A Calculation Method for Tooth Wear Depth Based on the Finite Element Method That Considers the Dynamic Mesh Force

The determination of meshing force and the load sharing ratio of gear teeth is critical to predict the dynamic behavior or the load capacity of gear transmissions. In the previous literature, the dynamic meshing force is usually calculated based on the traditional dynamic model, which ignores the different effects of the meshing characteristics of each pair of teeth on the dynamic behavior of the gear system. In this work, a new calculation method of dynamic meshing force is proposed based on the new dynamic model considering the meshing state of multiple pairs of teeth. The difference between the traditional calculation method and the new calculation method of dynamic meshing force is analyzed. Based on the new dynamic model and new calculation method of dynamic meshing force, the influence of different factors on dynamic response and dynamic meshing force are further discussed. The results show that, compared with the traditional calculation method, this new method can be used to effectively calculate the dynamic meshing force and the load sharing ratio of each pair of teeth with different meshing characteristics. The presented method for the calculation of the dynamic meshing force and the load sharing ratio provides an important reference for analyzing and predicting the dynamic behavior or the load capacity of spur gears, especially the high contact ratio (HCR) gears with contact ratio more than two.

Experimental investigation of the ring-planet gear meshing forces identification

In this study, the interactions between backlash, bearing clearance and bearing flexibility are studied in geared flexible rotors. For this purpose, the non-linear mathematical model developed in a recent previous study for geared rotors on flexible bearings is extended further to handle clearance nonlinearity in bearings. The model consists of elastic shafts on elastic bearings with clearance, and coupled by a non-linear gear mesh interface. Shafts are modeled by using finite elements. The mathematical model includes the axial loading on shafts, rotary inertia, material damping of shafts, backlash nonlinearity of gear mesh and flexible bearings with clearances nonlinearity. The model considers also the effects of gear errors and profile modifications. The excitation effect of time varying mesh stiffness is indirectly included into the analysis through a periodic displacement representing loaded static transmission error (STE). STE input is modeled by the highest n harmonic terms of the Fourier series representation of the measured or calculated data. The model developed is believed to be one of the most advanced ones: It includes several linear and non-linear effects and capable of handling various different gear-rotor-bearing configurations. Through several case studies, the combined effects of backlash, bearing clearance, and bearing flexibility on dynamic mesh force and bearing loads are studied.

Face gear dynamics is addressed by many scholars. However, dynamic behaviors of face gear drives associated with pinion dedendum fatigue cracks are yet to be investigated. Thus, in the study, a calculation solution of static transmission errors (STE) of face gear drives associated with pinion dedendum fatigue cracks is constructed, based on the proposed equivalent face gear drives, and a four-degree-of-freedom (DOF) dynamic model of face gear drives is formulated. The dynamic behaviors of an example case of face gear drives associated with pinion dedendum fatigue cracks are investigated. The simulation results indicate natural frequencies of face gear drives would be decreased with pinion dedendum fatigue crack extensions; the relationship between accelerations and velocities of face gear drives, dynamic mesh forces, and vibration accelerations at bearings would also be changed. However, pinion dedendum fatigue cracks of face gear drives are detected difficultly, due to simulation result similarities. Therefore, a determination solution for pinion dedendum fatigue cracks of face gear drives is proposed. The result shows a trend of vibration accelerations at bearings, namely, a first decreasing and then increasing tendency, which could be beneficial to detecting pinion dedendum fatigue cracks of face gear drives. These contributions would improve engineering applications of face gear drives in the future.

NON-LINEAR DYNAMIC RESPONSE OF A SPUR GEAR PAIR: MODELLING AND EXPERIMENTAL COMPARISONS

Since gears transmit motion by meshing teeth, it is hard to avoid problems which wear and damage to the tooth surface. Because Tooth surface wear is difficult to predict, the approach to the wear problem still often relies on empirical rules and symptomatic measures. If the undermining by wear can be predicted by mathematical formulas, it may allow engineers to solve wear issues in model-based development, or to consider wear resistance from the design state. Therefore, the development of wear prediction formulas seemed like a very valuable goal, so the authors decided to work on it. In this paper, as a step for the final goal, the gear driving test was conducted using multiple gears with different tooth surface roughness. Then to detect changes in tooth profile, the tooth profile error was measured, and wear depth was calculated. Moreover, surface roughness on teeth was measured, and the relationship between surface roughness and wear depth was investigated.

This work investigates dynamics of a three-stage gear system in a new power-driven turret. This paper mainly consists to develop two kinds of dynamic models of the main drive train. One is the dynamic contact model of the gear system to calculate the dynamic meshing forces and the tooth root stresses. The other is a virtual prototype model adequately incorporating the deformability of the bearings to analyze the dynamical response. The dynamic meshing force and the tooth root stress were compared with the theoretically calculated results. The vibration acceleration signals of the turret were gathered by tests. According to the experimental data, the simulation results of the virtual prototype model were given based on which the rationality of the virtual prototype model is verified.

Vibration-based updating of wear prediction for spur gears

Time-varying mesh stiffness is one of the main excitation sources of a gear system, and it is also considered as an important factor for the vibration and noise of gears. Thus, this excitation is usually taken as an input into the gear dynamic model to obtain the system dynamic responses. However, the mesh stiffness of a gear pair is actually nonlinear with respect to the dynamic mesh force (DMF) that fluctuates during the operation of gears. Therefore, the dynamic model of gears with the quasi-static mesh stiffness calculated under a constant load is not accurate sufficiently. In this paper, a dynamic model of spur gear is established with considering the effect of the force-dependent time-varying mesh stiffness, backlash and profile deviation. Due to the nonlinear relationship between the mesh stiffness and the load for each tooth pair, it needs first to determine the load sharing among tooth pairs and then calculate the overall mesh stiffness of the gear pair. As the mesh stiffness and DMF are related, the mesh stiffness is no longer directly taken into the gear dynamic model as an input, but is jointly solved with the numerical integration process using the gear dynamic model. Finally, the dynamic responses predicted from the established gear dynamic model are compared with the experimental results for validation and compared with the traditional models to reveal their differences. The results indicate that the established dynamic model of spur gear transmission has a wider application range than the traditional models.

In order to accurately describe the complex transmission systems and clarify the effects of internal and external excitation on the vibration characteristics for large complex transmission systems, a dynamics model considering gear meshing error and time-varying meshing stiffness is established at two-stage planetary gears systems. The coupling model is established by matrix set method for two-stage planetary gears and multistage spur gears transmission systems. The random load excitation verified by experiments is the external excitation in the system, and the excitation caused by the time-varying meshing stiffness and meshing error is the internal excitation. The research on the response of the vibration systems shows that the internal excitation caused by gear meshing stiffness and gear meshing error has important influence on the dynamic load behavior of spur gears transmission in complex systems. For the planetary system with heavy external load, the dynamic behavior mainly depends on the external load, and the influence of internal excitation on the dynamic meshing force is less than that of the external excitation for the planetary system. The external excitation mainly affects the low-frequency region of the dynamic meshing force for the system, and the internal excitation affects the high-frequency region of the dynamic force for the system. The frequency influence interval of the internal excitation is much larger than that of the external excitation.

For proactively arranged downtime and maintenance of the gearbox, the accurate assessment of residual useful life (RUL) for worn-surface gear is a critical problem that needs to be solved. In light of this, a novel dynamic assessment methodology for RUL of sun gear was proposed by coupling Archard’s wear model and a nonlinear dynamics model. By modifying the internal excitation model, the effect of tooth surface wear is included in the dynamics model. Then, the effect of dynamic behavior on the wear process of the tooth surface is further fed back by the dynamic meshing force. On this basis, the influence of input speed, torque load and initial wear depth on the RUL was analyzed using a planetary gear NGW11-6 as an example. The results indicate that the torque load has the greatest influence on the RUL, followed by the initial wear depth, and that the input speed has the least influence. The established methodology will provide a theoretical basis for the RUL assessment and scheduled maintenance of transmission systems with worn-surface gears.

Modeling and analysis of a spur gear pair considering multi-state mesh with time-varying parameters and backlash

To study the coupling influences of time-varying external load and gear wear on dynamics of spur gear systems, a lateral-rotational-coupled dynamic model is developed. The influence of gear wear is incorporated into the dynamical model through the Fourier formulations of meshing stiffness and static transmission error. An example gear system is used to predict dynamic characteristics with and without surface wear under constant, as well as time-varying external load conditions. The influences of wear and that of time-varying load on dynamics of the example system in time domain and frequency domain are studied at different speeds. Finally, the quantitative relationships between the dynamic factor and operating parameters as well as wear depth are revealed.

Gear tooth spalling is a typical tooth surface defects in gear transmission. The accurate calculation of the time-varying mesh stiffness of the gear pair with spalling defect under mixed elastohydrodynamic lubrication (EHL) plays an important role in tooth damage prediction and fault feature analysis. In this work, an improved time-varying mesh stiffness (TVMS) model of spalling faulty gear in mixed EHL line contact considering the coupled effect of rough surface topography is established. The proposed model considers the combined effect of spalling defect, surface roughness topography, and lubrication on the contact characteristics of the meshing interface to obtain a revised contact stiffness, which is further used to determine the comprehensive TVMS. The cylindrical contact coefficient for curved meshing interface with spalling fault is derived and incorporated into the statistical micro-contact Greenwood and Williamson model (GW model) to obtain the asperity contact stiffness. The revised contact stiffness at the rough faulty gear interface is obtained from the parallel action of the asperity contact stiffness and oil film stiffness under load-sharing concept. The dynamic responses of the gear system are analyzed using the established mesh stiffness model through the six degrees of freedom translational-torsional dynamic model, in which the dynamic meshing force is calculated iteratively using the transit surface topography and geometrical parameters along the line of contact instead of a quasi-static meshing force. Effects of multi-parameters as surface roughness, spalling fault size, and lubricant on the mesh characteristics and dynamic responses of the gear system are further investigated and discussed.

In response to the new requirements for face gear transmissions in the latest heavy-load helicopters, this paper investigates the impact of pressure angle variations on the geometric contact and performance of asymmetric face gears. Firstly, using meshing theory, the tooth surface equations for asymmetric rack cutter, shaper, and face gear are derived, and the parameterized 3D models are created. Secondly, the line of action (LOA), contact ratio, contact point curvature, and load sharing ratio considering multi-pair contact (MPC) under different pressure angles are analyzed. Then, a wear calculation model based on the Archard wear equation is developed, detailing the tooth surface wear depth calculation. Finally, a case study calculates the contact stress and wear depth under different pressure angles is presented. The results indicate that appropriately increasing the pressure angle effectively lower the contact stress and wear depth on tooth surface, although further increases degrade meshing performance. The gear pair with a pressure angle of 30° in the case study exhibits optimal performance. This study provides a reference for optimizing asymmetric face gear meshing performance under various pressure angles.