Wear modelling of bevel gears with biconvex-concave teeth

The article presents theoretical and practical approaches to increasing the service life of bevel gears with biconvex-concave (BCC) teeth by selecting optimal meshing parameters based on wear modeling. Theoretical studies involve the development of mathematical models: 1. Meshing of BCC teeth in bevel gears, which contains a description of the lateral three-dimensional surfaces of the teeth and enables the determination of geometric (radii of curvature of profiles), kinematic (rolling and sliding velocities) and strength (normal meshing force, contact stresses, and bending stresses) parameters for further wear; 2. Wear of BCC teeth in bevel gears, which takes into account the wear in the pole zone of engagement and the reduction in surface hardness due to the degradation of the cemented layer. This model ensures sufficient accuracy in calculating tooth surface wear (with a relative error not exceeding 10%).Analytical dependencies between wear and profile shift coefficients, modulus, number of teeth, width of the gear ring were established using mathematical models of meshing and wear. This allowed to substantiate rational meshing parameters (according to the criterion of wear resistance of the teeth) to ensure increased durability of the gears. A method for the experimental determination of wear in the BCC teeth of bevel gears has been developed. It is based on the laser scanning method and allows the coordinates of points on the lateral three-dimensional surface of the teeth to be determined with a maximum absolute error of 0.02 mm. The results of the experimental wear determination, conducted under production conditions, confirmed the accuracy and adequacy of the developed theoretical principles. A method for predicting the durability of bevel gears with BCC teeth at the design stage has been developed. It takes into account changes in their geometric, kinematic, and strength parameters resulting from the alteration of the side surface profile of the teeth due to wear after each load cycle. A program has been developed to select rational engagement parameters for bevel gears with BCC teeth. It is based on the durability-prediction method. Using this program rational parameters for bevel gears with biconvex-concave teeth have been selected, increasing their service life by a factor of 1.93.

In the present paper, the author has derived a method for calculating the stresses in herical gear teeth by applying the solution for the moments induced in a cantilever plate with infinite length subjected to a concentrated load. That is, the helical gear tooth was regarded as a cantilever plate with finite length and the distributed load on the contact line of tooth was considered as a group of concentrated loads per unit length. And then, from the further investigation for this calculating method, a simplified formula for the calculation of stresses in helical gear teeth has been developed. The stresses in gear teeth calculated by these methods are shown to be in good agreement with those measured by wire resistance strain gauges cemented in the actual gear teeth.

A new, innovative procedure called point load superposition for determining the contact stresses in mating gear teeth is presented. It is believed that this procedure will greatly extend both the range of applicability and the accuracy of gear contact stress analysis. Point load superposition is based upon fundamental solutions from the theory of elasticity. It is an iterative numerical procedure which has distinct advantages over the classical Hertz method, the finite element method, and over existing applications with the boundary element method. Specifically, friction and sliding effects, which are either excluded from or difficult to study with the classical methods, are routinely handled with the new procedure. Presented here are the basic theory and the algorithms. Several examples are given. Results are consistent with those of the classical theories. Applications to spur gears are discussed.

Dudley's Handbook of Practical Gear Design and Manufacture

Load-capacity has always been one of the performances that is paid much attention to in the development of bevel gear transmission applications. Consequently, the mathematical model of novel bevel gear with high load-capacity based on geometric elements is proposed in this paper, which could be applied to the aviation, aerospace and other fields. In parallel, the design principle and design method of the novel bevel gear are introduced in detail. Subsequently, the conditions for tooth surface continuity and non-interference are derived. Furthermore, the model of novel bevel gear is established. Finally, the load-bearing characteristics are analyzed, revealing that an increase in the number of contact points could significantly enhance the load capacity of the bevel gear pairs. When the load torque applied to bevel gear II is 100 Nm, the contact pressure endured by the bevel gear pair with five-point contact is decreased by 41.37% compared to the bevel gear pair with single-point contact. When the number of contact points is the same, increasing the distance between the contact points could also reduce the contact stress. This provides strong theoretical support for the application of the bevel gear based on the geometric elements.

An adhesive wear model for helical gears in line-contact mixed elastohydrodynamic lubrication

A procedure for calculating transmitted load distribution along face width as well as tooth stresses of straight bevel gears is introduced. The procedure is based upon the Tredgold assumption, which assumes that a straight bevel gear, when projected on a plane tangent to the back cone, can be approximated as a spur gear having a pitch radius equal to the back-cone radius and same pitch as the bevel gear. To increase the solution accuracy, the bevel gear is divided into a number of spur gears by a finite number of slicing planes that are parallel to the plane of projection. Each slice is then analyzed as a separate spur gear, and tooth stiffness, load, and stresses are determined separately. As a result, the load and stress distribution for the actual bevel gear are obtained. The procedure assumes that the sum of tooth deflection, profile modification, and manufacturing errors at the pairs of contacting slices of the pinion and gear are all equal, in order to avoid overlap and tooth interference. It is also assumed that the sum of the normal loads contributed by each pair of contacting slices is equal to the total normal load on the entire bevel gear, which is obtainable from the transmitted Power/torque. Once the normal loads for all slices representing the bevel gear are determined, fillet and Hertzian stresses are calculated from the applied loads and slice geometry. Consequently, the distribution of such stresses for the actual bevel gear are also calculated. An example is presented to explain the criterion. Experimental substantiation, using strain gauge measurements, is also presented to demonstrate the validity of the criterion.

Two kinds of stresses in the gear teeth are root bending stress and tooth contact stress. These two stresses results in the failure of gear teeth. The root bending stress results in fatigue failure and contact stress results in pitting failure at the contact surface. The stress analysis used to minimize gear failure in the design of helical gear. It is also optimize the design of helical gear on the transmission system of the truck motor vehicle, where the power transmission is required at heavy loads with smoother and noiseless operation. In this paper bending stress and contact stress estimated using analytical method while modeling of gears used the numerical solution. Method of beam strength based on modified Lewis calculation used to predict the bending strength of helical gears. Contact stress was estimated using related method of AGMA contact stress. Stress modeling of helical gears is done by ANSYS 14.5, which is a finite element analysis package. The results are then compared with both AGMA and ANSYS procedures. The values of bending strength and contact stress determined using AGMA method found to be compatible with ANSYS simulation.

ABSTRACTA thermal elastohydrodynamic lubrication model is combined with a wear model under mixed lubrication to investigate the lubrication performance and wear characteristics of double‐circular‐arc spiral bevel gears for nutation drive. Moreover, the effects of operating conditions on the characteristic parameters of the film are analysed under the mixed lubrication point‐contact conditions. Furthermore, the characteristics of gears in terms of friction coefficient and wear depth are discussed. According to the results, the performance of lubrication and wear during the mutual meshing of the convex tooth surface of the external bevel gear and the concave tooth surface of the inner bevel gear is better than that during the mutual meshing of the other pair of tooth surfaces. The minimum film thickness of the whole meshing process occurs near the inner of the bevel gear due to the joint action of the load and the end edge effect. Moreover, an increase in torque at a certain rotational speed is favourable to the lubrication performance of the meshing process. The wear depth in the double‐circular‐arc spiral bevel gears' meshing process is heavily influenced by the roughness of the tooth surface.

The calculation of tooth wear under mixed elastohydrodynamic lubrication is very complex and requires consideration of many conditions such as load distribution in the tooth meshing zone, micro-convex elastoplastic deformation and tooth surface temperature. The accurate calculation of tooth wear requires a lot of time and effort. In order to calculate tooth face wear under mixed elastomeric flow lubrication quickly and accurately, a new proxy model of tooth face wear is developed using the Kriging method. The pressure distribution required for the wear calculation was obtained utilizing the modified Reynolds equation and ZMC elasto-plastic model. The numerical calculation model of gear wear was derived using the modified Archard wear model. The Kriging model was used to construct a proxy model between gear parameters and tooth wear, and the degree of approximation and goodness of fit of the Kriging model were investigated. The results are as follows. The wear depth at each position is different, the smallest at the pitch, the largest near the tooth root, and the pinion has a larger wear depth than the gear. The Kriging model is highly efficient and accurate in its computation and overcomes the shortage of excessive time spent on the calculation of numerical calculation models.

Cheek teeth of some mammalian herbivores exhibit pronounced changes in occlusal size and shape through wear, purportedly caused by strong curvature. Such changes are extreme in the upper cheek teeth of extinct, dentally archaic lagomorphs. Morphologic and taxonomic turnover in lagomorphs suggests that these dentally archaic forms may have been unable to develop hypselodont (ever-growing) cheek teeth. This study investigates how the interaction of tooth shape and wear can cause occlusal size and shape changes, and potentially impose structural constraints on crown height. These constraints may help explain extinction of mammals with teeth like archaic lagomorphs, evolution and diversification of other mammalian herbivores during the late Miocene, and the relative paucity of hypsodont cheek tooth shapes in extant mammals.I first quantify two-dimensional curvature accounting for shape differences observed in hypsodont teeth, P4s of the archaic lagomorphs Russellagus and Hesperolagomys, which exhibit pronounced change with wear, and Ondatra lower incisors, which show minimal change with wear. Using this quantification, I generate theoretical curvature morphologies and describe a geometric model of tooth wear that generates values for qualitative and quantitative aspects of the occlusal surface at different wear stages. Modeled results of wear surface topography and dimensions closely correspond to observed patterns in Russellagus, Hesperolagomys, and Ondatra. Model results on wear in theoretical tooth morphologies identify two major shape factors influencing wear: orientation of the wear surface (incisor-like or cheek-tooth-like), and tooth curvature (“concentric” or “nonconcentric”). Modeled wear also suggests two geometric constraints on crown height. Teeth with nonconcentric curvatures can have crown height limited by potential tooth area. “Incomplete wear” in any tooth can present severe constraints on increasing crown height, causing structurally untenable morphologies in very tall-crowned to hypselodont teeth.

This paper presents comparable sets of the no-load power loss as a product of windage and churning behaviors of a family of various rotating parts (i.e., disc, spur gear, straight bevel gear, and orthogonal face gear). Experimental measurements were carried out under pure air only and under partial immersion in oil to qualify and quantify the windage and churning effects of no-load power losses of a family of spur, bevel, and face gears along with a representative disc as the baseline. Aiming at exploring the influence of gear teeth on the total no-load power losses, two different theoretical analytical approaches are introduced to account for the churning contributions, by which the total power losses are estimated. Both analytical approaches compare well with the experimental findings. Furthermore, a spatial intersecting cross-axis gear (e.g., straight bevel gear and orthogonal face gear) results in higher no-load power losses than that of a representative disc or a parallel-axes gear. The significance of gear teeth (gear vs. disc) on windage behavior is presented, as well as the gear windage effects on the churning phenomenon in a high-speed splash-lubricated gear.

The numerical prediction of spiral bevel gear contact dynamics is often hampered by the complexity of the gear tooth geometry and the three-dimensional (3D) nature of the motion transfer, which forms a major challenge for efficient contact detection and accurate contact force calculation. This works proposes a novel spiral bevel gear contact force element that enables the accurate prediction of 3D bevel gear contact force distributions in flexible multibody system dynamics simulations (MBDS). An efficient contact detection strategy, based on the surface of action, is used, while the accurate contact force calculation is obtained by inclusion of a new semi-analytical bevel gear contact stiffness model. This stiffness model relies on model order reduction (MOR) of finite element method (FEM)-based models to describe the bevel gear tooth and blank deformation, and analytical modeling to predict the tooth contact deformation. The solution can be used to model and analyze lightweight bevel gear dynamics. The method is validated against high-fidelity nonlinear finite element analysis-based contact simulations to evaluate the predicted transmission error, contact pattern, and contact pressure distribution. Finally, a use case is presented to demonstrate how the methodology enables detailed insights in spiral bevel gearbox dynamics, and noise and vibration (NVH) performance.

Spur gears with asymmetric teeth have a significant potential for some applications requiring extreme performance like in the aerospace industry. In this study, the influence of tooth wear on the dynamic behavior of involute spur gears with asymmetric teeth is analyzed. The Archard's wear model was adopted in formulating and accounting for wear. Effects of gear parameters such as gear contact ratio, tooth height, mesh stiffness, and pressure angles on tooth wear are considered. These parameters are used to describe the relationship between dynamic tooth load and tooth wear. A comparison of symmetric and asymmetric teeth is also presented with respect to tooth wear. Sample simulation results, which were obtained by using an in-house developed computer program, are illustrated with numerical examples. The numerical results match well with the practical and analytical results which are available in literature. For asymmetric teeth, it was shown that the wear depth decreased with increasing pressure angle on drive side.

<div class="htmlview paragraph">The design of any gearing system is a difficult, multifaceted process. When the system includes bevel gearing, the process is further complicated by the complex nature of the bevel gears themselves. In most cases, the design is based on an evaluation of the ratio required for the gear set, the overall envelope geometry, and the calculation of bending and contact stresses for the gear set to determine its load capacity. There are, however, a great many other parameters which must be addressed if the resultant gear system is to be truly optimum.</div> <div class="htmlview paragraph">A considerable body of data related to the optimal design of bevel gears has been developed by the aerospace gear design community in general and by the helicopter community in particular. This paper provides a summary of just a few design guidelines based on these data in an effort to provide some guidance in the design of bevel gearing so that maximum capacity may be obtained. The following factors, which may not normally be considered in the usual design practice, are presented and discussed in outline form: <ul class="list disc"> <li class="list-item"><div class="htmlview paragraph">Integrated gear/shaft/bearing systems</div></li> <li class="list-item"><div class="htmlview paragraph">Effects of rim thickness on gear tooth stresses</div></li> <li class="list-item"><div class="htmlview paragraph">Resonant response.</div></li> </ul></div>