Characteristics of trochoids and their application to

Mechanism and Machine Theory 35 (2000) 291±304

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Characteristics of trochoids and their application to determining gear teeth ®llet shapes Xiaogen Su*, Donald R. Houser Department of Mechanical Engineering, The Ohio State University, 206 W. 18th Ave., Columbus, OH 43210, USA Received 11 June 1997; received in revised form 13 April 1998

Abstract Some interesting characteristics of trochoids are discussed in this paper and a new concept of virtual involute is introduced. These characteristics prove very useful in determining the exact tooth ®llet shape generated by racks consisting of circular arcs and straight line segments. The ®llet curves generated by racks with a protuberance or with a chamfered tip are discussed in detail. A numerical example is given to illustrate their application. # 1999 Elsevier Science Ltd. All rights reserved.

1. Introduction Intensive studies have been done on involute gears [1±4]. Computer modeling and computer graphics have been used for generating gear teeth shapes [5,6]. However, the mechanism of the gear tooth generation, especially, of the formation of the tooth ®llet has not been well understood by gear designers until today. Some interesting and useful characteristics of trochoids are discussed in this paper. A new concept of virtual involute is introduced. These characteristics prove to be of great help in ®nding exact ®llet shapes generated by rack-type cutters. Based on these characteristics, a more intuitive proof of undercutting is given in this paper. The ®llet curves generated by racks with a protuberance or with a chamfered tip are discussed in detail. The location of the intersection point between two trochoids is also investigated. The developments presented in this paper make it possible to establish a more formal approach to determining the exact tooth shape generated by racks consisting of circular arcs and straight line segments. Such an approach is * Corresponding author. Tel.: +1-614-292-0900; fax: +1-614-292-3163. E-mail address: [email protected] (X. Su) 0094-114X/00/$ - see front matter # 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 9 4 - 1 1 4 X ( 9 9 ) 0 0 0 0 4 - X

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X. Su, D.R. Houser / Mechanism and Machine Theory 35 (2000) 291±304

helpful in the generation of boundary element meshes [7] and the generation of ®nite element models for gear root stress calculation [8], in the calculation of the geometry factor [7,9,10], and also to the reverse engineering in gear application. 2. Primary and secondary trochoids and their equations Most parallel axis gears used in industry are involute gears and hobbing is the most popular manufacturing technique used to rough cut these gears. The hobbing process is equivalent to the pure rolling of the pitch line of a rack against the pitch circle of the gear. Fig. 1 shows a rack with a round tip. When the pitch line rolls against the pitch circle, the straight side of the blade generates the involute portion of the pro®le. Every point on the rack traces out a curve which is termed as a primary trochoid by Khiralla [2]. Thus the track of the center of the round tip T0 is a primary trochoid. The round tip envelops another curve which is termed as a secondary trochoid in [2]. Because the left halves of the trochoids do not contribute to the ®nal formation of the tooth shape, only the right halves of the trochoids are shown in Fig. 1. A general point T on the primary trochoid is depicted in Fig. 1. The equation of the primary trochoid is: rp sinu rp u x T0 cosu yT0 sinu xT 1 rp 1 cosu yT rp u x T0 sinu yT0 cosu where angle u is the rolling parameter, rp is the radius of pitch circle, x T0 ,yT0 is the coordinate of point T0 in the ®xed coordinate system XPY. Based on the pure rolling motion, line TR is the normal direction of the primary trochoid at point T. The coordinate of the corresponding point F on the secondary trochoid can be expressed as: rp sinu rp u x T0 cosu yT0 sinu rt cosg u xF 2 rp 1 cosu rt sing u yF rp u x T0 sinu yT0 cosu where g TRM arctanyT0 =rp u x T0 and rt is the tip radius. Note that the primary

Fig. 1. The primary and secondary trochoids.


293

trochoid and the secondary trochoid are two equidistant curves. Point T and point F share the same normal line FTR. The normal of the two trochoids always intersects the pitch circle at the rolling point R between the pitch line and the pitch circle. 3. Characteristics of trochoids Fig. 2a shows a rack and Fig. 2b shows the generating process. Line P4 N parallel to the pitch line is drawn through the tangency point N (usually termed as the interference point) between the base circle and the line of action. It will be shown later that if the straight side of the blade intersects this line, undercutting happens; otherwise, undercutting does not happen. For this reason, line P4 N is termed as the interference line in this paper. In Fig. 2a, the interference line divides the side blade into two segments, s2 P2 P4 and s3 P4 P7 . Each point on the blade generates a trochoid. Those generated by P1 ,P2 ,P3 ,P4 ,P6 are shown and numbered as t1 ,t2 ,t3 ,t4 ,t6 in Fig. 2b. This family of trochoids have the following characteristics. Note that, except the ®rst one, each of the others applies only to the right halves of trochoids. 1. The normal at any point on the trochoid passes through the rolling point R between the pitch line and the pitch circle (refer to Fig. 1). The secondary trochoid generated by the round tip always lies beyond its corresponding primary trochoid. 2. Two points, P1 and P2 , on the same horizontal line trace out two identical trochoids t1 and t2 . If P2 is to the right of P1 , t2 stays to the right side of t1 . 3. A trochoid traced out by a point, such as point P6 , on segment s3 never penetrates to the right side of the involute generated by the segment s3 , but is always tangent to the involute. Trochoid t4 traced out by P4 is the limiting case. t4 is tangent to the involute at its starting point A0 on the base circle. 4. A trochoid traced out by a higher point, such as point P2 , on segment s2 stays to the right side of a trochoid traced out by a lower point, such as point P3 , on segment s2 at least until they reach the involute generated by s3 . An intuitive proof is given here. The area swept through by blade segment P3 P7 is covered completely by the area swept by P2 P7 in the ®llet region, thus t2 stays to the right side of t3 before they cross the involute. 5. If we draw another involute from the same starting point A0 on the base circle in the opposite direction to the involute pro®le, the trochoid traced out by any point on segment s2 never crosses this opposite involute and is always tangent to this opposite involute. Here the opposite involute is termed as the virtual involute because it will not actually be cut out. This will be proved in the following. It is established that the fundamental equation nvRG 0 holds at any generated point during the generation process (n is the common normal at the generated point and vRG is the relative velocity between the rack and the gear at this point, subscripts R and G denote the rack and the gear, respectively, nvRG is their dot product). In other words, the common normal at the cutting point must pass through the ®xed pitch point P [5,6]. Any point on a straight blade moves to such a position with its normal passing through the ®xed pitch point P when the point crosses the line of action. In this way, the blade

294


Fig. 2. Generating process.

segment P4 P7 generates the involute pro®le of the gear. Referring to Fig. 3, the blade segment P4 P2 above the interference line also generates an involute which is in the opposite direction to the gear pro®le. The opposite involute has the same starting point on the base circle as the involute pro®le because they share the same generating point P4 on the base circle. This


295

Fig. 3. The virtual involute.

opposite involute lies in the tooth space and is not actually formed. In the right side of Fig. 3, such a generating position is shown as a dashed line. The blade T4 T2 is tangent to the virtual involute at T3 which is the intersection of the blade with the line of action. In the coordinate system attached to the gear, the virtual involute is drawn as a thick solid line. Just as the trochoid t6 (Fig. 2b) never penetrates to the right side of the involute pro®le, the trochoid t3 (in Fig. 3) never penetrates to the left side of the virtual involute generated by the blade segment P2 P4 because the line segment P2 P4 always stays to the right side of the virtual involute during the generating process. There is one common point between the trochoid t3 and the virtual involute. This common point T3 is the point generated by P3 and it is the tangency point between t3 and the virtual involute. In the coordinate system attached to the gear, this point is G3 .

4. Application of the characteristics of trochoids 4.1. Proof of undercutting Zhao [11] presented an excellent proof of undercutting when the gear is generated by a gearshaped cutter. For a rack-type cutter, it has been stated [1,3,4] that when the straight side blade extends over the interference line, undercutting happens. Sometimes this statement seems a little elusive. A better proof can be given based on the characteristics of trochoids. As has been stated above, the intersection point P4 of the blade with the interference line traces out the trochoid t4 that is tangent to the involute pro®le at its starting point A0 . Any point above P4 traces out a trochoid which stays to the right side of t4 at least until it reaches the involute pro®le, thus it must intersect the involute pro®le at a point below A0 and causes undercutting. The trochoid t2 traced out by the highest point P2 stays to the farthest right side and causes the most undercutting. From Fig. 2b, it can be observed that the root ®llet will be identical to

296


the trochoid formed by the highest point P2 of the straight blade if a sharp tip is used. If the radius of the round rack tip is relatively small, the root ®llet would be very close to the primary trochoid generated by the highest point of the straight blade. This conclusion was also presented by Colbourne [3].

Fig. 4. Intersection between two trochoids.


297

4.2. Intersection point between two trochoids We have stated that trochoids t2 and t3 in Fig. 2 will not cross each other until they reach the involute pro®le. But they do cross each other at some position to the left of the involute pro®le. The question is, where is the point at which these two trochoids cross each other? The knowledge may be useful in de®ning a more complicated ®llet shape, for example, one formed by a rack with a protuberance. Fig. 4a shows a case. P3 and P0 are two points above the interference line. They trace out two trochoids t3 and t0 . The virtual involute (the opposite involute) is drawn. Both t3 and t0 are tangent to the virtual involute at G3 and G0 , respectively. The tangency points G3 and G0 are found by constructing two circles passing through points T3 and T0 (shown in Fig. 4a), respectively. To ®nd the intersection point between t3 and t0 , their parametric equations (1) can be written out. Matching both the x and y coordinates produces a system of two nonlinear equations and Newton's method can be used to solve them. The tangency points G3 and G0 can be used as starting points. Draw the common normal G3 E3 N3 at the tangency point G3 . This normal intersects the pitch circle at the rolling point E3 and is tangent to the base circle at point N3 . The corresponding value of the rolling parameter u3 for G3 is calculated as follows (refer to Fig. 4b) u3 POg E3 a3 ap inva3 invap a3 arccosrb1 =r3 N3 Og G3 inva3 tana3

a3 G3 Og A0

invap tanap

ap POg A0

3

where ap N3 Og E3 is the pressure angle at the pitch diameter and r3 is the radius at the point G3 . The corresponding value of the rolling parameter u0 for G0 can be calculated in the same manner. After we obtain the starting values u3 and u0 , an iterative process is started to ®nd the intersection point between t3 and t0 . The intersection point between t3 and t0 can be approximately located in the following way (Fig. 4a). Construct tangent lines G0 H0 and G3 H3 of the virtual involute (note the normals at G0 and G3 are tangent to the base circle and intersect the pitch circle at the rolling points). G0 H0 and G3 H3 have an intersection at point J. It is obvious that the intersection between t0 and t3 is in the triangular area enclosed by involute segment G0 G3 , line segments G0 J and G3 J. 4.3. Root ®llet formed by a rack with a protuberance If a rack without a protuberance is used to cut the gear, the ®nal root ®llet is a segment of a primary trochoid if the rack has a sharp tip, or is a segment of a secondary trochoid if the rack has a round tip. If a rack with a protuberance is used, the determination of the root ®llet is not so straightforward. Fig. 5 shows an exaggerated example. Clearly, the root ®llet consists of two trochoid segments traced out by two dierent points. Now, let us examine in Fig. 6 a

298


Fig. 5. Fillet generated by a rack with a big protuberance.

rack with a reasonable protuberance. Some dashed lines are added to locate points e,e1 ,e3 ,T1 and T3 . As has been proven, the higher point e1 on segment e1 P4 traces out a trochoid whose right branch lies to the right side of all of the trochoids traced by points between e1 and P4 until they reach the involute pro®le. It is also known that the right branch of trochoid t1 , traced out by S1 , lies to the right of the trochoid traced by e1 . It is known t3 , traced by P3 , stays to the right of the trochoid t1 , traced by S1 , until they reach the rotated involute pro®le (shown as a dashed line in Fig. 6a) which is tangent to P3 e. Thus t3 surely stays to the farthest right of all trochoids traced out by points on blade segments P4 S2 , S2 S1 and S1 P3 at least until t3 reaches the rotated involute pro®le. But we are not sure, between the involute pro®le and the rotated involute pro®le, whether or not t3 will stay to the right of t1 . Construct the virtual involute segment C3 C1 generated by segment P3 S1 and the virtual involute segment C3 C1 has the same starting point, D0 , on the base circle as the rotated involute pro®le. The end points C3 and C1 can be determined by constructing two circles passing through T3 and T1 , respectively (T3 ,T1 are the intersection points with the line of action shown in Fig. 6a). If the same triangular area is constructed as in Fig. 4a (not drawn in Fig. 6a), we ®nd the entire area lies to the left of the involute pro®le. This means that the intersection point between t3 and t1 lies to the left side of the involute pro®le. Thus, the trochoid traced out by S1 will not cross the trochoid traced out by point P3 until it reaches the involute pro®le. Finally, the secondary trochoid generated by the round tip surely stays to the right side of the trochoid t3 . The conclusion is that only the circular tip P2 P3 is involved with the ®nal tooth ®llet generation in our example. This root ®llet is a segment of the secondary trochoid generated by the round tip. Summarizing, if a protuberance exists, the virtual involute segment C1 C3 generated by blade segment P3 S1 needs to be found. If the end point C1 lies to the left of the involute pro®le, only the round tip contributes to the ®nal formation of the root ®llet. Now, the relation between the two dimensions h and k of the protuberance (shown in Fig. 6b, h is the amount of protuberance and k is the distance from point S1 to the pitch line) will be derived when the end point C1 of the virtual involute segment C1 C3 is on the involute pro®le. The involute pro®le and the virtual involute segment C3 C1 intersect each other at point C1 . From the characteristics of involutes [4], the following relation holds:


Fig. 6. Application of the virtual involute in determining the ®llet shape.

299

300


A0 Og C1 D0 Og C1

A0 Og D0 h invaC1 tanaC1 2 2rb

aC1

4

where aC1 arccosrb =Og C1 . Og C1 can be expressed as: 2 !2 31=2 2 k 5 Og C1 Og T1 4 rp k tanap where rp and ap are the radius and pressure angle at the pitch circle. If inequality h tanaC1 2rb

aC1

6

holds, C1 is to the right of the involute pro®le. The actual intersection point between t1 and the secondary trochoid generated by the round tip has to be found in an iterative way to exactly de®ne the ®nal ®llet shape. If the actual intersection lies to the right side of the involute pro®le, the ®nal ®llet also contains a small segment of t1 . The transition point from the secondary trochoid to the primary trochoid t1 forms a singular point on the ®llet curve. The curve w of point S1 satisfying h tanaC1 2rb

aC1

is drawn as a dashed line in Fig. 6b. It starts at point P4 . If S1 is to the left of w, the inequality (5) holds. The shape of w is determined by the pro®le angle of the rack. If segment P3 S1 is not parallel to P4 P, the same process still applies, but some changes need to be considered. 4.4. Determining the involute segment in the root ®llet area In some unusual cases, the root ®llet may contain an involute segment. In Fig. 7, a rack is simply chamfered at point P2 . Now we have two dierent side blades with dierent pro®le angles. We must now draw the two base circles, the two lines of action and the two interference lines. Blade P1 P2 is below the interference line 2 and it generates an involute segment J1 J2 (in thick solid line) in the root ®llet area. This involute segment ranges from the circle of radius r1 Og S1 Og J1 to the circle of radius r2 Og S2 Og J2 . The radii r1 and r2 can be found by drawing a horizontal line passing through P1 and a second horizontal line passing through P2 . These two lines intersect the line of action 2 at S1 and S2 , respectively. Points P1 and P2 trace the trochoids t1 and t2 , respectively. It is concluded that the involute segment smoothly connects the two trochoids because the trochoid traced out by any point


301

Fig. 7. Root ®llet containing involute segment.

below the interference line never penetrates to the right side of the involute generated by the blade segment and is always tangent to the involute. The formed tooth pro®le consists of a part of the root circle, part of trochoid t1 , an involute segment J1 J2 generated from base circle 2, part of trochoid t2 and another involute segment starting from the undercutting point J3 to the outside circle. Except at point J3 , all other junctures have at least C 1 continuity. The rolling parameter u1 corresponding to the transition point J1 is calculated from the following equations: u1 ap2

aJ1 invap2

invaJ1

s rb2

7

aJ1 arccosrb2 =rJ1 where ap2 is the rack pro®le angle corresponding to the blade segment P1 P2 and s is the distance from the pitch point P to the blade segment P1 P2 . Parameter rb2 is the radius of base circle 2 and rJ1 Og J1 Og S1 . The example below shows how one applies the earlier developments in determining the ®llet shape of a rack-generated gear.

5. A numerical example A numerical example is given in Fig. 8. The dimensions of the rack and the parameters of the gear are listed in Table 1. First, check the Eq. (4) (refer to Fig. 6b). Og C1 Og T1 rp k2 tankap 2 1=2 2 2 1=2 7:7525 8:3067; aC1 arccos 8:3067 0:3673 radian 21:04748; 8:25 1:9854 1:9854 tan208 h invaC1 tanaC1 aC1 0:0175; 2rb 0:0110. Inequality (5) holds, thus only the circular tip contributes to the ®nal ®llet formation. The curve w in Fig. 8a also con®rms this. Fig. 8b shows the trochoids traced out by point P3 and S1 . The secondary trochoid generated by the round tip is also given. These three trochoids are very close to each other near the undercutting point.

302


Fig. 8. A numerical example.


303

Table 1 The dimensions of the rack and the parameters of the gear Rack dimensions (dimensionless) Tip radius Protuberance dimension 1 Protuberance dimension 2 Pressure angle Point in ®xed system XPY P2 P3 S1 S2 P (pitch point) P5 C (center of round tip) Gear geometry parameters Gear parameter Tooth number Module Rack shift distance Pitch diameter Base diameter

r h k a x coordinate 1.2754 0.7116 0.5406 0.5406 0.0000 1.0919 1.2754

0.6000 0.1710 1.9854 208 y coordinate 2.8500 2.4552 1.9854 1.4854 0.0000 3.0000 2.2500

Value 11 1.5 0.0 16.5 15.5049

Based on the above calculation, we are con®dent that only the secondary trochoid contributes to the ®llet formation. The transition point from root circle to the secondary trochoid is Fm . The intersection between the secondary trochoid and the involute pro®le is J. The equations for all segments and the coordinates of all junctures are: x root 5:4cosy The root circle is: 8 yroot 5:4siny

Point Fm 0:8315,2:9144T . From Eq. (2), the secondary trochoid segment from Fm to J is: 8:25sinu 8:25u 1:2754 cosu 2:25sinu xF 8:25u 1:2754 sinu 2:25cosu 8:25 1 cosu yF 2 3 2:25 6 0:6cos arctan u 7 6 7 8:25u 1:2754 6 9 7 6 7 2:25 4 5 0:6sin arctan u 8:25u 1:2754 where the rolling parameter u belongs to (0.1546, 0.7923). Point J 0:0223,0:0637T . The involute pro®le segment from J to the addendum circle is:

304


2

x I1 yI1

cos sin

3 7:7525 sintana a 7 6 0 0:0149 sin 0:0149 6 cosa 7 6 7 8:25 0:0149 cos 0:0149 4 7:7525 5 costana a cosa

10

where the parameter a belongs to (0.3270, 0.6516). The found tooth shape is drawn in Fig. 8b. 6. Conclusions Five characteristics of trochoids have been discussed in this paper. The authors also introduced a new concept-virtual involute. It has been shown that the virtual involute could be used to ®nd the intersection point between two trochoids and clearly determine the root ®llet shapes generated by racks with a protuberance. Based on these characteristics, a better and simpler proof of undercutting was also given and an unusual ®llet curve which includes an involute segment was investigated. Similar procedures can be used to study the ®llet shapes generated by pinion-type cutters. Acknowledgments The authors would like to thank the sponsors of the Gear Dynamics and Gear Noise Research Laboratory at The Ohio State University for their ®nancial support to the research. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11]

E. Buckingham, Analytical Mechanics of Gears, Dover, New York, 1963. T.W. Khiralla, On the Geometry of External Involute Spur Gears, 1976. J.R. Colbourne, The Geometry of Involute Gears, Springer-Verlag, Berlin, 1987. H.H. Mabie, F.W. Ocvirk, Mechanisms and Dynamics of Machinery, Wiley, New York, 1978. H. Bai, M. Savage, R.J. Knorr, Computer modeling of rack-generated spur gears, Mechanisms and Machine Theory 20 (1985) 351±360. S.M. Vijayaker, B. Sarkar, D.R. Houser, Gear tooth pro®le determination from arbitrary rack geometry, Paper No. AGMA 87FTM4. S.M. Vijayaker, D.R. Houser, The use of boundary elements for the determination of the geometry factor, Paper No. AGMA 86FTM10. S.M. Vijayaker, A combined surface integral and ®nite element solution for a 3D contact problem, International Journal for Numerical Methods in Engineering 31 (1991) 525±546. R.G. Mitchiner, H.H. Mabie, The determination of the lewis form factor and the AGMA geometry factor J for external spur gear teeth, Paper No. ASME 80-DET-59. M.A. Lopez, R.T. Wheway, A method for determining the AGMA tooth form factor from equations for the generated tooth root ®llet, Paper No. ASME 85-DET-9. X. Zhao, Proof of the undercutting phenomenon for an involute tooth pro®le on a cylindrical gear, Mechanisms and Machine Theory 27 (1992) 93±95.