Mating Tooth Surfaces Research Articles

Multi-Point Control for Face-Milled Spiral Bevel Gears with a Predesigned Fourth-Order Motion Curve

This paper presents an ultimate motion methodology of a face-milling spiral bevel gear pair to synthesize the mating tooth surfaces with a predesigned fourth-order motion curve. The methodology is to control some contact points along the contact path in the process of tooth contact analysis via application of an extended local synthesis which permits some transmission errors rather than zero at the concerned contact point. The modified offset motion correction is selected to demonstrate the proposed methodology. Applied torque corresponding to an elastic approach of 0.00635 mm at the mean contact point is calculated and the loaded tooth contact analysis is performed. Numerical results show that the extended local synthesis can effectively control the transmission errors on the predesigned fourth-order motion curve at arbitrarily predesigned contact points along the contact path of the spiral bevel gear pair. The tooth contact pattern for the actual tooth pair is scattered into three segments since the rotational motion of the driven gear at any instant angular position is dependent on the tooth pair with the least transmission error among the three adjacent tooth pairs. The actual tooth contact patterns of the spiral bevel gear pair become continuous when meshing tooth surfaces are elastically deformed.

A Transient Mixed Elastohydrodynamic Lubrication Model for Helical Gear Contacts

Abstract In this study, a non-Newtonian, transient, isothermal, mixed elastohydrodynamic lubrication (EHL) model is proposed for helical gear contacts. The model accounts for nonelliptical contacts subject to spatially varying sliding and rolling velocity fields that are not aligned with any principal axis of the contact region, which is the case for helical gear contacts. The time-varying changes pertaining to key contact parameters and relative motion of roughness profiles on mating tooth surfaces are captured simultaneously to follow the contact from the root to the tip of a tooth while accounting for the transient effect due to relative motions of the roughness profiles. Actual tooth load distributions, contact kinematics, and compliances of helical gear contacts are provided to this model by an existing helical gear load distribution model. Measured three-dimensional roughness profiles covering the entire meshing zone are incorporated in the analyses to investigate its impact on the EHL conditions as well as mechanical power loss. Results of a parametric sensitivity study are presented to demonstrate the influence of operating conditions and surface roughness on the EHL behavior and the resultant gear mesh mechanical power loss of an example helical gear pair. The accuracy of the proposed mixed-EHL model is assessed by comparing the mechanical power loss predictions to available experimental results.

Improved mixed elastohydrodynamic lubrication of hypoid gears by the optimization of manufacture parameters

Extensive wear appears in the case of dry contacts, or when the lubrication of the contacting surfaces is not appropriate. The aim of this research is to improve the mixed elastohydrodynamic lubrication in hypoid gears by the optimization of manufacture parameters for tooth surface processing.A full numerical analysis of the thermal mixed EHL in hypoid gears is applied. The equation system and the numerical procedure are unified for a full coverage of all the lubrication regions including the full film, mixed, and boundary lubrication. In the hydrodynamically lubricated areas the calculation method employed is based on the simultaneous solution of the Reynolds, elasticity, energy, and Laplace's equations. In the asperity contact areas the Reynolds equation is reduced to an expression equivalent to the mathematical description of dry contact problem. The real geometry and kinematics of the gear pair based on the manufacturing procedure is applied, thus the exact geometrical separation of the mating tooth surfaces is included in the oil film shape, and the real velocities of these surfaces are used in the Reynolds and energy equations. The transient nature of gear tooth mesh is included. The oil viscosity variation with respect to pressure and temperature and the density variation with respect to pressure are included. The non-Newtonian behaviour of the lubricant is considered. Using this model, the pressures, film thickness, temperatures, and power losses in the mixed lubrication regime are predicted. By using the developed method, the influence of the manufacturing parameters on the conditions of mixed elastohydrodynamic lubrication is investigated. On the basis of the obtained results recommendations are formulated to improve the mixed EHL and the efficiency of face-milled hypoid gears.

Tolerance analysis of spur gears based on skin model shapes and a boundary element method

Tolerance analysis is an important approach for evaluating the accuracy and function of mechanical products. Research on tolerance analysis has mainly focused on the transfer and accumulation of geometric deviations during assembly, while the effects of contact behavior have not been considered in detail. Therefore, this paper presents a tolerance analysis approach to spur gears that takes into account the effects of contact behavior between mating tooth surfaces on transmission performance. Machining and assembly errors are represented based on the concept of skin model shapes. The contact behavior between mating tooth surfaces is calculated using the boundary element method (BEM) to perform tooth contact analysis (TCA). The proposed tolerance analysis approach is applied to a pair of spur gears as an example. The manner in which the transmission error, tooth contact pattern and contact pressure distribution are affected by machining and assembly errors is discussed. This research demonstrates the notable effects of contact behavior between mating tooth surfaces on transmission performance and provides a potential method for the tolerance analysis of other mechanisms.

Improvements in the mixed elastohydrodynamic lubrication and in the efficiency of hypoid gears

In this paper, the influence of the manufacturing parameters on the conditions of mixed elastohydrodynamic lubrication is investigated. On the basis of the obtained results, recommendations are formulated to improve the mixed EHL and the efficiency of face-milled hypoid gears. A full numerical analysis of the mixed EHL in hypoid gears is applied. The equation system and the numerical procedure are unified for a full coverage of all the lubrication regions, including the full film, mixed, and boundary lubrication. In the hydrodynamically lubricated areas, the calculation method employed is based on the simultaneous solution of the Reynolds, elasticity, energy, and Laplace's equations. In the asperity contact areas, the Reynolds equation is reduced to an expression equivalent to the mathematical description of dry contact problem. The real geometry and kinematics of the gear pair based on the manufacturing procedure are applied; thus, the exact geometrical separation of the mating tooth surfaces is included in the oil film shape, and the real velocities of these surfaces are used in the Reynolds and energy equations. The transient nature of gear tooth mesh is included. The oil viscosity variation with respect to pressure and temperature and the density variation with respect to pressure are included. The non-Newtonian behaviour of the lubricant is considered. Using this model, the pressures, film thickness, temperatures, and power losses in the mixed lubrication regime are predicted. The effectiveness of the presented method is demonstrated by using hypoid gear examples.

Pitch cone design and influence of misalignments on tooth contact behaviors of crossed beveloid gears

A pitch cone design theory assuming line contact is formulated to analyze the tooth engagement behavior of crossed beveloid gears. The analysis applies the spatial gearing model and beveloid gear manufacturing technique to determine the cutter parameters and geometry. Using the proposed procedure, a crossed beveloid gear pair with small shaft angle is studied. Also, the proposed formulation is capable of quantifying the influence of misalignments including offset and angular position errors on the contour and position of contact path and lines over a complete mesh cycle. The results show that both offset and angular position errors of the mating members worsen the conjugate action of the mating tooth surfaces. Furthermore, both the contact path and the transmission error are more sensitive to the angular error than the offset error. The loaded tooth contact characteristics are subsequently modeled using the finite element method for comparison purpose. The proposed model is partially validated by comparing the numerical simulation results.

A THEORETICAL BASIS FOR THE ANALYSIS OF LOGIX GEARS

A novel approach to gear design to produce so-called Logix gears is based on differential geometric methods and results in higher order contact parameters, i.e., reduced stress concentration. Profiles designed in this way admit concave-to-convex mating tooth surfaces and a simple two gear set with such properties is presented as an example. Once a pair of mating gears has been configured it is then shown how to design hobbing racks to cut these gears.

Load Distribution in Spiral Bevel Gears

Abstract A new approach for the computerized simulation of load distribution in mismatched spiral bevel gears with point contact is presented. The loaded tooth contact is treated in a special way: it is assumed that the point contact under load spreads over a surface along the “potential” contact line (Simon, 2006, Mech. and Machine Theory, in press), which line is made up of the points of the mating tooth surfaces in which the separations of these surfaces are minimal, instead of assuming the usually applied elliptical contact area. The bending and shearing deflections of gear teeth, the local contact deformations of mating surfaces, gear body bending and torsion, the deflections of supporting shafts, and the manufacturing and alignment errors of mating members are included. The tooth deflections of the pinion and gear teeth are calculated by the finite element method. As the equations governing the load sharing among the engaged tooth pairs and load distribution along the tooth face are nonlinear, an approximate and iterative technique is used to solve this system of equations. The method is implemented by a computer program. By using this program the load and tooth contact pressure distributions, the angular displacements of the driven gear and the stresses in the pinion and gear teeth are calculated. The influence of design data and transmitted torque on load distribution parameters and fillet stresses is investigated and discussed.

Tooth contact analysis of conical involute gears

The mathematical model of conical involute gears is developed based on the theory of gearing and the generating mechanism. Tooth contact analysis (TCA) is performed to examine the meshing and bearing contact of the conical involute gear pairs with intersected and crossed axes. In addition, the principal directions and curvatures of the gear surfaces are investigated and the contact ellipses of the mating tooth surfaces are also studied. Finally, the numerical illustrative examples are provided to demonstrate the computational results, test gears are made for tooth-bearing tests, and the conclusion is verified that the theory has the applicability.

Influence of tooth errors and shaft misalignments on loaded tooth contact in cylindrical worm gears

The influence of tooth errors and shaft misalignments on loaded tooth contact in different types of cylindrical worm gears is investigated. The geometry and kinematics of worm gears based on the generation of gear teeth by a hob is presented. A method for loaded tooth contact analysis of such worm gear pairs is developed. The method includes the determination of contact lines in fully conjugated worm gears with theoretical line contact, the determination of path of contact, potential contact lines and separations of mating teeth surfaces along these contact lines in modified worm gears with theoretical point contact, and the calculation of load distribution and transmission errors for different types of modified and non-modified worm gears. A computer program based on the theoretical background presented has been developed. Using this program the influence of tooth errors and shaft misalignments of mating members on tooth contact is investigated and discussed.

Fourth-Order Kinematic Synthesis for Face-Milling Spiral Bevel Gears With Modified Radial Motion (MRM) Correction

The use of a fourth-order motion curve is proposed by Stadtfeld and Gaiser to reduce the running noise of a bevel gear set recently. However, the methodology of synthesizing the tooth surfaces was not clearly shown in the literature. In this work, we proposed a methodology to synthesize the mating tooth surfaces of a face-milling spiral bevel gear set transmitting rotations with a predetermined fourth-order motion curve and contact path. A modified radial motion (MRM) correction in the machine plane of a computer numerical control (CNC) hypoid generator is introduced to modify the pinion tooth surface. With MRM correction, an arbitrary predetermined contact path on the pinion tooth surface with predetermined fourth-order motion curve can be achieved. Parameters of MRM correction are calculated according to the predetermined contact path and motion curve. As shown by the numerical examples, the contact path and the motion curve were obtained as expected by applying the MRM correction. The results of this work can be applied to the pinion, which is generated side-by-side (for example, fixed setting method, formate method, and Helixform method) and can be used as a basis for further study on the motion curve optimizations.

Contact characteristics of beveloid gears

This study investigates the contact characteristics of beveloid gear pairs with intersected, crossed and parallel axes. Tooth contact analysis (TCA) is performed to examine the meshing and bearing contact of the beveloid gear pairs composed of a pinion and a gear. In addition, the principal directions and curvatures of the pinion and gear surfaces are investigated and the contact ellipses of the mating tooth surfaces are also studied. Numerical illustrative examples are provided to demonstrate the computational results.

Bearing contact of a helical gear pair with involute teeth pinion and modified circular-arc teeth gear

AbstractThis paper investigates the contact ellipse of a helical gear set, composed of an involute pinion and a modified circular-arc gear based on curvature analysis. This gear drive exhibits point contact and parabolic transmission error due to the double crowning effect of the gear, i.e. crowning in both profile and longitudinal directions. The principal directions and curvatures of the generating tool surfaces were derived by means of differential geometry and Rodrigues’ equation. The principal directions and curvatures of the pinion and gear surfaces were obtained directly from the generating surfaces. Finally, the determination of the contact ellipses of the mating tooth surfaces was achieved. Numerical examples are also provided to demonstrate the computational results.

Bearing contact of a helical gear pair with involute teeth pinion and modified circular-arc teeth gear

This paper investigates the contact ellipse of a helical gear set, composed of an involute pinion and a modified circular-arc gear based on curvature analysis. This gear drive exhibits point contact and parabolic transmission error due to the double crowning effect of the gear, i.e. crowning in both profile and longitudinal directions. The principal directions and curvatures of the generating tool surfaces were derived by means of differential geometry and Rodrigues’ equation. The principal directions and curvatures of the pinion and gear surfaces were obtained directly from the generating surfaces. Finally, the determination of the contact ellipses of the mating tooth surfaces was achieved. Numerical examples are also provided to demonstrate the computational results.

Compensating analysis of a double circular-arc helical gear by computerized simulation of meshing

A mathematical model of a stepped double circular-arc helical tooth profile with two centre offsets is developed. The conditions of gear meshing that reflect manufacturing and assembly errors are simulated. The locations of bearing contact and the contact path pattern of mating tooth surfaces are determined by tooth contact analysis (TCA). By applying the proposed mathematical model and TCA, single error impact can be determined. To compensate for offset and angular misalignment, the authors propose an adjustable bearing whereby transmission errors can be minimized. The investigation is illustrated with several numerical examples.

Load Distribution in Hypoid Gears

A new approach for the computerized simulation of load distribution in mismatched hypoid gears with point contact is presented. The load distribution calculation is based on the bending and shearing deflections of gear teeth, on the local contact deformations of the mating surfaces, on gear body bending and torsion, on the deflections of the supporting shafts, and on the manufacturing and alignment errors of the mating members. The tooth deflections of the pinion and gear teeth are calculated by FEM, and the tooth contact is treated in a special way: it is assumed that the point contact under load spreads over a surface along the “potential” contact line, which line is made up of the points of the mating tooth surfaces in which the separations of these surfaces are minimal, instead of assuming an elliptical contact pattern. The system of governing equations is solved by approximations and by using the successive-over-relaxation method. The corresponding computer program is developed. The calculations, performed by this program, show that in the case of hypoid gears, the new approach gives a more realistic contact pattern and contact pressure than the usually assumed and applied elliptical contact approach, especially for the tooth pairs contacting on the toe and on the heel of teeth, and in the case of load distribution calculations made in misaligned gear pairs. By using this program the influence of design data on load distribution parameters is investigated and discussed. [S1050-0472(00)00504-3]

Computer simulation and stress analysis of helical gears with pinion circular arc teeth and gear involute teeth

The tooth contact analysis (TCA) technique and finite element method (FEM) have been applied to gear contact analysis and stress analysis. A mathematical model for helical gears with pinion circular arc teeth and gear involute teeth is assumed to be given by Litvin and Tsay. The geometry of the gears are described by the parameters of manufacturing. Simulation of the conditions of gear meshing, including the axes misalignment and center distance variation are performed. The locations of bearing contact and contact pattern of mating tooth surfaces are determined by the TCA. Results of the TCA provide the location and direction of applied loadings for the computer-aided FEM stress analysis. By applying the given mathematical model and the TCA techniques, a computer-aided stress analysis model for a new pair of gearing has been proposed. An automatic meshes generation (AMG) program has been developed for the FEM stress analysis. A three-dimensional stress analysis for this type of gearing has also been investigated and expressed by Von Mises stress contour distributions.

A simple method of bevel gear cutting (1st report, The basic conception of the cutting method)

In some cases of bevel gear services, the most important demands of them are the cost and the speed of the gear production. This paper proposes a new simplified cutting method. In the method, both the face-milling cutter and the bevel gear work rotate at a high speed and the cutter is fed only in the direction of the tooth depth. This cutting method gives the spiral bevel gears the tooth surfaces, which are ruled by the cutter edge and are eased off on the tip and the bottom of tooth profile, and which contact almost exactly along the tooth trace. The relation of the shape of the cutter edge and the tooth gap of the mating tooth surfaces is investigated theoretically. Plastic bevel gears are cut trially. With these trials, it was assured that the cutting method is available.

Effects of Hunting Gear Ratio Upon Surface Durability of Gear Teeth

It was clearly shown that the pitting limits of gear pairs with a hunting gear ratio were appreciably lower than those of gear pairs with a unity gear ratio. This was ascribed to the difference in effectiveness of running-in between the mating tooth surfaces. Test gears with hardnesses ranging from 185 HB to 850 Hv were finished by hobbing, skiving or grinding. The contact fatigue damage was estimated from work-hardening at the tooth surfaces. The critical plastic strain which might cause pitting cracks was shown in terms of Brinell hardness. The availability of face-width was calculated from the two different pitting limits for gear pairs with two different gear ratios. It ranged from 0.23 to 0.80 in the present experiments because pitting limits of gear pairs with a unity gear ratio were appreciably greater. This suggested that there was a strong possibility to increase the load carrying capacity of tooth surfaces of gear pairs having gear ratios other than unity.

The synthesis of tooth profile shapes and helical gears of high hydraulic performance for rotary type pumps

In this work a method is presented for the synthesis of high hydraulic performance of non involute helical gears and tooth profiles. Two gear capacity criteria are used in the synthesis: (1) the capacity based on optimal helical advance equal to the circular pitch for which perturbations of delivery speed and torque can be completely eliminated and (2) the capacity based on optimal tooth profile which consists substantially in circular arcs whose centers fall on the pitch point, and whose connecting part consists of a suitable curve which satisfies the condition for continuous contact without intersection of two mating teeth surfaces. The present paper deals with the theoretical basis and the comparison with test results. The tooth profile recommended as a result of this study has permitted size and cost reduction in a currently manufactured line of pumps with delivery pressure to 20 kg/cm 2.