# Gear Tooth-Forming Mechanism and Experimental Research of a Spiral Bevel Gear Rolling Process Using Local Induction Heating

Authors: Shitao Wang; Zhenshan Gao; Xiaozhong Deng
Journal: Mathematical Problems in Engineering
 (2023)

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## Abstract

The gear tooth near-net rolling process is a novel technique to enhance the tooth strength of spiral bevel gears. To study the gear tooth-forming mechanism, high-temperature rheological experiments of the material 20CrMnTiH were conducted using the Gleeble 1500D thermosimulation machine, and numerical simulations of the spiral bevel gear rolling forming process based on local induction heating were performed with the use of DEFORM-3D. According to the numerical and rolling experimental results, it is feasible to produce spiral bevel gears using the rolling process with local induction heating. In order to solve the uneven material flow of the gear tooth during the tests, the material compensation coefficientk for gear forming was proposed and the shape of the gear blank was optimized. The simulation results show that the effective height of the tooth heel-end and toe-end in the optimized design increases by about 1 mm.


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## Body

## 1. Introduction

In the field of intersecting shafts and misaligning shafts, spiral bevel gears are commonly utilized as significant power transmission elements. Spiral bevel gear transmission offers the benefits of smooth transmission, high load-bearing capacity, and high transmission efficiency when compared with other transmission types. As a result, spiral bevel gears are widely used in automobile drive axles, aircraft main transmission structures, and other sectors. At present, the process of making spiral bevel gears is mainly focused on tooth milling, which is not only of low productivity and low material utilization but also leads to cutting off the continuous metal flow line inside the components, resulting in a significant reduction of their strength and service life [1].In contrast to the conventional milling method, the service life of the gear processed by precision forging near-net forming technology has increased by nearly 70%, and the near-net formed gears only need tooth finish machining to be put into service [2]. However, due to the problems of difficulty in ejection and low die life in precision forging near-net forming, the machining of the pinion still adopts the tooth milling process. In order to realize the plastic-forming manufacturing of the pinion, the solution of near-net forming by rolling needs to be proposed. So far, especially in the processing of threads, splines, and cylindrical gears, the rolling forming technology has received extensive attention and in-depth research by many domestic and foreign scholars [3–5].Neugebauer et al. [6] explored an unconventional pitch design for gear dies and observed that round dies need extra kinematic compensation during rolling compared to rack dies. By applying this method, pitch accuracy is improved by up to 50%, and high-tooth gears (up to 10 mm in height and a tooth height coefficient larger than 2) can be rolled with greater accuracy. Sasaki et al. [7] optimized the surface rolling process of sintered steel helical gears by using a newly developed two-roll die transverse-type CNC forming rolling machine with tooth lead corrections. Wang et al. [8] systematically introduced the current status and advantages of gear-rolling technology and pointed out the shortcomings of gear-rolling technology, which are characterized by inaccurate tooth parting and differences in material fluidity during the rolling process, leading to the generation of tooth defects “rabbit ears.” Li et al. [9] conducted a numerical simulation of the rolling process and investigated the material flow at different locations during rolling and the stress-strain at different periods of rolling, explaining the causes of “rabbit ear.” Khodaee and Melander References [10, 11] evaluated the quality of gears by changing the rotation direction of the die and verified that the rabbit ear could be suppressed to a certain extent by the forward and reverse rotation experiments, and they studied the variation law of the radial, axial load, and torque of the die wheel during the rolling process. Li et al. [12] investigated the effect of different process parameters on the tooth defect “rabbit ear” using DEFORM-3D software and verified it by experiments.Kamouneh et al. [13] studied three types of tooth defects (rabbit ear, tooth asymmetry, and barreling) presented in the gear roll-forming process and proposed solutions. Li et al. [14] analyzed the effect of various process parameters on the left and right “rabbit ear,” and put forward multiple roll-forming plans to suppress the “rabbit ear.” Fu et al. [15] made a secondary development of DEFORM-3D software to control the temperature of the workpiece in the rolling process. The local induction heating of the workpiece process not only reduced the rolling force but also restrained defects in the rolling process such as inner hole expansion. Zhao et al. [16] investigated the kinematic relationship between the gear die and gear blank during the rolling process, and the results show that the meshing motion of the gear die and the workpiece could be guaranteed when the initial extrusion of the workpiece was not less than 1 mm. Ma et al. [17] discussed the blanks of straight cylindrical gears and proposed a new method for calculating the dimensions of preformed blanks that can be used for gear rolling, and numerical simulations and experimental studies were used to verify the feasibility of the aforementioned method.In conclusion, previous studies on rolling forming technology have mainly focused on splines and cylindrical gears. However, the application of the near-net rolling forming technology to spiral bevel gears has received less attention. The pinion of the spiral bevel gear is made as the research object in this paper, and the local induction heating of the workpiece is carried out before rolling. According to the numerical and rolling experimental results, it is feasible to machine the pinion using the rolling process. In order to solve the uneven material flow of the gear tooth during the tests, the mechanism of tooth defects in the forming process tests was investigated. Then, the rolling and forming process parameters were optimized, and the effectiveness of the optimized process parameters was verified through simulation results.


## 2. Materials and Methods

The magnitude of the forming load is mostly determined by the material’s plastic deformation resistance. It has a significant influence on the development of the forming process, the selection of forging equipment, the design of the die structure, and the die service life. It is used as a preliminary step for metal plastic forming research. The cylindrical bar utilized in this test is 10 mm × 15 mm in size with polished and parallel end faces. 20CrMnTiH steel is a high-performance gear steel. To decrease friction and guarantee consistent deformation during compression, the two end sides are coated with lubricant (70 percent graphite + 30 percent machine oil), and the composition of the selected material’s essential chemical elements is listed in Table1.Table 1
Chemical composition of 20CrMnTiH steel (unit: wt.%).
ElementCMnSiCrTiSPNumerical value0.17–0.230.80–1.100.17–0.371.00–1.390.04–0.1≤0.035≤0.035The effects of the deformation temperature, strain rate, and deformation volume on the rheological stress and organization during deformation are investigated using an isothermal constant strain rate hot compression specimen on a Gleeble 1500D dynamic thermal simulation tester. The specimens are heated at 1200°C at a rate of 10°C/s for 5 minutes to fully austenitize them, then cooled at the deformation temperatures of 900°C, 950°C, 1000°C, 1050°C, and 1100°C at a rate of 5°C/s for 15 seconds to ensure temperature homogenization, and deformed in isothermal compression at the deformation temperature, as shown in Figure1. The thermal simulation tester directly recorded pressure, temperature, and displacement during compression deformation, and the true stress and true strain curves were displayed. Argon gas safeguarded the entire compression deformation process. The true stress-true strain graphs of 20CrMnTiH at different deformation temperatures at strain rates of 0.01 S−1, 0.1 S−1, 1 S−1, and 5 S−1 are shown in Figures 2(a)–2(d). The true stress-true strain diagrams of 20CrMnTiH at different deformation rates of 900°C, 950°C, 1000°C, 1050°C, and 1100°C are shown in Figures 3(a)–3(e).Figure 1
Compression test process flow.Figure 2
Stress-strain diagram of 20CrMnTiH at different temperatures. (a) 0.01 S−1. (b) 0.1 S−1. (c) 1 S−1. (d) 5 S−1.
(a)(b)(c)(d)Figure 3
Stress-strain diagram of 20CrMnTiH at different deformation rates. (a) 900°C. (b) 950°C. (c) 1000°C. (d) 1050°C. (e) 1100°C.
(a)(b)(c)(d)(e)Under high-temperature plastic deformation conditions, the relationship among rheological stress, strain rate, and temperature for conventional thermal deformation can be expressed in the hyperbolic sinusoidal form containing deformation activation energyQ and absolute deformation temperature T proposed by Sellars and Tegar [18] as follows:(1)ε˙=A⋅Fσexp−QRT,(2)Fσ=σn,atlowstresslevelsασ<0.8,expβσ,athighstresslevelsασ>1.2,sinhασn,forallstresses,where ε˙ is the strain rate, A and α are the material constants, here α=β/n, R is the gas constant and takes the value 8.314J⋅mol−1⋅K−1, Fσ is the stress function, there are three expressions, and σ is the peak stress or steady rheological stress.Equations (1) and (2) can predict the high-temperature flow stress of 20CrMnTiH and are integrated into the DEFORM-3D preprocessor to obtain reliable simulation results.


## 3. Finite Element Modeling and Simulation

### 3.1. Finite Element Modeling

#### 3.1.1. Finite Element Model of Gear Dies

Table2 lists the basic parameters of the gear die and pinion. During the rolling process, the gear die and the pinion blank are a pair of completely conjugate tooth surfaces, and the tooth surface equation r2 of the gear die is derived from the tooth surface equation r1 of the pinion. The generated tooth surface point data were transferred to the UG software for solid modeling, allowing for the creation of a 3D model of the gear die.Table 2
The geometric parameters of the pinion and gear die.
PinionGear dieModulus (mm)5.69Number of teeth939Tooth width (mm)20Spiral angle (°)29.43Pressure angle (°)22.5Tooth top height factor0.825Top gap coefficient0.182Section cone angle (°)12.994677.0054In Figure4, Sm,Xm,Ym,Zm and Sn,Xn,Yn,Zn show the coordinate system of the meshing of the pinion with the gear die. E is the offset distance; for spiral bevel gears, E = 0 and the axis intersection angle τ = 90°. S1,X1,Y1,Z1 and S2,X2,Y2,Z2 are the coordinate systems of the pinion and the gear die, respectively. φ1 and φ2 are the current meshing angles of the pinion and the gear die, respectively.Figure 4
The coordinate system of gear meshing.(3)r2u,θg,ϕg=M2mMmnMn1r1,(4)n2u,θg,ϕg=L2mLmnLn1n1,(5)M2m=10000cosϕ2−sinϕ200sinϕ2cosϕ200001,(6)Mmn=cosτ0sinτ0010Esinτ0cosτ00001,(7)Mn1=10000cosφ1sinφ100−sinφ1cosφ100001,(8)nm⋅vm12=0,(9)vm12=ωm12×nm−ωm2×O2O1,(10)ωm12=ωm1−ωm2,(11)OmOq=0−E0T,(12)ωm1=−100T,(13)ωm2=m2100−1T,(14)nmS,θ=LmnLn1n1S,θ,(15)m21=Z1Z2=ω2ω1=φ2φ1.During the machining process, the meshing equation (4) is satisfied between the gear die and the pinion. In the formula, ω1,ω2,ϕ1,ϕ2, and Z1,Z2 represent the angular velocity, rotation angle, and number of teeth of the pinion and gear die, respectively. According to Formulas (3), (4), and (8), the parameters u can be eliminated, and the equation of the tooth surface with θg and ϕg as the surface coordinates can be obtained. L2m, Lmn, and Ln1 are obtained by removing the fourth row and fourth column from M2m, Mmn, and Mn1, respectively.The model of the gear rolling process is composed of a pinion blank and two gear dies, as shown in Figure5. The gear dies rotate during the feeding motion. The feed speed of the gear die is set to v1=0.2mm/s, and the rotational speed is ω1=50r/min. The gear ratio of the gear die to the target gear is ω2=39/9×ω1. The friction factor between the gear die and the pinion blank is set to f=0.2, the heat transfer coefficient is set to 5 N/sec/mm/C, the step length is defined as 0.01 according to the distance, and the number of grids is 100,000. The encryption scale factor is set to 0.01.Figure 5
Finite analysis model of the gear rolling process.


#### 3.1.2. Localized Induction Heating Simulation

Carrying high-frequency current, the coil generates an alternating magnetic field, which in turn induces eddy currents in the blank that dissipate energy and bring about heating. In the process of gear rolling based on high-frequency induction heating, the plastic deformation of the blank is much larger than its elastic deformation, so the elastic deformation can be ignored and the blank is set as a plastic body. During the deformation process, the deformation degree of the die and the splint is very small, which can be defined as a rigid body.The forming force can be significantly reduced using the hot-rolling method. Figure6 depicts the spiral bevel gear’s local induction heating principle. While extracting the coil, the gear dies approach the pinion blank to initiate rolling formation when the temperature reaches a predefined value. There are two heating models: whole heating with no thermal gradient and local heating with a thermal gradient. The effects of both overall and local heating on tooth surface formation were investigated in this study. Figures 7(a) and 7(b) illustrate the finite element model with copper as the coil material and 20CrMnTiH as the blank material. Tetrahedral mesh materials are used for the coil and the blank, with the coil having 50,000 cells and the blank having 100,000 meshes. For the rolling and forming of spiral bevel gears, the deformation area is located between the tooth heel-end and the tooth toe-end of the blank, and the mesh is refined for it. The rolling effect improved when the blank was heated at 950°C. The basic parameters of local induction heating are shown in Table 3.Figure 6
Local induction heating model.Figure 7
(a) Blank axial temperature gradient and (b) radial temperature gradient of blank.
(a)(b)Table 3
Basic parameters of local induction heating.
FrequencyPowerΩThermal emissivityAmbient temperaturekHzkWrad/sec°CCoils8803.140.720Pinion blank00.720


### 3.2. The Construction of Experimental Equipment for the Rolling Process

The gear-rolling system using local induction heating is shown in Figure10. The synchronization issue with the rotation of the two gear dies is resolved by using a set of spiral bevel gears with a tooth ratio of 4.333 without gear backlash as the guiding synchronization mechanism. The blank is inductively heated using the coil, and after it reached the desired temperature, the blank started rolling. A numerical control is used to control the blank rotation speed. When the two gear dies come into contact, the blank and gear dies rotate at a certain speed ratio until the entire rolling process is finished. The two gear dies move toward the direction of the blank at the same speed.Figure 8
Tooth surface defects in spiral bevel gears. (a) Overall heating and (b) local induction heating.Figure 9
The gear cross-section. (a) Overall heating cross-section. (b) Local induction heating cross-section.
(a)(b)Figure 10
The equipment of rolling forming with local induction heating.Figure11 shows the formed pinion obtained on the local induction heating equipment. The formed pinion is similar to the numerical simulation results within a certain error range. The defect analysis of the pinion based on local induction heating can not only improve the forming quality of the target gear but also avoid the adverse effects of other defects caused by the local deformation of the inner hole and the keyway slot.Figure 11
The formed pinion after the tooth toe-end and heel-end machining.


## 3.1. Finite Element Modeling

### 3.1.1. Finite Element Model of Gear Dies

Table2 lists the basic parameters of the gear die and pinion. During the rolling process, the gear die and the pinion blank are a pair of completely conjugate tooth surfaces, and the tooth surface equation r2 of the gear die is derived from the tooth surface equation r1 of the pinion. The generated tooth surface point data were transferred to the UG software for solid modeling, allowing for the creation of a 3D model of the gear die.Table 2
The geometric parameters of the pinion and gear die.
PinionGear dieModulus (mm)5.69Number of teeth939Tooth width (mm)20Spiral angle (°)29.43Pressure angle (°)22.5Tooth top height factor0.825Top gap coefficient0.182Section cone angle (°)12.994677.0054In Figure4, Sm,Xm,Ym,Zm and Sn,Xn,Yn,Zn show the coordinate system of the meshing of the pinion with the gear die. E is the offset distance; for spiral bevel gears, E = 0 and the axis intersection angle τ = 90°. S1,X1,Y1,Z1 and S2,X2,Y2,Z2 are the coordinate systems of the pinion and the gear die, respectively. φ1 and φ2 are the current meshing angles of the pinion and the gear die, respectively.Figure 4
The coordinate system of gear meshing.(3)r2u,θg,ϕg=M2mMmnMn1r1,(4)n2u,θg,ϕg=L2mLmnLn1n1,(5)M2m=10000cosϕ2−sinϕ200sinϕ2cosϕ200001,(6)Mmn=cosτ0sinτ0010Esinτ0cosτ00001,(7)Mn1=10000cosφ1sinφ100−sinφ1cosφ100001,(8)nm⋅vm12=0,(9)vm12=ωm12×nm−ωm2×O2O1,(10)ωm12=ωm1−ωm2,(11)OmOq=0−E0T,(12)ωm1=−100T,(13)ωm2=m2100−1T,(14)nmS,θ=LmnLn1n1S,θ,(15)m21=Z1Z2=ω2ω1=φ2φ1.During the machining process, the meshing equation (4) is satisfied between the gear die and the pinion. In the formula, ω1,ω2,ϕ1,ϕ2, and Z1,Z2 represent the angular velocity, rotation angle, and number of teeth of the pinion and gear die, respectively. According to Formulas (3), (4), and (8), the parameters u can be eliminated, and the equation of the tooth surface with θg and ϕg as the surface coordinates can be obtained. L2m, Lmn, and Ln1 are obtained by removing the fourth row and fourth column from M2m, Mmn, and Mn1, respectively.The model of the gear rolling process is composed of a pinion blank and two gear dies, as shown in Figure5. The gear dies rotate during the feeding motion. The feed speed of the gear die is set to v1=0.2mm/s, and the rotational speed is ω1=50r/min. The gear ratio of the gear die to the target gear is ω2=39/9×ω1. The friction factor between the gear die and the pinion blank is set to f=0.2, the heat transfer coefficient is set to 5 N/sec/mm/C, the step length is defined as 0.01 according to the distance, and the number of grids is 100,000. The encryption scale factor is set to 0.01.Figure 5
Finite analysis model of the gear rolling process.


### 3.1.2. Localized Induction Heating Simulation

Carrying high-frequency current, the coil generates an alternating magnetic field, which in turn induces eddy currents in the blank that dissipate energy and bring about heating. In the process of gear rolling based on high-frequency induction heating, the plastic deformation of the blank is much larger than its elastic deformation, so the elastic deformation can be ignored and the blank is set as a plastic body. During the deformation process, the deformation degree of the die and the splint is very small, which can be defined as a rigid body.The forming force can be significantly reduced using the hot-rolling method. Figure6 depicts the spiral bevel gear’s local induction heating principle. While extracting the coil, the gear dies approach the pinion blank to initiate rolling formation when the temperature reaches a predefined value. There are two heating models: whole heating with no thermal gradient and local heating with a thermal gradient. The effects of both overall and local heating on tooth surface formation were investigated in this study. Figures 7(a) and 7(b) illustrate the finite element model with copper as the coil material and 20CrMnTiH as the blank material. Tetrahedral mesh materials are used for the coil and the blank, with the coil having 50,000 cells and the blank having 100,000 meshes. For the rolling and forming of spiral bevel gears, the deformation area is located between the tooth heel-end and the tooth toe-end of the blank, and the mesh is refined for it. The rolling effect improved when the blank was heated at 950°C. The basic parameters of local induction heating are shown in Table 3.Figure 6
Local induction heating model.Figure 7
(a) Blank axial temperature gradient and (b) radial temperature gradient of blank.
(a)(b)Table 3
Basic parameters of local induction heating.
FrequencyPowerΩThermal emissivityAmbient temperaturekHzkWrad/sec°CCoils8803.140.720Pinion blank00.720


## 3.1.1. Finite Element Model of Gear Dies

Table2 lists the basic parameters of the gear die and pinion. During the rolling process, the gear die and the pinion blank are a pair of completely conjugate tooth surfaces, and the tooth surface equation r2 of the gear die is derived from the tooth surface equation r1 of the pinion. The generated tooth surface point data were transferred to the UG software for solid modeling, allowing for the creation of a 3D model of the gear die.Table 2
The geometric parameters of the pinion and gear die.
PinionGear dieModulus (mm)5.69Number of teeth939Tooth width (mm)20Spiral angle (°)29.43Pressure angle (°)22.5Tooth top height factor0.825Top gap coefficient0.182Section cone angle (°)12.994677.0054In Figure4, Sm,Xm,Ym,Zm and Sn,Xn,Yn,Zn show the coordinate system of the meshing of the pinion with the gear die. E is the offset distance; for spiral bevel gears, E = 0 and the axis intersection angle τ = 90°. S1,X1,Y1,Z1 and S2,X2,Y2,Z2 are the coordinate systems of the pinion and the gear die, respectively. φ1 and φ2 are the current meshing angles of the pinion and the gear die, respectively.Figure 4
The coordinate system of gear meshing.(3)r2u,θg,ϕg=M2mMmnMn1r1,(4)n2u,θg,ϕg=L2mLmnLn1n1,(5)M2m=10000cosϕ2−sinϕ200sinϕ2cosϕ200001,(6)Mmn=cosτ0sinτ0010Esinτ0cosτ00001,(7)Mn1=10000cosφ1sinφ100−sinφ1cosφ100001,(8)nm⋅vm12=0,(9)vm12=ωm12×nm−ωm2×O2O1,(10)ωm12=ωm1−ωm2,(11)OmOq=0−E0T,(12)ωm1=−100T,(13)ωm2=m2100−1T,(14)nmS,θ=LmnLn1n1S,θ,(15)m21=Z1Z2=ω2ω1=φ2φ1.During the machining process, the meshing equation (4) is satisfied between the gear die and the pinion. In the formula, ω1,ω2,ϕ1,ϕ2, and Z1,Z2 represent the angular velocity, rotation angle, and number of teeth of the pinion and gear die, respectively. According to Formulas (3), (4), and (8), the parameters u can be eliminated, and the equation of the tooth surface with θg and ϕg as the surface coordinates can be obtained. L2m, Lmn, and Ln1 are obtained by removing the fourth row and fourth column from M2m, Mmn, and Mn1, respectively.The model of the gear rolling process is composed of a pinion blank and two gear dies, as shown in Figure5. The gear dies rotate during the feeding motion. The feed speed of the gear die is set to v1=0.2mm/s, and the rotational speed is ω1=50r/min. The gear ratio of the gear die to the target gear is ω2=39/9×ω1. The friction factor between the gear die and the pinion blank is set to f=0.2, the heat transfer coefficient is set to 5 N/sec/mm/C, the step length is defined as 0.01 according to the distance, and the number of grids is 100,000. The encryption scale factor is set to 0.01.Figure 5
Finite analysis model of the gear rolling process.


## 3.1.2. Localized Induction Heating Simulation

Carrying high-frequency current, the coil generates an alternating magnetic field, which in turn induces eddy currents in the blank that dissipate energy and bring about heating. In the process of gear rolling based on high-frequency induction heating, the plastic deformation of the blank is much larger than its elastic deformation, so the elastic deformation can be ignored and the blank is set as a plastic body. During the deformation process, the deformation degree of the die and the splint is very small, which can be defined as a rigid body.The forming force can be significantly reduced using the hot-rolling method. Figure6 depicts the spiral bevel gear’s local induction heating principle. While extracting the coil, the gear dies approach the pinion blank to initiate rolling formation when the temperature reaches a predefined value. There are two heating models: whole heating with no thermal gradient and local heating with a thermal gradient. The effects of both overall and local heating on tooth surface formation were investigated in this study. Figures 7(a) and 7(b) illustrate the finite element model with copper as the coil material and 20CrMnTiH as the blank material. Tetrahedral mesh materials are used for the coil and the blank, with the coil having 50,000 cells and the blank having 100,000 meshes. For the rolling and forming of spiral bevel gears, the deformation area is located between the tooth heel-end and the tooth toe-end of the blank, and the mesh is refined for it. The rolling effect improved when the blank was heated at 950°C. The basic parameters of local induction heating are shown in Table 3.Figure 6
Local induction heating model.Figure 7
(a) Blank axial temperature gradient and (b) radial temperature gradient of blank.
(a)(b)Table 3
Basic parameters of local induction heating.
FrequencyPowerΩThermal emissivityAmbient temperaturekHzkWrad/sec°CCoils8803.140.720Pinion blank00.720


## 3.2. The Construction of Experimental Equipment for the Rolling Process

The gear-rolling system using local induction heating is shown in Figure10. The synchronization issue with the rotation of the two gear dies is resolved by using a set of spiral bevel gears with a tooth ratio of 4.333 without gear backlash as the guiding synchronization mechanism. The blank is inductively heated using the coil, and after it reached the desired temperature, the blank started rolling. A numerical control is used to control the blank rotation speed. When the two gear dies come into contact, the blank and gear dies rotate at a certain speed ratio until the entire rolling process is finished. The two gear dies move toward the direction of the blank at the same speed.Figure 8
Tooth surface defects in spiral bevel gears. (a) Overall heating and (b) local induction heating.Figure 9
The gear cross-section. (a) Overall heating cross-section. (b) Local induction heating cross-section.
(a)(b)Figure 10
The equipment of rolling forming with local induction heating.Figure11 shows the formed pinion obtained on the local induction heating equipment. The formed pinion is similar to the numerical simulation results within a certain error range. The defect analysis of the pinion based on local induction heating can not only improve the forming quality of the target gear but also avoid the adverse effects of other defects caused by the local deformation of the inner hole and the keyway slot.Figure 11
The formed pinion after the tooth toe-end and heel-end machining.


## 4. Results and Discussion

### 4.1. Analysis of Gear Tooth-Forming Defects

#### 4.1.1. Tooth Top Defect Analysis

During the rolling process of spiral bevel gears, there is a “rabbit ear” defect, as shown in Figure12, which means that the two sides of the tooth are higher than the middle ones. At a later stage of rolling, when the top of the target gear is in contact with the root of the gear die, a “folding” defect is formed at the top of the tooth. These two defects significantly affect the forming quality of the gear teeth, as shown in Figure 13.Figure 12
Equivalent strain distribution cloud of the pinion.Figure 13
Tooth defect type and effective tooth height of the forming pinion. (a, c) Rabbit ear defects. (b, d) Folding defects.
(a)(b)(c)(d)When the thickness of the tooth is large, it generally forms a “rabbit ear” defect, as shown in Figures13(a) and 13(c). The reversal direction changes in each rolling cycle and generally forms defects, as shown in Figure 13(a). The gear die rotates in a single direction, which generally results in defects, as shown in Figure 13(c). When the thickness is relatively small, the left and right ends of the teeth are more likely to come in contact with each other and form “fold” defects.In contrast to the straight cylindrical gear tooth structure, spiral bevel gears exist in the tooth heel-end and tooth toe-end of the gear teeth, and the tooth heel-end of the tooth thickness is greater than that of the tooth toe-end. As the tooth thickness increases, the distance between the two ends of the tooth-pulling tip position increases, and the left and right ends of the rabbit ear are not easily in contact with each other. Then, the tooth heel-end generally forms rabbit ear defects easily, as shown in Figures 13(a) and 13(c). Thinner tooth thickness at the tooth toe-end is more likely to form folding defects, as shown in Figures 13(b) and 13(d). To better illustrate the formation of the entire tooth of the spiral bevel gear, four sections perpendicular to the axis were taken along the direction of the width of the spiral bevel gear teeth at intervals of 4 mm, as shown in Figure14. Figure 15 shows the gear teeth formation of the four cross-sections after roll forming, which can be seen from the middle of the spiral bevel gear tooth width near the tooth toe-end of the cross-section of the top of the tooth-folding defects and the middle of the tooth width near the tooth heel-end of the location of the rabbit ear defects.Figure 14
The cutting positions of the blank.Figure 15
Tooth top defects of the pinion. (a) 4 mm. (b) 8 mm. (c) 12 mm. (d) 16 mm.
(a)(b)(c)(d)


#### 4.1.2. Effective Tooth Heightha

During the rolling process, the material mainly flows in the radial direction to form the tooth profile, but there is some material movement along the axis near the tooth heel-end and tooth toe-end. The axial flow of the material not only reduces the forming tooth profile by radial flow but also forms flaps on both the end faces of the teeth, as shown in Figure15. The spiral bevel gear has an angle between the generatrix and the axis, and it is difficult to restrict the axial flow of the material by installing baffles on the mold. Therefore, this study investigated the deformation law of the material during the rolling process to improve the forming quality of the target gear.Five points, such asP1, P2, P3, P4, and P5, are taken at equal intervals from the tooth toe-end to the tooth heel-end to track and observe their strains, as shown in Figure 16. Here, the maximum strain in the middle part is 25 mm/mm, which is larger than the two ends. The strain in the tooth toe-end was larger than that in the tooth heel-end. The axial flow of the material and the tooth thickness have a significant influence on the strains of P1, P2, P3, P4, and P5 when the blank is constant. The end flow of the material in the middle part of the gear teeth is weaker than that in the tooth heel-end and tooth toe-end; the strain at this position is greater than that at the two ends. Therefore, during the rolling process of the spiral bevel gear, the tooth tip of the middle part of the blank first comes in contact with the tooth root of the die, and the height of the tooth increases the fastest. Material axial flow exists at both the tooth heel-end and tooth toe-end of a spiral bevel gear. The tooth thickness of the tooth toe-end is less than that of the tooth heel-end, which increases the height of the tooth toe-end faster; thus, the strain at the tooth toe-end is greater than that at the tooth heel-end.Figure 16
Effective strain in different positions of one tooth.
(a)(b)


### 4.2. Optimized Pinion Blanks

In the rolling process, assuming that the metal is incompressible, according to the equal volume method, the tooth heel-end size of the pinion blank can be obtained. There is a taper between the tooth heel-end and tooth toe-end of the spiral bevel gear blank. If the cone angle of the blank is too large, the tooth height at tooth toe-end will grow slower than expected. If the cone angle is too small, the tooth heel-end tooth height will grow slower and vice versa. According to previous experience, the forming effect of the target gear is better when the taper is near the pitch angle.As shown in Figure18, under the influence of end flow, when the material of the gear die extruding blank is A2, the material flowing to the direction of the target gear tooth height is A1. According to Formulas (16), (20), and (21), A2 can be obtained and A1 can be obtained by UG measurement. k is the material compensation coefficient. Figure 19 shows the material compensation coefficient k at different positions from the tooth heel-end to the tooth toe-end.(16)r2=θg,ϕg,(17)X=xθg,ϕg,(18)Y=yθg,ϕg,(19)Z=zθg,ϕg,(20)Z=fX,Y,(21)A2=∬f2X,Ydσ,(22)A2=A1+k.Figure 17
Effective tooth height from the tooth heel-end to the toe-end.Figure 18
Forming process of one tooth.Figure 19
The coefficient of compensation from the tooth heel-end to the toe-end.From the abovementioned analysis, it can be seen that, in order to change the folding degree and effective tooth heightha of the different areas of the gear teeth, the strain at different positions of the gear teeth can be changed by changing the shape of the blank, thereby improving its quality. Figure 20(a) shows the change in the taper of the tooth heel-end and tooth toe-end to adjust their effective tooth height ha. Figure 20(c) shows the change in the transition curve between the tooth heel-end and the tooth toe-end. This is to compensate for the material loss caused by the end flow and to adjust the strain of the intermediate material to make the material deformation more uniform during the rolling process. Figure 20(b) shows the height of the tooth heel-end of the pinion blank, which is adjusted to ensure that the volume of the blank is slightly larger than the volume of the target gear. Table 4 gives different values for the external geometric parameters of the pinion blank.Figure 20
The optimized shape of the pinion blank. (a) The taper between the tooth heel-end and tooth toe-end end of the blank. (b) The height of the tooth heel-end of the pinion blank. (c) The transition curve between the tooth heel-end and the tooth toe end.
(a)(b)(c)Figure17 shows that the effective height ha of the tooth heel-end, middle, and tooth toe-end of the spiral bevel gear after formation is measured. As shown in Figure 17, the effective tooth height at the middle part of the spiral bevel gear is greater than that at the two ends. The effective tooth height and the taper (slope) between the tooth heel-end and tooth toe-end are all smaller than the theoretical value, which is consistent with the direction P1 >P5 in Figure 16(a).Table 4
Geometric parameters of the blank.
L (°)d (mm)R (mm)11.38600.531.512.9946131.5151.531.5In this experiment, the effective tooth height of the spiral bevel gear is used as the research basis, and the optimal blank parameters are determined based on the degree of agreement between the effective tooth heightha and the theoretical tooth height under the different blank parameters L and d. The R value affects the tip circle diameter of the tooth heel-end and is an extremely important parameter in the rolling process. In this experiment, the effective height of the spiral bevel gear is studied. The size of the R does not affect the effective tooth height of the spiral bevel gear; therefore, it is set as a fixed value.Nine different blank parameters are selected for the numerical simulation, and the effective tooth heights of the tooth heel-end, middle end, and tooth toe-end of the target spiral bevel gear are measured. As shown in Figure21, by optimizing the shape of the blank, the phenomenon in which the effective height does not match the theoretical value is avoided to a certain extent, which is caused by the large difference in material strain between the tooth heel-end and tooth toe-end.Figure 21
Effective tooth height of rolling forming gear teeth. (a) Effective tooth height of forming gear teeth under different optimization parameters. (b) The tooth height of the two schemes.
(a)(b)There are two main types of tooth surface defects in the near-net rolling forming process of the pinion: tooth surface folding and tooth surface scraping. In Figure 8, it is visible that local induction can contribute to improving the quality of the tooth surface to a certain extent, but as shown in Figure23(a), there are still some folding defects on the tooth surface. Compared with Figures 23(b) and 23(c), it can be seen that the numerical simulation of the folding and scratching of the tooth surface are better than the actual machining of the tooth surface, which may be due to the following reasons: (1) In the actual processing process, with the increase in temperature, the blank and oxygen in the air undergo a chemical reaction; (2) there is wear on the surface of the die wheel, or there is oxidation on the surface of the die during rolling; (3) there are process errors in the use of the equipment.Figure 22
Effective strains in different positions of one teeth.Figure 23
Surface quality of the pinion teeth. (a) Tooth surface scratches and top folding defects. (b) Tooth surface folding defect. (c) Flash formed by end flow of metal material.
(a)(b)(c)From Figure21(a), it can be seen that, when L = 15° and d = 1 mm, the taper of the tooth heel-end and tooth toe-end is almost parallel to the theoretical value, and the effective tooth height ha at the middle part of the gear is more reasonable. Figures 21(b) and 22 show the comparison of the tooth shape before and after optimization. It can be seen that the rolling experiments on the blanks after optimization can make deformation from the tooth heel-end and tooth toe-end more uniform. The effective tooth height of the target gear tends to be parallel to the theoretical values of L = 15° and d = 1 mm. The difference in height can be improved by deepening the root of the die to accommodate more rabbit and folded defects on the top part of the target gear teeth and by increasing the effective height of the target gear. When the effective height reaches the theoretical value, the defects at the top of the tooth can be removed by machining.


#### 4.2.1. Rolling Force

The magnitude of the radial feed force in the roll-forming process is mainly related to the temperature and contact area. By changing the transition curve of the spiral bevel gear, as shown in Figure24(a), the contact area between the gear die and target gear is reduced, thereby reducing the feed force of the die and increasing its service life. Figure 24 shows the transition curve that reduces the rolling force of the pinion roll forming.Figure 24
Radial-rolling force. (a) Pinion blank parameters:L = 15°, d = 0 mm, and R = 31.5 mm. (b) Pinion blank parameters: L = 15°, d = 1 mm, and R = 31.5 mm.
(a)(b)In the initial stage, the die gear only comes into contact with the tooth heel-end and tooth toe-end of the blank, which reduces the contact area between the die and the blank, thereby reducing the rolling force. In the middle stages of gear rolling, a certain degree of fluctuation is exhibited between the rolling forces of the two blanks, with a significant difference in the fluctuation degree between them. The peak value of Figure24(a) is approximately 3×105N and that of Figure 24(b) is approximately 3.7×104N. In the later stage of roll forming, there is relatively less excess material in the contact between the tooth tip at the center of the modified blank and the tooth root of the die blank, and no large stress is generated at the tooth root of the die.


## 4.1. Analysis of Gear Tooth-Forming Defects

### 4.1.1. Tooth Top Defect Analysis

During the rolling process of spiral bevel gears, there is a “rabbit ear” defect, as shown in Figure12, which means that the two sides of the tooth are higher than the middle ones. At a later stage of rolling, when the top of the target gear is in contact with the root of the gear die, a “folding” defect is formed at the top of the tooth. These two defects significantly affect the forming quality of the gear teeth, as shown in Figure 13.Figure 12
Equivalent strain distribution cloud of the pinion.Figure 13
Tooth defect type and effective tooth height of the forming pinion. (a, c) Rabbit ear defects. (b, d) Folding defects.
(a)(b)(c)(d)When the thickness of the tooth is large, it generally forms a “rabbit ear” defect, as shown in Figures13(a) and 13(c). The reversal direction changes in each rolling cycle and generally forms defects, as shown in Figure 13(a). The gear die rotates in a single direction, which generally results in defects, as shown in Figure 13(c). When the thickness is relatively small, the left and right ends of the teeth are more likely to come in contact with each other and form “fold” defects.In contrast to the straight cylindrical gear tooth structure, spiral bevel gears exist in the tooth heel-end and tooth toe-end of the gear teeth, and the tooth heel-end of the tooth thickness is greater than that of the tooth toe-end. As the tooth thickness increases, the distance between the two ends of the tooth-pulling tip position increases, and the left and right ends of the rabbit ear are not easily in contact with each other. Then, the tooth heel-end generally forms rabbit ear defects easily, as shown in Figures 13(a) and 13(c). Thinner tooth thickness at the tooth toe-end is more likely to form folding defects, as shown in Figures 13(b) and 13(d). To better illustrate the formation of the entire tooth of the spiral bevel gear, four sections perpendicular to the axis were taken along the direction of the width of the spiral bevel gear teeth at intervals of 4 mm, as shown in Figure14. Figure 15 shows the gear teeth formation of the four cross-sections after roll forming, which can be seen from the middle of the spiral bevel gear tooth width near the tooth toe-end of the cross-section of the top of the tooth-folding defects and the middle of the tooth width near the tooth heel-end of the location of the rabbit ear defects.Figure 14
The cutting positions of the blank.Figure 15
Tooth top defects of the pinion. (a) 4 mm. (b) 8 mm. (c) 12 mm. (d) 16 mm.
(a)(b)(c)(d)


### 4.1.2. Effective Tooth Heightha

During the rolling process, the material mainly flows in the radial direction to form the tooth profile, but there is some material movement along the axis near the tooth heel-end and tooth toe-end. The axial flow of the material not only reduces the forming tooth profile by radial flow but also forms flaps on both the end faces of the teeth, as shown in Figure15. The spiral bevel gear has an angle between the generatrix and the axis, and it is difficult to restrict the axial flow of the material by installing baffles on the mold. Therefore, this study investigated the deformation law of the material during the rolling process to improve the forming quality of the target gear.Five points, such asP1, P2, P3, P4, and P5, are taken at equal intervals from the tooth toe-end to the tooth heel-end to track and observe their strains, as shown in Figure 16. Here, the maximum strain in the middle part is 25 mm/mm, which is larger than the two ends. The strain in the tooth toe-end was larger than that in the tooth heel-end. The axial flow of the material and the tooth thickness have a significant influence on the strains of P1, P2, P3, P4, and P5 when the blank is constant. The end flow of the material in the middle part of the gear teeth is weaker than that in the tooth heel-end and tooth toe-end; the strain at this position is greater than that at the two ends. Therefore, during the rolling process of the spiral bevel gear, the tooth tip of the middle part of the blank first comes in contact with the tooth root of the die, and the height of the tooth increases the fastest. Material axial flow exists at both the tooth heel-end and tooth toe-end of a spiral bevel gear. The tooth thickness of the tooth toe-end is less than that of the tooth heel-end, which increases the height of the tooth toe-end faster; thus, the strain at the tooth toe-end is greater than that at the tooth heel-end.Figure 16
Effective strain in different positions of one tooth.
(a)(b)


## 4.1.1. Tooth Top Defect Analysis

During the rolling process of spiral bevel gears, there is a “rabbit ear” defect, as shown in Figure12, which means that the two sides of the tooth are higher than the middle ones. At a later stage of rolling, when the top of the target gear is in contact with the root of the gear die, a “folding” defect is formed at the top of the tooth. These two defects significantly affect the forming quality of the gear teeth, as shown in Figure 13.Figure 12
Equivalent strain distribution cloud of the pinion.Figure 13
Tooth defect type and effective tooth height of the forming pinion. (a, c) Rabbit ear defects. (b, d) Folding defects.
(a)(b)(c)(d)When the thickness of the tooth is large, it generally forms a “rabbit ear” defect, as shown in Figures13(a) and 13(c). The reversal direction changes in each rolling cycle and generally forms defects, as shown in Figure 13(a). The gear die rotates in a single direction, which generally results in defects, as shown in Figure 13(c). When the thickness is relatively small, the left and right ends of the teeth are more likely to come in contact with each other and form “fold” defects.In contrast to the straight cylindrical gear tooth structure, spiral bevel gears exist in the tooth heel-end and tooth toe-end of the gear teeth, and the tooth heel-end of the tooth thickness is greater than that of the tooth toe-end. As the tooth thickness increases, the distance between the two ends of the tooth-pulling tip position increases, and the left and right ends of the rabbit ear are not easily in contact with each other. Then, the tooth heel-end generally forms rabbit ear defects easily, as shown in Figures 13(a) and 13(c). Thinner tooth thickness at the tooth toe-end is more likely to form folding defects, as shown in Figures 13(b) and 13(d). To better illustrate the formation of the entire tooth of the spiral bevel gear, four sections perpendicular to the axis were taken along the direction of the width of the spiral bevel gear teeth at intervals of 4 mm, as shown in Figure14. Figure 15 shows the gear teeth formation of the four cross-sections after roll forming, which can be seen from the middle of the spiral bevel gear tooth width near the tooth toe-end of the cross-section of the top of the tooth-folding defects and the middle of the tooth width near the tooth heel-end of the location of the rabbit ear defects.Figure 14
The cutting positions of the blank.Figure 15
Tooth top defects of the pinion. (a) 4 mm. (b) 8 mm. (c) 12 mm. (d) 16 mm.
(a)(b)(c)(d)


## 4.1.2. Effective Tooth Heightha

During the rolling process, the material mainly flows in the radial direction to form the tooth profile, but there is some material movement along the axis near the tooth heel-end and tooth toe-end. The axial flow of the material not only reduces the forming tooth profile by radial flow but also forms flaps on both the end faces of the teeth, as shown in Figure15. The spiral bevel gear has an angle between the generatrix and the axis, and it is difficult to restrict the axial flow of the material by installing baffles on the mold. Therefore, this study investigated the deformation law of the material during the rolling process to improve the forming quality of the target gear.Five points, such asP1, P2, P3, P4, and P5, are taken at equal intervals from the tooth toe-end to the tooth heel-end to track and observe their strains, as shown in Figure 16. Here, the maximum strain in the middle part is 25 mm/mm, which is larger than the two ends. The strain in the tooth toe-end was larger than that in the tooth heel-end. The axial flow of the material and the tooth thickness have a significant influence on the strains of P1, P2, P3, P4, and P5 when the blank is constant. The end flow of the material in the middle part of the gear teeth is weaker than that in the tooth heel-end and tooth toe-end; the strain at this position is greater than that at the two ends. Therefore, during the rolling process of the spiral bevel gear, the tooth tip of the middle part of the blank first comes in contact with the tooth root of the die, and the height of the tooth increases the fastest. Material axial flow exists at both the tooth heel-end and tooth toe-end of a spiral bevel gear. The tooth thickness of the tooth toe-end is less than that of the tooth heel-end, which increases the height of the tooth toe-end faster; thus, the strain at the tooth toe-end is greater than that at the tooth heel-end.Figure 16
Effective strain in different positions of one tooth.
(a)(b)


## 4.2. Optimized Pinion Blanks

In the rolling process, assuming that the metal is incompressible, according to the equal volume method, the tooth heel-end size of the pinion blank can be obtained. There is a taper between the tooth heel-end and tooth toe-end of the spiral bevel gear blank. If the cone angle of the blank is too large, the tooth height at tooth toe-end will grow slower than expected. If the cone angle is too small, the tooth heel-end tooth height will grow slower and vice versa. According to previous experience, the forming effect of the target gear is better when the taper is near the pitch angle.As shown in Figure18, under the influence of end flow, when the material of the gear die extruding blank is A2, the material flowing to the direction of the target gear tooth height is A1. According to Formulas (16), (20), and (21), A2 can be obtained and A1 can be obtained by UG measurement. k is the material compensation coefficient. Figure 19 shows the material compensation coefficient k at different positions from the tooth heel-end to the tooth toe-end.(16)r2=θg,ϕg,(17)X=xθg,ϕg,(18)Y=yθg,ϕg,(19)Z=zθg,ϕg,(20)Z=fX,Y,(21)A2=∬f2X,Ydσ,(22)A2=A1+k.Figure 17
Effective tooth height from the tooth heel-end to the toe-end.Figure 18
Forming process of one tooth.Figure 19
The coefficient of compensation from the tooth heel-end to the toe-end.From the abovementioned analysis, it can be seen that, in order to change the folding degree and effective tooth heightha of the different areas of the gear teeth, the strain at different positions of the gear teeth can be changed by changing the shape of the blank, thereby improving its quality. Figure 20(a) shows the change in the taper of the tooth heel-end and tooth toe-end to adjust their effective tooth height ha. Figure 20(c) shows the change in the transition curve between the tooth heel-end and the tooth toe-end. This is to compensate for the material loss caused by the end flow and to adjust the strain of the intermediate material to make the material deformation more uniform during the rolling process. Figure 20(b) shows the height of the tooth heel-end of the pinion blank, which is adjusted to ensure that the volume of the blank is slightly larger than the volume of the target gear. Table 4 gives different values for the external geometric parameters of the pinion blank.Figure 20
The optimized shape of the pinion blank. (a) The taper between the tooth heel-end and tooth toe-end end of the blank. (b) The height of the tooth heel-end of the pinion blank. (c) The transition curve between the tooth heel-end and the tooth toe end.
(a)(b)(c)Figure17 shows that the effective height ha of the tooth heel-end, middle, and tooth toe-end of the spiral bevel gear after formation is measured. As shown in Figure 17, the effective tooth height at the middle part of the spiral bevel gear is greater than that at the two ends. The effective tooth height and the taper (slope) between the tooth heel-end and tooth toe-end are all smaller than the theoretical value, which is consistent with the direction P1 >P5 in Figure 16(a).Table 4
Geometric parameters of the blank.
L (°)d (mm)R (mm)11.38600.531.512.9946131.5151.531.5In this experiment, the effective tooth height of the spiral bevel gear is used as the research basis, and the optimal blank parameters are determined based on the degree of agreement between the effective tooth heightha and the theoretical tooth height under the different blank parameters L and d. The R value affects the tip circle diameter of the tooth heel-end and is an extremely important parameter in the rolling process. In this experiment, the effective height of the spiral bevel gear is studied. The size of the R does not affect the effective tooth height of the spiral bevel gear; therefore, it is set as a fixed value.Nine different blank parameters are selected for the numerical simulation, and the effective tooth heights of the tooth heel-end, middle end, and tooth toe-end of the target spiral bevel gear are measured. As shown in Figure21, by optimizing the shape of the blank, the phenomenon in which the effective height does not match the theoretical value is avoided to a certain extent, which is caused by the large difference in material strain between the tooth heel-end and tooth toe-end.Figure 21
Effective tooth height of rolling forming gear teeth. (a) Effective tooth height of forming gear teeth under different optimization parameters. (b) The tooth height of the two schemes.
(a)(b)There are two main types of tooth surface defects in the near-net rolling forming process of the pinion: tooth surface folding and tooth surface scraping. In Figure 8, it is visible that local induction can contribute to improving the quality of the tooth surface to a certain extent, but as shown in Figure23(a), there are still some folding defects on the tooth surface. Compared with Figures 23(b) and 23(c), it can be seen that the numerical simulation of the folding and scratching of the tooth surface are better than the actual machining of the tooth surface, which may be due to the following reasons: (1) In the actual processing process, with the increase in temperature, the blank and oxygen in the air undergo a chemical reaction; (2) there is wear on the surface of the die wheel, or there is oxidation on the surface of the die during rolling; (3) there are process errors in the use of the equipment.Figure 22
Effective strains in different positions of one teeth.Figure 23
Surface quality of the pinion teeth. (a) Tooth surface scratches and top folding defects. (b) Tooth surface folding defect. (c) Flash formed by end flow of metal material.
(a)(b)(c)From Figure21(a), it can be seen that, when L = 15° and d = 1 mm, the taper of the tooth heel-end and tooth toe-end is almost parallel to the theoretical value, and the effective tooth height ha at the middle part of the gear is more reasonable. Figures 21(b) and 22 show the comparison of the tooth shape before and after optimization. It can be seen that the rolling experiments on the blanks after optimization can make deformation from the tooth heel-end and tooth toe-end more uniform. The effective tooth height of the target gear tends to be parallel to the theoretical values of L = 15° and d = 1 mm. The difference in height can be improved by deepening the root of the die to accommodate more rabbit and folded defects on the top part of the target gear teeth and by increasing the effective height of the target gear. When the effective height reaches the theoretical value, the defects at the top of the tooth can be removed by machining.


### 4.2.1. Rolling Force

The magnitude of the radial feed force in the roll-forming process is mainly related to the temperature and contact area. By changing the transition curve of the spiral bevel gear, as shown in Figure24(a), the contact area between the gear die and target gear is reduced, thereby reducing the feed force of the die and increasing its service life. Figure 24 shows the transition curve that reduces the rolling force of the pinion roll forming.Figure 24
Radial-rolling force. (a) Pinion blank parameters:L = 15°, d = 0 mm, and R = 31.5 mm. (b) Pinion blank parameters: L = 15°, d = 1 mm, and R = 31.5 mm.
(a)(b)In the initial stage, the die gear only comes into contact with the tooth heel-end and tooth toe-end of the blank, which reduces the contact area between the die and the blank, thereby reducing the rolling force. In the middle stages of gear rolling, a certain degree of fluctuation is exhibited between the rolling forces of the two blanks, with a significant difference in the fluctuation degree between them. The peak value of Figure24(a) is approximately 3×105N and that of Figure 24(b) is approximately 3.7×104N. In the later stage of roll forming, there is relatively less excess material in the contact between the tooth tip at the center of the modified blank and the tooth root of the die blank, and no large stress is generated at the tooth root of the die.


## 4.2.1. Rolling Force

The magnitude of the radial feed force in the roll-forming process is mainly related to the temperature and contact area. By changing the transition curve of the spiral bevel gear, as shown in Figure24(a), the contact area between the gear die and target gear is reduced, thereby reducing the feed force of the die and increasing its service life. Figure 24 shows the transition curve that reduces the rolling force of the pinion roll forming.Figure 24
Radial-rolling force. (a) Pinion blank parameters:L = 15°, d = 0 mm, and R = 31.5 mm. (b) Pinion blank parameters: L = 15°, d = 1 mm, and R = 31.5 mm.
(a)(b)In the initial stage, the die gear only comes into contact with the tooth heel-end and tooth toe-end of the blank, which reduces the contact area between the die and the blank, thereby reducing the rolling force. In the middle stages of gear rolling, a certain degree of fluctuation is exhibited between the rolling forces of the two blanks, with a significant difference in the fluctuation degree between them. The peak value of Figure24(a) is approximately 3×105N and that of Figure 24(b) is approximately 3.7×104N. In the later stage of roll forming, there is relatively less excess material in the contact between the tooth tip at the center of the modified blank and the tooth root of the die blank, and no large stress is generated at the tooth root of the die.


## 5. Conclusion

The use of near-net rolling forming technology to process the pinion of spiral bevel gears conforms to the category of green manufacturing. Through numerical simulations and experiments, it was verified that local heating could improve the tooth surface quality. Simultaneously, an in-depth analysis of the tooth shape defects existing in the pinion during roll forming was carried out. The main research results are as follows:(1)
The high-temperature deformation behaviour of gear steel 20CrMnTiH is studied experimentally to provide a basis for numerical simulations.(2)
The overall heating of spiral bevel gear pinion blanks and the local induction heating of pinion blanks are studied experimentally, and it is found that local induction heating could improve the tooth surface quality by comparison.(3)
Through experiments, it is found that after the spiral bevel gear is roll-formed, the tooth heel-end and tooth toe-end tooth tops will have “pulling tip” and “folding” defects, respectively. Because of the axial flow of the material, the tooth heel-end and tooth toe-end of the pinion will have an insufficient tooth top filling, and the effective tooth height will be lower than that of the theoretical value. For the abovementioned forming defects, an optimization method for the spiral bevel gear blank is proposed, and the suppression mechanism of defect formation is analyzed by the numerical simulation, which provides a theoretical basis for perfecting the roll forming of the pinion.



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*Source: 8920567-2023-01-11.xml*