1、05FTM10Finite Element Study of the IkonaGear Tooth Profileby: J.R. Colbourne, University of Alberga, S. Liu, IkonaGear InternationalTECHNICAL PAPERAmerican Gear Manufacturers AssociationFinite Element Study of the Ikona Gear Tooth ProfileJohn R. Colbourne, University of Alberta and Shubin Liu, Ikona
2、 GearInternationalThe statements and opinions contained herein are those of the author and should not be construed as anofficial action or opinion of the American Gear Manufacturers Association.AbstractThe Ikona gear tooth profile is a patented non-involute tooth profile for internal gear pairs. Gea
3、rs with thisprofile have the following properties: the teeth are conjugate; the contact ratio is very high; there is no tipinterference, even when only a one-tooth difference between the pinion and internal gear; there is minimalbacklash; and the gears can be cut on conventional gear-cutting machine
4、s. Large reduction ratios can beachieved by a single gear pair and a high contact ratio results in lower tooth stresses than for a similar involutegear. Plus, minimal backlash makes the Ikona profile ideal for many applications, such as servo-drives,medical prostheses, and robots. Stress analysis of
5、 these gears assumes that the contact force is equal ateach contacting tooth pair. Finite element results demonstrate how the number of tooth pairs in contact mayincrease under load. Finally, an estimate will be presented, showing the variation of tooth force between thecontacting teeth.Copyright 20
6、05American Gear Manufacturers Association500 Montgomery Street, Suite 350Alexandria, Virginia, 22314October, 2005ISBN: 1-55589-858-0Finite Element Study of the Ikona Gear Tooth Profile John R. Colbourne, Professor Emeritus, University of Alberta, Canada Shubin Liu, Senior Mechanical Engineer, Ikona
7、Gear International, Coquitlam, BC, Canada Tip Interference The Ikona tooth profile was originally developed to eliminate tip interference. For involute internal gear pairs, there is generally tip interference if the difference in the tooth numbers is less than six. The purpose of the Ikona design wa
8、s to create tooth profiles in which there would be no tip interference, even when the tooth number difference was as low as one. Consider a gear pair in which the internal gear has one tooth more than the pinion. The internal gear is fixed, while the pinion is mounted on an eccentric. In a conventio
9、nal design, either involute or non-involute, the eccentricity (ie the center distance) would be about 0.5 modules. During one revolution of the eccentric, the axis of the pinion would move up and down about 1.0 modules. A tooth of the pinion, initially in contact with a tooth of the internal gear, w
10、ould be required to move from one tooth space of the internal gear, around one of the teeth, and into the adjacent tooth space. However, in a conventional design the working depth of the teeth would be at least 1.6 modules. Thus the pinion tooth would inevitably collide with the internal gear tooth,
11、 and tip interference would occur. In the Ikona gear pair with a one-tooth difference, the center distance is increased to about 1.0 modules, so that the pinion tooth moves from one tooth space to the next without hitting the internal gear tooth. The pitch circle radii Rp1and Rp2of the pinion and th
12、e internal gear are given by Rp1= N1C / (N2 N1) (1) Rp2= N2C / (N2 N1) (2) where N1and N2are the tooth numbers and C is the center distance. With a one-tooth difference, and a center distance of 1.0 modules, the pitch circle radii are equal to N1m and N2m, where m is the module. For a conventional p
13、inion, the radius of the tip circle is approximately (0.5N1+1)m, so the pitch circle now lies entirely outside the pinion. It is advantageous to redefine the module, as applied to the Ikona tooth profile. The module is generally defined for involute gears as the reference circle diameter of either g
14、ear, divided by the number of teeth. For non-involute gears, where only one pitch circle exists, the module would be the diameter of the pitch circle divided by the number of teeth. In either case, the module is then used as a measure of the tooth size. In the case of the Ikona profile, the pitch ci
15、rcle diameter is now much larger than the tip circle diameter. If the conventional definition of the module is used, the module would be unrelated to the tooth size. For this reason, the module is now defined in terms of the pinion tip circle diameter Dtip1: m = Dtip1/ (N1+ 2) (3) This definition me
16、ans that an Ikona gear of a given module will have teeth of approximately the same pitch as an involute gear of the same module. Path of Contact and Tooth Profiles Figure 1 shows the tip circles of an Ikona gear pair, with the line of centers vertical, and the gear centers at fixed points C1and C2.
17、Any contact between the teeth must take place in the crescent-shaped area where the tip circles overlap. The pitch point lies on the line of centers, but well outside the tip circles. The Law of Gearing states that at any contact point, the common normal to the gear teeth must pass through the pitch
18、 point. This implies that the path of contact crosses the line of centers at the pitch point. Since the path of contact must lie within the crescent-shaped area, and the pitch point lies outside it, it follows that the path of contact does not intersect the line of centers, but lies entirely on one
19、side. Figure 1 Crescent-shaped area formed by the pinion and gear tip circles, and the path of contact. The path of contact will start at a point on the internal gear tip circle, and will end at a point on the pinion tip circle. At any point on the path of contact, the common normal to the tooth pro
20、files must pass through the pitch point. Therefore, we know the direction of the common tangent, and hence we can calculate the pressure angles of the tooth profiles at that point. For the Ikona gear, the initial point of the path of contact is generally chosen as the point on the internal gear tip
21、circle giving a pressure angle of 8 degrees. The final point is generally chosen as the point on the pinion tip circle giving a pressure angle of 35 degrees. For the path of contact between these two points, we can choose any curve whose radius of curvature is approximately equal to the tip circle r
22、adius of the pinion, so that the curve remains within the crescent-shaped area. The curve chosen for the Ikona gear is an Archimedes spiral. This has the advantage that the radius, measured from either gear center, increases monotonically as we move along the curve. Hence the contact point moves con
23、tinuously along the tooth profiles, so that there is no possibility of interference, or of undercutting when the gears are cut. Since the radius and the pressure angle of each profile are known at every point of the path of contact, the tooth profiles can be constructed using finite differences. The
24、 pinion teeth can be hobbed by any hobbing machine, using a specially shaped hob, and the internal gear teeth can be cut using a shaper cutter. Contact Ratio and Backlash The pinion position is known at the start and end of the path of contact. The difference between the angular positions at these t
25、wo points is divided by the angular pitch, to give the contact ratio. Because of the curved path of contact, the contact ratios are very much higher than those of involute gear pairs. For example, the contact ratio of a 45-46 tooth gear pair is 3.24, and the contact ratio of a 78-79 tooth gear pair
26、is 5.21. Figure 2 An Ikona gear pair with 45-46 Teeth. Figure 2 shows a gear pair with the center-lines of a pinion tooth and a tooth space of the internal gear coinciding with the line of centers. To the left of the line of centers, the right-hand faces of several pinion teeth are in contact with t
27、he internal gear teeth, each of the contact points lying on the path of contact. However, since both gears are symmetric about the line of centers, there is another path of contact lying to the right of the line of centers, where the left-hand faces of the pinion teeth are in contact with the intern
28、al gear teeth. If the pinion is driving and turning clockwise, the left-hand path of contact is active, while small gaps occur at the contact points to the right of the line of centers, due to the tooth flexibility. If the pinion turns counter-clockwise, the situation is reversed. If there is no bac
29、klash in an involute gear pair, at least one of the pinion teeth will make contact on both faces with the internal gear. 2Thus any error in the profiles, or a change in the center distance, will cause binding in the teeth. The gear pair must therefore be designed with adequate backlash. In an Ikona
30、gear pair, by contrast, there are no pinion teeth making double contact with the internal gear, even when there is no backlash. Therefore a small error in the profiles, or a change in the center distance, will cause minor bending in the teeth, but no binding. In principle, an Ikona gear pair can be
31、designed with no backlash. In practice, it has been found preferable to thin the gears slightly, giving minimal backlash, because otherwise the gears are difficult to assemble. But the amount of backlash is very much less than is necessary for an involute gear pair. Tooth Stresses For design purpose
32、s, it has been assumed that the tooth forces are the same at each contact point. This is thought to be reasonably accurate, since the combined flexibility of each tooth pair is approximately constant. The tooth flexibility of each tooth increases towards the tip, and at each contact point one tooth
33、is loaded nearer to its form diameter, and the meshing tooth is loaded nearer to its tip. With this assumption, it is not difficult to calculate the tooth force at each contact point, corresponding to any specified input torque. The tooth stresses can then be found, using procedures similar to those
34、 used for involute gears. The maximum contact stress is found using Hertz line contact theory, based on the load intensity and the maximum relative curvature. For the fillet stresses, we apply the tooth force at the highest point of multiple tooth contact. For example, if the contact ratio is 3.24,
35、we find the highest point of 3-tooth contact. We calculate a stress concentration factor in the same manner as ANSI/AGMA 2001, based on the minimum radius of curvature in the fillet. Then the maximum fillet stress is calculated in the same manner as for involute gears. As a precaution, the fillet st
36、resses are also found for loading at the tooth tip, using the reduced contact force since one extra tooth pair is in contact. So far, no cases have been found where this situation has caused higher fillet stresses than the traditional loading case. The method described here for calculating the tooth
37、 stresses depends on the assumption, stated earlier, that the tooth forces are the same at each contacting tooth pair. The purpose of this paper is to use finite element analysis (FEA) to verify this assumption. Finite Element Model for Ikona Gear Study The finite element code used is the ANSYS 8.1,
38、 see Reference2, because both 2D and 3D elastic contact problem solution capabilities had been made available in it. The quadratic elements and the contact problem analysis algorithm of Lagrange multiplier, such as explained in Reference3, were used in both the 2D and 3D analyses. Validation of the
39、code had been made for these capabilities by the authors before they were used in the analysis of the contact problem analysis of Ikona gear teeth. Figure 3 shows the mechanics model in 2D for the current study. The pinion could be freely rotated with respect to its support. The support was assumed
40、to be rigid such that the deflection of the support was neglected. The friction was also neglected. The gear was fixed at the outer circular boundary. Pure torque was applied on the pinion at the locations, such as the hole, far away from the contact zones between teeth. The material for both the pi
41、nion and gear was taken to be steel, with the modulus of elasticity equal to 2.07 x 105MPa and Poissons ratio equal to 0.3, respectively. Figure 3 Contact Mechanics Model for Ikona Gears 3The center distance in the FEA model was taken to be that based on which the active profiles of the pinion and g
42、ear were created in conjugate action. The effects of the manufacturing accuracy on the geometry were also neglected. Effects of Applied Loads on Load Sharing among Teeth Without tip relief, more pairs of teeth than those predicted by the contact ratio defined above, may come into contact if the appl
43、ied load causes significant elastic deformation to close the initial gap between the two teeth approaching each other. For the first example of the study, the general dimensions of pinion and the gear are listed in Table 1. Table 1 General Dimensions for the 1st Ikona Gear Pair Number of teeth for p
44、inion 42 Number of teeth for gear 44 Diameter of tip circle for pinion 61.0 mm Diameter of tip circle for gear 61.2 mm Diameter of root circle for pinion 55.9 mm Diameter of root circle for gear 65.8 mm Face width of pinion 5.1 mm Face width of gear 5.1 mm Center distance 2.0 mm Contact ratio 3.3 In
45、itially let four pairs of teeth come into contact prior to applying three different torques equal to 100 Nm, 200 Nm and 300 Nm, respectively such that the maximum bending stress in the pinion fillet reaches 923 MPa; the resulting three different contact stress distributions, as shown in Figure 4, Fi
46、gure 5 and Figure 6, respectively, could be obtained via the 2D plane stress finite element analysis. With the torque equal to 100 Nm, it can be seen from Figure 4 that four pairs of teeth remain into contact. However, with the torque equal to 200 Nm or 300 Nm, it can be seen from Figure 5 and Figur
47、e 6 that six pairs of teeth came into contact. The finite element mesh used in these three analyses is shown in Figure 7, and this mesh is considered to be a fine mesh, especially near the contact zones between teeth. Figure 8, 9 and 10 show the contour plots for the bending stresses in the fillets
48、of the pinion and gear, with the torque equal to 100 Nm, 200 Nm and 300 Nm, respectively. Notice that the directions of the 1stprincipal stresses at the points on the surface of the fillet, coincide with the directions of the tangents of the surface of the fillet. Therefore on the surface of the fil
49、let, the 1ststresses are equal to the bending stresses. The maximum bending stress, with the torque equal to 100 Nm, 200 Nm and 300 Nm, is equal to 356 MPa, 656 MPa, and 923 MPa, respectively. Figure 4 Contact stress contour plot for the torque equal to 100 Nm. Figure 5 Contact stress contour plot for the torque equal to 200 Nm. 4Figure 6 Contact stress contour plot for the torque equal to 300 Nm. Figure 7 Finite element mesh in 2D. Figure 8 Bending stresses in the fillets for the torque equal to 100 Nm. Figure 9 Bending stresses in the fillets for the
