Maximum Tooth Contact Pressure Research Articles

Multi-objective optimization of hypoid gears to improve operating characteristics

In this paper a multi-objective optimization method of hypoid gears correlating to the operating characteristics is presented. Optimal design of hypoid gears demands that multiple objectives be simultaneously achieved. Four objectives considered in this study are the minimization of the maximum tooth contact pressure, transmission error and the average temperature in the gear mesh, and the maximization of the mechanical efficiency of the gear pair. The goals of the optimization are achieved by the optimal modification of meshing teeth surfaces. In practice, these modifications are introduced by applying the appropriate machine tool setting for the manufacture of the pinion and the gear, and/or by using a tool with an optimized profile. The proposed optimization procedure relies heavily on the loaded tooth contact analysis for the prediction of tooth contact pressure distribution and transmission errors, and on the mixed elastohydrodynamic analysis of lubrication to determine temperature and efficiency. A fast elitist nondominated sorting genetic algorithm (NSGA-II) is applied to solve the model. The effectiveness of the method is demonstrated by using hypoid gear examples. The obtained results have shown that by the optimization considerable improvements in the operating characteristics of the gear pair are achieved.

Optimization of Gear Design and Manufacture

In this study a method is proposed for the optimization of the design and manufacture of gears. Optimal modifications are introduced into gear tooth surfaces in order to improve the operating characteristics of the gear pair, namely, to reduce the tooth contact pressure and the transmission errors, and to decrease the sensitivity of the gear pair to errors in tooth surfaces and to the relative positions of the mating members. The optimal modifications of gear tooth surfaces are introduced by the application of the adequate machine tool settings and tool geometry. An optimization methodology is applied to systematically define optimal tool geometry and machine tool settings to simultaneously minimize tooth contact pressures and angular displacement error of the driven gear. The use of a CNC gear manufacture machine makes it possible to perform nonlinear correction motions for cutting the optimized gears. Effectiveness of the method was demonstrated by using a spiral bevel gear example. Significant reductions in the maximum tooth contact pressure and in the transmission errors were obtained.

Optimal machine tool settings for face-hobbed hypoid gears manufactured on CNC hypoid generator

In this study, a method is proposed for the determination of optimal machine tool setting for the manufacture of face-hobbed hypoid gears on CNC hypoid generators. The optimal head-cutter geometry and machine tool settings are determined to introduce the optimal tooth modifications into the teeth of face-hobbed hypoid gears. The goals of the tooth surface modifications are to reduce the tooth contact pressure and the transmission errors and to decrease the sensitivity of the gear pair to errors in tooth surfaces and to the relative positions of the mating members. An optimization methodology is applied to systematically define the optimal head-cutter geometry and machine tool settings to simultaneously minimize tooth contact pressures and angular displacement error of the driven gear. The method is based on the machine tool setting variation on the cradle-type generator conducted by polynomial functions of fifth order. An algorithm is developed for the execution of motions on the CNC hypoid generator using the optimal relations on the cradle-type machine. Effectiveness of the method was demonstrated by using a face-hobbed hypoid gear example. Significant reductions in the maximum tooth contact pressure and in the transmission errors were obtained.

Optimal Machine-Tool Settings for the Manufacture of Face-Hobbed Spiral Bevel Gears

In this study, an optimization methodology is proposed to systematically define the optimal head-cutter geometry and machine-tool settings to simultaneously minimize the tooth contact pressure and angular displacement error of the driven gear (the transmission error), and to reduce the sensitivity of face-hobbed spiral bevel gears to the misalignments. The proposed optimization procedure relies heavily on the loaded tooth contact analysis for the prediction of tooth contact pressure distribution and transmission errors influenced by the misalignments inherent in the gear pair. The load distribution and transmission error calculation method employed in this study were developed by the author of this paper. The targeted optimization problem is a nonlinear constrained optimization problem, belonging to the framework of nonlinear programming. In addition, the objective function and the constraints are not available analytically, but they are computable, i.e., they exist numerically through the loaded tooth contact analysis. For these reasons, a nonderivative method is selected to solve this particular optimization problem. That is the reason that the core algorithm of the proposed nonlinear programming procedure is based on a direct search method. The Hooke and Jeeves pattern search method is applied. The effectiveness of this optimization was demonstrated on a face-hobbed spiral bevel gear example. Drastic reductions in the maximum tooth contact pressure (62%) and in the transmission errors (70%) were obtained.

Manufacture of Optimized Face-Hobbed Spiral Bevel Gears on Computer Numerical Control Hypoid Generator

In this study, a method is proposed for the advanced manufacture of face-hobbed spiral bevel gears on CNC hypoid generators with optimized tooth surface geometry. An optimization methodology is applied to systematically define optimal head-cutter geometry and machine tool settings to introduce optimal tooth modifications. The goal of the optimization is to simultaneously minimize tooth contact pressures and angular displacement error of the driven gear (the transmission error). The optimization is based on machine tool setting variation on the cradle-type generator conducted by optimal polynomial functions. An algorithm is developed for the execution of motions on the CNC hypoid generator using the relations on the cradle-type machine. Effectiveness of the method was demonstrated by using a face-hobbed spiral bevel gear example. Significant reductions in the maximum tooth contact pressure and in the transmission errors were obtained.

Optimization of face-hobbed hypoid gears

In this study, an optimization methodology is presented to systematically define head-cutter geometry and machine tool settings to introduce optimal tooth modifications in face-hobbed hypoid gears. The goal of the optimization is to simultaneously minimize tooth contact pressures and angular displacement error of the driven gear (the transmission error), while concurrently confining the loaded tooth contact pattern within the tooth boundaries and avoiding any edge- or corner-contact conditions. The proposed optimization procedure relies heavily on the loaded tooth contact analysis for the prediction of tooth contact pressure distribution and transmission errors developed by the author of this paper. The targeted optimization problem is a nonlinear constrained optimization problem, belonging to the framework of nonlinear programming. In addition, the objective function and the constraints are not available analytically, but they are computable, i.e., they exist numerically through the loaded tooth contact analysis. For these reasons, a nonderivative method is selected to solve this particular optimization problem. The effectiveness of this optimization was demonstrated by using a face-hobbed hypoid gear example. Considerable reductions in the maximum tooth contact pressure and in the transmission errors were obtained.

Optimal Tooth Surface Modifications in Face-Hobbed Hypoid Gears

In this study, an optimization methodology is proposed to systematically define head-cutter geometry and machine tool settings to introduce optimal tooth modifications in face-hobbed hypoid gears. The goal of the optimization is to simultaneously minimize tooth contact pressures and angular displacement error of the driven gear, while concurrently confining the loaded contact pattern within the tooth boundaries. The proposed optimization procedure relies heavily on a loaded tooth contact analysis for the prediction of tooth contact pressure distribution and transmission errors. The objective function and the constraints are not available analytically, but they are computable, i.e., they exist numerically through the loaded tooth contact analysis. The core algorithm of the proposed nonlinear programming procedure is based on a direct search method. Effectiveness of this optimization was demonstrated by using a face-hobbed hypoid gear example. Considerable reductions in the maximum tooth contact pressure and in the transmission errors were obtained.

Design of face-hobbed spiral bevel gears with reduced maximum tooth contact pressure and transmission errors

The aim of this study is to define optimal tooth modifications, introduced by appropriately chosen head-cutter geometry and machine tool setting, to simultaneously minimize tooth contact pressure and angular displacement error of the driven gear (transmission error) of face-hobbed spiral bevel gears. As a result of these modifications, the gear pair becomes mismatched, and a point contact replaces the theoretical line contact. In the applied loaded tooth contact analysis it is assumed that the point contact under load is spreading over a surface along the whole or part of the “potential” contact line. A computer program was developed to implement the formulation provided above. By using this program the influence of tooth modifications introduced by the variation in machine tool settings and in head cutter data on load and pressure distributions, transmission errors, and fillet stresses is investigated and discussed. The correlation between the ease-off obtained by pinion tooth modifications and the corresponding tooth contact pressure distribution is investigated and the obtained results are presented.

Head-cutter for optimal tooth modifications in spiral bevel gears

Abstract A method for the determination of optimal tooth modifications in spiral bevel gears based on improved load distribution and reduced maximum tooth contact pressure and transmission errors is presented. The modifications are introduced into the pinion tooth-surface by using a head-cutter with bicircular profile and with optimal diameter. In the optimization of tool parameters the influence of relative position errors of the mating gears is included. By using the computer program developed for load distribution calculation, the influence of head-cutter parameters and relative position errors of the mating gears on tooth contact and transmission errors is investigated. Based on the results obtained, the radii and the position of the circular tool profile arcs and the cutter diameter for pinion teeth finishing were optimized. By applying the optimal tool parameters, the maximum tooth contact pressure is reduced by 45% and the maximum angular position error of the driven gear by 60%, in regard to the spiral bevel gear pair with a pinion manufactured by a cutter of straight-sided profile and of diameter determined by the commonly used methods.

Machine-Tool Settings to Reduce the Sensitivity of Spiral Bevel Gears to Tooth Errors and Misalignments

The method for loaded tooth contact analysis is applied for the investigation of the combined influence of machine-tool settings for pinion teeth finishing and misalignments of the mating members on load distribution and transmission errors in mismatched spiral bevel gears. By using the corresponding computer program, the influence of pinion’s offset and axial adjustment error, angular position error of the pinion axis, tooth spacing error, and machine-tool setting correction for pinion teeth finishing, on tooth contact pressure, tooth root stresses, and angular displacement of the driven gear member from the theoretically exact position based on the ratio of the numbers of teeth is investigated. On the basis of the obtained results, the optimal combination of machine-tool settings is determined. By the use of this set of machine-tool settings, the maximum tooth contact pressure and transmission errors can be significantly reduced. However, in some cases, by the use of appropriate machine-tool settings for the reduction of tooth contact pressure, the angular displacement of the driven gear increases. Therefore, different optimized combinations of machine-tool settings for pinion tooth finishing for the reduction of the sensitivity of gears to misalignments in regard to maximum tooth contact pressure and transmission errors should be applied. By the use of the combination of machine-tool settings to reduce the sensitivity of gears to misalignments in regard to transmission errors, a slight reduction of maximal tooth contact pressure is achieved, too.

Optimal Tooth Modifications in Hypoid Gears

A method for the determination of optimal tooth modifications in hypoid gears based on improved load distribution and reduced transmission errors is presented. The modifications are introduced into the pinion tooth surface by using a cutter with bicircular profile and optimal diameter. In the optimization of tool parameters the influence of shaft misalignments of the mating members is included. As the result of these modifications a point contact of the meshed teeth surfaces appears instead of line contact; the hypoid gear pair becomes mismatched. By using the method presented in (Simon, V., 2000, “Load Distribution in Hypoid Gears,” ASME J. Mech. Des., 122, pp. 529–535) the influence of tooth modifications introduced on tooth contact and transmission errors is investigated. Based on the results that was obtained the radii and position of circular tool profile arcs and the diameter of the cutter for pinion teeth generation were optimized. By applying the optimal tool parameters, the maximum tooth contact pressure is reduced by 16.22% and the angular position error of the driven gear by 178.72%, in regard to the hypoid gear pair with a pinion manufactured by a cutter of straight-sided profile and of diameter determined by the commonly used methods.

Optimal Machine Tool Setting for Hypoid Gears Improving Load Distribution

A method for the determination of optimal machine tool setting for manufacturing modified (mismatched) hypoid gears based on improved load distribution and reduced transmission errors is presented. The applied load distribution calculation is based on the conditions that the total angular position errors of the gear teeth being instantaneously in contact under load must be the same, and along the contact line of every tooth pair instantaneously in contact, the composite displacements of tooth surface points—as the sums of tooth deformations, geometrical surface separations, gear body bending and torsion, deflections of the supporting shafts, misalignments, and composite tooth errors—should correspond to the angular position of the gear. The tooth deformations consists of the bending and shearing deflections of gear teeth and of the local contact deformations of the mating surfaces. The tooth deflections are calculated by the finite element method. As the equations governing the load sharing and load distribution are nonlinear, an approximate and iterative technique is used to solve this system of equations. The method is implemented by a computer program. Using the program that was developed the influence of machine tool setting parameters for pinion manufacture on maximum tooth contact pressure, load distribution factor, and transmission errors is investigated. By successively choosing the optimal value for every machine tool setting parameter, and by applying the optimal set of these parameters, the maximum tooth contact pressure is reduced by 5.8%, the load distribution factor by 5.9%, and the angular position error of the driven gear by 65.4%, in regard to the hypoid gear pair manufactured by the machine tool setting determined by the commonly used method.