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A study on the mathematical models and contact ratios of extended cycloid and cycloid bevel gear sets
Pergamon
Mech. Mach. Theory Vol. 32, No. I, pp. 39-50, 1997 Copyright © 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved PII: S0094-114X(96)00032-8 0094-114X/97 $15.00 + 0.00
A STUDY ON THE MATHEMATICAL MODELS AND CONTACT RATIOS OF EXTENDED CYCLOID AND CYCLOID BEVEL GEAR SETS L. M. SUNG and Y. C. TSAI Department of Mechanical Engineering, National Sun Yat-Sen University, Kaohsiung, Taiwan 80424, Republic of China
(Received 20 March 1995; in revised form 28 May 1996)
Abstract--In this paper, the general mathematical models of conjugate tooth profiles and constraints for bevel gear sets are presented. The tooth profiles of involute bevel gears, extended cycloid bevel gears and cycloid bevel gears are derived from the general mathematical models, The contact ratios for the extended cycloid and cycloid bevel gear sets are investigated. The design charts comparing the contact ratios of cycloid bevel gear sets with those of involute bevel gear sets are provided. They are useful for the engineers to design the bevel gear sets of high contact ratio. Copyright © 1996 Elsevier Science Ltd
1. I N T R O D U C T I O N
Recently, the tooth profiles and generations of bevel gears have been studied. Huston and Coy [1] have presented the parametric representations of the ideal spiral bevel gears. The surface geometry of circular-cut spiral bevel gears has been developed by Huston and Coy [2]. The variations of tooth profiles in the transverse planes of circular-cut, spiral-bevel crown gears have been analyzed by Huston et al. [3]. Based on the principle for the performance of parallel motion of a straight line, the method for generation of spiral bevel gears has been proposed by Litvin et al. [4]. Tsai and Chin [5] have derived the mathematical models of the ideal tooth profiles of straight and spiral bevel gears. Litvin et aL [6] have studied the settings of a tilted head cutter for generating spiral bevel gears. The mathematical model that describes the parametric conjugate tooth profiles of bevel gears has been developed by Chang and Tsai [7]. The tooth geometry of circular-cut spiral bevel gears has been proposed by Fong and Tsay [8]. For the investigation on the kinematic analysis and synthesis of bevel-gear trains, Yang and Freudenstein [9] have analyzed the kinematics, statics and inertia-forces of the epicyclic bevel-gear trains by the dual matrices. The analytical method for the determination of the kinematic errors in spiral-bevel gear trains has been derived by Litvin et aL [10]. The kinematic design and analysis of coupled planetary bevel-gear trains have been presented by Day et aL [11]. Freudenstein et aL [12] have proposed a general procedure for the kinematic analysis of robotic bevel-gear trains. Additionally, Baxter [13] has expressed the lattice contact in generated spiral bevel gears. The kinematic synthesis of spherical two-gear drives with prescribed dwell characteristics has been presented by Lee and Akbil [14]. Winter and Paul [15] have described the influence of relative displacements between the pinion and the gear on tooth root stresses of spiral bevel gears. The generalized transmission error of spiral bevel gears has been proposed by Mark [16]. AI-Shareedah and Alawi [17] have analyzed the reliability of bevel gears with and without back support. The reliability model of bevel gear reductions has been described by Savage et al. [18]. Sung and Tsai [19] have studied the tooth proportions and misalignments of involute bevel gear sets. For the investigation on the parametric conjugate tooth profiles, Chang and Tsai [20] have presented the mathematical models of spur gears. Tsai and Sung[21] have developed the mathematical models of parametric conjugate tooth profiles and constraints for the gear sets with skew axes. In this paper, the general mathematical models and constraints of conjugate tooth profiles of bevel gear sets will be proposed. The tooth profiles of involute bevel gear sets, extended cycloid bevel gear sets and cycloid bevel gear sets will be studied. Charts for comparing the contact ratios of cycloid bevel gear sets with those of involute bevel gear sets will be provided. 39
40
L.M. Sung and Y. C. Tsai 2. M A T H E M A T I C A L M O D E L S AND C O N S T R A I N T S OF GEAR SETS W I T H SKEW AXES
In Fig. 1, the rotating axes of gear 1 and gear 2, a gear set with skew axes, are II and 12, respectively. The notations C and T are separately denoted as the shortest distance and twisting angle of these two rotating axes. Vectors ol and o~2 are the angular velocities of these two gears when rotating angles are ~ and ~2. These are two Cartesian systems S(x, y, z) and S'(x', y', z'). Coordinate systems St(xt, y~, z~) and $2(x2, y2, z2) are rigidly connected with gear 1 and gear 2, respectively. The x-axis and xt-axis coincide with the rotating axis Ii. The x'-axis and x2-axis coincide with the rotating axis/2. The z-axis and z'-axis coincide with the common normal of axes 1~ and/2. The speed ratio of gear 2 to gear 1 is m. The position parmeters of the instantaneous screw axis are 6 and u as shown in Fig. 1. The parameters 6 and u are the twisting angle and shortest distance of rotating axis l~ and instantaneous screw axis. The N-plane proposed by Tsai and Sung [21] is shown in Fig. 2. It contains the contact point P. The vector V~, the relative velocity of the conjugate tooth profiles at the contact point P, is normal to the N-plane. The line I, is perpendicular to the pitch line and Vre~. The unit vector n, is the unit direction vector of the pitch line. The unit vector n~ that lies on the plane z = u is perpendicular to the unit vector n,. The unit vectors n2 and n3 are the unit normal vector of the N-plane and the unit direction vector of the line In respectively. The parameters 1, ~ and fl are the position parameters of the N-plane. The notation p is the distance from the contact point P to pitch line. Tsai and Sung [21] presented the mathematical model of tooth profile of gear 1 in coordinate system Sl(xt, yl, z~) as follows: r I cos ~ - p cos ct sin ~ r,(ct, p, 1, ~bl) = ] Ul COS ~bl + VI sin ~bl / ' [_- U, sin ~b, + V, cos ~lJ
(1)
Z, Z °
O', O tO 2
•
,./
l2
/
C)/u ~tantaneous
~ X
O, Oj ,XI/ x_y L-8
I
Y1
Fig. 1. Skew axes and coordinate systems.
Screw Axis
Extended cycloid and cycloid bevel gear sets
41
Z
r,2 /
/rN-P' e
" ~-~=
~
nl
n2 - ~ ~ - - -
n % Pitch Line
=Y Ii --,
~ V ,'el X
Fig. 2. Relative velocity and pitch line.
where Uj = lsin 6 + p cos ~ cos 6, V~ = u + p s i n ct.
The mathematical model of tooth profile in gear 2 in coordinate system $2(x2, y2, z2) is described as
r2(a, p, 1, ~b2) =
l cos(6 - 7) - P cos ~ sin(6 - 7)-] U2 cos ~b2 + V2 sin ~b2 [, - U2 sin ~b2 + V2cos ~b2 J
I
(2)
where U2 = I sin(6 - 7) + P cos ct cosO - 7), V2 = u - C + p sin ct.
The constraint for the parametric conjugate tooth profiles of gear set with skew axes is proposed by Tsai and Sung [21] as act ap']
Ko
Ol
Ko = p
sin fl,
,
Op
ap
~c¢
Ogp, - ~ j ~ K, - ~ + K2 -~ + K3 ' ~ + K4 = 0,
where
K~ = - c o s fl,
(3)
42
L.M. Sung and Y. C. Tsai K2 = W, cos a sin fl + Wffsin a sin fl cos 3 + cos fl sin 6), K3 = - W~p sin a sin fl + W2p cos ~ sin fl cos 5,
K4 = (l sin a sin 6 - u cos a cos 3)cos fl, W~ = l sin 5 + p cos a cos 6, Wz = u + p sin a.
In equation (3), a and p are functions of l and 4), i.e. a = off/, 4)1) and p = p(l, 4)0. 3. GEAR SETS WITH INTERSECTING AXES
The bevel gear set is the o f its rotating axes is equal F o r the bevel gear set, the S,(Xl, yl, z~) is degenerated
degenerated case of a gear set with skew axes. The shortest distance to zero, i.e. C = 0. The included angle between the rotating axes is y. mathematical model of tooth profile o f gear 1 in coordinated system to
r,(r, 4),) = r
COS 2 COS 6 - sin 2 cos 0c sin 6-] Si cos 4), + T, sin 4), J , --SI sin 4)1 + T, cos 4),
(4)
where S, = cos 2 sin 3 + sin 2 cos ~ cos 3, T, = sin 2 sin ~. In equation (4), tan 2 = p / l and r = l sec 2. Similarly, the m a t h e m a t i c a l model o f tooth profile of gear 2 in coordinate system Sdx2, y2, z:) is r2(r, 4)2) = r
cos 2 cos(3 - y) - sin 2 cos ~ sin(3 - ?)-] $2 cos 4)5 + T2 sin 4)2 ] , - $2 sin 4)5 + T2 cos 4)2
(5)
where $2 = cos 2 sin(6 - 7) + sin 2 cos ~ cos(3 - ?), T2 =
sin 2 sin ~.
T h e constraints for the parametric conjugate tooth profiles of bevel gear set are p r o p o s e d by Sung and Tsai [19]. 02 ( 02) 04), - sin ct sin 6 1 + l tan 2 - ~ ,
(6)
92 ( 02) 04)2 - sin a sin(6 - ?) 1 + l tan 2 -5/ ,
(7)
Equations (6) and (7) are two quasi-linear partial differential equations o f order one. In this paper, assume that the p a r a m e t e r ~ is a function of 4), (or 4)2), i.e. ~ = ~t(4),) (or a = ~(4)2)). T h e solutions o f equations (6) and (7) m a y be derived as 2 = sin 6
fsin fsin
2 = sin(6 - 7)
+f,(r),
(8)
d4)2 + f f f r ) ,
(9)
d4)l
Extended cycloid and cycloid bevel gear sets
43
The parameter 2 is formulated as a function of r and ~b, (or ~b2), i.e. 2 = 2(r, ~b,) (or 2 = 2(r, ~b2)). Partial differentiating equations (8) and (9) yields 02 Oq~----~= sin 6 sin ~(tp,),
(10)
~2 8~b----~= sin(6 - 7)sin a($2),
(I 1)
d Or - dr ~,(r)] =
(12)
~2(r)].
4. INVOLUTE BEVEL GEARS The involute bevel gear set is the special case in which the parameter a is a constant. Assume that the parameter a is a constant and the initial values of q~, and ~b2 are equal to zero. F r o m equations (4), (5), (8) and (9), the mathematical model of tooth profile of gear 1 in coordinate system S,(x,,y,, z,) is deduced to [cos 2 cos 6 - sin 2 cos a sin 6 ] r,(r, ~b,) = r [ S, cos ~b, + T, sin 4~, J , [ - S , sin ~b, + T, cos ~b,
(13)
where S, = cos 2 sin 6 + sin 2 cos a cos 6, T, = sin 2 sin a, 2 = ~bl sin a sin 6 +f(r). Similarly, the mathematical model of tooth profile of gear 2 in coordinate system $2(x2, y2, z2) is described as r2(r, ~b2) = r
cos 2 cos(6 - 7) - sin 2 cos ct sin(6 - 7)-] $2 cos ~b2 + T2 sin ~b2 J , - $2 sin tp2 + T2 cos ~b2
(14)
where $2 = cos 2 sin(6 - 7) + sin 2 cos ~ cos(6 - 7), T2 = sin 2 sin a, 2 = ~b~sin ~ sin(6 - 7) + f ( r ) . 5. E X T E N D E D CYCLOID BEVEL GEARS
Sung and Tsai [19] propose that the pitch line of bevel gear set lies on the N-plane of contact point. For the bevel gear set, if the common normal of meshing tooth profiles at contact point passes through the pitch line, the meshing tooth profiles are conjugate. In Fig. 3, the rotating axes of cone 1, cone 2 and pitch cones of gear 1 and gear 2 are lc~, lc2,/, and 12, respectively. The pitch line is their common contact line. These four cones roll at the pitch line without slipping. The rotating angles of axes It, and lc2 are ~bc, and ~ba. The angles of cone 1 and cone 2 are ~Ib~and E~b~. The angle PrOPe is 0 ebb. The points C, and C2 are rigidly connected with cone 1 and cone 2, respectively. When cone 1 and pitch cones of gear 1 and gear 2 roll without slipping at the pitch line, the point C, generates separately the tooth profiles of gear 1 and gear 2. The common normal of meshing tooth profiles at contact point C, passes through the pitch line. Therefore, the tooth profiles of gear 1 and gear 2 are conjugate. These generated conjugate tooth profiles are the
44
L.M. Sung and Y. C. Tsai
extended epicycloid for gear 1 and extended hypocycloid for gear 2. The spherical triangle is shown in Fig. 3. By the law of cosines, 2 = cos-tFcos Elb) cos *rib' + sin E~b' sin *r~b,cos/l¢,lsinlal~l \ sin *Ib' , ] J ' L
P,CtP,t (15)
where and cos g =
- c o s E?~cosl~.l + cos(E?' - 0 *~') sin Elb' sinl;tl
(16)
By the law of sines, equation (17) is deduced to sin(c~b) -- 0,b,)sin(~b, \ ~ sinlSl'~ } sin a =
sin[21
(17)
From equations (15)-(17), the parameter • is a function of ~b, (or q~2), i.e. m = ~ ( ~ 1 ) (or ~ = a(q~2)). Substituting equations (15)-(17) into (4) and (5) yields the mathematical models of the extended epicycloid gear 1 and extended hypocycloid gear 2. Similarly, when cone 2 and pitch cones of gear 1 and gear 2 roll at the pitch line, point C2 generates the conjugate tooth profiles of gear 1 and gear 2. The generated conjugate tooth profiles are the extended hypocycloid for gear 1 and extended epicycloid for gear 2. The spherical triangle PpC2P¢2is shown in Fig. 3. By the geometry of spherical triangle, 2
=
cos-'[cos E[b) cos a[b) + sin e~b,sin a[b)
fl~b,lsinl~['~-]
k
(18)
)I'
where a[b) = E[b' + 0% and cos 0c =
- c o s E[b>cosl21 + cos(Elb' + 0
(19)
sin(E[b' + Oeb))sin(~bl k, sinsinl6l'~ E~b' J sin • =
sinl21
(20)
Cone 2 -,
ll(b)
/I
""L Pe P"
/
J
°
~
l~l
%
•2 Fig. 3. Generating points of extended cycloid bevel gear set.
Cone i'"
Extended cycloidand cycloidbevel gear sets
/
45
-
"'-- Extended Cycloid
Fig. 4. Comparison of tooth profiles between extended cycloid and involute bevel gears.
Substituting equations (18)-(20) into (4) and (5) yields the mathematical models of the extended hypocyloid gear 1 and extended epicycloid gear 2. Figure 4 is the comparison of the tooth profiles between extended cycloid and involute bevel gears. Figures 5 and 6 are the extended cycloid and involute bevel gears cut by the CNC machine center, respectively. In Fig. 7, the notations ~to~and ,,lb~ to2 are separately denoted as the angles of addendum cones of gear 1 and gear 2. The angle P,., of spherical triangle P~C,P,.~ is ~b~bl. By the law of cosines, el/
COS- I COS 7 2 -- COS(el b~ -- 0~b~)cos(161 + ~I~')
(21)
The angle Pc2 of spherical triangle P2C2;P,.2 is ~blb]. ~,~
_ , [ c o s 7~2 - c o s ( C ~ + O~)cos(16 - ~'1 + 4~9.]
Fig. 5. An extended cycloid bevel gear.
(22)
46
L.M. Sung and Y. C. Tsai (b)
Cone 2.
7ol
lI
////
\N "~'-~
' , ~ (6)
"/2
\N C o n e 1"-
Fig. 6. Straight involute bevel gear.
T h e contact ratio C R ~b) of extended cycloid bevel gear set is formulated as CRIb ~ _ -/)(b)('4~(b) \~tlcl l sin elb) + ..k(b) ~tlc21 sin e~b)) ,
(23)
where plb~__
N] bl _ N~b~ 2 sinl6l 2 sin[6 -- 71'
In equation (23), the notations N~b~ and N~zb~ are the numbers of teeth in gear 1 and gear 2, respectively.
Fig. 7. Addendum cones of extended cycloid bevel gear set.
Extended cycloid and cycloid bevel gear sets 6. CYCLOID
4-l
BEVEL GEARS
If the angle ecbJshown in Fig. 3 is equal to zero, the extended cycloid bevel gear set becomes the cycloid bevel gear set. Substituting that rYbJ= 0 into equations (15x17) yields R = cos-’ cos2 EIbJ+ sin2 eibJcos (!$$Y)], [
(24)
-cos EIbJcos(l( + cos cibJ sin ~1~) sinllJ ’
(25)
cos a =
sin cIbJsin w ( sin a = sir@)
I >
(26)
.
The mathematical models of the epicycloid gear 1 and hypocycloid gear 2 are derived by substituting equations (24x26) into (4) and (5). Similarly, from equations (l&o-(O) and that (YbJ= 0, equations (27x29) are obtained. 1= cos-‘[co$
cos a =
EjbJ + sin2 LjbJcos(&&i$$)], - cos e$“Jcoslll + cos cf’ - sin 6;“)sinll I ’
(27)
(28)
41 sin161 sin c$“’sin Y ( sin 61”)> sin a = sin);ll ’
(29)
The mathematical models of the hypocycloid gear 1 and epicycloid gear 2 are formulated by substituting equations (27x29) into (4) and (5). The contact ratio CRcbJof cycloid bevel gear set is expressed from equations (21x23) and that @bJ = 0. (-JRcb’=
P(“)(f#Qjsin tlbJ+ $$j sin ~4~)) 7 71
(30)
where hq"'
)XbJ =
2
NJb’ = 2 sin(S - yl ’
1’ 1*
q$I = cos-’
cos y$J - cos 61”’cos(lbl + 61”‘) sin c\“’sin@1 + c\“)) [
f#$]= cos-’
cos $J - cos 8’ cos()b - yJ + EibJ) sin ~1~) sin@ - yl + ~1”)) [
The contact ratio of involute bevel gear set is investigated by Sung and Tsai [ 191.Assume that speed ratios, included angles of rotating axes, addendum cones of pinions, addendum cones of gears and Pb) for the cycloid and involute bevel gear sets are the same. Let the contact ratio of cycloid bevel gear set divided by that of involute bevel gear set be v$“. (31)
48
L. M. Sung and Y. C. Tsai 1.3
@
~0
m i | m ~ m m
m m m n m l m m m ~ ~
m m m m m m m n m l l l l l l l n m i l l l l l n m m m m m m n m ~ ~ m m m m m ~ ~ m m m ~ m ~ n m
l
m m m l l l l m m m m m ~ m m m m
l l
m m l l m m m ~ ~
1.2
mm~mm~
1.1
~ d m ~ m m m m ~
:~INENNNNNNE~E
o
m
~
m
~
p
~
~
m
m
m
~
m ~ ~ m ~ ~ m m m ~ P ~ N I ~ N ~ N N ~ N N N ~ . ~ m
1
~
m
m
~
o
m
m m
m m
i
:..~ m m
=~=~
nun mm nm
m u m u m m u nm
.~ 0.g o
0.8 20
25
30
35
40
iii
45
Pitch Angle of Pinion (deg.) Fig. 8. Comparison between contact ratios of cycloid and involute bevel gear sets for different pressure angles. where
4,(b) _ cos [-cos %q' - cos e?)cos(151 + e?)l ' <"-
c21
- L COS--
1
7ic,-~<7;~T8
¥~>7
COS ~(ob) -- COS e~b) c o s @ - ?i + @))
_,/cos Corm
x,~>=~os t ~ 7 - ~ o ~ Y(b) : C O S - ' ( : O : ~ )
_,(cosl6i "~
t~J'
-- COS-I(COSiI~Tbb?" " C ~O S )
In equation (31), la[ is the complement of pressure angle of involute bevel gear set. The notations ?(b~) and 7~b~>are the angles o f base cones o f involute bevel gear set. To prevent undercutting for the involute bevel gear sets, equations (32) and (33) are proposed by Sung and Tsai [19].
c o ~ -C, ( ~ (b)] < cos-,/¢°si~l~ cos-(cosi,~- ~i~
t, c--~7~ )'
(32)
(b) c o ~\cos -C, ( ~~'~,7/
~ + co~,(\ ~°~ ~J~ cos ~'b: )
(33)
tcos ~,~,,,/
t, cos ~,<~,,)+
cos- ,/~°~ I ~ I
\cos ~,~,]
Extended cycloid and cycloid bevel gear sets
49
1.2
~ib)=0.50 il
r,~ r'~_ 1.1
P
~
d
H
I H i
j ~.(b)= 0.45
~.
¢.(t,)= 0.40
°° R
) _-;_ --
'
. o .~
C(b)= 0.35 __p____...
__
~[.-,
I i I H P ~ ~ -.,,,,,m I
0.9
l l I
~ - , - i i l mini I
. . . .
0.30
l
0.8
25
30
35
40
45
Pitch Angle of Pinion (deg.) Fig. 9. Comparison between contact ratios o f cycloid and involute bevel gear sets for different values o r~ ,~(b).
Assume that the ratio of E~b) to 161 and ratio of Elb) to 16 - Yl are the same and ~(b) is the ratio of E~b) to 161. ~b> _ E?' El~' J6l - 16 - 71"
(34)
In general, the included angle between the rotating axes of a bevel gear set is 90 °. Figure 8 shows the relation between r/¢(~> and pitch angle of pinion for different pressure angles of involute bevel gear sets, where ¢(b) is 0.4 and the addendum angle of bevel gear is 4 °. Figure 9 is the chart representing the relation between r/~,~) and pitch angle of pinion for different values of ~(b), where the pressure angle of involute bevel gear set is 20 ° and the addendum angle of bevel gear is 4 ° . From Figs 8 and 9, it is found that, in low speed ratios, the contact ratios of cycloid bevel gear sets are greater than those of involute bevel gear sets.
7. CONCLUSION In this paper, the general mathematical models of conjugate tooth profiles and constraints for bevel gear sets have been investigated. The tooth profiles of involute bevel gears, extended cycloid bevel gears and cycloid bevel gear have been deduced from the general mathematical models. The contact ratios for the extended cycloid and cycloid bevel gear sets have been described. The charts comparing the contact ratios of cycloid bevel gear sets with those of involute bevel gear sets have been provided. In low speed ratios, the contact ratios of cycloid bevel gear sets are greater than those of involute bevel gear sets. The design charts are useful for the engineers to design the bevel gear sets of high contact ratio. Additionally, the presented mathematical models provide a basis "for further investigations such as the tooth geometries, computer-aided manufacturing, computer-aided tooth contact analysis and finite element stress analysis of involute, extended cycloid and cycloid bevel gear sets. Acknowledgments--The authors are thankful to the National Science Council, R. O. C. for their grant, no. NSC 83-0401-E-110-011.
50
L.M. Sung and Y. C. Tsai
REFERENCES 1. Huston, R. L. and Coy, J. J., Transactions of ASME Journal of Mechanics and Design, 1981, 103, 127. 2. Huston, R. L. and Coy, J. J., Transactions of ASME Journal of Mechanics and Design, 1982, 104, 743. 3. Huston, R. L., Lin, Y. and Coy, J. J., Transactionsof ASME Journal of Mechanics and Transmission Automation Design, 1983, 105, 132. 4. Litvin, F. L., Tsung, W. J., Coy, J. J. and Heine, C., Transactions of ASME Journal of Mechanics and Transmission Automation Design, 1987, 109, 163. 5. Tsai, Y. C. and Chin, P. C., Transactions of ASME Journal of Mechanics and Transmission Automation Design, 1987, 109, 443. 6. Litvin, F. L, Zhang, Y., Lundy, M. and Heine, C., Transactions of ASME Journal of Mechanics and Transmission Automation Design, 1988, 110, 495. 7. Chang, H. L. and Tsai, Y. C., Proceedings of the 1990 ASME Design TechnicalConferences--2nd Conference in Flexible Assembly Systems, Chicago, 1990, p. 5. 8. Fong, Z. H. and Tsay, C.-B., Transactions of ASME Journal of Mechanics Design, 1991, 113, 174. 9. Yang, A. T. and Freudenstein, F., Transactions of ASME Journal of Engineering Industry, 1973, 95, 497. 10. Litvin, F. L., Goldrich, R. N., Coy, J. J. and Zaretsky, E. V., Transactions of ASME Journal of Mechanics and Transmission Automation Design, 1983, 105, 310. 11. Day, C. P., Akeel, H. A. and Gutkowski, L. J., Transactions of ASME Journal of Mechanics and Transmission Automation Design, 1983, 105, 441. 12. Freudenstein, F., Longman, R. W. and Chen, C.-K., Transactions of ASME Journal of Mechanics and Transmission Automation Design, 1984, 106, 371. 13. Baxter, Jr, M. L., Transactions of ASME Journal of Mechanics Design, 1978, 100, 41. 14. Lee, T. W. and Akbil, E., Transactions of ASME Journal of Mechanics and Transmission Automation Design, 1983, 105, 663. 15. Winter, H. and Paul, M., Transactions of ASME Journal of Mechanics and Transmission Automation Design, 1985, 107, 43. 16. Mark, W. D., Transactions of ASME Journal of Mechanics and Transmission Automation Design, 1987, 109, 275. 17. AI-Shareedah, E. M. and Alawi, H., Mechanics Machine Theory, 1987, 22, 13. 18. Savage, M., Brikmanis, C., Lewicki, D. G. and Coy, J. J., Transactionsof ASME Journal of Mechanics and Transmission Automation Design, 1988, 110, 189. 19. Sung, L. M. and Tsai, Y. C., Journal of the Chinese Society of Mechanical Engineers, 1994, 15, 273. 20. Chang, H. L. and Tsai, Y. C., Transactions of ASME Journal of Mechanics and Transmission Automation Design, 1992, 114, 8. 21. Tsai, Y. C. and Sung, L. M., Journal of Applied Mechanisms and Robotics, 1993, 1, 36.
Mech. Mach. Theory Vol. 32, No. I, pp. 39-50, 1997 Copyright © 1996 Elsevier Science Ltd Printed in Great Britain. All rights reserved PII: S0094-114X(96)00032-8 0094-114X/97 $15.00 + 0.00
A STUDY ON THE MATHEMATICAL MODELS AND CONTACT RATIOS OF EXTENDED CYCLOID AND CYCLOID BEVEL GEAR SETS L. M. SUNG and Y. C. TSAI Department of Mechanical Engineering, National Sun Yat-Sen University, Kaohsiung, Taiwan 80424, Republic of China
(Received 20 March 1995; in revised form 28 May 1996)
Abstract--In this paper, the general mathematical models of conjugate tooth profiles and constraints for bevel gear sets are presented. The tooth profiles of involute bevel gears, extended cycloid bevel gears and cycloid bevel gears are derived from the general mathematical models, The contact ratios for the extended cycloid and cycloid bevel gear sets are investigated. The design charts comparing the contact ratios of cycloid bevel gear sets with those of involute bevel gear sets are provided. They are useful for the engineers to design the bevel gear sets of high contact ratio. Copyright © 1996 Elsevier Science Ltd
1. I N T R O D U C T I O N
Recently, the tooth profiles and generations of bevel gears have been studied. Huston and Coy [1] have presented the parametric representations of the ideal spiral bevel gears. The surface geometry of circular-cut spiral bevel gears has been developed by Huston and Coy [2]. The variations of tooth profiles in the transverse planes of circular-cut, spiral-bevel crown gears have been analyzed by Huston et al. [3]. Based on the principle for the performance of parallel motion of a straight line, the method for generation of spiral bevel gears has been proposed by Litvin et al. [4]. Tsai and Chin [5] have derived the mathematical models of the ideal tooth profiles of straight and spiral bevel gears. Litvin et aL [6] have studied the settings of a tilted head cutter for generating spiral bevel gears. The mathematical model that describes the parametric conjugate tooth profiles of bevel gears has been developed by Chang and Tsai [7]. The tooth geometry of circular-cut spiral bevel gears has been proposed by Fong and Tsay [8]. For the investigation on the kinematic analysis and synthesis of bevel-gear trains, Yang and Freudenstein [9] have analyzed the kinematics, statics and inertia-forces of the epicyclic bevel-gear trains by the dual matrices. The analytical method for the determination of the kinematic errors in spiral-bevel gear trains has been derived by Litvin et aL [10]. The kinematic design and analysis of coupled planetary bevel-gear trains have been presented by Day et aL [11]. Freudenstein et aL [12] have proposed a general procedure for the kinematic analysis of robotic bevel-gear trains. Additionally, Baxter [13] has expressed the lattice contact in generated spiral bevel gears. The kinematic synthesis of spherical two-gear drives with prescribed dwell characteristics has been presented by Lee and Akbil [14]. Winter and Paul [15] have described the influence of relative displacements between the pinion and the gear on tooth root stresses of spiral bevel gears. The generalized transmission error of spiral bevel gears has been proposed by Mark [16]. AI-Shareedah and Alawi [17] have analyzed the reliability of bevel gears with and without back support. The reliability model of bevel gear reductions has been described by Savage et al. [18]. Sung and Tsai [19] have studied the tooth proportions and misalignments of involute bevel gear sets. For the investigation on the parametric conjugate tooth profiles, Chang and Tsai [20] have presented the mathematical models of spur gears. Tsai and Sung[21] have developed the mathematical models of parametric conjugate tooth profiles and constraints for the gear sets with skew axes. In this paper, the general mathematical models and constraints of conjugate tooth profiles of bevel gear sets will be proposed. The tooth profiles of involute bevel gear sets, extended cycloid bevel gear sets and cycloid bevel gear sets will be studied. Charts for comparing the contact ratios of cycloid bevel gear sets with those of involute bevel gear sets will be provided. 39
40
L.M. Sung and Y. C. Tsai 2. M A T H E M A T I C A L M O D E L S AND C O N S T R A I N T S OF GEAR SETS W I T H SKEW AXES
In Fig. 1, the rotating axes of gear 1 and gear 2, a gear set with skew axes, are II and 12, respectively. The notations C and T are separately denoted as the shortest distance and twisting angle of these two rotating axes. Vectors ol and o~2 are the angular velocities of these two gears when rotating angles are ~ and ~2. These are two Cartesian systems S(x, y, z) and S'(x', y', z'). Coordinate systems St(xt, y~, z~) and $2(x2, y2, z2) are rigidly connected with gear 1 and gear 2, respectively. The x-axis and xt-axis coincide with the rotating axis Ii. The x'-axis and x2-axis coincide with the rotating axis/2. The z-axis and z'-axis coincide with the common normal of axes 1~ and/2. The speed ratio of gear 2 to gear 1 is m. The position parmeters of the instantaneous screw axis are 6 and u as shown in Fig. 1. The parameters 6 and u are the twisting angle and shortest distance of rotating axis l~ and instantaneous screw axis. The N-plane proposed by Tsai and Sung [21] is shown in Fig. 2. It contains the contact point P. The vector V~, the relative velocity of the conjugate tooth profiles at the contact point P, is normal to the N-plane. The line I, is perpendicular to the pitch line and Vre~. The unit vector n, is the unit direction vector of the pitch line. The unit vector n~ that lies on the plane z = u is perpendicular to the unit vector n,. The unit vectors n2 and n3 are the unit normal vector of the N-plane and the unit direction vector of the line In respectively. The parameters 1, ~ and fl are the position parameters of the N-plane. The notation p is the distance from the contact point P to pitch line. Tsai and Sung [21] presented the mathematical model of tooth profile of gear 1 in coordinate system Sl(xt, yl, z~) as follows: r I cos ~ - p cos ct sin ~ r,(ct, p, 1, ~bl) = ] Ul COS ~bl + VI sin ~bl / ' [_- U, sin ~b, + V, cos ~lJ
(1)
Z, Z °
O', O tO 2
•
,./
l2
/
C)/u ~tantaneous
~ X
O, Oj ,XI/ x_y L-8
I
Y1
Fig. 1. Skew axes and coordinate systems.
Screw Axis
Extended cycloid and cycloid bevel gear sets
41
Z
r,2 /
/rN-P' e
" ~-~=
~
nl
n2 - ~ ~ - - -
n % Pitch Line
=Y Ii --,
~ V ,'el X
Fig. 2. Relative velocity and pitch line.
where Uj = lsin 6 + p cos ~ cos 6, V~ = u + p s i n ct.
The mathematical model of tooth profile in gear 2 in coordinate system $2(x2, y2, z2) is described as
r2(a, p, 1, ~b2) =
l cos(6 - 7) - P cos ~ sin(6 - 7)-] U2 cos ~b2 + V2 sin ~b2 [, - U2 sin ~b2 + V2cos ~b2 J
I
(2)
where U2 = I sin(6 - 7) + P cos ct cosO - 7), V2 = u - C + p sin ct.
The constraint for the parametric conjugate tooth profiles of gear set with skew axes is proposed by Tsai and Sung [21] as act ap']
Ko
Ol
Ko = p
sin fl,
,
Op
ap
~c¢
Ogp, - ~ j ~ K, - ~ + K2 -~ + K3 ' ~ + K4 = 0,
where
K~ = - c o s fl,
(3)
42
L.M. Sung and Y. C. Tsai K2 = W, cos a sin fl + Wffsin a sin fl cos 3 + cos fl sin 6), K3 = - W~p sin a sin fl + W2p cos ~ sin fl cos 5,
K4 = (l sin a sin 6 - u cos a cos 3)cos fl, W~ = l sin 5 + p cos a cos 6, Wz = u + p sin a.
In equation (3), a and p are functions of l and 4), i.e. a = off/, 4)1) and p = p(l, 4)0. 3. GEAR SETS WITH INTERSECTING AXES
The bevel gear set is the o f its rotating axes is equal F o r the bevel gear set, the S,(Xl, yl, z~) is degenerated
degenerated case of a gear set with skew axes. The shortest distance to zero, i.e. C = 0. The included angle between the rotating axes is y. mathematical model of tooth profile o f gear 1 in coordinated system to
r,(r, 4),) = r
COS 2 COS 6 - sin 2 cos 0c sin 6-] Si cos 4), + T, sin 4), J , --SI sin 4)1 + T, cos 4),
(4)
where S, = cos 2 sin 3 + sin 2 cos ~ cos 3, T, = sin 2 sin ~. In equation (4), tan 2 = p / l and r = l sec 2. Similarly, the m a t h e m a t i c a l model o f tooth profile of gear 2 in coordinate system Sdx2, y2, z:) is r2(r, 4)2) = r
cos 2 cos(3 - y) - sin 2 cos ~ sin(3 - ?)-] $2 cos 4)5 + T2 sin 4)2 ] , - $2 sin 4)5 + T2 cos 4)2
(5)
where $2 = cos 2 sin(6 - 7) + sin 2 cos ~ cos(3 - ?), T2 =
sin 2 sin ~.
T h e constraints for the parametric conjugate tooth profiles of bevel gear set are p r o p o s e d by Sung and Tsai [19]. 02 ( 02) 04), - sin ct sin 6 1 + l tan 2 - ~ ,
(6)
92 ( 02) 04)2 - sin a sin(6 - ?) 1 + l tan 2 -5/ ,
(7)
Equations (6) and (7) are two quasi-linear partial differential equations o f order one. In this paper, assume that the p a r a m e t e r ~ is a function of 4), (or 4)2), i.e. ~ = ~t(4),) (or a = ~(4)2)). T h e solutions o f equations (6) and (7) m a y be derived as 2 = sin 6
fsin fsin
2 = sin(6 - 7)
+f,(r),
(8)
d4)2 + f f f r ) ,
(9)
d4)l
Extended cycloid and cycloid bevel gear sets
43
The parameter 2 is formulated as a function of r and ~b, (or ~b2), i.e. 2 = 2(r, ~b,) (or 2 = 2(r, ~b2)). Partial differentiating equations (8) and (9) yields 02 Oq~----~= sin 6 sin ~(tp,),
(10)
~2 8~b----~= sin(6 - 7)sin a($2),
(I 1)
d Or - dr ~,(r)] =
(12)
~2(r)].
4. INVOLUTE BEVEL GEARS The involute bevel gear set is the special case in which the parameter a is a constant. Assume that the parameter a is a constant and the initial values of q~, and ~b2 are equal to zero. F r o m equations (4), (5), (8) and (9), the mathematical model of tooth profile of gear 1 in coordinate system S,(x,,y,, z,) is deduced to [cos 2 cos 6 - sin 2 cos a sin 6 ] r,(r, ~b,) = r [ S, cos ~b, + T, sin 4~, J , [ - S , sin ~b, + T, cos ~b,
(13)
where S, = cos 2 sin 6 + sin 2 cos a cos 6, T, = sin 2 sin a, 2 = ~bl sin a sin 6 +f(r). Similarly, the mathematical model of tooth profile of gear 2 in coordinate system $2(x2, y2, z2) is described as r2(r, ~b2) = r
cos 2 cos(6 - 7) - sin 2 cos ct sin(6 - 7)-] $2 cos ~b2 + T2 sin ~b2 J , - $2 sin tp2 + T2 cos ~b2
(14)
where $2 = cos 2 sin(6 - 7) + sin 2 cos ~ cos(6 - 7), T2 = sin 2 sin a, 2 = ~b~sin ~ sin(6 - 7) + f ( r ) . 5. E X T E N D E D CYCLOID BEVEL GEARS
Sung and Tsai [19] propose that the pitch line of bevel gear set lies on the N-plane of contact point. For the bevel gear set, if the common normal of meshing tooth profiles at contact point passes through the pitch line, the meshing tooth profiles are conjugate. In Fig. 3, the rotating axes of cone 1, cone 2 and pitch cones of gear 1 and gear 2 are lc~, lc2,/, and 12, respectively. The pitch line is their common contact line. These four cones roll at the pitch line without slipping. The rotating angles of axes It, and lc2 are ~bc, and ~ba. The angles of cone 1 and cone 2 are ~Ib~and E~b~. The angle PrOPe is 0 ebb. The points C, and C2 are rigidly connected with cone 1 and cone 2, respectively. When cone 1 and pitch cones of gear 1 and gear 2 roll without slipping at the pitch line, the point C, generates separately the tooth profiles of gear 1 and gear 2. The common normal of meshing tooth profiles at contact point C, passes through the pitch line. Therefore, the tooth profiles of gear 1 and gear 2 are conjugate. These generated conjugate tooth profiles are the
44
L.M. Sung and Y. C. Tsai
extended epicycloid for gear 1 and extended hypocycloid for gear 2. The spherical triangle is shown in Fig. 3. By the law of cosines, 2 = cos-tFcos Elb) cos *rib' + sin E~b' sin *r~b,cos/l¢,lsinlal~l \ sin *Ib' , ] J ' L
P,CtP,t (15)
where and cos g =
- c o s E?~cosl~.l + cos(E?' - 0 *~') sin Elb' sinl;tl
(16)
By the law of sines, equation (17) is deduced to sin(c~b) -- 0,b,)sin(~b, \ ~ sinlSl'~ } sin a =
sin[21
(17)
From equations (15)-(17), the parameter • is a function of ~b, (or q~2), i.e. m = ~ ( ~ 1 ) (or ~ = a(q~2)). Substituting equations (15)-(17) into (4) and (5) yields the mathematical models of the extended epicycloid gear 1 and extended hypocycloid gear 2. Similarly, when cone 2 and pitch cones of gear 1 and gear 2 roll at the pitch line, point C2 generates the conjugate tooth profiles of gear 1 and gear 2. The generated conjugate tooth profiles are the extended hypocycloid for gear 1 and extended epicycloid for gear 2. The spherical triangle PpC2P¢2is shown in Fig. 3. By the geometry of spherical triangle, 2
=
cos-'[cos E[b) cos a[b) + sin e~b,sin a[b)
fl~b,lsinl~['~-]
k
(18)
)I'
where a[b) = E[b' + 0% and cos 0c =
- c o s E[b>cosl21 + cos(Elb' + 0
(19)
sin(E[b' + Oeb))sin(~bl k, sinsinl6l'~ E~b' J sin • =
sinl21
(20)
Cone 2 -,
ll(b)
/I
""L Pe P"
/
J
°
~
l~l
%
•2 Fig. 3. Generating points of extended cycloid bevel gear set.
Cone i'"
Extended cycloidand cycloidbevel gear sets
/
45
-
"'-- Extended Cycloid
Fig. 4. Comparison of tooth profiles between extended cycloid and involute bevel gears.
Substituting equations (18)-(20) into (4) and (5) yields the mathematical models of the extended hypocyloid gear 1 and extended epicycloid gear 2. Figure 4 is the comparison of the tooth profiles between extended cycloid and involute bevel gears. Figures 5 and 6 are the extended cycloid and involute bevel gears cut by the CNC machine center, respectively. In Fig. 7, the notations ~to~and ,,lb~ to2 are separately denoted as the angles of addendum cones of gear 1 and gear 2. The angle P,., of spherical triangle P~C,P,.~ is ~b~bl. By the law of cosines, el/
COS- I COS 7 2 -- COS(el b~ -- 0~b~)cos(161 + ~I~')
(21)
The angle Pc2 of spherical triangle P2C2;P,.2 is ~blb]. ~,~
_ , [ c o s 7~2 - c o s ( C ~ + O~)cos(16 - ~'1 + 4~9.]
Fig. 5. An extended cycloid bevel gear.
(22)
46
L.M. Sung and Y. C. Tsai (b)
Cone 2.
7ol
lI
////
\N "~'-~
' , ~ (6)
"/2
\N C o n e 1"-
Fig. 6. Straight involute bevel gear.
T h e contact ratio C R ~b) of extended cycloid bevel gear set is formulated as CRIb ~ _ -/)(b)('4~(b) \~tlcl l sin elb) + ..k(b) ~tlc21 sin e~b)) ,
(23)
where plb~__
N] bl _ N~b~ 2 sinl6l 2 sin[6 -- 71'
In equation (23), the notations N~b~ and N~zb~ are the numbers of teeth in gear 1 and gear 2, respectively.
Fig. 7. Addendum cones of extended cycloid bevel gear set.
Extended cycloid and cycloid bevel gear sets 6. CYCLOID
4-l
BEVEL GEARS
If the angle ecbJshown in Fig. 3 is equal to zero, the extended cycloid bevel gear set becomes the cycloid bevel gear set. Substituting that rYbJ= 0 into equations (15x17) yields R = cos-’ cos2 EIbJ+ sin2 eibJcos (!$$Y)], [
(24)
-cos EIbJcos(l( + cos cibJ sin ~1~) sinllJ ’
(25)
cos a =
sin cIbJsin w ( sin a = sir@)
I >
(26)
.
The mathematical models of the epicycloid gear 1 and hypocycloid gear 2 are derived by substituting equations (24x26) into (4) and (5). Similarly, from equations (l&o-(O) and that (YbJ= 0, equations (27x29) are obtained. 1= cos-‘[co$
cos a =
EjbJ + sin2 LjbJcos(&&i$$)], - cos e$“Jcoslll + cos cf’ - sin 6;“)sinll I ’
(27)
(28)
41 sin161 sin c$“’sin Y ( sin 61”)> sin a = sin);ll ’
(29)
The mathematical models of the hypocycloid gear 1 and epicycloid gear 2 are formulated by substituting equations (27x29) into (4) and (5). The contact ratio CRcbJof cycloid bevel gear set is expressed from equations (21x23) and that @bJ = 0. (-JRcb’=
P(“)(f#Qjsin tlbJ+ $$j sin ~4~)) 7 71
(30)
where hq"'
)XbJ =
2
NJb’ = 2 sin(S - yl ’
1’ 1*
q$I = cos-’
cos y$J - cos 61”’cos(lbl + 61”‘) sin c\“’sin@1 + c\“)) [
f#$]= cos-’
cos $J - cos 8’ cos()b - yJ + EibJ) sin ~1~) sin@ - yl + ~1”)) [
The contact ratio of involute bevel gear set is investigated by Sung and Tsai [ 191.Assume that speed ratios, included angles of rotating axes, addendum cones of pinions, addendum cones of gears and Pb) for the cycloid and involute bevel gear sets are the same. Let the contact ratio of cycloid bevel gear set divided by that of involute bevel gear set be v$“. (31)
48
L. M. Sung and Y. C. Tsai 1.3
@
~0
m i | m ~ m m
m m m n m l m m m ~ ~
m m m m m m m n m l l l l l l l n m i l l l l l n m m m m m m n m ~ ~ m m m m m ~ ~ m m m ~ m ~ n m
l
m m m l l l l m m m m m ~ m m m m
l l
m m l l m m m ~ ~
1.2
mm~mm~
1.1
~ d m ~ m m m m ~
:~INENNNNNNE~E
o
m
~
m
~
p
~
~
m
m
m
~
m ~ ~ m ~ ~ m m m ~ P ~ N I ~ N ~ N N ~ N N N ~ . ~ m
1
~
m
m
~
o
m
m m
m m
i
:..~ m m
=~=~
nun mm nm
m u m u m m u nm
.~ 0.g o
0.8 20
25
30
35
40
iii
45
Pitch Angle of Pinion (deg.) Fig. 8. Comparison between contact ratios of cycloid and involute bevel gear sets for different pressure angles. where
4,(b) _ cos [-cos %q' - cos e?)cos(151 + e?)l ' <"-
c21
- L COS--
1
7ic,-~<7;~T8
¥~>7
COS ~(ob) -- COS e~b) c o s @ - ?i + @))
_,/cos Corm
x,~>=~os t ~ 7 - ~ o ~ Y(b) : C O S - ' ( : O : ~ )
_,(cosl6i "~
t~J'
-- COS-I(COSiI~Tbb?" " C ~O S )
In equation (31), la[ is the complement of pressure angle of involute bevel gear set. The notations ?(b~) and 7~b~>are the angles o f base cones o f involute bevel gear set. To prevent undercutting for the involute bevel gear sets, equations (32) and (33) are proposed by Sung and Tsai [19].
c o ~ -C, ( ~ (b)] < cos-,/¢°si~l~ cos-(cosi,~- ~i~
t, c--~7~ )'
(32)
(b) c o ~\cos -C, ( ~~'~,7/
~ + co~,(\ ~°~ ~J~ cos ~'b: )
(33)
tcos ~,~,,,/
t, cos ~,<~,,)+
cos- ,/~°~ I ~ I
\cos ~,~,]
Extended cycloid and cycloid bevel gear sets
49
1.2
~ib)=0.50 il
r,~ r'~_ 1.1
P
~
d
H
I H i
j ~.(b)= 0.45
~.
¢.(t,)= 0.40
°° R
) _-;_ --
'
. o .~
C(b)= 0.35 __p____...
__
~[.-,
I i I H P ~ ~ -.,,,,,m I
0.9
l l I
~ - , - i i l mini I
. . . .
0.30
l
0.8
25
30
35
40
45
Pitch Angle of Pinion (deg.) Fig. 9. Comparison between contact ratios o f cycloid and involute bevel gear sets for different values o r~ ,~(b).
Assume that the ratio of E~b) to 161 and ratio of Elb) to 16 - Yl are the same and ~(b) is the ratio of E~b) to 161. ~b> _ E?' El~' J6l - 16 - 71"
(34)
In general, the included angle between the rotating axes of a bevel gear set is 90 °. Figure 8 shows the relation between r/¢(~> and pitch angle of pinion for different pressure angles of involute bevel gear sets, where ¢(b) is 0.4 and the addendum angle of bevel gear is 4 °. Figure 9 is the chart representing the relation between r/~,~) and pitch angle of pinion for different values of ~(b), where the pressure angle of involute bevel gear set is 20 ° and the addendum angle of bevel gear is 4 ° . From Figs 8 and 9, it is found that, in low speed ratios, the contact ratios of cycloid bevel gear sets are greater than those of involute bevel gear sets.
7. CONCLUSION In this paper, the general mathematical models of conjugate tooth profiles and constraints for bevel gear sets have been investigated. The tooth profiles of involute bevel gears, extended cycloid bevel gears and cycloid bevel gear have been deduced from the general mathematical models. The contact ratios for the extended cycloid and cycloid bevel gear sets have been described. The charts comparing the contact ratios of cycloid bevel gear sets with those of involute bevel gear sets have been provided. In low speed ratios, the contact ratios of cycloid bevel gear sets are greater than those of involute bevel gear sets. The design charts are useful for the engineers to design the bevel gear sets of high contact ratio. Additionally, the presented mathematical models provide a basis "for further investigations such as the tooth geometries, computer-aided manufacturing, computer-aided tooth contact analysis and finite element stress analysis of involute, extended cycloid and cycloid bevel gear sets. Acknowledgments--The authors are thankful to the National Science Council, R. O. C. for their grant, no. NSC 83-0401-E-110-011.
50
L.M. Sung and Y. C. Tsai
REFERENCES 1. Huston, R. L. and Coy, J. J., Transactions of ASME Journal of Mechanics and Design, 1981, 103, 127. 2. Huston, R. L. and Coy, J. J., Transactions of ASME Journal of Mechanics and Design, 1982, 104, 743. 3. Huston, R. L., Lin, Y. and Coy, J. J., Transactionsof ASME Journal of Mechanics and Transmission Automation Design, 1983, 105, 132. 4. Litvin, F. L., Tsung, W. J., Coy, J. J. and Heine, C., Transactions of ASME Journal of Mechanics and Transmission Automation Design, 1987, 109, 163. 5. Tsai, Y. C. and Chin, P. C., Transactions of ASME Journal of Mechanics and Transmission Automation Design, 1987, 109, 443. 6. Litvin, F. L, Zhang, Y., Lundy, M. and Heine, C., Transactions of ASME Journal of Mechanics and Transmission Automation Design, 1988, 110, 495. 7. Chang, H. L. and Tsai, Y. C., Proceedings of the 1990 ASME Design TechnicalConferences--2nd Conference in Flexible Assembly Systems, Chicago, 1990, p. 5. 8. Fong, Z. H. and Tsay, C.-B., Transactions of ASME Journal of Mechanics Design, 1991, 113, 174. 9. Yang, A. T. and Freudenstein, F., Transactions of ASME Journal of Engineering Industry, 1973, 95, 497. 10. Litvin, F. L., Goldrich, R. N., Coy, J. J. and Zaretsky, E. V., Transactions of ASME Journal of Mechanics and Transmission Automation Design, 1983, 105, 310. 11. Day, C. P., Akeel, H. A. and Gutkowski, L. J., Transactions of ASME Journal of Mechanics and Transmission Automation Design, 1983, 105, 441. 12. Freudenstein, F., Longman, R. W. and Chen, C.-K., Transactions of ASME Journal of Mechanics and Transmission Automation Design, 1984, 106, 371. 13. Baxter, Jr, M. L., Transactions of ASME Journal of Mechanics Design, 1978, 100, 41. 14. Lee, T. W. and Akbil, E., Transactions of ASME Journal of Mechanics and Transmission Automation Design, 1983, 105, 663. 15. Winter, H. and Paul, M., Transactions of ASME Journal of Mechanics and Transmission Automation Design, 1985, 107, 43. 16. Mark, W. D., Transactions of ASME Journal of Mechanics and Transmission Automation Design, 1987, 109, 275. 17. AI-Shareedah, E. M. and Alawi, H., Mechanics Machine Theory, 1987, 22, 13. 18. Savage, M., Brikmanis, C., Lewicki, D. G. and Coy, J. J., Transactionsof ASME Journal of Mechanics and Transmission Automation Design, 1988, 110, 189. 19. Sung, L. M. and Tsai, Y. C., Journal of the Chinese Society of Mechanical Engineers, 1994, 15, 273. 20. Chang, H. L. and Tsai, Y. C., Transactions of ASME Journal of Mechanics and Transmission Automation Design, 1992, 114, 8. 21. Tsai, Y. C. and Sung, L. M., Journal of Applied Mechanisms and Robotics, 1993, 1, 36.