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Investigation on gear rolling process using conical gear rollers and design method of the conical gear roller
Journal of Materials Processing Tech. 259 (2018) 141–149
Contents lists available at ScienceDirect
Journal of Materials Processing Tech. journal homepage: www.elsevier.com/locate/jmatprotec
Investigation on gear rolling process using conical gear rollers and design method of the conical gear roller
T
⁎
Tao Wua, Guangchun Wanga, , Jin Lib, Ke Yana a b
Key Laboratory for Liquid-Solid Structural Evolution and Processing of Materials (Ministry of Education), Shandong University, Jinan, Shandong 250061, PR China School of Mechanical and Electrical Engineering, Shandong Jianzhu University, Jinan, Shandong 250101, PR China
A R T I C LE I N FO
A B S T R A C T
Keywords: Gear Rolling Conical gear roller Structure design
A type of gear rolling process using conical gear rollers was firstly proposed in this study. Different with the cylindrical gear rollers used in the traditional gear rolling process, the roller which looks like a conical gear with different tooth profile along the axial direction was adopted in the rolling process. Besides, the radial feed applied on the roller in traditional rolling process was replaced by the axial feed applied on the blank. The proposed rolling process was divided into three stages, viz., tooth graduation, tooth forming and tooth finishing. According to the function of the roller in each stage, the structure of the conical gear roller was designed. Importantly, the detailed formulas to calculate the essential parameters of the gear roller’s tooth in tooth forming stage were established, such as the diameter of addendum circle, the radius of addendum tip, the pressure angle, and the cone angle. Through the numerical simulation, the variation and forming mechanism of the tooth profile at different stages were analyzed. The results revealed that the proposed process achieves better uniform tooth graduation and refining of the tooth profile, which increase the forming accuracy of the tooth to some extent in rolling process. Moreover, the experimental device was developed and the feasibility of the conical gear rolling process was verified.
1. Introduction Gears are widely used as mechanical components which play the role of transmitting motion and power. The forming precision and quality of gears are of great importance to improve the performance of mechanical equipment. Cutting is a traditional method to produce gears, and it can be used to produce most of the gears. Based on the type and size of gear, the selection of an appropriate cutting process can improve the productivity and forming precision. However, the gear cutting is a way of material removal, which causes a large amount of material waste. The rolling process is another economic alternative method of producing highly exact gears, which has attracted much attention from researchers all over the world. Based on the principle of gear meshing, the roller should be designed to have similar shape of the gear. During the rolling process, the gear roller contacts and extrudes blank with a feed in the radial direction. The materials at the outer part of the blank gradually form the tooth shape through its generating motion with the gear roller, as shown in Fig. 1. This kind of precision forming technology has many advantages, such as high material utilization, high productivity and small forming force. The formed gears also have better
⁎
mechanical properties than those obtained from cutting process. However, the complexity of gear rolling process brings great difficulty to the theoretical analysis, parameter calculation and numerical simulation. It is difficult to find a reliable and effective solution to avoid defects in the gear rolling process. In practical production, the process parameters of gear rolling are mainly determined based on the experiences, as well as trial and error methods. It not only lacks scientific basis, but also wastes raw materials and prolongs the production cycle (Klepikov and Bodrov, 2003). In recent years, the research on gear rolling have been carried out by means of experimental or numerical methods. (Kamouneh et al. (2007a, b) studied the involute helical gear rolling process by means of numerical simulation, and explained the enhancement of gear rolling by analyzing changes in grain on the surface of the gear. Neugebauer et al. (2007a, b, 2008) firstly designed a gear roller with variable pitch, established the rolling model, and came to a conclusion that the forming accuracy could be improved by 50%. Pater et al. (2011) simulated the rolling process with DEFORM-3D, proved the feasibility of the gear rolling process, and analyzed the distribution of the temperature field, stress field and rolling force during the rolling process. Yu et al. (2011) presented a method which formed the shaft and the teeth
Corresponding author. E-mail address: [email protected] (G. Wang).
https://doi.org/10.1016/j.jmatprotec.2018.04.034 Received 7 November 2017; Received in revised form 28 March 2018; Accepted 20 April 2018 Available online 22 April 2018 0924-0136/ © 2018 Elsevier B.V. All rights reserved.
Journal of Materials Processing Tech. 259 (2018) 141–149
T. Wu et al.
Fig. 1. Schematic diagram of gear rolling.
get larger and larger along axial direction. In the last tooth finishing stage, the size of the gear roller’s addendum decreases to the standard size gradually along axial direction, while blank changes into meshing motion with the gear rollers, and the outer achieves the objective gear tooth eventually. Based on the above three deformation stages, the design methods and formulas of gear roller structure parameters in each stage were presented in this paper. Moreover, the paper verified the rationality of the proposed process and design through the method of numerical simulation and experiment.
at the same time. Alireza and Melander (2013) provided a method using finite element simulation as a tool to evaluate gear quality after gear rolling. Kadashevich et al. (2015) investigated the thermal distribution and geometric deviations in gear rolling. Brecher et al. (2015) employed regression analysis to design the optimization of gear hobbing processes with different gear specifications based on numerical simulations. Li et al. (2016, 2017) discussed the slipping phenomenon and the formation mechanism of rabbit ear in gear rolling using numerical simulation and experimental methods. In traditional gear rolling process, a standard gear is always used as the roller to extrude the blank with a feed in radial direction. In this way, the radial clearance could not be cut out because of the shortage of working depth, which results into a larger dedendum circle in the objective gear and a lower dimensional precision. Moreover, the rabbit ear defect cannot be avoided. In the present study, to solve the aforementioned problems in traditional gear rolling process, a novel method is firstly proposed by a rolling feed in the axial direction instead of the radial direction in the traditional methods. The conical gear roller with different pitches in axial direction is shown in Fig. 2. The distance between the axis of roller and blank remains constant, and the rotation speed ratio of roller to blank is always unchanged during the whole rolling process. The rolling process is divided into three stages. In the first tooth graduation stage, the blank is pushed into the die along axial direction through an entry angle structure, and then the uniform tooth graduation is realized. In the second tooth forming stage, the blank is formed with gear roller which looks like a conical gear, afterwards the forming tooth shape will
2. Design method of conical gear roller In order to ensure the forming precision, the tooth number of gear roller should be as much as possible. Considering the space limitations, the distance between the axial centers of paired gear rollers is supposed to be less than the one of rolling equipment. The maximum tooth number of zmax can be obtained by Eq. (1) (He, 2001), and the result should be translated to integer.
zmax = (cmax − 2h 0 − dz )/ m − (3 ∼ 5)
(1)
where, cmax is the maximum distance between the axial centers of the paired gear roller, h0 is the whole dedendum depth of the objective gear, dz is the diameter of the objective gear, m is the module. The structural parameters of the conical gear roller are respectively determined according to the three different deformation stages. In the first tooth graduation stage, an entry angle structure is designed in order to ensure that the blank extrudes into the die successfully. The radius of addendum circle at the entrance ra’, the blank’s radius rz, and the distance between the axial center of gear roller and blank a should meet the relationship of ra’ + rz < a. The blank’s radius rz can be determined by equal volume of the blank before and after rolling process. Besides, there is a minimum feed in order to realize uniform tooth graduation, which means that the addendum circle of the gear roller has a minimum radius. The minimum radius of addendum circle is calculated as follows: β
a2 + rz2 − 2arz cos 2
⎧ ra2 ≥ ⎨β = ⎩
360 z
(2)
where, ra2 is the minimum radius of addendum circle in tooth graduation stage, β is the radius angle corresponding chord length of blank, z is the tooth number of the objective gear. In the second tooth forming stage, the roller is designed as conical shape with teeth which have increasingly higher addendum along axial direction. The gear roller can be divided into large and small end face which are called heel and toe in bevel gear, respectively. The toe connects with the roller in tooth graduation stage. The radius of the addendum circle and the tooth profile at heel should be determined. In order to rolling the clearance on the objective gear, the
Fig. 2. Schematic diagram of the conical gear rolling method. 142
Journal of Materials Processing Tech. 259 (2018) 141–149
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Fig. 4. Schematic diagram of the gear roller with cone angle in tooth forming stage. Fig. 3. Schematic diagram of calculating the minimum tip of the addendum.
Then, the pressure angle in pitch circle of the gear roller can be calculated by considering Eqs. (8) and (9). The cone angle of the gear roller in tooth forming stage has an important influence on the metal flow and forming load during gear rolling process, and thus affects the objective gear’s forming quality and the design of rolling device. Fig. 4 shows the gear roller with cone angle in tooth forming stage. The relation between the axial feed distance, the radial feed distance and the cone angle is obtained from Fig. 4 as follows:
addendum of gear roller at heel is more than that of standard gear c*m, where c* is the coefficient of addendum’s radial clearance, m is module. The formula for calculating the diameter of gear roller is
da1 = m1 (z 0 + 2ha* + 2c *)
(3)
where, da1 is the diameter of the gear roller at heel, m1 is the module of the gear roller, z0 is tooth number of the gear roller, ha* is the coefficient of addendum. The addendum of the gear roller higher than standard can lead to undercut because the extension epicycloid envelope formed by the addendum angle of the gear roller invades the dedendum transition curve of the objective gear. A round corner on the top of tooth, usually called tip, which can increase involute start point of the objective gear, is designed at the addendum of the gear roller. There is a minimum tip for avoiding undercut when the end point of addendum on the gear roller’s involute coincides with the involute start point of the objective gear, as shown in Fig. 3. In Fig. 3, △AOD is a right triangle, and the relationship of r’a12=rb12 + AD2 is followed, where r’a1 is the radius of the involute end point of the gear roller with tip, rb1 is the base radius at the heal of the gear roller in tooth forming. Moreover, the values of r’a1 and AD can be calculated using the following equations.
r ′a1 = a − rb
(4)
AD2 = (CD + rc )2
(5)
CD =
(ra1 − rc )2 − rb21
tan μ =
vb =
(10)
b vl l
(11)
where, vb is the radial feed speed; vl is the axial feed speed. From Eqs. (10) and (11), the cone angle μ is inversely proportional to the axial feed distance and the radial effective rolling force, but proportional to the radial feed speed and the rolling efficiency. Hence, the range of its value should consider various factors such as forming quality, rolling efficiency, equipment capability and installation space. In the final tooth finishing stage, gear roller needs a radial fallback movement to correct the tooth profile of the objective gear. That means the meshing state between roller and blank gradually changes from without clearance in tooth forming stage to with clearance under gear drive (Table 1).
(6)
Table 1 Parameters of the gear roller in numerical simulation.
(7)
With the continuous rolling process, the pressure and friction force on the objective gear tooth are varied, which results in smaller pressure angle in pitch circle of the objective gear than the expected value. To correct the reduction, the pitch pressure angles of gear roller α0 and objective gear α should meet the formula as follows:
α 0 = α − Δα
b l
where, μ is the cone angle in tooth forming stage, b is the radial feed distance, l is the axial feed distance. The relation between the gear roller’s axial feed speed and radial feed speed is
where, a is the distance between the axis of the gear roller and blank, rb is the base radius, rc is the tip radius of the gear roller. Therefore, the minimum tip radius rcmin can be obtained by Eq. (7).
(a − rb )2 = rb21 + ( (ra1 − rc )2 − rb21 + rc )2
(9)
Δα = 3α − 45
(8)
Based on the empirical data summarized in the previous literature (He, 2001), while α = 30°, Δα = 30′∼45′; while α = 35°, Δα = 45′∼1°; while α = 45°, Δα = 1°∼1°15′. The quantitative relationship between α and Δα is obtained by fitting method as follows: 143
Parameters
Value
Tooth number z0 Module m Distance between the axial center of gear roller and blank a Radius of the dedendum circle rf Radius of the addendum circle in tooth graduation stage ra2 Radius of the addendum circle at heel in tooth forming stage ra1 Tip radius at heel in tooth forming stage rc Pitch Pressure angle at heel in tooth forming stage α0 Cone angle of the gear roller in tooth forming stage μ Length of gear roller in tooth forming stage l
61 1 mm 46 mm 29.25 mm 30.75 mm 31.75 mm 0.4 mm 19.75° 10° 5.67 mm
Journal of Materials Processing Tech. 259 (2018) 141–149
T. Wu et al.
Fig. 5. Schematic diagram of the size of (a) blank and (b) gear roller in simulation.
Fig. 6. FEM model of conical gear rolling process with a feed in axial direction.
Fig. 7. Tooth profile in tooth graduation stage with (a) axial feed, (b) radial feed.
Fig. 8. Schematic diagram of the tooth graduation with (a) axial feed, (b) radial feed.
3. Numerical simulation
of blank is designated to 0.5 mm. In addition, since the deformation in gear rolling occurs only in outer part of the blank, the blank is designed to have a circular ring shape. Moreover, due to the symmetry of the deformation, half blank is selected to further improve simulation efficiency. Depending on the calculation formulas of the gear roller’s parameters described above, the parameters in various stages are obtained or selected as follows. Based on the dimension of blank and gear roller, the radial feed
3.1. Numerical modeling In this study, the tooth number, module, and pressure angle of the objective gear is selected as 31, 1 mm, and 20°, respectively. Then the radius rz = 15.4 mm of the blank can be obtained. In numerical simulation, the axial flow of the blank can be restrained by applying an axial constraint or baffle. In order to save computational time, the thickness 144
Journal of Materials Processing Tech. 259 (2018) 141–149
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Fig. 9. Tooth profile in tooth forming stage at the moment of (a) feed reaching 1/3, (b) feed reaching 2/3 and (c) feed completed.
Fig. 10. Position of tracking points in radial direction at the moment of (a) before rolling, (b) feed reaching 1/3, (c) feed reaching 2/3 and (d) feed completing.
symmetry constraint. (2) Two baffles are placed at both end faces of the blank to prevent axial flow of metal. (3) In order to avoid the eccentricity of the blank during the rolling process, the inner circle of the blank was fixed and the gear roller rotated. (4) The axial feeding of the blank was converted into the axial feeding of the gear roller. The friction type was set to shear friction, and the friction factor is 0.12 since the proposed process is in the category of cold forming. According to the test simulation results, the axial feed speed of the gear roller was set to 0.1 mm/s. The rotation of the blank was converted into the revolution of the gear roller due to the blank remained stationary during the rolling process. In order to keep rolling action between the gear roller and the blank, their linear velocity should be equal. The angular velocity of the gear roller’s revolution and rotation is set to 30 rpm and 15.246 rpm, respectively. Based on the above description, a simulation model of gear rolling using a conical gear roller and an axial feed of the blank is shown in Fig. 6.
Fig. 11. Axial displacement of track point P2, P14 and P24.
3.2. Simulated results
distance is 1 mm in tooth forming stage and 1.15 mm in whole stage. The schematic diagram of the blank and gear roller in simulation basing on the design parameters described above are shown in Fig. 5. The numerical simulation was carried out using DEFORM-3D. Pure aluminum which has good plasticity and formability is selected as blank material. The corresponding brand of pure aluminum in DEFORM is Al1100. And its flow behavior in simulation is defined as following plastic law. The total number of blank’s finite elements is set to 12000, and the outer part is refined in proportion to 2:1. The boundary conditions are set as follows: (1) The axis symmetry plane of the blank was set to a
In tooth graduation stage, the outer profile of the blank using conical gear rolling and traditional gear rolling are compared, as shown in Fig. 7. The outer circle evenly distributed with shallow grooves, which indicate that the uniform tooth graduation was realized in both processes. However, comparing Fig. 7(a) and (b) (Liu, 2013), the grooves using axial feed way are obviously smaller than those using radial feed way, since the teeth of the conical roller are smaller while they are fixed in traditional roller, as shown in Fig. 8. It indicates that the proposed process improves the graduation accuracy in tooth graduation stage. 145
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Fig. 12. Tooth profile (a) at the beginning and (b) at the ending of tooth finishing stage.
Fig. 13. Front and back view of tooth profile comparison between obtained gear and objective gear (standard gear) at the ending of (a) tooth forming stage and (b) tooth finishing stage.
Fig. 14. Experimental device of conical gear rolling, (a) experiment device, (b) enlarged view of the device.
roller increases along the axial direction. On the other hand, the material needed in the process of tooth growing up is getting less and less. In addition, in Fig. 9 also a tiny tip appears on the right corner of the objective gear when feed reaches 1/3. It is a defect in gear rolling process which is usually called rabbit ear. With the continuous deformation, the rabbit ear becomes larger. Fig. 10 shows the metal flow in radial direction on the surface
In tooth forming stage, the tooth profile of the blank at different steps is shown in Fig. 9. It can be seen that the tooth height of the objective gear increases gradually with the axial feed carrying on, but the growth speed of the tooth height is not linearly increased with the feed's speed. This phenomenon is partly because the volume of the gear roller penetrating into the blank increases gradually in the condition of unchanged axial feed speed in unit time, since the tooth of the gear
146
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Fig. 15. Lead samples obtained by using the conical gear rollers corresponding to the moment of (a) feed reaching 1/3, (b) feed reaching 2/3, (c) feed completed in tooth forming stage and (d) tooth finishing stage completed.
bends and flows to the other side, resulting in both sides of the rabbit ear are folded to refill. It improves the status of tooth filling. Moreover, due to the metal materials flow to the addendum’s radial clearance, the final gear has a higher addendum and a smaller thickness than standard. Figs. 13(a) and (b) shows the comparison between the obtained gear and objective gear (standard gear) at the ending of tooth forming stage and tooth finishing stage, respectively. It can be found that the tooth thickness of the objective gear agrees well with the objective gear, when the forming stage is finished. However, the tooth is not fully filled because of the occurrence of rabbit ear. In tooth finishing stage, more and more materials flow to the top of the tooth, and the under-filling of tooth can be improved to some extent. The tooth profile is also refined in this process. However, the rabbit ear on the top of the tooth is compressed by the dedendum of the gear roller, which leads to the forming a fold defect. Since the addendum of formed tooth exceeds that of the standard gear, it may be considered to be removed by subsequent machining. The simulation results reveal that the objective gear can be rolled out by the proposed conical gear rolling process. Based on the deformation characteristics of different stages, various tooth profiles are arranged on the gear roller along the axial direction. The proposed process achieves better uniform tooth graduation and refining of the tooth profile to some extent, and makes the tooth filling more full than traditional process. Importantly, in tooth finishing stage, the tooth profile precision of the objective gear can be basically assured by the way of reducing the addendum diameter of the gear roller.
during the tooth forming stage by setting tracking points at the center cross section of the blank. As is seen, the shape variation of the adjacent two teeth exhibits similar tendency. In the gear rolling process, the side of blank starting to contact the roller is called active side, while the other side of blank where roller departs from is called driven side. At the ending stage of tooth forming as shown in Fig. 10(d), the tracing points of blank surface are basically gathered on the top of the tooth, while a small amount is left at the driven side. Moreover, there is almost no tracking point at the active side as well as at the root of the tooth. This phenomenon indicates that the metal on the active side flows to the top of the tooth, since the bearing extrusion pressure and friction force pointing to the outer radial direction. However, the material flow is less dramatic at driven side, since the friction direction is opposite during the biting in and biting out of the roller. Fig. 11 shows the axial displacement of tracking points P2, P14, and P24. P14 at the top of the tooth does not occur axial displacement because it doesn't contact the roller. However, P2 and P24 locate on both sides of the tooth profile, and they have minor displacements along the axial feed direction since bearing the shear friction with the roller. It indicates that metal mainly deforms along the radial direction, and the deformation along axial direction is slight. In the last tooth finishing stage, as the meshing state of roller and blank changes from without clearance to with clearance gradually, the tooth profile of the blank is refined by meshing with the roller. The tooth profiles at the beginning and at the ending of tooth finishing stage are shown in Fig. 12. In this stage, rabbit ear gets larger continuously until it contacts the dedendum of the gear roller. Afterwards, rabbit ear 147
Journal of Materials Processing Tech. 259 (2018) 141–149
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Fig. 16. The comparison of experimental tooth shape to (a) simulated tooth shape and (b) standard tooth shape at the moment of tooth forming stage completed and tooth finishing stage completed.
Fig. 17. Comparison of tooth profile between the proposed and traditional processes, (a) enlarged tooth profile, (b) radial view and (c) axial view of the obtained gears by traditional process (top row) and proposed process (bottom row). The rabbit ear is marked with red rectangle, and the material extruded out and folded is marked with yellow circle (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article).
4. Experimental verification
the bottom. There is a difficulty to manufacture the conical gear roller because of its complex structure. Considering the cost and experimental purpose, the ABS engineering plastics is used as the roller material, and fused deposition modeling (FDM) technology was employed to manufacture the roller. The replacement of the roller’s material reduces the forming force of the experimental device, so that the material of blank
Fig. 14 shows the developed experimental device of conical gear rolling process. The conical gear roller and blank are respectively fixed on each end of the axis, which can realize rotary motion under the driving of the motor within a settled velocity. The blank can be pushed into the die along the axial direction through feed slide device fixed on 148
Journal of Materials Processing Tech. 259 (2018) 141–149
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the radius of addendum tip, the pressure angle and the cone angle. (2) The simulation results reveal that the desirable objective gear can be obtained using the conical gear rolling process. The tooth profile precision of the objective gear can be basically assured by the way of reducing the addendum diameter of the gear roller in the tooth finishing stage. (3) The forming experiment of conical gear rolling was carried out. During the experimental process, the gear roller meshed with the blank stably. Gear samples with better tooth shape and appearance were rolled out. The experimental results show that the model established by design and numerical simulation is reliable and it proves that the proposed process is feasible.
also needs to be changed. Lead with a purity of 99.99% is used as the blank material in the experiment. The tooth number of the objective gear and the gear roller are 31 and 61, respectively. During experiment, the speed of gear roller is set to be 10 rpm, while the speed ratio between the gear roller and the objective gear is 31/61. Additionally, the axial feeding speed is set to be 1 mm/s. Fig. 15 shows the obtained gears at different stages of rolling. It can be seen that the tooth number of the sample is accurate, and the tooth profile is orderly and clear. The average measured values of the diameter of addendum circle and dedendum circle are 32.37 mm and 28.93 mm, respectively. Fig. 16 shows the tooth shape obtained from experiment and simulation, and also the standard one at two moments in the process. As can be seen, the trend of tooth deformation are in agreement with the simulation results. Because the materials used in the process of experiment and simulation are different, the tooth height under the two condition is not exactly the same. The rabbit ear is compressed by the way of reducing addendum diameter of the roller in tooth finishing stage. The tooth finishing stage has some positive effects on the tooth profile modification. It proves that the conical gear roller designed in this study is reliable, and the gear can be formed using the process in axial feed method. However, the tooth height of gear sample is still smaller than standard at present, which indicates that the dedendum of the conical gear roller needs further revision design. Fig. 17 shows the comparison of the gears obtained from the proposed and traditional processes. Although the restrictions of axial movement were applied on both ends of the blank in traditional process, the severe rabbit ear and fold of material in the end face can still be observed. These defects are difficult to be avoided, and they occurred throughout the whole forming process, which have negative effects on the calculation of the initial blank size, and also the precision of the final tooth shape. In the proposed process, the materials flowing to the end face only occurs at the finished moment when the blank departs from the gear rollers. This ensures the stability of the material flow in the radial direction during the most time of the forming process. Consequently, the rabbit ear becomes smaller and the fold disappears, as can been seen form Figs. 17(a) and (b).
Acknowledgement The authors would like to acknowledge the financial support from National Natural Science Foundation of China (Grant No. 51475271). References Alireza, K., Melander, A., 2013. Finite element simulation as a tool to evaluate gear quality after gear rolling. Key Eng. Mater. 554–557, 300–306. Brecher, C., Brumm, M., Krömer, M., 2015. Design of gear hobbing processes using simulations and empirical data. Proc. CIRP 33, 484–489. He, F., 2001. Design of cold rolling tools for small module involute spline shafts. Tool. Eng. 35, 23–25. Kadashevich, I., Beutner, M., Karpuschewski, B., Halle, T., 2015. A novel simulation approach to determine thermally induced geometric deviations in dry gear hobbing. Proc. CIRP 31, 483–488. Kamouneh, A.A., Ni, J., Stephenson, D., Vriesen, R., Degrace, G., 2007a. Diagnosis of involutometric issues in flat rolling of external helical gears through the use of finiteelement models. Int. J. Mach. Tool. Manuf. 47, 1257–1262. Kamouneh, A.A., Ni, J., Stephenson, D., Vriesen, R., 2007b. Investigation of work hardening of flat-rolled helical-involute gears through grain-flow analysis, FE-modeling, and strain signature. Int. J. Mach. Tool. Manuf. 47, 1285–1291. Klepikov, V.V., Bodrov, A.N., 2003. Precise shaping of spliend shafts in automobile manufacturing. Russian Eng. Res. 23, 37–40. Li, J., Wang, G.C., Wu, T., 2016. Numerical simulation and experimental study of the slippage phenomenon in gear rolling. J. Mater. Process. Technol. 234, 280–289. Li, J., Wang, G.C., Wu, T., 2017. Numerical-experimental investigation on the rabbit ear formation mechanism in gear rolling. Int. J. Adv. Manuf. Technol. 91, 3551–3559. Liu, H.M., 2013. Process Design & Numerical Simulation of Spur Gear Rolling. MD Thesis. Shandong University. Neugebauer, R., Putz, M., Hellfritzsch, U., 2007a. Improved process design and quality for gear manufacturing with flat and round rolling. CIRP Ann. Manuf. Technol. 56, 307–312. Neugebauer, R., Klug, D., Hellfritzsch, U., 2007b. Description of the interactions during gear rolling as a basis for a method for the prognosis of the attainable quality parameters. Prod. Eng. 1, 253–257. Neugebauer, R., Hellfritzsch, U., Lahl, M., 2008. Advanced process limits by rolling of helical gears. Int. J. Mater. Form. 1, 1183–1186. Pater, Z., Gontarz, A., Tofil, A., 2011. Analysis of the cross-wedge rolling process of toothed shafts made from 2618 aluminum alloy. J. Shanghai Jiaotong Univ. 16, 162–166. Yu, J., Wang, B.Y., Hu, Z.H., 2011. Die design and experiment for forming the teeth of shafts by rolling. J. Univ. Sci. Technol. Beijing 33, 1544–1549.
5. Conclusions A type of conical rolling process was proposed in this paper. The whole deformation process can be divided into tooth graduation, tooth forming, tooth finishing stages along the axial direction. A combination of theoretical calculation, numerical simulation and experimental verification were conducted to study the roller design and the rolling process. The following conclusions are drawn, (1) According to the action of the conical gear roller in each stage, the formulae of roller design were provided. Especially, detailed formulas to calculate the important parameters of the gear roller’s tooth in the forming stage were given, such as the diameter of addendum circle,
149
Contents lists available at ScienceDirect
Journal of Materials Processing Tech. journal homepage: www.elsevier.com/locate/jmatprotec
Investigation on gear rolling process using conical gear rollers and design method of the conical gear roller
T
⁎
Tao Wua, Guangchun Wanga, , Jin Lib, Ke Yana a b
Key Laboratory for Liquid-Solid Structural Evolution and Processing of Materials (Ministry of Education), Shandong University, Jinan, Shandong 250061, PR China School of Mechanical and Electrical Engineering, Shandong Jianzhu University, Jinan, Shandong 250101, PR China
A R T I C LE I N FO
A B S T R A C T
Keywords: Gear Rolling Conical gear roller Structure design
A type of gear rolling process using conical gear rollers was firstly proposed in this study. Different with the cylindrical gear rollers used in the traditional gear rolling process, the roller which looks like a conical gear with different tooth profile along the axial direction was adopted in the rolling process. Besides, the radial feed applied on the roller in traditional rolling process was replaced by the axial feed applied on the blank. The proposed rolling process was divided into three stages, viz., tooth graduation, tooth forming and tooth finishing. According to the function of the roller in each stage, the structure of the conical gear roller was designed. Importantly, the detailed formulas to calculate the essential parameters of the gear roller’s tooth in tooth forming stage were established, such as the diameter of addendum circle, the radius of addendum tip, the pressure angle, and the cone angle. Through the numerical simulation, the variation and forming mechanism of the tooth profile at different stages were analyzed. The results revealed that the proposed process achieves better uniform tooth graduation and refining of the tooth profile, which increase the forming accuracy of the tooth to some extent in rolling process. Moreover, the experimental device was developed and the feasibility of the conical gear rolling process was verified.
1. Introduction Gears are widely used as mechanical components which play the role of transmitting motion and power. The forming precision and quality of gears are of great importance to improve the performance of mechanical equipment. Cutting is a traditional method to produce gears, and it can be used to produce most of the gears. Based on the type and size of gear, the selection of an appropriate cutting process can improve the productivity and forming precision. However, the gear cutting is a way of material removal, which causes a large amount of material waste. The rolling process is another economic alternative method of producing highly exact gears, which has attracted much attention from researchers all over the world. Based on the principle of gear meshing, the roller should be designed to have similar shape of the gear. During the rolling process, the gear roller contacts and extrudes blank with a feed in the radial direction. The materials at the outer part of the blank gradually form the tooth shape through its generating motion with the gear roller, as shown in Fig. 1. This kind of precision forming technology has many advantages, such as high material utilization, high productivity and small forming force. The formed gears also have better
⁎
mechanical properties than those obtained from cutting process. However, the complexity of gear rolling process brings great difficulty to the theoretical analysis, parameter calculation and numerical simulation. It is difficult to find a reliable and effective solution to avoid defects in the gear rolling process. In practical production, the process parameters of gear rolling are mainly determined based on the experiences, as well as trial and error methods. It not only lacks scientific basis, but also wastes raw materials and prolongs the production cycle (Klepikov and Bodrov, 2003). In recent years, the research on gear rolling have been carried out by means of experimental or numerical methods. (Kamouneh et al. (2007a, b) studied the involute helical gear rolling process by means of numerical simulation, and explained the enhancement of gear rolling by analyzing changes in grain on the surface of the gear. Neugebauer et al. (2007a, b, 2008) firstly designed a gear roller with variable pitch, established the rolling model, and came to a conclusion that the forming accuracy could be improved by 50%. Pater et al. (2011) simulated the rolling process with DEFORM-3D, proved the feasibility of the gear rolling process, and analyzed the distribution of the temperature field, stress field and rolling force during the rolling process. Yu et al. (2011) presented a method which formed the shaft and the teeth
Corresponding author. E-mail address: [email protected] (G. Wang).
https://doi.org/10.1016/j.jmatprotec.2018.04.034 Received 7 November 2017; Received in revised form 28 March 2018; Accepted 20 April 2018 Available online 22 April 2018 0924-0136/ © 2018 Elsevier B.V. All rights reserved.
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Fig. 1. Schematic diagram of gear rolling.
get larger and larger along axial direction. In the last tooth finishing stage, the size of the gear roller’s addendum decreases to the standard size gradually along axial direction, while blank changes into meshing motion with the gear rollers, and the outer achieves the objective gear tooth eventually. Based on the above three deformation stages, the design methods and formulas of gear roller structure parameters in each stage were presented in this paper. Moreover, the paper verified the rationality of the proposed process and design through the method of numerical simulation and experiment.
at the same time. Alireza and Melander (2013) provided a method using finite element simulation as a tool to evaluate gear quality after gear rolling. Kadashevich et al. (2015) investigated the thermal distribution and geometric deviations in gear rolling. Brecher et al. (2015) employed regression analysis to design the optimization of gear hobbing processes with different gear specifications based on numerical simulations. Li et al. (2016, 2017) discussed the slipping phenomenon and the formation mechanism of rabbit ear in gear rolling using numerical simulation and experimental methods. In traditional gear rolling process, a standard gear is always used as the roller to extrude the blank with a feed in radial direction. In this way, the radial clearance could not be cut out because of the shortage of working depth, which results into a larger dedendum circle in the objective gear and a lower dimensional precision. Moreover, the rabbit ear defect cannot be avoided. In the present study, to solve the aforementioned problems in traditional gear rolling process, a novel method is firstly proposed by a rolling feed in the axial direction instead of the radial direction in the traditional methods. The conical gear roller with different pitches in axial direction is shown in Fig. 2. The distance between the axis of roller and blank remains constant, and the rotation speed ratio of roller to blank is always unchanged during the whole rolling process. The rolling process is divided into three stages. In the first tooth graduation stage, the blank is pushed into the die along axial direction through an entry angle structure, and then the uniform tooth graduation is realized. In the second tooth forming stage, the blank is formed with gear roller which looks like a conical gear, afterwards the forming tooth shape will
2. Design method of conical gear roller In order to ensure the forming precision, the tooth number of gear roller should be as much as possible. Considering the space limitations, the distance between the axial centers of paired gear rollers is supposed to be less than the one of rolling equipment. The maximum tooth number of zmax can be obtained by Eq. (1) (He, 2001), and the result should be translated to integer.
zmax = (cmax − 2h 0 − dz )/ m − (3 ∼ 5)
(1)
where, cmax is the maximum distance between the axial centers of the paired gear roller, h0 is the whole dedendum depth of the objective gear, dz is the diameter of the objective gear, m is the module. The structural parameters of the conical gear roller are respectively determined according to the three different deformation stages. In the first tooth graduation stage, an entry angle structure is designed in order to ensure that the blank extrudes into the die successfully. The radius of addendum circle at the entrance ra’, the blank’s radius rz, and the distance between the axial center of gear roller and blank a should meet the relationship of ra’ + rz < a. The blank’s radius rz can be determined by equal volume of the blank before and after rolling process. Besides, there is a minimum feed in order to realize uniform tooth graduation, which means that the addendum circle of the gear roller has a minimum radius. The minimum radius of addendum circle is calculated as follows: β
a2 + rz2 − 2arz cos 2
⎧ ra2 ≥ ⎨β = ⎩
360 z
(2)
where, ra2 is the minimum radius of addendum circle in tooth graduation stage, β is the radius angle corresponding chord length of blank, z is the tooth number of the objective gear. In the second tooth forming stage, the roller is designed as conical shape with teeth which have increasingly higher addendum along axial direction. The gear roller can be divided into large and small end face which are called heel and toe in bevel gear, respectively. The toe connects with the roller in tooth graduation stage. The radius of the addendum circle and the tooth profile at heel should be determined. In order to rolling the clearance on the objective gear, the
Fig. 2. Schematic diagram of the conical gear rolling method. 142
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Fig. 4. Schematic diagram of the gear roller with cone angle in tooth forming stage. Fig. 3. Schematic diagram of calculating the minimum tip of the addendum.
Then, the pressure angle in pitch circle of the gear roller can be calculated by considering Eqs. (8) and (9). The cone angle of the gear roller in tooth forming stage has an important influence on the metal flow and forming load during gear rolling process, and thus affects the objective gear’s forming quality and the design of rolling device. Fig. 4 shows the gear roller with cone angle in tooth forming stage. The relation between the axial feed distance, the radial feed distance and the cone angle is obtained from Fig. 4 as follows:
addendum of gear roller at heel is more than that of standard gear c*m, where c* is the coefficient of addendum’s radial clearance, m is module. The formula for calculating the diameter of gear roller is
da1 = m1 (z 0 + 2ha* + 2c *)
(3)
where, da1 is the diameter of the gear roller at heel, m1 is the module of the gear roller, z0 is tooth number of the gear roller, ha* is the coefficient of addendum. The addendum of the gear roller higher than standard can lead to undercut because the extension epicycloid envelope formed by the addendum angle of the gear roller invades the dedendum transition curve of the objective gear. A round corner on the top of tooth, usually called tip, which can increase involute start point of the objective gear, is designed at the addendum of the gear roller. There is a minimum tip for avoiding undercut when the end point of addendum on the gear roller’s involute coincides with the involute start point of the objective gear, as shown in Fig. 3. In Fig. 3, △AOD is a right triangle, and the relationship of r’a12=rb12 + AD2 is followed, where r’a1 is the radius of the involute end point of the gear roller with tip, rb1 is the base radius at the heal of the gear roller in tooth forming. Moreover, the values of r’a1 and AD can be calculated using the following equations.
r ′a1 = a − rb
(4)
AD2 = (CD + rc )2
(5)
CD =
(ra1 − rc )2 − rb21
tan μ =
vb =
(10)
b vl l
(11)
where, vb is the radial feed speed; vl is the axial feed speed. From Eqs. (10) and (11), the cone angle μ is inversely proportional to the axial feed distance and the radial effective rolling force, but proportional to the radial feed speed and the rolling efficiency. Hence, the range of its value should consider various factors such as forming quality, rolling efficiency, equipment capability and installation space. In the final tooth finishing stage, gear roller needs a radial fallback movement to correct the tooth profile of the objective gear. That means the meshing state between roller and blank gradually changes from without clearance in tooth forming stage to with clearance under gear drive (Table 1).
(6)
Table 1 Parameters of the gear roller in numerical simulation.
(7)
With the continuous rolling process, the pressure and friction force on the objective gear tooth are varied, which results in smaller pressure angle in pitch circle of the objective gear than the expected value. To correct the reduction, the pitch pressure angles of gear roller α0 and objective gear α should meet the formula as follows:
α 0 = α − Δα
b l
where, μ is the cone angle in tooth forming stage, b is the radial feed distance, l is the axial feed distance. The relation between the gear roller’s axial feed speed and radial feed speed is
where, a is the distance between the axis of the gear roller and blank, rb is the base radius, rc is the tip radius of the gear roller. Therefore, the minimum tip radius rcmin can be obtained by Eq. (7).
(a − rb )2 = rb21 + ( (ra1 − rc )2 − rb21 + rc )2
(9)
Δα = 3α − 45
(8)
Based on the empirical data summarized in the previous literature (He, 2001), while α = 30°, Δα = 30′∼45′; while α = 35°, Δα = 45′∼1°; while α = 45°, Δα = 1°∼1°15′. The quantitative relationship between α and Δα is obtained by fitting method as follows: 143
Parameters
Value
Tooth number z0 Module m Distance between the axial center of gear roller and blank a Radius of the dedendum circle rf Radius of the addendum circle in tooth graduation stage ra2 Radius of the addendum circle at heel in tooth forming stage ra1 Tip radius at heel in tooth forming stage rc Pitch Pressure angle at heel in tooth forming stage α0 Cone angle of the gear roller in tooth forming stage μ Length of gear roller in tooth forming stage l
61 1 mm 46 mm 29.25 mm 30.75 mm 31.75 mm 0.4 mm 19.75° 10° 5.67 mm
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Fig. 5. Schematic diagram of the size of (a) blank and (b) gear roller in simulation.
Fig. 6. FEM model of conical gear rolling process with a feed in axial direction.
Fig. 7. Tooth profile in tooth graduation stage with (a) axial feed, (b) radial feed.
Fig. 8. Schematic diagram of the tooth graduation with (a) axial feed, (b) radial feed.
3. Numerical simulation
of blank is designated to 0.5 mm. In addition, since the deformation in gear rolling occurs only in outer part of the blank, the blank is designed to have a circular ring shape. Moreover, due to the symmetry of the deformation, half blank is selected to further improve simulation efficiency. Depending on the calculation formulas of the gear roller’s parameters described above, the parameters in various stages are obtained or selected as follows. Based on the dimension of blank and gear roller, the radial feed
3.1. Numerical modeling In this study, the tooth number, module, and pressure angle of the objective gear is selected as 31, 1 mm, and 20°, respectively. Then the radius rz = 15.4 mm of the blank can be obtained. In numerical simulation, the axial flow of the blank can be restrained by applying an axial constraint or baffle. In order to save computational time, the thickness 144
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Fig. 9. Tooth profile in tooth forming stage at the moment of (a) feed reaching 1/3, (b) feed reaching 2/3 and (c) feed completed.
Fig. 10. Position of tracking points in radial direction at the moment of (a) before rolling, (b) feed reaching 1/3, (c) feed reaching 2/3 and (d) feed completing.
symmetry constraint. (2) Two baffles are placed at both end faces of the blank to prevent axial flow of metal. (3) In order to avoid the eccentricity of the blank during the rolling process, the inner circle of the blank was fixed and the gear roller rotated. (4) The axial feeding of the blank was converted into the axial feeding of the gear roller. The friction type was set to shear friction, and the friction factor is 0.12 since the proposed process is in the category of cold forming. According to the test simulation results, the axial feed speed of the gear roller was set to 0.1 mm/s. The rotation of the blank was converted into the revolution of the gear roller due to the blank remained stationary during the rolling process. In order to keep rolling action between the gear roller and the blank, their linear velocity should be equal. The angular velocity of the gear roller’s revolution and rotation is set to 30 rpm and 15.246 rpm, respectively. Based on the above description, a simulation model of gear rolling using a conical gear roller and an axial feed of the blank is shown in Fig. 6.
Fig. 11. Axial displacement of track point P2, P14 and P24.
3.2. Simulated results
distance is 1 mm in tooth forming stage and 1.15 mm in whole stage. The schematic diagram of the blank and gear roller in simulation basing on the design parameters described above are shown in Fig. 5. The numerical simulation was carried out using DEFORM-3D. Pure aluminum which has good plasticity and formability is selected as blank material. The corresponding brand of pure aluminum in DEFORM is Al1100. And its flow behavior in simulation is defined as following plastic law. The total number of blank’s finite elements is set to 12000, and the outer part is refined in proportion to 2:1. The boundary conditions are set as follows: (1) The axis symmetry plane of the blank was set to a
In tooth graduation stage, the outer profile of the blank using conical gear rolling and traditional gear rolling are compared, as shown in Fig. 7. The outer circle evenly distributed with shallow grooves, which indicate that the uniform tooth graduation was realized in both processes. However, comparing Fig. 7(a) and (b) (Liu, 2013), the grooves using axial feed way are obviously smaller than those using radial feed way, since the teeth of the conical roller are smaller while they are fixed in traditional roller, as shown in Fig. 8. It indicates that the proposed process improves the graduation accuracy in tooth graduation stage. 145
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Fig. 12. Tooth profile (a) at the beginning and (b) at the ending of tooth finishing stage.
Fig. 13. Front and back view of tooth profile comparison between obtained gear and objective gear (standard gear) at the ending of (a) tooth forming stage and (b) tooth finishing stage.
Fig. 14. Experimental device of conical gear rolling, (a) experiment device, (b) enlarged view of the device.
roller increases along the axial direction. On the other hand, the material needed in the process of tooth growing up is getting less and less. In addition, in Fig. 9 also a tiny tip appears on the right corner of the objective gear when feed reaches 1/3. It is a defect in gear rolling process which is usually called rabbit ear. With the continuous deformation, the rabbit ear becomes larger. Fig. 10 shows the metal flow in radial direction on the surface
In tooth forming stage, the tooth profile of the blank at different steps is shown in Fig. 9. It can be seen that the tooth height of the objective gear increases gradually with the axial feed carrying on, but the growth speed of the tooth height is not linearly increased with the feed's speed. This phenomenon is partly because the volume of the gear roller penetrating into the blank increases gradually in the condition of unchanged axial feed speed in unit time, since the tooth of the gear
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Fig. 15. Lead samples obtained by using the conical gear rollers corresponding to the moment of (a) feed reaching 1/3, (b) feed reaching 2/3, (c) feed completed in tooth forming stage and (d) tooth finishing stage completed.
bends and flows to the other side, resulting in both sides of the rabbit ear are folded to refill. It improves the status of tooth filling. Moreover, due to the metal materials flow to the addendum’s radial clearance, the final gear has a higher addendum and a smaller thickness than standard. Figs. 13(a) and (b) shows the comparison between the obtained gear and objective gear (standard gear) at the ending of tooth forming stage and tooth finishing stage, respectively. It can be found that the tooth thickness of the objective gear agrees well with the objective gear, when the forming stage is finished. However, the tooth is not fully filled because of the occurrence of rabbit ear. In tooth finishing stage, more and more materials flow to the top of the tooth, and the under-filling of tooth can be improved to some extent. The tooth profile is also refined in this process. However, the rabbit ear on the top of the tooth is compressed by the dedendum of the gear roller, which leads to the forming a fold defect. Since the addendum of formed tooth exceeds that of the standard gear, it may be considered to be removed by subsequent machining. The simulation results reveal that the objective gear can be rolled out by the proposed conical gear rolling process. Based on the deformation characteristics of different stages, various tooth profiles are arranged on the gear roller along the axial direction. The proposed process achieves better uniform tooth graduation and refining of the tooth profile to some extent, and makes the tooth filling more full than traditional process. Importantly, in tooth finishing stage, the tooth profile precision of the objective gear can be basically assured by the way of reducing the addendum diameter of the gear roller.
during the tooth forming stage by setting tracking points at the center cross section of the blank. As is seen, the shape variation of the adjacent two teeth exhibits similar tendency. In the gear rolling process, the side of blank starting to contact the roller is called active side, while the other side of blank where roller departs from is called driven side. At the ending stage of tooth forming as shown in Fig. 10(d), the tracing points of blank surface are basically gathered on the top of the tooth, while a small amount is left at the driven side. Moreover, there is almost no tracking point at the active side as well as at the root of the tooth. This phenomenon indicates that the metal on the active side flows to the top of the tooth, since the bearing extrusion pressure and friction force pointing to the outer radial direction. However, the material flow is less dramatic at driven side, since the friction direction is opposite during the biting in and biting out of the roller. Fig. 11 shows the axial displacement of tracking points P2, P14, and P24. P14 at the top of the tooth does not occur axial displacement because it doesn't contact the roller. However, P2 and P24 locate on both sides of the tooth profile, and they have minor displacements along the axial feed direction since bearing the shear friction with the roller. It indicates that metal mainly deforms along the radial direction, and the deformation along axial direction is slight. In the last tooth finishing stage, as the meshing state of roller and blank changes from without clearance to with clearance gradually, the tooth profile of the blank is refined by meshing with the roller. The tooth profiles at the beginning and at the ending of tooth finishing stage are shown in Fig. 12. In this stage, rabbit ear gets larger continuously until it contacts the dedendum of the gear roller. Afterwards, rabbit ear 147
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Fig. 16. The comparison of experimental tooth shape to (a) simulated tooth shape and (b) standard tooth shape at the moment of tooth forming stage completed and tooth finishing stage completed.
Fig. 17. Comparison of tooth profile between the proposed and traditional processes, (a) enlarged tooth profile, (b) radial view and (c) axial view of the obtained gears by traditional process (top row) and proposed process (bottom row). The rabbit ear is marked with red rectangle, and the material extruded out and folded is marked with yellow circle (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article).
4. Experimental verification
the bottom. There is a difficulty to manufacture the conical gear roller because of its complex structure. Considering the cost and experimental purpose, the ABS engineering plastics is used as the roller material, and fused deposition modeling (FDM) technology was employed to manufacture the roller. The replacement of the roller’s material reduces the forming force of the experimental device, so that the material of blank
Fig. 14 shows the developed experimental device of conical gear rolling process. The conical gear roller and blank are respectively fixed on each end of the axis, which can realize rotary motion under the driving of the motor within a settled velocity. The blank can be pushed into the die along the axial direction through feed slide device fixed on 148
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the radius of addendum tip, the pressure angle and the cone angle. (2) The simulation results reveal that the desirable objective gear can be obtained using the conical gear rolling process. The tooth profile precision of the objective gear can be basically assured by the way of reducing the addendum diameter of the gear roller in the tooth finishing stage. (3) The forming experiment of conical gear rolling was carried out. During the experimental process, the gear roller meshed with the blank stably. Gear samples with better tooth shape and appearance were rolled out. The experimental results show that the model established by design and numerical simulation is reliable and it proves that the proposed process is feasible.
also needs to be changed. Lead with a purity of 99.99% is used as the blank material in the experiment. The tooth number of the objective gear and the gear roller are 31 and 61, respectively. During experiment, the speed of gear roller is set to be 10 rpm, while the speed ratio between the gear roller and the objective gear is 31/61. Additionally, the axial feeding speed is set to be 1 mm/s. Fig. 15 shows the obtained gears at different stages of rolling. It can be seen that the tooth number of the sample is accurate, and the tooth profile is orderly and clear. The average measured values of the diameter of addendum circle and dedendum circle are 32.37 mm and 28.93 mm, respectively. Fig. 16 shows the tooth shape obtained from experiment and simulation, and also the standard one at two moments in the process. As can be seen, the trend of tooth deformation are in agreement with the simulation results. Because the materials used in the process of experiment and simulation are different, the tooth height under the two condition is not exactly the same. The rabbit ear is compressed by the way of reducing addendum diameter of the roller in tooth finishing stage. The tooth finishing stage has some positive effects on the tooth profile modification. It proves that the conical gear roller designed in this study is reliable, and the gear can be formed using the process in axial feed method. However, the tooth height of gear sample is still smaller than standard at present, which indicates that the dedendum of the conical gear roller needs further revision design. Fig. 17 shows the comparison of the gears obtained from the proposed and traditional processes. Although the restrictions of axial movement were applied on both ends of the blank in traditional process, the severe rabbit ear and fold of material in the end face can still be observed. These defects are difficult to be avoided, and they occurred throughout the whole forming process, which have negative effects on the calculation of the initial blank size, and also the precision of the final tooth shape. In the proposed process, the materials flowing to the end face only occurs at the finished moment when the blank departs from the gear rollers. This ensures the stability of the material flow in the radial direction during the most time of the forming process. Consequently, the rabbit ear becomes smaller and the fold disappears, as can been seen form Figs. 17(a) and (b).
Acknowledgement The authors would like to acknowledge the financial support from National Natural Science Foundation of China (Grant No. 51475271). References Alireza, K., Melander, A., 2013. Finite element simulation as a tool to evaluate gear quality after gear rolling. Key Eng. Mater. 554–557, 300–306. Brecher, C., Brumm, M., Krömer, M., 2015. Design of gear hobbing processes using simulations and empirical data. Proc. CIRP 33, 484–489. He, F., 2001. Design of cold rolling tools for small module involute spline shafts. Tool. Eng. 35, 23–25. Kadashevich, I., Beutner, M., Karpuschewski, B., Halle, T., 2015. A novel simulation approach to determine thermally induced geometric deviations in dry gear hobbing. Proc. CIRP 31, 483–488. Kamouneh, A.A., Ni, J., Stephenson, D., Vriesen, R., Degrace, G., 2007a. Diagnosis of involutometric issues in flat rolling of external helical gears through the use of finiteelement models. Int. J. Mach. Tool. Manuf. 47, 1257–1262. Kamouneh, A.A., Ni, J., Stephenson, D., Vriesen, R., 2007b. Investigation of work hardening of flat-rolled helical-involute gears through grain-flow analysis, FE-modeling, and strain signature. Int. J. Mach. Tool. Manuf. 47, 1285–1291. Klepikov, V.V., Bodrov, A.N., 2003. Precise shaping of spliend shafts in automobile manufacturing. Russian Eng. Res. 23, 37–40. Li, J., Wang, G.C., Wu, T., 2016. Numerical simulation and experimental study of the slippage phenomenon in gear rolling. J. Mater. Process. Technol. 234, 280–289. Li, J., Wang, G.C., Wu, T., 2017. Numerical-experimental investigation on the rabbit ear formation mechanism in gear rolling. Int. J. Adv. Manuf. Technol. 91, 3551–3559. Liu, H.M., 2013. Process Design & Numerical Simulation of Spur Gear Rolling. MD Thesis. Shandong University. Neugebauer, R., Putz, M., Hellfritzsch, U., 2007a. Improved process design and quality for gear manufacturing with flat and round rolling. CIRP Ann. Manuf. Technol. 56, 307–312. Neugebauer, R., Klug, D., Hellfritzsch, U., 2007b. Description of the interactions during gear rolling as a basis for a method for the prognosis of the attainable quality parameters. Prod. Eng. 1, 253–257. Neugebauer, R., Hellfritzsch, U., Lahl, M., 2008. Advanced process limits by rolling of helical gears. Int. J. Mater. Form. 1, 1183–1186. Pater, Z., Gontarz, A., Tofil, A., 2011. Analysis of the cross-wedge rolling process of toothed shafts made from 2618 aluminum alloy. J. Shanghai Jiaotong Univ. 16, 162–166. Yu, J., Wang, B.Y., Hu, Z.H., 2011. Die design and experiment for forming the teeth of shafts by rolling. J. Univ. Sci. Technol. Beijing 33, 1544–1549.
5. Conclusions A type of conical rolling process was proposed in this paper. The whole deformation process can be divided into tooth graduation, tooth forming, tooth finishing stages along the axial direction. A combination of theoretical calculation, numerical simulation and experimental verification were conducted to study the roller design and the rolling process. The following conclusions are drawn, (1) According to the action of the conical gear roller in each stage, the formulae of roller design were provided. Especially, detailed formulas to calculate the important parameters of the gear roller’s tooth in the forming stage were given, such as the diameter of addendum circle,
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