angle gear dynamics

The modeling of elastohydrodynamic lubrication friction and the analysis of its dynamic effect on right-angle gears, such as hypoid and spiral bevel types are performed in the present study. Unlike the classically applied empirical constant coefficient of friction at the contacting tooth surfaces, the enhanced physics-based gear mesh friction model is both spatial and time-varying. The underlying formulation assumes mixed elastohydrodynamic lubrication (EHL) condition in which the division and load distribution between the full film and asperity contact zones are determined by the film thickness ratio and load sharing coefficient. In the proposed time-varying friction model, the calculation of friction coefficient is performed at each contact grid inside the instantaneous contact area that is being subjected to mineral oil lubrication. The effective friction coefficient and directional parameters synthesized from the net frictional and normal contact forces are then incorporated into a nonlinear time-varying right-angle gear dynamic model. Using this model, the effect of friction on the gear dynamic response due to the transmission error and mesh excitations is analyzed. Also, parametric studies are performed by varying torque, surface roughness and lubrication properties to understand the salient role of tooth sliding friction in gear dynamics. The simulation results are included. But experimental verification is needed.

Research on the Dynamic Response of a Combining Gear Drive System

Gear dynamics model and 16-degree of freedom nonlinear vibration equations of the combining gear drive system are established considering backlash, transmission error and time-varying mesh stiffness, the equations are solved thanks to the fourth-order Runge-Kutta method, and the gear dynamic responses are analyzed, the vibration performances of this system affected by the dimensionless vibration frequency are investigated. The results show when the dimensionless frequency is 0.687, the system appears harmonic response; when the dimensionless frequency is 0.868, the system appears quasi-periodic response; when the dimensionless frequency is 0.889, the system appears chaos.

A model for the simulation of the interactions between dynamic tooth loads and contact fatigue in spur gears

The main objective of the paper is to study the possible interactions between contact fatigue and dynamic tooth loads on gears. A specific 3D dynamic gear model is combined to contact fatigue models accounting for crack initiation and propagation. The numerical findings compare well with the experimental evidence from a back-to-back test rig. Three characteristic points on a tooth profile are analysed and it is shown that contact fatigue on spur gears clearly depends on dynamic phenomena. Finally, the introduction of profile relief is discussed and its positive influence on the risk of failures at engagement is emphasised.

A Frequency Domain Finite Element Approach for Three-Dimensional Gear Dynamics

A finite element formulation for the dynamic response of gear pairs is proposed. Following an established approach in lumped parameter gear dynamic models, the static solution is used as the excitation in a frequency domain solution of the finite element vibration model. The nonlinear finite element/contact mechanics formulation provides an accurate calculation of the static solution and average mesh stiffness that are used in the dynamic simulation. The frequency domain finite element calculation of dynamic response compares well with numerically integrated (time domain) finite element dynamic results and previously published experimental results. Simulation time with the proposed formulation is two orders of magnitude lower than numerically integrated dynamic results. This formulation admits system level dynamic gearbox response, which may include multiple gear meshes, flexible shafts, rolling element bearings, housing structures, and other deformable components.

An efficient finite element solution for gear dynamics

A finite element formulation for the dynamic response of gear pairs is proposed. Following an established approach in lumped parameter gear dynamic models, the static solution is used as the excitation in a frequency domain solution of the finite element vibration model. The nonlinear finite element/contact mechanics formulation provides accurate calculation of the static solution and average mesh stiffness that are used in the dynamic simulation. The frequency domain finite element calculation of dynamic response compares well with numerically integrated (time domain) finite element dynamic results and previously published experimental results. Simulation time with the proposed formulation is two orders of magnitude lower than numerically integrated dynamic results. This formulation admits system level dynamic gearbox response, which may include multiple gear meshes, flexible shafts, rolling element bearings, housing structures, and other deformable components.

Construction of Semianalytical Solutions to Spur Gear Dynamics Given Periodic Mesh Stiffness and Sliding Friction Functions

Gear dynamic models with time-varying mesh stiffness, viscous mesh damping, and sliding friction forces and moments lead to complex periodic differential equations. For example, the multiplicative effect generates higher mesh harmonics. In prior studies, time-domain integration and fast Fourier transform analysis have been utilized, but these methods are computationally sensitive. Therefore, semianalytical single- and multiterm harmonic balance methods are developed for an efficient construction of the frequency responses. First, an analytical single-degree-of-freedom, linear time-varying system model is developed for a spur gear pair in terms of the dynamic transmission error. Harmonic solutions are then derived and validated by comparing with numerical integration results. Next, harmonic solutions are extended to a six-degree-of-freedom system model for the prediction of (normal) mesh loads, friction forces, and pinion/gear displacements (in both line-of-action and off-line-of-action directions). Semianalytical predictions compare well with numerical simulations under nonresonant conditions and provide insights into the interaction between sliding friction and mesh stiffness.

Nonlinear dynamics of a two-stage gear system with mesh stiffness fluctuation, bearing flexibility and backlash

Abstract This work investigates dynamics of a two-stage gear system involving backlash and time-dependent mesh stiffness. This paper mainly consists to develop a 12 degree of freedom gear dynamic model. The model consists of a two-stage spur gear, three shafts, and two inertias representing load and prime mover and three bearings. Gear contact is characterised by a periodically changing stiffness and a backlash which can lead to loss of the contact. The nonlinear dynamic response of the system is studied thanks to a technique of linearization which decomposes the nonlinear system into some linear systems satisfying some conditions. Each system is solved thanks to the Newmark iterative algorithm. The results obtained appear to be the phenomenon of loss of teeth contact because of the discontinuity of the kinematic movement between the motives and receiving components of the system.

Optimized tooth profile based on identified gear dynamic model

A new method to determine the dynamic model of a practical gear system is proposed by the identification of the relation between the rotational movement of gears and the acoustical noise signal measured on the gear tester. The rotational movement alone the line of action of gears is treated as the input of the dynamic gear system and the acoustical noise signal as the output. The parameters of the model are estimated by the LSM (least square method). By an example that the gear pair works under a certain design load and rotational speed, the identified model is verified and the optimized gear tooth profile modification is derived based on the identified model. The tested results show that the profile modification can greatly reduce gear noise.

Dynamic modelling of spur gear pair and application of empirical mode decomposition-based statistical analysis for early detection of localized tooth defect

Gears are one of the most common and important machine components in many advanced machines. An improved understanding of vibration signal is required for the early detection of incipient gear failure to achieve high reliability. This paper mainly consists of two parts: in the first part, a 6-degree-of-freedom gear dynamic model including localized tooth defect has been developed. The model consists of a spur gear pair, two shafts, two inertias representing load and prime mover and bearings. The model incorporates the effects of time-varying mesh stiffness and damping, backlash, excitation due to gear errors and profile modifications. The second part consists of signal processing of simulated and experimental signals. Empirical mode decomposition (EMD) is a method of breaking down a signal without leaving a time domain. The process is useful for analysing non-stationary and nonlinear signals. EMD decomposes a signal into some individual, nearly monocomponent signals, named as intrinsic mode function (IMF). Crest factor and kurtosis have been calculated of these IMFs. EMD pre-processed kurtosis and crest factor give early detection of pitting as compared to raw signal.

Comparison of localised spalling and crack damage from dynamic modelling of spur gear vibrations

This paper presents a 26 degree of freedom gear dynamic model of three shafts and two pairs of spur gears in mesh for comparison of localised tooth spalling and damage. This paper details how tooth spalling and cracks can be included in the model by using the combined torsional mesh stiffness of the gears. The FEA models developed for calculation of the torsional stiffness and tooth load sharing ratio of the gears in mesh with the spalling and crack damage are also described. The dynamic simulation results of vibration from the gearbox were obtained by using Matlab and Simulink models, which were developed from the equations of motion. The simulation results were found to be consistent with results from previously published mathematical analysis and experimental investigations. The difference and comparison between the vibration signals with the tooth crack and spalling damage are discussed by investigating some of the common diagnostic functions and changes to the frequency spectra results. The result of this paper indicates that the amplitude and phase modulation of the coherent time synchronous vibration signal average can be effective in indicating the difference between localised tooth spalling and crack damage.

Gear fatigue crack prognosis using embedded model, gear dynamic model and fracture mechanics

This paper presents a model-based method that predicts remaining useful life of a gear with a fatigue crack. The method consists of an embedded model to identify gear meshing stiffness from measured gear torsional vibration, an inverse method to estimate crack size from the estimated meshing stiffness; a gear dynamic model to simulate gear meshing dynamics and determine the dynamic load on the cracked tooth; and a fast crack propagation model to forecast the remaining useful life based on the estimated crack size and dynamic load. The fast crack propagation model was established to avoid repeated calculations of FEM and facilitate field deployment of the proposed method. Experimental studies were conducted to validate and demonstrate the feasibility of the proposed method for prognosis of a cracked gear.

Spur Gear Dynamic Models Including Defects: A Review

A large number of gear models for simulating the dynamic behavior of gears have appeared in the literature since 1920. In this paper we present a review of the dynamic modeling of spur gears including defects. We give the classification of gear dynamic models by different authors, and we explain various terms that are used in gear dynamic modeling. Gear dynamic models without any defects for single-, two-, and multi-degree-of-freedom systems have been discussed. The effects of various nonlinearities, such as variable mesh stiffness, mesh damping, gear errors, and backlash, on gear dynamic behavior have been discussed. Many researchers are actively developing advanced dynamic models of gear case vibration to ascertain the effect of different types of gear damage. Dynamic models including the effect of friction, wear and spalls on gear dynamic behavior have been considered in detail. The aim of this paper is to emphasize the importance of defect models for a clear understanding of gear vibrations and also to explore areas for introducing other defects in existing gear dynamic models.