1、14FTM11 AGMA Technical Paper Mathematical Modeling for the Design of Spiroid, Helical, Spiral Bevel and Worm Gears By Dr. G. Kazkaz, Gearometry, Inc. 2 14FTM11 Mathematical Modeling for the Design of Spiroid, Helical, Spiral Bevel and Worm Gears Dr. Ghaffar Kazkaz, Gearometry, Inc. The statements an
2、d opinions contained herein are those of the author and should not be construed as an official action or opinion of the American Gear Manufacturers Association. Abstract Spiroid and worm gears have superior advantages for high torque and miniaturization applications, and for this reason, they are pa
3、rticularly preferred in aerospace, robotic and medical applications. They are typically manufactured by hobbing technology, a process that has had an overall lead time of 4 to 14 weeks. Besides the relatively lengthy lead time, and despite the fact that the tooth profile is defined through its press
4、ure angles, 3D drawings of the gears cannot be produced. This is due to the difficulty of capturing the entire curvature of the gear face from the outside to the inside diameter. Because of this difficulty, 3D quality control and FEA analysis under load are difficult and must be accomplished through
5、 classical analysis that incorporate pinion bending stresses, gear tooth shear stresses and compressive stresses between pinion and gear teeth. Due to some of these challenges, these gears have maintained a niche status with the industry. This paper will present a novel work for Spiroid and worm gea
6、rs that mathematically calculates the gear tooth profile in terms of the geometry of the cutting tool (hob) and the machining setup. Because of their similarity, the work was also expanded to spiral bevel gears. We have developed software to plot the gear tooth when the parameters of the geometry of
7、 the tool and machining setup are entered. The gear tooth shape can then be altered and optimized by manipulating the input parameters until a desired tooth profile is produced. In effect, the result will be designing the hob and machining setup for best gear tooth profile on the computer. Afterward
8、, the generated gear tooth data are entered into CAD software to generate a true 3D model of the gear. The tool path will also be generated from the data for CNC machining beside hobbing. This mathematical modeling allows for direct CNC machining rather than hobbing and may reduce the prototype lead
9、 time from weeks to hours. The pinion can be designed in a similar process and its tooth can be graphed inside the gears groove, showing the contact points and the clearance between the two surfaces. This novel work has already resulted in the invention of a new gear type combining Spiroid and worm
10、gear in a single gear driven by the same pinion. This provides significant increase in torque capability. Mathematical modeling is presented in this paper as a tool design to reduce the lead time and cost for designing Spiroid, worm and spiral bevel gears. Mathematical modeling is based on mathemati
11、cally calculating the 3D gear tooth profile in terms of the cutting tool (hob) geometry, the machining setup and the gear size (inside and outside diameters). Software has been developed to allow designers to enter values of these parameters and observe the resulting 3D gear tooth profile. The desig
12、ner can observe the resulting tooth profile on the computer and adjust the input parameter values to obtain the desired profile. The software also generates the profile in numerical xyz points which is necessary to produce 3D drawings of the gear, conduct direct quality control and perform FEA analy
13、sis under load. It was also demonstrated that mathematical modeling can be a tool for gear innovation. It has already resulted in the invention of a new gear type, Spiroid/worm hybrid that combines a double Spiroid gear and a worm gear in one. It more than doubled the gear torque capability in compa
14、rison to a single Spiroid gear with a minimal increase in size or weight. Copyright 2014 American Gear Manufacturers Association 1001 N. Fairfax Street, Suite 500 Alexandria, Virginia 22314 October 2014 ISBN: 978-1-61481-103-9 3 14FTM11 Mathematical Modeling for the Design of Spiroid1), Helical, Spi
15、ral Bevel and Worm Gears Dr. Ghaffar Kazkaz, Gearometry, Inc. Spiroid gears Oliver Sari invented Spiroid gears in 1954 while working for ITW and the division ITW Spiroid was created. The gear set consists of a gear and a helical pinion assembled as shown in Figure 1. It is similar to spiral bevel ge
16、ar set, except the pinion axis is moved a certain distance off the gear axis. This distance is called offset or center distance. It makes it possible to hold the pinion shaft on both ends to increase leverage and stability. Spiroid gears are special gears with a wide range of RPM ratio from as littl
17、e as 4 to as many as a few hundred. This facilitates single stage designs that reduce size and increase efficiency. They can also have a high contact ratio which makes them suitable for high torque and low noise requirement. They are used in robotics, aerospace, medical and industrial applications.
18、There are three Spiroid gear types shown in Figure 2. They are: the flat face, the skewed angle with cylindrical pinion and the skewed angle with tapered pinion. Figure 1. Spiroid gear/pinion assembly or gear/hob setup Figure 2. The three types of Spiroid gear 1)Spiroid is a registered trademark of
19、ITW. The views expressed herein are those of the author alone and do not necessarily reflect the views of ITW. 4 14FTM11 The pinion can be manufactured by any method for making helical gears, including: grinding wheel, shaping or CNC milling but the grinding wheel is the dominant method. The gear is
20、 manufactured in a hobbing process using a cutting tool called hob. The hob geometry is similar to the pinion geometry, except its tooth sides in the axial direction are straight as shown in Figure 3. It has a certain number of gashes in the direction perpendicular to the helix that are sharpened fo
21、r cutting. During machining the gear and the hob will be rotating with an RPM ratio equals to the ratio of their teeth numbers. A cylindrical hob and a tapered hob are shown in Figure 4. There are a total of nine basic geometry parameters. They are: outside diameter, tooth height, number of teeth (s
22、tarts), lead (axial pitch), high side pressure angle, low pressure angle, tooth width on top in axial direction and the start and the end of its tooth along the shaft with respect to centerline. The taper angle is an additional parameter for the tapered hob. Other parameters that have to be taken in
23、to account when designing a hob are: gear inside radius, gear outside radius, gear number of teeth and the center distance (offset) between the gear axis and the hob axis. Also in the skew angle case the skew angle is another additional parameter. This large number of parameters, 13 - 15, makes it h
24、ard to predict the gear tooth shape and profile. Designing a hob for a new gear set has the potential to be a trial and error process. The design objective is to have a gear tooth that is free of gouging on its sides or clipping of its top. It also should have the desired width and a balance of pres
25、sure angles and land area at the root and at the top. After selecting the geometry, the hob is manufactured by an outside machine shop with a lead time of eight to ten weeks. Then the gear is cut in-house. It can be difficult during this process to realize the precise design objectives of the desire
26、d gear tooth geometry. Figure 3. Axial cross-section of a cylindrical hob Figure 4. Cylindrical and tapered hobs 5 14FTM11 Initially the hob is placed over the gear blank as shown in the Figure 1. During machining the gear and the hob will be rotating. Machining will be complete when the hob penetra
27、tes a distance equal to the tooth height. In the case of the skew angle gear, the hob axis will have an incline angle over the gear plane. After cutting a certain number of gears, the hob becomes dull. The hob will be sharpened and its diameter will be reduced by a few thousands of an inch each time
28、. This will have some effect on the gear tooth profile and its contact with the pinion. In spite of their torque capability advantages, Spiroid gears have traditionally been a niche industry. Because of the lead time and costs associated with making the hob, their use is sometimes reserved for cases
29、 where high torque is a must. Many manufactured gears are smaller than 5 inches in diameter. Since the full-length tooth profile of the manufactured gear remains unknown, true 3D models of the gear cannot be produced, and it can be difficult to determine FEA and tooth strength analysis under load ou
30、tside of classical analytical methods. Mathematical modeling solution Gearometry has developed math equations and processes to accurately calculate the gear 3D, xyz, tooth profile in terms of the hob geometry, the machining setup and the gear parameters. We also developed software in the form of Exc
31、el spread sheets where values of the parameters are entered and the 3D profile of the gear tooth or gear groove is produced in the form of sketches, as in Figures 5 through 7, and in the form of xyz point data organized data arrays. Figure 5. 3D sketch of an optimized flat face Spiroid gear groove s
32、howing the high and low sides Figure 6. Gear groove profile for a gear inside radius of 1.25” 6 14FTM11 Figure 7. Gear groove profile with enhanced design using Gearometry software For gear tooth profile representation the selection of an appropriate coordinate system is very important. The z axis o
33、f our coordinate system is always aligned with the gear axis. For the flat face gear, z = 0 is at the root of the tooth. The x axis is parallel to the center distance and the y axis is parallel to the pinion/hob axis. In the case of the flat face gear, the xyz points of the tooth profile are arrange
34、d in 11 horizontal lines for the high side (parallel to the x-y plane) and 11 lines for the low side. Each line has 21 points. Therefore, each side has 231 points. The lowest line of each side is at the root where z = 0. The highest line is at the top where z = tooth height. In Figures 5 through 7 t
35、he eleven lines in the southeast corner represent the low side and the eleven lines in the northwest corner represent the high side. The two heavy black lines in the center are at the root and two heavy red lines are located at the tooth top. Therefore, the sketches in these Figures represent the ge
36、ar groove where a tooth of a hob or pinion fits. The last heavy red line in the northwest corner is the top of the low side of next groove. The area between this line and the line before it is the land area of the tooth top. Each pair of lines of the same color represents a horizontal cross-section,
37、 in the x-y plane, of the gear groove. The cross-sections are equally spaced vertically. The separation distance between them is one tenth of the tooth height. As in the conventional hob design, at the beginning, one enters values of the design parameters. But in our software, the designer will be a
38、ble to manipulate the entered values in the spread sheet and observe how the tooth profile will change until the desired design is obtained. This optimization process can typically take an hour or two depending on individual cases and user satisfaction. At the end, this process will result in the de
39、signing of the hob geometry and determining of the optimum machining setup parameters to produce the perfect gear. For illustration we will explain the process of designing a set of Spiroid gear and its pinion. We will consider the flat face gear set of Figure 2. This set was designed and manufactur
40、ed in the conventional method. The values of its 13 design parameters are listed in Table 1. They were estimated using reverse engineering of the hob geometry from the available pinion geometry. We used our software to sketch the gear tooth profile with the parameters in Table 1 (also shown in in Fi
41、gure 5). At this point, we ask the question whether the gear tooth profile is truly optimized, or is it merely an example of an “acceptable defect free“ profile. When designing a gear set, the objective is to maximize the torque capability and the contact ratio without increasing the set size. This
42、means extending the gear inside radius as much as possible toward the center. Using our software, we extended the gear inside radius down to 1.25” from 1.375” while keeping all other parameters values the same. Figure 6 shows that the tooth profile starts to have defects when inside radius is below
43、1.30”. However, the tooth will be defect free for inside radius of 1.31”. Extending the gear inside radius down to 1.31” can significantly increase the contact ratio, increase the tooth strength and increase the gear set torque capability. The drawing in Figure 7 shows that a perfect gear tooth prof
44、ile can be obtained for an inside radius of 1.22 inches when the hob lead is reduced from 1.50 to 1.32 inches, the high pressure angle is reduced from 40 to 35, the low pressure angle is reduced from 25 to 23 and the tooth width on the top is 7 14FTM11 reduced from 0.025” to 0.01”. The gear outside
45、radius, the center distance, the tooth height and the RPM ratio are kept the same. With this new design the contact ratio is increased from 1 to 2. The gear tooth is significantly increased in circumferential width and in radial width. However, this improvement is at the expense of the pinion tooth.
46、 Using a narrower hob tooth, the pinion tooth width at tooth mid height will be reduced by about 17.8%. But now we have two teeth in contact at any time instead of one, therefore the load per pinion tooth is reduced by 50%. Depending on the application, the designer has to make decisions on whether
47、to enhance the gear tooth, the pinion tooth, the contact ratio or some other component of the gear set. This software provides a wide range of options to the designer with accurate information for decision making. In fact, more good options could be further explored by manipulating the tooth height
48、value. Smaller heights would allow wider groove or larger pressure angles. Both would lead to a wider pinion tooth. In addition, the xyz data points of the curves in the last Figure can be transported into engineering CAD software to generate the true 3D model of the gear. One model is shown in Figu
49、re 8 for a 35 tooth gear. Such models can now be used to perform accurate FEA analysis under load that will help in making design decisions. Figure 9 shows FEA results on the Spiroid/worm hybrid gear of Figure 21 where the 3D model of is generated from xyz data points. Table 1. Parameters and their optimized values of the flat face gear design Hob thread start 0.250” Hob thread end 1.750” Hob radius 0.410” Hob tooth height 0.190” Hob high pressure angle 40 Hob low pressure angle 25 Hob tooth width on top 0.025” Hob number of teeth 4 Hob lead 1.500” Gear inside
