A Study on Noncircular Gears with Non-Uniform Teeth

A Study on Noncircular Gears with Non-Uniform Teeth Kazushi Kumagai* 1 and Tetsuya Oizumi* *1 Department of Infomation System, Sendai National College of Technology 4-16-1 Ayashi-Chuo, Aoba-ku, Sendai 989-318, JAPAN ckuma@sendai-nct.ac.jp * Department of Intelligent Electronics, Sendai National College of Technology 4-16-1 Ayashi-Chuo, Aoba-ku, Sendai 989-318, JAPAN ooizumi@sendai-nct.ac.jp Abstract The authors propose a concept of noncircular gears with non-uniform teeth NTG. Noncircular gears are usually designed by the uniform design parameters: pitch; tooth height; pressure angle; and so on. Because they are usually manufactured by conventional gear cutting machine using cutter for the standardized circular gears, which have uniform design parameters. If we abandon the use of the conventional gear cutting machine, we can vary the design parameters along the pitch curve. Hereby, it is possible to optimize the design parameters for each tooth, and the noncircular gear may be given excellent characteristics. Therefore, we may apply NTG to the new field where we could not apply them before. In this paper, the authors briefly introduce the concept of NTG and discuss conditions for the design parameters of NTG that have to be satisfied. As an example, a pair of NTG with functionalized module non-uniform module are shown. It was designed under the discussed conditions and manufactured by a wire cut electric discharge processing machine wirecut EDM. Moreover, as an applied example of NTG, a noncircular gear set paired with a circular gear on the one side, which is used to cancel out error of rotational angle is mentioned. 1. Introduction Gears are very important elements in mechanism and are used at various places in machine. Usually, gears are circular gears with a constant angular speed ratio. Also, we have noncircular gears. Its angular speed ratio varies and so it may be useful in various application. But the use of them is rare. As the reason, there are difficulties in designing and cutting them. At the designing, the teeth become thin partly and furthermore are even undercut at the large curvature of the pitch curve. In addition, if the gear has concave shape where the curvature of the pitch curve is negative, it must be examined on internal-gear interference. Consequently, in order to design noncircular gears with enough strength and no interference, designer has to modulate even the transmission function given as the primary specification. At the cutting, on the other hand, the generating motion is complex, that is, position of the pitch-point and speed of the generating motion fluctuate. Therefore, the cutting force is not uniform and the cutting accuracy is not monotonous. Moreover, the gear with negative curvature of the pitch curve can not be cut by hobbing machine. Even by gear shaper, some trimming interference must be examined. When usual hobbing machine or usual gear shaper is applied to machining noncircular gears, ordinary cutting tool for circular gear is used. Both the hobbing machine and the gear shaper are build fundamentally to generate the generating motion for circular gear cutting, and the cutting tool is also constructed for circular gear of uniform teeth. Therefore, there are some restrictions in such machining for noncircular gears. The restrictions are added to the difficulties of designing noncircular gears. In order to remove the above-mentioned weak points, we propose to apply wirecut EDM to cutting noncircular gears. Because the wirecut EDM uses a wire as cutting tool, any tooth profile can be cut as if it were drawn by plotter, that is, there are few restrictions for designing noncircular gears. We proposed a method in which an idea of non-uniform pitch was applied to solving the problem of thin teeth of composite noncircular gears 1. The composite noncircular gear is a noncircular gear which has two different pitch curves for forward and backward rotation, respectively. The two pitch curves have different length and also the pitch is different from each other. Therefore tooth thickness varies along both pitch curves. Tooth thickness, however, has to be uniformalized as possible, because of homogeneous strength. We proposed to vary the pitch instead of the tooth thickness along the pitch curve. Therefore, the composite noncircular gear can not be cut by usual gear cutting machine tools. It can be cut by wirecut EDM. The idea of non-uniform pitch overthrows the limitation of designing noncircular gears. In this paper, we extend the above-mentioned

idea, and reconsider design conditions such as pitch, module, pressure angle, and so on. From these, we propose a concept of noncircular gear with non-uniform teeth NTG which dose not possess uniform design conditions. As an example, we introduce a gear with functionalized module FMG which varies the pitch and the tooth height functionally, and describe the designing results and cutting results. As another example, we also introduce a noncircular gearset paired with a circular gear and a noncircular gear. This gearset is applied to canceling out angular displacement errors in worm drive for work table of machine tool.. A concept of noncircular gear with non-uniform teeth.1 Design conditions of gear According to Japanese Industrial Standards, JIS B 010, "gear is the machine element which transmits movement by teeth meshing succeedingly." In the kinematics of machinery, however, gear is defined as "the machine element which realizes rolling movement of a pair of pitch curves by teeth meshing succeedingly." Accurately to realize the rolling movement, the tooth profiles must satisfy the fundamental law of gearing: "The profile of the teeth of a gear must be such that the common normal at the point of contact between two teeth always passes through a fixed point on the line of centers" 3. In the first step of tooth profile discussion, it is required only how to satisfy the fundamental law of gearing, and there is no concept of the pitch as an interval of teeth. The intervals of teeth become important from the next step in which meshing is maintained succeedingly. But a concept of constant pitch is not introduced in the definition of gear. Here, we try to consider an ordinary circular gear with the involute tooth profile, that is, spur gear. At the design of the gear, we implicitly use the following conditions: 1 constant gear ratio truly circular pitch curve uniform pitch 3 uniform module 4 uniform pressure angle 5 uniform addendum modification coefficient 6 uniform tooth height These conditions are considered in the cross-section including the pitch circle. Also they are considered on bevel gear. The condition 1 is most essential in the kinematics, and means that the transmission function is constant. On the other hand, the conditions -6 are not essential, but are required from viewpoints of the economic efficiency: exchangeability; uniformity; standardization; easily manufacturing; and so on. An ordinary noncircular gear is the gear from which the condition 1 is removed to adapt to some particular purpose. Therefore noncircular gears are not used in wide and various purposes so much, and their standardization is rarely required. For the noncircular gears, the conditions -6 are remained in order to manufacture them using ordinary cutting tools and machine tools of circular gears. Therefore, if not using those tools, the noncircular gears are able to be designed without restrictions based on the conditions -6. This means that the parameters concerned with the conditions -6 can be varied and optimized to adapt to the purpose of the gears, that is, it is possible to design gears under higher degrees of freedom. Consequently, we named such noncircular gears "non-uniform teeth noncircular gears" in the meaning that parameters of tooth differ on each tooth. Of course, circular gears with non-uniform teeth is also possible to be designed and manufactured.. Non-uniform pitch Even though the condition of uniform pitch is removed, the more essential condition should be satisfied to mesh succeedingly. Though the pitch is the uniform interval between teeth, we redefine that pitch is length on the pitch curve which is allotted to each tooth, in other words, a pitch is a section. Since the number of teeth is integer, pitch has discrete value. Thus, the relationship between the whole length S Z of the pitch curve and the pitch p i of the tooth No.i is obtained as Eq. 1. S Z = π 0 r p + r p ' dθ = i= 1 1 where, Z is the number of a tooth, r p is radius vector pointing at a point on the pitch curve, θ is polar angle, r p ' is showing the derivative of r p θ. Equation 1 is the most essential condition that not only non-uniform pitch gears but also every gears should satisfy..3 Non-uniform pressure angle Assuming that the tooth profile is involute curve, we can consider that there are two ways to vary the pressure angle as following: 1 To change the pressure angle on each tooth. To vary the pressure angle along the tooth profile. The way 1 means that each tooth possesses a different line of action individually. The way means that the tooth profile is not involute curve..4 Non-uniform tooth height By changing the tooth height on each tooth, the contact length is changed. Therefore, the contact ratio can be changed by each tooth. But it is necessary to examine on undercut, interference, and so on. Z p i

3. Cutting gears by wirecut EDM We are conforming to use positively wirecut EDM as a way of cutting the noncircular gears, and then evaluating the characteristics of the cutting. 3.1 Manufacturable gear by wirecut EDM Wirecut EDM performs electric discharge cutting by a stretched wire in accordance with its name, and is able to cut only ruled surface which is generated by a moving straight line. Therefore, the gear which is formed of the ruled surface can be cut by wirecut EDM. For example, spur gear, helical gear, straight bevel gear, helical bevel gear, of course, noncircular spur gear, noncircular helical gear, and so on are formed by ruled surface. However, there is a possibility that even such gears can not be cut, because wirecut EDM possesses wire supporting dies of which size is much larger than tooth space, and its range of taper cutting angle is limited. 3. Cutting accuracy Cutting accuracies of the wirecut EDM equipped in Sendai National College of Technology are 3 µ m of positioning accuracy, and 5µ mrmax of surface roughness according to inspections by the manufacturer of the EDM. Thereby, we may expect to make gears of 0 or 1st class accuracy in JIS. But the surface roughness is not enough to use as tooth surface. To reduce the influence of the electric discharge cracks to the tooth strength, it is necessary to finish the tooth surface by adequate finishing method. 3.3 Machining rate It is the largest weak point that the wirecut EDM takes a lot of time to cut. Though accurate cutting speed depends on material of work, usual cutting speed is about 50 mm /min. Therefore, in a case of a spur gear of the pitch circle diameter 100mm, module, and face width 10mm, only the first cut takes -3 hours. Including the second cut and finishing, the whole process takes 5-6 hours. Even if it can be automatically operated for all-night using the automatic wire tying, it takes more than 10 hours to cut one pair of the gears. Consequently, wire EDM is not appropriate for a machine tool to cut many gears in process lines of mass production. If plastic injection molds, die casting, or punch press is applied to gear manufacturing, wirecut EDM is appropriate to cutting the mold, the die, or the punch. 4. Gear with functionalized module As an example of non-uniform teeth gears, we designed a gearset of which pitch and tooth height varied functionally. In the case of standard spur gear, using module m, the whole tooth height is.5m and circular pitch is π m. Both the pitch and the tooth height are functions of the module m. Hereat, if the module m is varied functionally along the pitch circle, the gear possesses teeth of non-uniform pitch and non-uniform tooth height Fig. 1 Fig. Define of non-uniform pitch p i Define of non-uniform pitch q j which vary functionally. we define such a gear "gear with functionalized module". 4.1 Pitch Since pitch is the length of pitch curve which is allotted to one tooth, the pitch p i has a start point and an end point. If considering the individuality of one tooth, the pitches should be defined like Fig. 1. But here we define new pitch which is allotted to each tooth profile like Fig., because of more smoothly varying the teeth size. Pitch q j of tooth profile No.j is defined like Fig.. The module m is assumed as some function of pitch curve length S. When S j is defined a length from the origin on the pitch curve to the median point of pitch q j, Pitch q j is expressed as π q j = m( S j ) From Fig. and Eq., Eq. 3 is obtained as follows. 1 S j+ 1 S j = ( q j+ 1 + q j ) 3 From Eq. and Eq. 3, Eq. 4 is obtained. π S j+ 1 S j = ( m( S j+ 1) + m( S j )) 4 4 Since point S j is the median point of pitch q j, point S j is considered to be the intersection of the tooth profile No.j and the pitch curve, that is, the position of tooth profile. Point S j is defined as mean point to locate the tooth profile. Now, if the value of S 1 is given as an initial value, tooth profile position S j is obtained succeedingly from Eq. 4. If Eq. 4 can not be rewritten as an explicit function of S j+1, we obtain S j from Eq. 4 iteratively by Newton-Raphson method.

Thus, since each side profile of one tooth is considered to be independent and mean point S j is the intersection of tooth profile and pitch curve, mean point of another gear in mesh is also such intersection. In this case, following Eq. 5 is defined from the relationship of Eq. 1. π 0 S Z = rp + rp ' dθ = Z 5 4. Addendum curve and dedendum curve When radius of pitch curve r p is given, radius of addendum curve r a and radius of dedendum r d are expressed as Eq. 6 and Eq. 7, respectively. ra = rp + m(s) 6 r 7 d = rp 1.5m( S) 4.3 Example gear To explain the concept of non-uniform teeth gear, we tried to design and cut a pair of circular gear which has constant gear ratio :3 and non-uniform module. Because of circular gears, the relationship between pitch curve length S and polar angle of radius θ r p is obtained as Eq. 8. S = r p θ 8 The module m is given as a function of θ, and then expressed as Eq. 9. m = m0 + ma cos( nθ ) 9 where m 0 is the average value, m a is an amplitude, and n is a number of leaves. The specifications for gear 1 and gear are shown in Table 1. In Table 1, the amplitude m a is already adjusted to satisfy Eq. 5. Designed gears are shown in Fig. 3 and Fig. 4, and the gearset, as a result of machining by wirecut EDM, are shown in Fig. 5. j= 1 Table 1 Design conditions of gear 1, gear No. of gear 1 Rad. of pitch curve r p mm 40 60 Average of m m 0 3 3 Amplitude of m m a 1.6607 1.6607 Number of leaves 3 Initial position S 1 0 0 Fig. 3 Profile of gear 1 q j Fig. 5 Fig. 4 Profile of gear A gear pair of functionalized module As applications of this gearset, we consider flow meter, and particular gearset in which contact ratio is partly improved to adapt to fluctuating load by rotating angle of gear. 5. Noncircular gearset paired with a true circular gear 5.1 Phase adjustment of noncircular gearset Noncircular gearset needs to adjust two phases between gears and between gear and axis of output shaft at assembly. On compensating angular displacement error of index table on machine tool, the error is not so large, because the worm gearing in the index table is designed to rotate on a constant gear ratio at high degree of accuracy. This means that the noncircular gear which should be used for the compensation becomes near to a true circular gear. In addition, for the addendum and dedendum curve of the noncircular gear, we substitute addendum circle and dedendum circle of a corresponding circular gear, because the composite noncircular gear has two different addendum curves and two different dedendum curves. On account of these, the phases adjustment may become more difficult. To reduce the difficulty, we proposed a gearset in which a non-uniform teeth noncircular gear was paired with a true circular gear.

Hereby, it becomes unnecessary to adjust the phase between gears. Consequently, the difficulty is reduced drastically. On the noncircular gear which meshes with a true circular gear, the pitch, pressure angle and tooth height do not become uniform. Therefore, the concept of non-uniform teeth noncircular gear is applied to this case as a practical use. 5. Application of NTG Because parameters of each tooth can be chosen freely in non-uniform teeth noncircular gear, it is possible to design noncircular gear which mesh with true circular gear. On the other hand, true circular gear possesses uniform pitch, uniform pressure angle, uniform profile shift, when it is used on a pitch circle of which center coincides with the gear center. Conversely, on the noncircular pitch curve given by transmission function to cancel angular displacement error, even true circular gear naturally has non-uniform pitch, non-uniform pressure angle, non-uniform profile shift. This means that the circular gear can be treated as a noncircular gear with non-uniform teeth, and the noncircular gear which meshes with circular gear must be a non-uniform teeth noncircular gear. 5.3 Merits of noncircular gearset paired with circular gear There are some merits in noncircular gearset paired with circular gear in addition to reducing difficulty of phase adjustment. The major merits are listed below. 1 A circular gear is ordinary enough in the market of machine parts. Therefore, its cost may be reduced. Using the circular gear on the motor side, its maintenance may be easy. 3 Making the circular gear of not hardened material, abrasion of noncircular gear may be prevented. These above merits are from a economical viewpoint. In addition, the following merit is obtained from mechanical viewpoint. 4 It is unnecessary that the gear ratio of noncircular gear to circular gear is a simple integer ratio. This means that any value such as 5.96 can be used as the gear ratio, and this fact can't be thought on former noncircular gears. It is a very strange merit. 5.4 Design of the gearset Now, we assume that the gearset is meshing in the given transmission function, the driver is a circular gear, and the follower is a noncircular gear. Thus, by enveloping loci of the driver's tooth profile in the follower coordinate system, the follower's tooth profile is generated. To make the composite noncircular gear, it is Fig. 6 Profiles of composite noncircular gearset paired with a circular gear Fig. 7 Photograph of composite noncircular gearset paired with a circular gear necessary to generate the forward and backward tooth profiles for the forward and backward pitch curves respectively, and then we must examine on interference of backward tooth profile in meshing. As the result, we must obtain the gearset which cause no interference at all. 5.5 Example gear Based on angular displacement error of a worm gearing which is the experimental equipment for error compensation, we obtained the transmission function, and then we designed a gearset which was paired with a circular gear and a non-uniform teeth noncircular gear. The result are shown in Fig. 6. The whole angular displacement error which the gearset should compensate is 40 seconds in total on the worm wheel axis. This total error consists of a medium periodic component 8 seconds and a residual component 18 seconds. The medium period corresponds to one rotation of the worm. The residual component consists of shorter periodic components. Since the gear ratio of the worm gearing is 60, the whole error is 40 minutes on worm axis. Consequently, we designed the gearset to compensate the error of 40 minutes. The gearset machined by wirecut EDM are shown in Fig. 7. 6. Conclusions The authors reconsidered gear design conditions: uniform pitch; uniform pressure angle; uniform tooth height; and so on, and proposed a concept of noncircular gear with non-uniform teeth. At this viewpoint, the design conditions to satisfy

were discussed. In addition, we proposed and discussed using wirecut EDM, instead of conventional gear cutting machines, as a machine tool for noncircular gears. As an example of non-uniform teeth gears, we introduced a gearset with functionalized module of which pitch and tooth height varies with module functionally, designed it, and manufactured by wirecut EDM. This is an example that we applied the concept of non-uniform teeth gear to true circular gears. At the first stage of this research, the concept was developed on purpose to overthrow the limits of noncircular gear design. When the limits are overthrown, noncircular gears may be applied to new fields of use. As an example of noncircular gears with non-uniform teeth, we introduced a gearset which was paired with a noncircular gear with non-uniform teeth and a true circular gear. The gearset is developed on purpose to compensate displacement error of worm gearing in an index table. The results of design and manufacturing were described, and the usefulness of the concept for non-uniform teeth noncircular gears were shown. By the way, though we are using wirecut EDM to cut noncircular gears, it may be possible to use also form carving EDM for gear manufacturing. Since there are some merits: no back force component; accurately generating of each tooth surface with pinion form electrode; and so on, we will try to use the EDM for cutting noncircular gears in the near future. Acknowledgments We thank Dr. T. Sakai and Dr. T. Emura, the emeritus professors of Tohoku University, for there encouragements and effective suggestions. References 1 Kumagai, Emura, Arakawa, Oizumi, "A Study on High Precision Servomechanism Using Noncircular Gear, nd report", JSME, No.941-, 1994, p.7. Kumagai, Oizumi, Emura, Arakawa, "A Study on High Precision Servomechanism Using Noncircular Gear, 4th report", JSME, No.951-, 1995, p.169. 3 Wilson, C.E., Sadler, J.P., Michels, W.J., Kinematics and Dynamics of Machinery, Harper Collins Publishers, 1983