Tooth Contact Analysis of Herringbone Rack Gears of an Impulse Continuously Variable Transmission

Rack gears are widely used in many areas, such as automobile, robotics, and renewable energy industry [1-3]. Impulse continuously variable transmissions (ICVTs) can provide reliable power conversion from a prime mover, such as an engine and an electric motor, to a driven part, such as a wheel and a chain, with a continuous output-to-input speed ratio [4,5]. Gears have various types according to their tooth profiles and tooth widths, and have a wide range of dimensions as small as the ones in small appliances to the very large gears used in heavy-duty applications [6,7]. Generally, gears are manufactured via hobbing [8] or forming cutting [9-11] based on the theory of gearing. For some gears with special tooth profiles, e.g., concave-convex and spiral tooth profiles, their manufacturing methods and machine-tools are complex. Since meshing performances of these gears with special tooth profiles are highly sensitive to manufacturing errors [12,13], high manufacturing accuracy of gear machine-tools is required for these gears [1416]. Contact patterns and transmission errors are two typical methods for meshing performances evaluation of gear systems [17-19]. A tooth profile modeling method was developed to improve accuracy of tooth contact analysis for gear tooth profiles [20]. Some other meshing performances, e.g., power losses, can also be evaluated based tooth contact analysis [21,22]. Since these gears have convex-concave tooth profiles, they cannot be manufactured via standard gear manufacturing methods. During a manufacture in this way, for each of the gear modules and the radius of curvature, a different blade size and gear holder is needed. However, it's clear that these gears have many advantages, if they can be produced sufficiently in the industry [23,24]. Since these gears have better load-bearing capabilities, have a balancing feature for the axial forces, quiet operation feature and their lubrication characteristics is better than herringbone gears and spur gears. It's noteworthy that there are number of studies carried out recently in relation to these gears. Rack gears are modeled with a computer-aided design (CAD) program in order to eliminate these problems, and in another study, it was emphasized that these gears can be manufactured in computer numerical control (CNC) milling machines using two different methods, and by making use of these presented methods, the manufacturing codes of the gears was created with CAD programs Research Article


Introduction
Rack gears are widely used in many areas, such as automobile, robotics, and renewable energy industry [1][2][3]. Impulse continuously variable transmissions (ICVTs) can provide reliable power conversion from a prime mover, such as an engine and an electric motor, to a driven part, such as a wheel and a chain, with a continuous output-to-input speed ratio [4,5]. Gears have various types according to their tooth profiles and tooth widths, and have a wide range of dimensions as small as the ones in small appliances to the very large gears used in heavy-duty applications [6,7]. Generally, gears are manufactured via hobbing [8] or forming cutting [9][10][11] based on the theory of gearing. For some gears with special tooth profiles, e.g., concave-convex and spiral tooth profiles, their manufacturing methods and machine-tools are complex. Since meshing performances of these gears with special tooth profiles are highly sensitive to manufacturing errors [12,13], high manufacturing accuracy of gear machine-tools is required for these gears [14][15][16].
Contact patterns and transmission errors are two typical methods for meshing performances evaluation of gear systems [17][18][19].
A tooth profile modeling method was developed to improve accuracy of tooth contact analysis for gear tooth profiles [20]. Some other meshing performances, e.g., power losses, can also be evaluated based tooth contact analysis [21,22]. Since these gears have convex-concave tooth profiles, they cannot be manufactured via standard gear manufacturing methods. During a manufacture in this way, for each of the gear modules and the radius of curvature, a different blade size and gear holder is needed. However, it's clear that these gears have many advantages, if they can be produced sufficiently in the industry [23,24]. Since these gears have better load-bearing capabilities, have a balancing feature for the axial forces, quiet operation feature and their lubrication characteristics is better than herringbone gears and spur gears. It's noteworthy that there are number of studies carried out recently in relation to these gears.
Rack gears are modeled with a computer-aided design (CAD) program in order to eliminate these problems, and in another study, it was emphasized that these gears can be manufactured in computer numerical control (CNC) milling machines using two different methods, and by making use of these presented methods, the manufacturing codes of the gears was created with CAD programs first time in the literature, and the manufacture of gears were carried out in an error-free manner using different materials. In this study, in order to determine the performance characteristics of these gears, manufactured correctly, for the applicability in the industry; three-dimensional (3D) tooth contact analysis of the gears have been carried out by using ANSYS.
The remaining part of this paper is organized as follows: a load distribution model of rack gears of the ICVT is introduced in Sec. 2. Some finite element analysis of the rack and pinion system are performed in Sec. 3. In order to verify operation performance of the rack and pinion system, some simulation results are discussed in Sec. 4. Finally, some conclusions from this study are presented in Sec. 5.

Load Distribution Model of Rack Gears
A 3D model of the rack gear for tooth contact analysis, as shown in Fig. 1, which are created by using CATIA. The rack gear is designed with a herringbone and curvilinear involute gear profile, as shown in  A tooth model of an involute rack gear is used to describe its geometric parameters [25], as shown in Fig. 3. The base central angle γb of a tooth can be represented as where z is the number of teeth, αt and αn are the transverse pressure angle and the normal pressure angle, respectively, and χ is the addendum coefficient. The pressure angle αc at the contact point can be represented as where ξc is the profile parameter of the contact point that can be represented as in which rb is the base radius and rc is the profile radius of the contact point. The tooth central angle γ(y) can be represented as where v(y) is the tooth profile angle.
A load distribution model of involute rack gears is used in this study based on the minimum elastic potential energy theory. The elastic potential energy of an involute rack gear tooth is composed of a bending component Ux, a compressive component Un and a shear component Us: where Ux, Un and Us can be represented as  (6) in which F is the normal load, Suppose that there are n pairs of meshing teeth at a time instant; the normal load acting on the î th tooth surface can be represented as where î U and ˆj U are elastic potential energies of the î th and ˆj th meshing tooth surfaces, respectively, and î V and ˆj V are inverses of elastic potential energies of the î th and ˆj th meshing tooth surfaces, respectively. The elastic potential energy î U of the î th meshing tooth surface can be represented as where ĉ i l and ĉ j l are lengths of contact lines of the î th and ˆj th meshing tooth surfaces, respectively. The involute rack gear tooth is divided into slices with unit lengths. The load acting on the k th slice can be represented as where εβ is the face contact ratio, βb is the base spiral angle and k V is the inverse of the elastic potential energy of the k th slice.

Finite Element Analysis of Rack and Pinion System
In this study, first the modeling of the gear was performed with CATIA, by using the analytical expressions that determine the gear profile. After the rack and spur gear pairs have been created, again using CATIA, the gears are mounted and the gear pairs have been created. Then the stress analyses of the gears have been carried out by ANSYS, after the gear pairs have been created with CATIA. When using ANSYS to perform stress analysis in gears by researchers, the gears have been assumed two-dimensional and the force has been applied to the nodes externally to perform the stress analysis [26,27]. And in this analysis, a tooth profile of a threedimensional solid model was taken as the reference, and the maximum stress distribution and the clutch state of that maximum stress was determined for all gears during clutching, starting from the moment of first contact of a tooth to the last contact. The pinion gear rotates 13.173 deg, from the moment of first contact of a tooth to the last contact. This angle of rotation was divided into 6 equal pieces in order to better determine the maximum stress in this clutching period, and 8 Nm load was applied to the pinion gear at each moment of rotation, under the flow limit, to observe the distribution of stresses produced on these three different gears, and the maximum Von Mises stresses and the maximum deformations were determined. Before starting finite element analysis, the strength values of the PEEK material used in the manufacturing of the gears were input to ANSYS, then model file created in CATIA was opened and the rack-pinion system was divided into finite elements with the help of the mesh menu of CATIA, as shown in Fig. 4. Since the tensions in regions other than the contact region are not important much, these areas were divided into finite elements automatically from the menu of the program. However, the finite element dimensions in the contact regions have been manually set to 0.55 mm, for a fine grain mesh of the gear, since the stress in this area is important. The total number of elements of the finite element model of the rack-pinion system is 7,320,150. Some local segmentation of meshes of gear teeth are performed to improve finite element analysis accuracy and convergent to a stable condition. In order to apply load on to the pinion, the boundary conditions of the rack-pinion system were entered. In this step, which is prior to the final step, the almost realistic boundary condition values were applied to the finite element model of the rack-pinion system. The boundary conditions for the tooth model of the pinion are to locate the tooth force and the supports [7]. For this purpose, the rack gear was supported at one of its tips (Fig. 5, Fig. 6). The load on gear tooth surfaces has been applied in the form of torque on to the pinion, in accordance with the studies performed in the study. The contacting tooth surfaces have been defined in the rack-pinion system, and after defining that the rack and the pinion will undergo a linear elastic deformation, a static load applied to gear tooth surface of the rack-pinion system, as shown in Fig. 7. The maximum Von Mises stresses have been identified because of this tension.

Simulation Results and Discussion
After achieving the necessary boundary conditions required for the analysis of the gears, the stress values for each angle of rotation were calculated. In the results of these stress analysis, carried out by the ANSYS, it has been observed that the weakest gear is the spur gear and the highest strength has been observed in herringbone gear, in the strength comparisons of the analyzed gear models. It has also been observed that the gear with concave-convex profile has stress values close to the herringbone gear.
Considering stress distribution, it has been observed that the stresses occurred on the concave-convex profile gears are smoother and the maximum stresses are at the center of the gear bow, as shown in Fig. 8. In the herringbone gear, however, the stresses are occurred at outer parts of the tooth profile in general, and as the clutch progresses the stresses shift to the center of the gear, as shown in Fig. 9. For spur gears, the maximum stresses were observed on a straight line along the bottom of gear teeth, as shown in Fig. 10.

Conclusions
In this study, a spur gear, a herringbone gear, and a concave-convex profile rack gear, having same module and number of teeth, were subjected to analysis. In the results of these stress analysis, carried out by using ANSYS, it has been observed that the weakest gear is the spur gear and the highest strength has been observed in herringbone gear, in the strength comparisons of the analyzed gear models. It has also been observed that the gear with concave-convex profile has stress values close to the herringbone gear, in terms of the strength. This will then shed light on the experimental studies to be performed on this type of gears.
Considering the stress distribution, it has been observed that the stresses occurred on the concave-convex profile gears are smoother and the maximum stresses are at the center of the gear bow. In the herringbone gear, however, the stresses are occurred at outer parts of the tooth profile in general, and as the clutch progresses the stresses shift to the center of the gear. And in spur gears, the maximum stresses were observed on a straight line along the bottom of gear teeth. Concave-convex gears will be more advantageous than the spur gears and herringbone gears, after finding the optimum radius through a research to be conducted on the curvature radii of the concave-convex gears. It's suggested that use of these gears will be more advantageous in places where load resistance is necessary and in the pumps with gears in particular. In order to ensure wider use of these gears in industry, other operating characteristics such as wear characteristics and operating temperatures should be determined. : Base radius of the contact point (mm)