FE-Analytical Contact Modeling for the 3D Simulation of Bevel Gear Dynamics

High-performance tooth flank collaborative optimization model for spiral bevel and hypoid gears

In this chapter, a general survey is first done on the tooth bending strength of bevel gears in the broadest meaning, including straight, helical (or skew), Zerol, spiral bevel and hypoid gears and the main factors affecting the tooth root stress state are defined. Attention is paid to the fatigue breakage at the tooth root of these gears and safety factor to be used for their design. The theoretical bases of calculation of tooth bending strength, already discussed in previous chapters, are then recalled, particular those concerning the Lewis constant strength parabola, 30° tangent and cantilever rectangular plate with constant thickness subject to bending load, with which the gear tooth is simulated. The methods of adaption of these theoretical bases to bevel gears are then described. A large part of the chapter is reserved for the procedures to calculate the tooth bending strength of these types of gears in accordance with the ISO standards, highlighting when deemed necessary as the relationships used by the same ISO are founded on the theoretical bases previously recalled.

<div class="htmlview paragraph">There has always been a great deal of misconception about the development of hypoid and spiral bevel contact patterns. Certain design criteria, no load and loaded contact pattern checks, test machine comparisons, and simple deflection tests can be performed quickly to give an idea if the initial contact pattern development is correct. Development time can be shortened and the correct contact pattern can be determined before the long term field and dynamometer testing starts.</div>

PurposeThis paper aims to present a comparison of numerical methods for determining the contact pattern of Gleason-type bevel gears. The mathematical model of tooth contact analysis and the finite element method were taken into consideration. Conclusions have been drawn regarding the usefulness of the considered methods and the compatibility of results. The object of the analysis was a bevel gear characterised by an 18:43 gear ratio and arc tooth line, and manufactured according to the spiral generated modified-roll method.Design/methodology/approachThe mathematical model of tooth contact analysis consists of both the mathematical model of tooth generating and the mathematical model of operating gear set. The first model is used to generate tooth flanks of the pinion and the ring gear in the form of grids of points. Then, such tooth surfaces are used for the tooth contact analysis performed with the other model. It corresponds to the no-load gear meshing condition. The finite element method model was built on the basis of the same tooth flanks obtained with the former model. The commercial finite element method software Abaqus was used to perform two instances of the contact analysis: a very light load, corresponding to the former no-load condition, and the operating load condition. The results obtained using the two models, in the form of the contact pattern for no-load condition, were compared. The effect of heavy load on contact pattern position, shape and size was shown and discussed.FindingsThe mathematical models correctly reproduce the shape, position and size of the contact pattern; thus, they can be reliably used to assess the quality of the bevel gear at the early stage of its design.Practical implicationsDetermination of the correct geometry of the flank surfaces of the gear and pinion teeth through the observation of contact pattern is a fundamental step in designing of a new aircraft bevel gear.Originality/valueA possibility of the independent use of the mathematical analysis of the contact pattern has been shown, which, thanks to the compatibility of the results, does not have to be verified experimentally.

<div class="htmlview paragraph">Current and future motor vehicle development demands improvement in the areas of economy, environmental acceptability and noise reduction. The performance and manufacture of bevel and hypoid gear sets will also be subject to the same improvement requirements.</div> <div class="htmlview paragraph">The above issues will be discussed from a viewpoint of an industrial mass production environment and highlight new developments in gear manufacturing with special emphasis on finish grinding of bevel gear sets. Additionally, new procedures and methods, permitting objective gear quality measurement and noise performance, as well as direct correlation to vehicle performance will be taken under consideration.</div> <div class="htmlview paragraph">The integration of manufacturing and test procedures in a closed loop production process will show the best approach to fulfill future OEM demands.</div>

A dynamic stiff string torque and drag (T&D) model is presented that assumes steady state motion of the drillstring as its basis for calculations. Results are compared to previously published T&D models that are based on static equilibrium. The novelty of the new dynamic model is the ability to solve T&D operations of the entire drillstring from bit to top drive in reasonable time using standard engineering computers. The new approach is based on a 3D dynamic model of drillstring and BHA in an elastic borehole. It considers bending stiffness, torsional stiffness, contact forces, and friction with localization of contact points. A numerical method is described that has proven to have excellent convergence. Complete governing equations are provided and the method is described in detail to permit readers to replicate results. The dynamic model is compared to two static stiff string models. Comparisons are also provided for three conventional soft string models including the Lubinski-Paslay-Cernocky bending stress magnification factor. Three field case studies are presented for horizontal wells. One well is short radius with dogleg severity over 50 deg/100 ft and two wells are unconventional shale wells with doglegs up to 15 deg/100 ft. Predictions for surface torque and drag up and down for the new dynamic stiff string model are compared to the static stiff and soft string models. In many situations modeled the top-level results for surface torque and drag up/down are close enough for all six models to be within the uncertainty range associated with the commonly used, lumped-parameter friction factor. However, some major differences in hook load for sliding and slack off operations are observed, which are shown to be caused by differences in location and magnitude of contact force between the drillstring and wellbore. Further, significantly lower surface torque is predicted by the new dynamic stiff string compared to other models for one case history because of lower contact forces in the vertical section of the well. In fact, the key finding of this paper is that major differences are observed for contact forces for the new dynamic stiff string model compared to all five other models, including the two static stiff string models. These differences in contact forces are most significant when the drillstring has helically buckled or when doglegs in the wellbore are high. Contact forces have a large impact on local stress behavior, which is important for predictions of casing and drillpipe wear, drillstring fatigue, and failure points in the drillstring. Although several previous papers have published stiff string models there is no industry standard formulation. The main problem holding back the development of an industry standard stiff model is perhaps the complexity of the numerical algorithm and substantial running time. To address this problem, some previous stiff string models account for bending stiffness of the drillstring but not for radial clearance while others appear to model only portions of the drillstring as stiff. The new stiff string model accounts for bending stiffness and radial clearance for the entire drillstring while still giving reasonable computational times. For stiff string models the advantage of using a dynamic approach to solve the steady state position of the drillstring is mainly related to superior convergence of the numerical algorithm compared to static stiff string models because calculation of contact points is faster.

The dynamics of spiral bevel gears have gained increasing importance due to concerns relating to noise and durability. This is because the mesh force acting on the gear teeth is amplified under dynamic conditions, potentially reducing the fatigue life of the gears. Furthermore, a sizable dynamic force can be transmitted to the housing, inducing structure-born gear whine. The elasticity of the bearings can influence the dynamics of spiral bevel gears. In this article, the finite element formulation of a spiral bevel geared rotor dynamic system is applied to investigate the influence of bearing elasticity on the dynamics of spiral bevel gears. The designs and configurations of rear axles are modeled and analyzed for real-world applications, to gain an enhanced practical understanding of the effect of bearing stiffness on spiral bevel gear dynamics.

To address the issue of increased gear noise in electric vehicle drivetrains due to higher rotational speeds, a differential tooth surface modification method for spiral bevel and hypoid gears is proposed. A mathematical model for spiral bevel and hypoid gears is established using the vector method. Based on this model, a finite element dynamic model of the gears is developed through secondary development using Adams software. A tooth surface modification approach involving parameters (bias factor and profile separation factor) varying according to a sinusoidal function is investigated, alongside its impact on micro-topography of the tooth surface. A comparative simulation analysis is performed to evaluate the sinusoidal parameter design method against traditional modification methods, emphasizing gear angular acceleration and meshing force under various operating conditions. The results demonstrate that the differential modification method achieves a significant reduction in the first three orders of meshing frequencies under almost all conditions, with maximum reductions in the first-order frequency amplitudes of the gear angular acceleration and meshing force reaching 22.98 % and 36.05 %, respectively. This confirms the method effectiveness in reducing gear vibration and noise. The proposed differential modification method for spiral bevel and hypoid gears offers a novel approach for vibration and noise mitigation, offering valuable technical support for designing and manufacturing of high-performance electric vehicles.

Loaded contact pressure distribution prediction for spiral bevel gear

The representation and manipulation of finite rotations and the modelling of flexible bodies are at the heart of flexible multi-body systems dynamics simulations. When dealing with finite element formulations, the interpolation of finite-rotation fields becomes a critical issue. Objectivity, i.e. the invariance of interpolated strain fields to the addition of a rigid body motion, has been an important problem. Furthermore, the manipulation of finite rotations and the unavoidable associated singularities must be dealt in a rational manner. The various approaches to these two problems are reviewed in this article. Simple algorithms that achieve objectivity are proposed. Furthermore, these algorithms can deal with finite rotations of arbitrary large magnitude through a simple rescaling operation. The choice of unknowns, total versus incremental, is also discussed as it affects the performance of the computational process. Numerical examples are presented, which demonstrate the ability of the proposed approach to handle finite rotations of arbitrary magnitudes in complex, flexible multi-body systems.

The Gleason face hobbing process has been widely applied by the gear industry. But so far, few papers have been found regarding exact modeling and simulation of the tooth surface generations and tooth contact analysis (TCA) of face hobbed spiral bevel and hypoid gear sets. This paper presents the generalized theory of the face hobbing generation method, mathematic models of tooth surface generations, and the simulation of meshing of face hobbed spiral bevel and hypoid gears. The face hobbing indexing motion is described and visualized. A generalized description of the cutting blades is introduced by considering four sections of the blade edge geometry. A kinematical model is developed and analyzed by breaking down the machine tool settings and the relative motions of the machine elemental units and applying coordinate transformations of the elemental motions. The developed face hobbing generation model is directly related to a physical bevel gear generator. A generalized and enhanced TCA algorithm is proposed. The face hobbing process has two categories, non-generated (Formate®) and generated methods, applied to the tooth surface generation of the gear. In both categories, the pinion is always finished with the generated method. The developed tooth surface generation model covers both categories with left-hand and right-hand members. Based upon the developed theory, an advanced tooth surface generation and TCA program is developed and integrated into Gleason CAGE™for Windows Software. Two numerical examples are provided to illustrate the implementation of the developed mathematic models.

In this paper, we present a method for the global optimization of the tooth contact pattern and transmission error of spiral bevel and hypoid gears, which includes three optimization objectives, three control parameters, and a complex-constrain genetic algorithm solving method. A new set of fundamental equations for pitch cone parameters of hypoid gear drives are established, as well as the relationships between pitch cone and curvature parameters. Based on this theory, three control parameters are selected to determine the pinion tooth surface. A hypoid gear drive is chosen for case studies. The results verify that the optimization methodology can achieve the expected optimization objectives and has good convergence. Correlations between optimization objectives and control parameters are discussed. Furthermore, a finite element model of a simplified hypoid gear drive system is established and its quasi-static meshing characteristics analyzed. The results again confirm the correctness of the optimization method. The effects of torque load on the contact pattern and transmission error are discussed. The results provide a theoretical reference for geometric calculations, quasi-static analysis, and optimal design of spiral bevel and hypoid gears.

Because the contact patterns of spiral bevel and hypoid gears are highly sensitive to tooth flank geometry, it is desirable to reduce the flank deviations caused by machine errors and heat treatment deformation. Several methods already proposed for flank correction are based on the cutter parameters, machine settings, and kinematical flank motion parameters of a cradle-type universal generator, which are modulated according to the measured flank topographic deviations. However, because of the recently developed six-axis Cartesian-type computer numerical control (CNC) hypoid generator, both face-milling and face-hobbing cutting methods can be implemented on the same machine using a corresponding cutter head and NC code. Nevertheless, the machine settings and flank corrections of most commercial Cartesian-type machines are still translated from the virtual cradle-type universal hypoid generator. In contrast, this paper proposes a flank-correction methodology derived directly from the six-axis Cartesian-type CNC hypoid generator in which high-order correction is easily achieved through direct control of the CNC axis motion. The validity of this flank-correction method is demonstrated using a numerical example of Oerlikon Spirac face-hobbing hypoid gears made by the proposed Cartesian-type CNC machine.

Dudley's Handbook of Practical Gear Design and Manufacture

Deproximating Tredgold's Approximation