1、14FTM18 AGMA Technical Paper Precision Bevel Gears with Low Tooth Count By S.P. Radzevich and V.V. Irigireddy, Apex Tool Group, LLC2 14FTM18 Precision Bevel Gears with Low Tooth Count Stephen P. Radzevich and Vishnu V. Irigireddy, Apex Tool Group, LLC The statements and opinions contained herein are
2、 those of the author and should not be construed as an official action or opinion of the American Gear Manufacturers Association. Abstract The paper deals with the geometry and kinematics of right-angle bevel gears that feature low tooth count. Gears that have 12 teeth and fewer are referred to as t
3、he low-tooth-count-gears (or just gears, for simplicity). It should be stressed from the very beginning that gears are not covered by either AGMA, or by any other national/international standards on gearing. This is mostly because of the kinematics and geometry of gears is not profoundly investigate
4、d yet. When operating, right-angle bevel gears generate vibration and produce an excessive noise. Dynamic loading of the gear teeth can result in the tooth failure. These problems become more severe in bevel gearings with low tooth count. Variation of the base pitch in bevel gearing is the root caus
5、e for the vibration generation, an excessive noise excitation, and unfavorable dynamic loading of the gear teeth. It has been proved 1 that the equality of: - The base pitch of the gear - The base pitch of the mating pinion - The operating base pitch is the fundamental requirement for the operating
6、of right-angle bevel gears that feature low tooth count. In this paper, the main features of design and machining of bevel gears are discussed. The proposed approach is based on the triple equality: Base pitch of the gear = Base pitch of the pinion = The operating base pitch Once the base pitches ar
7、e equal, then the root cause for vibration generation, noise excitation, and unfavorable dynamic loading of the gear teeth is eliminated. The consideration is focused mainly on right-angle bevel gearing. However, the reported results of the research are applicable for bevel gearings with different s
8、haft angles. Copyright 2014 American Gear Manufacturers Association 1001 N. Fairfax Street, Suite 500 Alexandria, Virginia 22314 October 2014 ISBN: 978-1-61481-110-7 3 14FTM18 Precision Bevel Gears with Low Tooth Count Stephen P. Radzevich and Vishnu V. Irigireddy, Apex Tool Group, LLC The paper dea
9、ls with the geometry and the kinematics of right-angle bevel gears that feature low tooth count. Right-angle bevel gears are a particular case of intersected-axis gearing (further Ia-gearing) with an arbitrary value of the shaft angle. Commonly, bevel gears with the base cone angle (bfor the bevel g
10、ear and bfor the bevel pinion) larger than the root cone angle (ffor the bevel gear and ffor the bevel pinion), that is, when the inequalities b fand b fare observed, are referred to as low-tooth-count gears, LTC-gears1. The geometry and the kinematics of gears that have 12 teeth and fewer are the m
11、ain focus of this paper2. All of the equations derived for LTC-gears are valid for gears with an arbitrary tooth count, and not only for gears with a large tooth count. When operating, right-angle bevel gears often generate vibration and produce an excessive noise. Dynamic loading of the gear teeth
12、can result in the tooth failure. These problems become more severe in bevel gearings with low tooth count. The performed analysis shows that inequality of base pitches of the gear and mating pinion is the root cause for insufficient performance of LTC-gears. In most applications, the main purpose of
13、 Ia-gearing is to smoothly transmit a rotation and torque between two intersected axes. Gear pairs that are capable of transmitting a uniform rotation from the driving shaft to the driven shaft are referred to as the geometrically accurate intersected-axis gear pairs (or, in other words, the ideal i
14、ntersected-axis gear pairs). Three requirements need to be fulfilled in order to a bevel gear pair can be referred to as the geometrically accurate bevel gear pair: - The geometry of the tooth flanks of a bevel gear and a mating bevel pinion has to obey the condition of contact. The condition of con
15、tact can be analytically represented in the form of dot product n V= 0 of the unit vector n of a common perpendicular at point of contact of tooth flanks G and P of the gear and the mating pinion, and the vector of the velocity of the relative motion of the tooth flanks G and P. The equation of cont
16、act n V= 0 is commonly referred to as Shishkovs equation of contact 1, 2. This equation was proposed by Shishkov as early as 1948 (or even earlier). - The geometry of the tooth flanks of a bevel gear and a mating bevel pinion has to obey the condition of conjugacy. To meet this requirement, common p
17、erpendicular at every point of contact of the tooth flanks G and P must intersect the axis of instant rotation (the pitch line, in other words). The shaft angle of a bevel gear pair is subdivided by the pitch line in a proportion that corresponds to gear ration of the bevel gear pair (see equations
18、4 through 8). - The geometry of the tooth flanks of a bevel gear and a mating bevel pinion has to ensure equal base pitches of (a) the gear, (b) the pinion, and (c) the operating base pitch of the gear pair, that is, these three base pitches must be equal to one another at every instant of time. A f
19、ew comments regarding the aforementioned requirements, which ideal bevel gearing has to obey, immediately follow. First, the necessity to meet the condition of contact, n V= 0, is obvious. If the condition of contact is violated, this immediately results either in the interference of the tooth flank
20、s G and P into each other or in departure of the tooth flanks G and P from one another. None of these two scenarios is valid in gearing. 1It is instructive to note here that in case of Pagearing, e.g., spur gearing, an equation dbg= dfgcan be composed. After the base diameter, dbg, and the root diam
21、eter, dfg, of a gear are expressed in terms of the module, m, (or diametral pitch, Pd), the tooth count, Ng, and the transverse profile angle, t, the solution to the equation dbg= dfgwith respect to Ngreturns Ng= 41.6. Therefore, Pa-gears with the standard tooth profile and the tooth count Ng 41 are
22、 referred to as LTC-gears. In general sense, a similar is valid with respect to Ia-gearing with low tooth count. In a narrower sense, LTC-gears are viewed as those with the tooth count Ng 12. 2It should be stressed here that LTC-gears are covered by neither AGMA, nor by any other national/internatio
23、nal standards on gearing. This is mostly because the kinematics and the geometry of LTC-gears are not profoundly investigated yet. 4 14FTM18 Second, the condition of conjugacy of the tooth flanks G and P of the bevel gear and pinion is an equivalent to the well-known Willis theorem 3. The Willis the
24、orem relates to parallel-axis gears (Pa-gears, for simplicity). No condition of conjugacy of the tooth flanks G and P in the cases of Ia-gearing as well as Ca-gearing (that is, for the case crossed-axis gearing) is known so far. Below, the condition of conjugacy is enhanced to the case of Ia-gearing
25、. Third, it should be noted here that the cycle of meshing of only one pair of gear teeth is covered by the condition of conjugacy of the tooth flanks G and P of the bevel gear and pinion. Contact ratio in all gearings is always greater than one. Therefore, at a certain instant of time, two or more
26、pairs of teeth are engaged in mesh simultaneously. To make the multiple contacts feasible, the equality of base pitches of (a) the gear, (b) the pinion, and (c) the operating base pitch is a must. Here and below the angular distance between each two adjacent desirable lines of contact LCidesand LCi+
27、1desis specified by the operating base pitch bop(equation 19); here i is an integer number. The angle bopis measured within the plane of action, PA, of the gear pair. This angle centered at the common apex Ag= Ap= Apa4. The operating base pitch, bop, of a bevel gear pair is illustrated in Figure 5.
28、Equality of three base pitches: bg= bp= bop(see items a) through c) is referred to as the fundamental law of gearing. Determination of the design parameters for ideal intersected-axis gearings is considered below. Elements of the kinematics in Ia-gearing Consider an Ia-gear pair shown in Figure 1. T
29、he driving pinion is rotated about its axis, Op, with a certain angular velocity, p. The driven gear is rotated about its axis, Og, with a certain angular velocity, g. The axes of rotation, Ogand Op, intersect at point Apa. These axes form an angle, . Commonly, this angle is equal to a right angle (
30、that is, = 90); however, Ia-gearings either with an angle, 90, or with an angle 0) or negative (adv 0). The operating base pitch angle, bop, is measured within the plane of action, PA. This is the central angle between two corresponding points within the lines of contact for two adjacent pairs of te
31、eth. For example, in Figure 5, the angle bopis shown between two points u5and v5that are located within a circular arc of an arbitrary radius rypa(centered at the apex Apa) and the lines of contact LCiand LCi+1for two adjacent pairs of teeth. 10 14FTM18 Figure 5. Contact ratio in Ia-gearing 4 The op
32、erating base pitch angle can be calculated from: 2sin2sinbop g bpbNN (19) where Ng, Npare tooth counts of the gear and the pinion correspondingly. Tooth proportions in Ia-gearing The discussed results of study of Ia-gearing enable one to calculate tooth proportions in a geometrically accurate gear a
33、nd pinion. Tooth thickness, space width, and backlash, are convenient to specify within the pitch plane, PP, of a gear pair. When a gear pair operates, rotations of: (a) the gear; (b) the pinion; and (c) the pitch plane are synchronized with one another. Therefore, when a gear (and a mating pinion)
34、turns through one tooth, the pitch plane also turns through one tooth, that is, the PP turns through an angle N. The angle Nis calculated as shown in equation 20 (see Figure 6): 2sin2sinNgpNN (20) where is pitch cone angle of the gear; is pitch cone angle of the mating pinion. Once the angle Nis det
35、ermined, the angular tooth thickness, t, and the angular space width, w, can be calculated. By definition, the following equality is valid: Ntw (21) When designing a pinion, it is common to set the angular tooth thickness equal to the angular space width, that is: 0.5tw N (22) 11 14FTM18 Figure 6. D
36、efinition of tooth thickness, t, and space width, w, in Ia-gearing (measured within the pitch plane, PP) 4 When designing a gear, the gear tooth thickness is decreased by backlash, B, that is Btw (23) Other proportions among the design parameters t, wand B, can be observed as well. It should be stre
37、ssed here again that there is no slippage between the pitch cones of the gear/pinion and the pitch plane when the gears rotate. Therefore, it can be imagined that the pitch plane wraps on (unwraps from) the corresponding pitch cones. Because of this, the design parameters measured within the pitch p
38、lane, correspond to the arc (and not to the chordal) design parameters of the gear and the pinion. Addendum and dedendum of a bevel gear also can be specified as the angular addendum and the angular dedendum of the gear. The angular tooth addendum in Ia-gearing is specified by the angular distance b
39、etween the pitch cone of the gear and the gear top-land cone (outer cone) of the gear. For bevel gears with standard tooth proportions, the tooth height of a bevel gear is set equal to module, m. This makes it possible to calculate the angular addendum, a, of the gear from the expression: sin-1aoppm
40、r (24) In a similar manner, the angular dedendum is specified. For bevel gears with standard tooth proportions, the dedendum is greater the addendum at a clearance, c. Therefore, the angular dedendum, d, of the gear is calculated as follows: sin-1doppmcr (25) The angular addendum, a, and the angular
41、 dedendum, d, of the gear tooth together specify the angular tooth height, h, of the gear (see Figure 1): had (26) Formulas similar to those aforementioned: sin-1aoppmr (27) 12 14FTM18 sin-1doppmcr (28) had (29) are valid for the calculation of the angular addendum, a, the angular dedendum, d, as we
42、ll as the angular tooth height, h, of a standard bevel pinion (Figure 1). The aforementioned design parameters in intersected-axis gearing correlate to corresponding design parameters in parallel-axis gearing. Conclusion An approach for designing geometrically accurate (ideal) bevel gearing is discu
43、ssed in the paper. Three issues are critical to achieve the goal. First, the condition of contact, n V= 0, between the tooth flanks G and P of the gear and the mating pinion needs to be fulfilled. The condition of contact represented in the form of a dot product n V= 0 is commonly referred to as the
44、 Shishkovs equation of contact. Second, geometrically accurate intersected-axis gearing must obey the condition of conjugacy. For this purpose, a new theorem is formulated for the case of Ia-gearing. This theorem is an equivalent of the well-known Willis theorem that is valid only for Pa-gearing. Th
45、e discussed approach shows how to design bevel gears that meet the requirements imposed by the condition of conjugacy. Third, as two (or more) pairs of teeth can be engaged in mesh simultaneously, geometrically accurate intersected-axis gearing must obey the fundamental law of gearing. This means th
46、at the triple equality must be fulfilled: Base pitch of the gear Base pitch of the pinion The operating base pitch The consideration in the paper is focused mainly on right-angle low-tooth-count bevel gearing. However, the reported results of the research are applicable for bevel gearings with diffe
47、rent shaft angles and tooth counts. It is the right point to stress here that intersected-axis LTC-gearing deserves more attention because of a numerous reasons. Inevitably broader application of LTC gears in the future is one of the reasons. High-power-density gear trains need in use of LTC-gearing
48、. All the gearings are evolving towards the highest possible power density being transmitted. This entails a broader application of LTC-gearing in the future. References 1. Shishkov, V.A., Elements of Kinematics of Generating and Conjugating in Gearing, in: Theory and Calculation of Gears, Vol. 6, L
49、eningrad: LONITOMASH, 1948. 2. Shishkov, V.A., Generation of Surfaces in Continuously Indexing Methods of Surface Machining, Moscow, Mashgiz, 1951. 3. Willis, R., Principles of Mechanisms, Designed for the Use of Students in the Universities and for Engineering Students Generally, London, John W. Parker, West Stand, Cambridge: J. & J.J. Deighton, 1841. 4. Radzevich, S.P., Theory of Gearing: Kinematics, Geometry, and Synthesis, C
