Study on the optimum standard parameters of hob optimization for reducing gear tooth root stress

Highlights

• The tip contour of a protuberance hob is defined in detail by a Bezier curve with six parameters.

• Hob tip contour is optimized for increasing the bending capacity of gear tooth root.

• A best set of Bezier curve parameters for usually standard hobs is found out for reducing the tooth root stress.

• A best set of Bezier curve parameters for protuberance hobs is found out for reducing the tooth root stress.

• These two sets of parameters are widely applicable to all common gear cases.

Abstract

Optimising the hob used to manufacture gears is an attractive approach for minimizing the gear tooth root stress. However, currently available approaches often require modifying the hob for every gear case, which limits their practical application and increases design and manufacturing costs. In this study, two sets of Bezier curve parameters for gear hobs were developed to minimize the tooth root stress: one for standard hobs and the other for protuberance hobs. The two parameter sets can be used to modify any hob and effectively reduce the gear bending stress, and they are widely applicable to all common gear cases. Calculation with the finite element method indicated that the first-best parameter set for standard hobs reduced the stress of most gear cases by 6%–10%. The second-best parameter set for protuberance hobs reduced the stress of almost all gear cases by 7%–15%. The average optimisation potential was 10.73%. Meanwhile, the manufacturing cost is very low because an optimised hob can be applied to producing other gears. These two parameter sets are a significant contribution to the optimisation and manufacture of gears.

Introduction

Research and development of gearboxes have long been focused on meeting the demand for a higher load-carrying capacity, lighter weight, and more power. However, ensuring a low manufacturing cost is also a significant factor. Several approaches have been considered: novel materials [1], [2], manufacturing techniques [3], and gear tooth shapes [4], [5]. Optimising the gear tooth profile can significantly reduce stress concentrations and improve the gear tooth strength [6]. Reducing the stress would obviously improve the bending fatigue life of the gear, and both the size and weight of the gear can be reduced while the same load capacity is maintained.

The tooth root stress needs to be accurately estimated to ensure an effective optimisation. There are three main approaches to obtaining the stress: using standards such as ISO 6336 [7] or DIN 3960 [8], experimental verification, and computer-aided simulation. The conventional calculation methods provided by standards for spur gears are only valid for standard gearwheel geometries. When gear teeth have nonstandard geometry, the maximum stress of the tooth root does not occur at the 30° tangent of the tooth root fillet. Experimental verification can effectively obtain the bending stress of nonstandard gears, such as through using a resistance wire strain gauge. Photoelastic measurements can be used to evaluate the gear stress [9]. However, experimental verification has limited practical application because of the high cost and long time required. The finite element method (FEM) is most often used to calculate the bending stress, strain, and deformation [10], [11], [12], [13], [14]. FEM can be used to observe the stresses on a gear tooth in detail. The results can be used to optimise the tooth root geometry and reduce the tooth root stress.

There are two approaches to optimising the tooth root geometry and increasing the gear tooth strength [15], [16]: optimising the tooth root geometry directly, and optimising the hob. For the first approach, an asymmetric tooth profile can be used instead of the conventional symmetric involute profile to increase the tooth root strength [17], [18]. However, an asymmetric gear can only be driven in one direction, so the reverse transmission capability is limited, and asymmetric gears are only applicable to specific cases. The conventional involute gear uses two convex curves rolling and sliding together; the Convoloid gear [19] has an innovative tooth profile that meshes convex curves with concave curves, which greatly reduces the contact and bending stresses at the same time. However, this innovative tooth profile is not widely applicable at present because the manufacturing procedure and vibration characteristics need to be further researched. The most common method for improving the tooth strength is to optimise the tooth profile based on the standard involute tooth. The most common solution is to use the circular fillet instead of a trochoidal fillet [4].Some researchers have used a genetic algorithm (GA) to directly optimise the tooth tip and root by modifying the coefficients of two meshing spur gears to reduce the tooth root stress and gear weight [20], [21]. Extensions of this approach include replacing the tooth root fillet with a parabola, ellipsis, chain curve, or other curve to reduce the bending stress [22], [23]. Pedersen optimised the tooth root shape by using a double-distorted super ellipse [24]; the primary advantages are the simplicity (only three design parameters at most) and that the shape is given analytically. Pedersen also showed how the optimised shape can be used to find a cutting tool shape that can produce the gear. Zou et al. proposed using a nonparametric cubic spline to design the tooth profile [25], [26]. Schneider et al. used a Bezier curve to design the tooth profile [27]. However, a good optimisation design requires the manufacture of the tooth profile to be easy, efficient, and economical. The first approach of optimising the tooth root geometry directly often requires five-axis milling and profile grinding, which is antithetical to large-scale production and unsuitable for general gear design.

The second optimisation approach of focusing on the hob contours is more convenient and economical. The most common method is to use a rack with a full tip radius for gear hobbing [28]. Spitas et al. explored the effect of optimising both the hob tip radius and addendum to reduced the maximum tooth bending stress [29]. Spitas et al. also investigated using different pressure angles and nonstandard module cutters to increase the fillet strength [30]. Math et al. considered four kinds of hob tip contours to improve the transition between the spur gear root and involute tooth profile [31]. Pedersen used a distorted super-elliptical curve to optimise the contour of the hob tip, which reduced the tooth root stress by up to 14.4% [32]. Zhao et al. used a quadratic rational Bezier curve to describe the cutter tip of a hob and reduced the tooth root stress by up to 11.8% [33]. Uelpenich et al. used a Bezier curve to optimise the hob contours and proposed an analytical equation to reduce the optimisation time [34]. However, they did not provide a detailed definition for the tip contour of a hob. Dong et al. used a genetic algorithm and FEM to optimise the contour of standard hobs with a Bezier curve and reduce the tooth root stress [35]. However, the above methods require modifying the hob for every gear case, which increases the design time and manufacture cost.

Previous research can be used to obtain an optimum group of Bezier curve parameters for each gear case to reduce the tooth root stress [34], [35]. However, these sets of Bezier curve parameters differ from each other, so the optimised hob geometry differs for each gear case. The present study focused on determining parameter sets that are widely applicable to reducing the gear tooth stress for all common gear cases. The main contributions are as follows:

• A Bezier curve with six parameters was established for optimising the tip contour of protuberance hobs. The protuberance hob is widely used in the gear manufacturing, but there has been little research on its optimisation.

• Two sets of Bezier curve parameters were obtained for optimising the tooth root stress and gear manufacture; one is applicable to standard hobs, and the other is applicable to protuberance hobs. The two parameter sets are independent of the number of teeth, shift coefficient, and pressure angle, so they are widely applicable to all common gear cases. The two parameter sets can be used to optimise any hob and reduce the tooth bending stress.

The rest of this paper is organized as follows. Section 2 describes how the tip corner contour of a protuberance hob can be defined in detail with a Bezier curve and how the parameters affect the tooth root stress. Section 3 discusses the best set of Bezier curve parameters for standard hobs. Section 4 discusses the best set of Bezier curve parameters for protuberance hobs and the stress reduction results when the best parameter set is applied to other protuberance hobs with different protuberance angles. Section 5 presents the conclusions.