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Study on the generalized mathematical model of noncircular gears
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SCleNCE~DIReCT •
MATHEMATICAL
AND
COMPUTER MODELLING
I~EVIt~R
Mathematical and Computer Modelling 41 (2005) 555-569 www.elsevier.com/locate/mcm
S t u d y on t h e G e n e r a l i z e d M a t h e m a t i c a l M o d e l of N o n c i r c u l a r G e a r s MING-FENG TSAY AND ZHANG-HUA FONG D e p a r t m e n t of Mechanical Engineering National C h u n g - C h e n g University 160 San-Hsing, Ming-Hsiung, Chia-Yi, 621, Taiwan, R.O.C.
(Received April 2003; revised and accepted August 2003)
A b s t r a c t - - T h e pitch curve of noncircular gears is usually determined by kinematical requirements such as position, velocity, and acceleration. However, it is usually difficult to obtain a tooth profile that complies with kinematical requirements. A Fourier series approximation method is proposed to approximate the pitch curve. Based on the approximated pitch curve, a mathematical model of the tooth profile of noncircular gears is derived. The proposed mathematical model is very flexible with sufficient accuracy and can also be used to process the reverse engineering problem of the existing noncircular gears. (~) 2005 Elsevier Ltd. All rights reserved.
Keywords--Noncircular
gears, Mathematical model, Fourier series.
NOMENCLATURE ASp
a~ ) b(j) Bp
arc length NI measured from angle zero to angle Cp as shown in Figure 2 the coefficients of Fourier cosine series, n = 0, 1,2 .... n the coefficients of Fourier sine series, n = 0, I, 2 . . . , n
n np
tooth number unit normal vector at the point on the pitch curve in coordinate system Sp
Op
gear rotation center in Sp
rp
centrode of gear in Sp
l~i)
position vector for curves i in Sp
tangential shift parameter of rack cutter
Sp
coordinate systems p
C
distance between gear rotation centers O1 and 02
Pc Tp
circular pitch
F (j)
the Fourier series
Gp
position vector of the pitch curves in coordinate system Sp
tp u(~)
unit tangent vector of Tp
hk
addendum
ra
module of gear
mlj
ratio of angular velocities wj to wz
V
tangent vector at the point on the pitch curve in Sp curve variable for curves i of the rack the relative velocity pressure angle of gear
The author would like to thank the National Science Council of R.O.C. for financially supporting this work under Contract No. NSC 86-2212-E-194-007. 0895-7177/05/$ - see front matter © 2005 Elsevier Ltd. All rights reserved. doi: 10.1016/j.mcm.2003.08.010
Typeset by J 4 j ~ - ~
M.-F. TSAYAND Z.-H. FONG
556
8(i) Cp wp q, p
curve variable of the root fillet of the rack rotation angles in Sp angular velocity in Sp angle between the tangent vector and the Xp direction
superscript i superscript j subscript p
radius of curvature of the pitch
curve i of rack 1 Case (a) 2 Case (b) 1 Gear 1 2 Gear 2 c rack f frame
calve
1. I N T R O D U C T I O N The kinematical behavior of noncircular gears is similar to the cam-follower mechanism except that the noncircular gear set is a positive transmission. Noncircular gear set can be used to transmit varied gear ratio with good accuracy, higher power density, ease for dynamic balance, and compact in size. However, the mathematical model for tooth profile of noncircular gears is rather difficult to derive. The most typical examples of application of noncircular gears are as follows, (i) as the deriving mechanism for a linkage to modify the displacement function of the velocity function, and (ii) for the generation of a prescribed function. The applications of noncircular gear include the Geneva mechanism with modified angular velocity, the Bopp and Reuter meters for the measurement of the discharge of liquid, and the automatic planting machine for agriculture. Litvin [1,2] proposed a methodology by extending from tooth evolute to generate tooth profile of noncircular gears. Kuczewski [3] used spur gears to locally mimic ellipse gears. Chang [4,5] developed a mathematical model for modified elliptical gears. Most mathematical models proposed are limited to certain type of pitch curve [6,7]. In this paper, a methodology is proposed to establish tooth profile of noncircular gears. The pitch curve or motion relationship between gears (a motion program) is approximated by applying Fourier series technique. The numerical accuracy of the pitch curve can be controlled by effective terms ofthe Fourier series in the cost of computation time. Based on the equation of approximated pitch curve, generalized equations of tooth profile and design limits of noncircular gears are derived. Computer-aided design software for noncircular gears is developed based on the proposed methodology. Similar methodology can be used to process the reverse engineering problem of the existing noncircular gears. The proposed mathematical model is very flexible and with sufficient accuracy as shown by the presented numerical examples.
2. A P P R O X I M A T E THE CENTRODE OF NONCIRCULAR GEAR BY FOURIER SERIES In order to improve the flexibility in the design of pitch curve of noncircular gear, the Fourier series is applied to approximate either the pitch curve or the instantaneous gear ratio program. Applied coordinate systems are shown in Figure 1. Coordinate systems Sl(xl,yl, zl) and $2(x2, Y2, z2) are rigidly attached to Gear 1 and Gear 2, respectively, while the coordinate system Sf(xf,yf, zf) is rigidly attached to the frame. There are two cases; one is for the known centrode curve and the other is for the known instantaneous gear ratio program as follows. (a) If the centrode of noncircular Gear 1 is known, the centrode of Gear 1 can be represented by the Fourier series F(1)(¢1) as
r~ (¢1) = F ~ (¢1).
(1)
The centrode of Gear 2, which is conjugated with the centrode of Gear 1, is represented in terms of ¢1 as
r2 (¢5 (¢1))
=
c
-
~
(¢1).
(2)
Mathematical Model of Noncircular Gears
557
I
t ~b2
t k
_
_
_
l
r,(~,)
'
C i
i
o, Figure 1. The coordinate system applied to the noncircular gear set.
The instantaneous ratio of angular velocities can be represented as wl d¢1 r2 m12 - w2 - d¢2 = rl"
(3)
The angular relationship between rotation angles ¢1 and ¢2 are shown as the following equation, ¢2 = f0 ¢1 d¢1 ,
(4)
m12
where C is the distance between gear rotation centers O1 and 02, and 42 is the rotation angle of Gear 2. (b) If the ratio of angular velocities m21(¢1) is known, gear ratio can be represented by the Fourier series F (2) (¢1) as m21 ~-
1
_ F(2)(¢1),
(5)
m12
and the pitch curve of Gears 1 and 2 as C
~1 = - - - - Z '
(8)
m12 + m12
~2 = ~ C .
(7)
m12 + 1
There are two approaches to obtain the coefficients of Fourier series as follows. (1) Assume the exact function F(J)(¢I) is known, the orthogonal trigonometric system is applied to obtain the coefficients as follows [8], OO
F (j) ( ¢ 1 ) = a ( J ) + ~
[a(nj) cos ( n ¢ 1 ) + b(nj) sin (n¢1)]
j=
1,2,
n=l
a~j) = ~1 fo 2'~ F(J) (¢1) d¢1,
(S)
a(nj) _-- 1¢r --/2, F (j) (41) cos (n¢l) d¢1, J0
b~ ) -= _1 f2~r F(j)(¢1) sin (n¢1) d¢l, 7r J0
n -- 1,2,3,4,...,
where a 0(j) , a (j), 1 . .. , a~ ) , b~j), ... , b~ ) E R are the coefficients of Fourier series.
558
M.-F. TSAY AND Z.-H. FONG
(2) If the coordinate of tack points ((at, F(J)(al)), n = 1, 2, 3, 4 , . . . , ) are known, where w is the number of tack points, numerical method [9] is applied to obtain the coefficients as follows, h
F (j) ( a t ) = a(J) + E
[a~) c°s (nat) + b(nJ)sin (nal)] '
j = 1, 2, k = 2h + 1,
[Q]~xl = [M]~xk [D]kxl,
(9) (10)
where h is the number of effective terms of Fourier series, k is the total numbers of the unknown coefficients, [M] is the matrix of orthogonal trigonometric functions cos (nat) and sin (naz), [Q] is the matrix of numerical values FJ (a~), and [D] is the matrix of Fourier series coefficients a 00) , a (J) 1 , . . . ~ u n _(J)~ b~), . . . ~ n ~(J) • [Q]T = [F j (al)
[DiT=[4 Ii
F j (a2)
a(
"-"
FJ (a~)].
(ii)
at b2 . .
(12)
c°sal sinai cos a2 sin a2
c°s2al sin2al cos 2a 2 sin 2a2
"
"
cos aw sin aw
cos 2aw sin 2aw
• .. • ..
cos kal sin kal cos ka2 sin ka2
•
[M] =
o
(13)
• ..
coska,~ sinka~
If w = k, then coefficient matrix can be obtained by [D] = [M] -1 [Q].
(14)
If w > k, the least square method shall be used to obtain the approximated results,
[D1 = ([M] T [M]) -1 [M] T [Q].
(15)
3. B A S I C G E O M E T R I C A L C O N S I D E R A T I O N S OF T H E N O N C I R C U L A R P I T C H C U R V E The position vector Gp(¢p) of the pitch curve rp is represented in the Cartesian coordinate system Sp as shown in Figure 2.
Y
tp
" Xp
Figure 2. Geometrical properties of the noncirculax pitch curve•
Mathematical Model of Noncircular Gears
559
GP(¢P):{xP}={rpc°sCp} " y prpsinCp
(16)
The tangent vector Wp at the point I on the pitch curve can be obtained by differentiation,
dxp } Tp =
Tpy
(17)
dyp "
=
The unit tangent vector tp can be derived as follows,
(18)
(19)
n P = { - t PtYp }= As shown in Figure 2, the unit tangent vector at point I can be simplified as
sin ~/
tpu
(20)
'
where 7 can be obtained from equation (20) as follows, 7 = tan-1
(tp~,tpy).
(21)
The arc length A¢, measured from point N to point I on the pitch curve can be calculated by the following equation,
=
+
\ dep ]
de,.
(22)
The circumference arc length A2~r should equal to integer multiples to the pitch of rack cutter so as to keep all tooth profile complete, therefore,
A2~r= ~rrnn,
(23)
where m is the module of gear, and n is the tooth number. 4. M A T H E M A T I C A L
MODEL
OF
THE
RACK
CUTTER
Assume the noncircular gear is generated by a rack cutter. There are some limitations on the concave portion of noncircular gear by using the rack cutter. However, the working part of tooth profile (the involute part) is virtually the same whether the gear is generated by a rack cutter or a shaper. A standard rack cutter composes five curves as shown in Figure 3. The parameters of the rack cutter include pressure angle ~, tip radius re, circular pitch Pc, and addendum hk. As shown in Figure 3, Curves 1 and 5 of the rack cutter are straight lines that generate the involute working part on tooth surface. Curves 2 and 4 are circular arcs that generate the root fillet of tooth surface. Curve 3 is straight line that generates the bottom land of noncircular gear. Tangential
560
M.-F. TSAYAND Z.-H. FONG
l~
Xc
II ~,1I
Bp
r
.....
.......
i ........
(2~ -
13)
7----
.....
Figure 3. The coordinate system applied to the rack cutter.
shift parameter Bp is the distance between the rack center and the origin Oc of coordinate system Sc.
Bp=
0,
if p = 1,
0.5Pc,
if p = 2 ,
(24)
where subscript p indicate which gear is being generated, p = 1 for Gear 1 and p = 2 for Gear 2. Tangential shift parameter is applied to synchronize the initial generating point of Gear 1 and Gear 2.
Bp
4.1. T h e M a t h e m a t i c a l
Equations for Curves 1 and 5 of the Rack Cutter
Curves 1 and 5 are used to generate opposite sides of the working tooth profiles of noncircular gears. Parameter u (i) is the curve variable for Curves 1 and 5. The position vector for Curves 1 and 5 of rack cutter can be obtained by the following equations, =
=
Pc+u
- h k sec ~0 _< u (i) < hk sec ~
(1) s i n ~
'
(i = 1, 5),
and the corresponding unit normal vectors are
{~(1)}
n(1)
{ sin~
,oxc
:
~(1)
=
-- C O S ~0
} '
(26) = 4.2. T h e M a t h e m a t i c a l
(5) 1~yc
=
-- COS~
"
:Equations for Curves 2 and 4 of the Rack Cutter
Curves 2 and 4 are used to generate opposite sides of root fillets of generated t o o t h profile.
Parameter 0 (i) is the curve variable of the root fillet. The position vectors and unit vectors for Curves 2 and 4 of rack cutter can be obtained by the following equations,
yc(2)
R(4) -=--
y(4)
-Bp + ~
=-
- hk tan ~0 -- rc cos ~ + rc sin 0 (2)
'
Pc + hk tan ~o + re cos ~ -- rc sin 0 (4) - B p -- -~-
'
Mathematical Model of Noncircular Gears 0 _> 0 (i) _> ~~ - ~
561
(i = 2, 4 ) ,
(27)
and the corresponding unit normal vectors are
n (2)
n(2) =
-- cos 0 (2)
,(2) po y c
=
sin 8 (2)
'
(28) n~4)=
n(y~)
(sinO(4)
"
4.3. T h e M a t h e m a t i c a l E q u a t i o n for C u r v e 3 o f t h e R a c k
Cutter
Curve 3 is used to generate the bottom land of the generated tooth profile of noncircular gears. Parameter u (3) is the curve variable of the Curve 3. The position vector and unit normal vector for Curve 3 of the rack cutter surface can be obtained by
n(3) yc
=
.
5. T H E M A T H E M A T I C M O D E L O F GENERATED NONCIRCULAR GEARS Assume the noncircular gear is generated by a rack cutter. As shown in Figure 4, the pitch line of the rack cutter performs pure rolling on the pitch curve of noncircular gears during the generating process. The coordinate system Sc(X,, Yc), S f ( X f , Y f ) , and Sp(Xp, Yp) are rigidly attached to the rack cutter, frame, and the generated gear, respectively. The generated gear rotates about a fixed center Op while the rack cutter translates without rotation in the plane. The relative velocity is zero at the instantaneous center I and the parameter Acp is the translational distance from the rack cutter origin point O~ to instantaneous center of rotation I. The coordinate transformation matrix from coordinate system S¢ to Sp is shown as follows,
[Mpc]=
sin 7
cos 7
rp cos Cp + Acp COS7 ]
-cos7
sin7
-rpsin¢p+A¢
0
0
sin7
.
(31)
1
Therefore, the locus of the rack cutter represented in coordinate system of the generated gear is R(~ ) = [Mpc] R (1)
(i = 1 , 2 , . . . , 5 ) .
(32)
According to the theory of gearing, the unit normal at the contact point must pass through the instantaneous center of rotation I. Equation of meshing may be represented in coordinate
system S~ as follows,
n(O
,(i)
(i -- 1 , 2 , . . . , 5 ) ,
(33)
where (X (i) , y(O) ___(0, -Acp) is the position vector of the instantaneous center I while (x (0, y(i)) and (n ~c,,oy¢ (i) ~(i)~j are the position vector and unit normal vector of the contact point, respectively.
562
M.-F. TSAY AND Z.-H. FONG
Figure 4. The coordinate systems applied to the gear generation procedure.
The equation for the tooth profile of generated gear can be obtained by solving equations (32) and (33), simultaneously. The simplified equations for complete tooth profile of noncircular gears are listed as follows. According to equations (32), (33), (25), and (26), the working tooth profiles of generated noncircular gear can be represented by x (~) = u (1) cos~vsin~ +
( -Bp + -~Pc+u(1)sin~)c°sT+rpC°SCp+A~ c°sT'
( Pc y(1) =_u(1) cos~ocosT÷ - B p + ~ - +
A¢~ = ( Bp Pc4
u(1)) sin~0 s i n ~ - r v s i n C v + A ~ sinT,
u(Ocsc ~o),
(34)
x(pS)=u(S)cos~osinT+(-Bp-(-~+u(5)sin~o))cosT+rpcosCp+A¢ cosT, y(pS)=u(5)cos~cosv+(-Bp-(-~+u(5)sin~v))sinv-rvsinCp+A, sinv,
-hksec~o < u (i) < hk sec~
(i ----1,5).
Mathematical Model of Noncircular Gears
563
According to equations (32), (33), (27), and (28), the root fillet of generated tooth profile can be represented by X(p2) = ( - h k + r c s i n ~ - rccos0 (2)) sin 7
+ (-Bp + (-~ - hk tan~- rccos~) + rcsinO(2)) cosT + rpCOS¢, + A¢, cosT, y(2)
=--(-hk
+ rc sin~ - rceos0(2)) cos~'
+ (-Bp + ( ~ - hk tan ~ - re eos ~) + rc sin O(2)) sin ~ - rp sin Cp + Acp sin T, Acp = Bp - ( ~ - hk tan~- rcCOS~+ (-hk + r~sin~)cot O(2)) ,
(36)
X(p4) = (--hk + r e s i n s - - r ~ c o s 0 (4)) sin7
-Bp-(~-hktan~-recos~-resinO(4))cos~+rpCOSCp+A¢pcos% yp(4) = _
(--hk +
+ (-B,-
rc s i n ~ - rc cos0 (4)) cosq'
(-~ - hk tan~- r~cos~) - rcsinO(4)) sinT- rpsin¢, + A¢,sin%
A¢ = B, + (-~ - hk tan~- rccos~) + (-hk + r~sin~)cot O(4), 0 _< 0 (~) _< ~~r-
~
(37)
(i = 2, 4 ) .
According to equations (32), (33), (29), and (30), the bottom land of generated tooth profile of noncircular gears can be represented by X(p3) =
(-hk + r~ sin ~
- re) sin 3, + (-Bp
yp(3) = ( h k - r~ sin ~ + r~) cos~/+
A¢ = Bp - u (s),
(-Bp u(3)
+ u (3)) cos ~/+ rp cos Cp +
ACp cos %
+ u (3)) sin~/- rp sin Cp + A¢~ sin%
<_ -~-hktan~--rcCOS~].
(38)
6. E Q U A T I O N OF U N D E R C U T T I N G A N D D I S C U S S I O N ON THE D E S I G N LIMITS According to the theory of gearing [1], undercutting occur when singular point exists on generated tooth profile. The mathematical definition of singularity of the tooth profile is represented by the following equation, V(2) --- V(1) Jr- V (12) ~-- 0. (39) Equation (39) implies the following two determinants are equal to zero,
of
of&,
Ou
0~/ dt
=
of
Ofd~
Ou
07 dt
= o.
(40)
Suppose that the undercutting points on the generated tooth profile of noncircular gears are generated by the working Curve 1 of the rack cutter. The relative velocity between the gear blank and the rack cutter, represented in coordinate system S¢, is shown as follows, V(12)
= wp (-Bp + -~Pe+u(1)sin~+ A¢,)ic+W,(-uO)cos~)jc.
(41)
564
M.-F. TSAY AND Z.-H. FONG
Equation of meshing f , derived from equation (33), corresponding to Curve 1 is simplified as Pc
f = u + ~- sin q0 - Bp sin qo + Acp sin ~.
(42)
The angular velocity of noncircular gears is assumed to be Wp = dd-~t . Substitute equations (41) and (42) into equation (40), the condition of undercutting is simplified as OA4~ 2 u cos 9~ + - ~ sin ~ = 0.
(43)
Parameters u = ~=hk sec ~0 at the lowest/highest end points on Curve 1 of the rack cutter. Assume the addendum parameter hk = 1.0 m. Equation (43) implies the limiting values of the gear module m as '/nma x ~
OAcp sin2 ~ = [ppsin 2 ~[
(44)
~
The minimum number of teeth thus is derived from the pitch curve circumference A2~ = ~rmn. n m i n _~ ~
S•Tr
nrain E N.
,
(45)
71"?TI,m a x
This result is substantially the same as that obtained by Wu [10] and Chang [5] with Pv(¢p) is the radius of curvature of the pitch curve [1].
PP = - dnp~
--
dnpy
--
II o
r~ + 2r~ - rprp
(46)
The maximum gear module of the rack cutter is limited by the minimum curvature radius [Pp[min of pitch curve. The minimum or maximum pitch radius (]Pp[min or [Pp]max) can be located when equation (47) is observed. dpp = O.
d¢v 7. E X A M P L E S
(47)
AND DISCUSSIONS
7.1. E x a m p l e 1 The basic data of this example is the same as provided by Chang [5]. As shown in Chang's paper [6], an elliptical gear with module m = 5.0 mm, number of teeth n -= 45, pressure angle ~c = 20 °, radius of circular arc rc = 0.3m, and eccentricity e -= 0.608, major semi-axis a -- 125 mm, and the elliptical gear pitch curve, rl = (al (1 - e12))/(1 + el cos ¢1) is explicitly defined. Fourier series is used to simulate the elliptical gear's pitch curve. Since the function of elliptical pitch curve is an even function, the function can be simplified as r l = ~n=o[a~ cos(n¢l)]. Fourteen items of above-mentioned series are used to approximate the pitch curve. The coefficients a3o,a~ . . . . , a~3 are calculated by equation (9) and the calculated values are listed in Table 1. The maximum deviation of the pitch curves between elliptical gear and approximated gear is less than 10 -a mm at ¢1 = zr as shown in Figure 5. The tooth profile based on approximated pitch curve is shown in Figure 6. The normal deviation of tooth surface between approximated gear and exact gear obtained by Chang's method is below 2 x 10 -5 mm as shown in Figure 7 in three typical teeth located at the major axis, minor axis, and ¢1 = 0.25r. The numerical results suggest that
Mathematical Model of Noncircular Gears
565
T a b l e 1. T h e Fourier series coefficients of E x a m p l e 1.
Coefficmnt a0 al a2
Data Coefficient 99.2608483 a7 -67.2555863 as 22.7849850 a9
Data -0.1016835 0.0344486 --0.0116706
a3
--7.7191439
al0
0.0039538
a4 a5
2.6115107 --0.8859514
all a12
--0.0013395 0.0004538
a6
0.3001444
a13
0.0001576
0.000075 0.00005 0.000025
,.AA.AA,
-0.000025 -0.00005 -0.000075
Ra~us Devia~on(mm)
Figure 5. The radius deviation of approximatedpRch curve of Example 1.
rl
" X1
Figure 6. The elliptical gear generated based on approximated pitch curve. approximated pitch curve and corresponding tooth profiles are fairly close to the theoretical ones and the deviations are controllable. The undercutting of this example is checked by equations (43)-(46). Figure 8 shows the radius of elliptical pitch curve and the corresponding curvature approximated by Fourier series. The minimum curvature [Pt Imin occurred at both end points of major axis of the ellipse and [Pl Imin = 78.8218 ram. The limiting value mmax of the gear module calculated by equation (44) is 9.22 ram, which is identical to the result provided by Chang [5]. Since the addendum of the rack cutter is just 5.0 mm, which is smaller than the limiting value of 9.22 mm. Therefore, undercutting will not occur on the tooth profile of the generated gear. 7.2. E x a m p l e 2
Assume a pitch curve (Gear 1) of cam with seven tack points are given as listed in Table 2. Based on the given pitch curve of Gear 1, we designed two conjugated gears with total gear ratios ~ equal to 1 and 2. Gear 1 rotates ~ revolutions per revolution of Gear 2. Assume that pressure angle ~ = 20 °, tip radius rc = 0.25m. The center distance C of the gear pair is obtained by equation (4).
566
M.-F. TSAY AND Z.-H. FONG mr~
0 . 00002
/
0.00001
-
-
!
the region 1
?/lm
-2
o. o-~oor----"--
workingsurfacen (ram) the region 5
--0 . 00002
(a). The normal deviation of the tooth surface at the major semi-axis. 0.00002
i'TII
the region I 0.00001
2
\
-0.00001
4
workingsurfaceu (ram)
theregion5
-0.00002 (b). The normal deviation of the tooth surface at the minor semi-axis.
0.00002
ml~
theregion1
F
t
\ the region 5
working surfaceu (ram)
(c). The normal deviation of the tooth surface at ¢1 = 45 °. Figure 7. The normal deviation on tooth surface generated by Curves 1 and 5 of rack cutter of Example 1.
(mm)
14o1~°2°°18° ~curve
r (¢0
lOO 8o 2
3
4
5 ~
el(Degree)
Figure 8. The radius of approximated elliptical pitch curve and curvature of Example 1.
Mathematical Model of Noncircular Gears
567
Table 2. The tack points of cam. Radian
Radius (ram)
Radian
Radius (mm)
0 0.9425 1.8401 2.7378
4o 48 52 57
3.6353 4.5329
54 47
5.43O5
43
20(
15(
IOC
5C rain 1
2
3
4
5
6
Figure 9. Curvatures of noncircular gears of Example 2.
Y,
- r , , t v v
"
"
'
'
V
:
|
,
Figure 10. Noneireular gears of Example 2.
Apply Fourier series to approximate the pitch curve of Gear 1 and use the approximated pitch curve r l to generate the pitch curve r2 of Gear 2. The curvature radius pp calculated by equation (46) is shown in Figure 9. [P2[min is 33.7457mm on Gear 2 when total gear ratio v = 1. T h e m a x i m u m module m is 3.93116mm as calculated by equation (23). Thus, the minimum teeth number of gears 1 and two corresponding conjugate gears (v = 1 and 2) are 25, 25, and 50, respectively. The center distances C = 98.7228mm when v -= 1, while C = 147.158mm when v -- 2. Figure 10 shows the approximated Gear 1 and the conjugated noncircular gears with total gear ratios v equal to 1 or 2.
7.3. E x a m p l e 3 Design a gear set with center distance C -- 1 5 0 m m and Gear 2 rotates at a predetermined angular velocity ratio, d¢2 d¢1 W2 ~-- dt
=m21
dt '
and a predetermined angular acceleration,
dm21 d¢1 ~2---
d2¢1 d--~ d~ ÷ m 2 1 dt 2 ,
568
M.-F. TsAY AND Z.-H. FONG
1.5 1.25 1 0.75
(2.) 1
0.25 f 0 -0.25
1
2 ~ 4
5
~/
Figure 11. The modified trapezoidal function.
\ Figure 12. The noncircular gear set of Example 3.
with the modified trapezoidal function [11] as shown in Figure 11. Apply the Fourier series to simulate the modified trapezoidal function and calculate the instantaneous ratio of angular velocities m21 by equation (5). The design limits can be obtained by equations (44)-(46). Figure 12 shows the tooth profiles of the noncircular gear pair with n = 40 and module m -- 3.7135 mm. 7.4. E x a m p l e 4 This example is aimed to check whether the proposed m a t h e m a t i c a l model can be applied to simulate the concave pitch curve or not. Figure 13 shows the concave noncircular gear pair with pressure angle ~ -- 20 °, the module m = 0.9489mm, the teeth number of gears n = 100, the radius of circular arc rc = 0.15 m, and the center distance C = 90.921 m m . The numerical results show t h a t m a t h e m a t i c a l model of working part of the tooth profile is acceptable while tooth profile in the root fillet m a y have some troubles.
Figure 13. The noncircular gear pair of Example 4.
Mathematical Model of Noncircular Gears
569
8. C O N C L U S I O N S Fourier series and numerical techniques are applied to approximate the pitch curves or instantaneous gear ratio. Based on the approximated pitch curves, the mathematical model of the tooth profiles of noncircular gear is proposed. Equations for the undercutting are used to determine the design limitations of noncircular gears. According to the presented numerical results, following conclusions are drawn. 1. The proposed pitch curve approximation method is effective and can be used to design the tooth profile of noncircular gear with sufficient accuracy. 2. There are two possible ways to design noncircular gear set with discrete data. Either pitch curve or instantaneous gear ratio can be approximated by the discrete data set. According to the approximated pitch curve or instantaneous gear ratio, the tooth profiles of the noncircular gear can be easily derived. The proposed mathematical model is very flexible. 3. Undercutting equation is very helpful to determine the design parameters. 4. It is possible to design noncircular gear without exact equations for the tooth profile. This is very important to do the reverse engineering for existing noncircular gear set.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.
F.L. Litivin, Theory of Gearing, NASA Publication, Washington, DC, (1989). F.L. Litivin, Gear Geometry and Applied Theory, Prentice-Hall, New Jersey, (1994). M. Kuczwski, Designing elliptical gears, Machine Design, 166-168, (1988). S.-L. Chang, Designing Involute Elliptical Gears, Master's Thesis, National Tsing Hua University, Taiwan, (1992). S.-L. Chang and C.-B. Tsay, Mathematical model and undercutting analysis of elliptical gears generated by rack cutters, Mech. Ma~h. Theory 31 (7), 879-890, (1996). T. Bernett, Elliptical gears for irregular motion, Mechanical Engineering 89 (6), 33-39, (1967). S.H. Tong and C.H. Yang, Generation of identical noncircular pitch curves, ASME, Journal of Mechanical Design 120 (2), 337-341, (1998). E. Kreyszig, Advanced Engineering Mathematics, Seventh Edition, John Wiley and Sons, Inc., (1993). B. Faires, Numerical Analysis, Sixth Edition, Brooks/Cole Publishing Company, (1997). X. Wu, S. Wang, G. Wang and A. Yang, Theory Mach. Prague, In Proe. 8~h Wld Congr., pp. 391-394, Czechoslovakia, (1991). J. John, Design of Machinery, McGraw-Hill International Editions, (1992).
SCleNCE~DIReCT •
MATHEMATICAL
AND
COMPUTER MODELLING
I~EVIt~R
Mathematical and Computer Modelling 41 (2005) 555-569 www.elsevier.com/locate/mcm
S t u d y on t h e G e n e r a l i z e d M a t h e m a t i c a l M o d e l of N o n c i r c u l a r G e a r s MING-FENG TSAY AND ZHANG-HUA FONG D e p a r t m e n t of Mechanical Engineering National C h u n g - C h e n g University 160 San-Hsing, Ming-Hsiung, Chia-Yi, 621, Taiwan, R.O.C.
(Received April 2003; revised and accepted August 2003)
A b s t r a c t - - T h e pitch curve of noncircular gears is usually determined by kinematical requirements such as position, velocity, and acceleration. However, it is usually difficult to obtain a tooth profile that complies with kinematical requirements. A Fourier series approximation method is proposed to approximate the pitch curve. Based on the approximated pitch curve, a mathematical model of the tooth profile of noncircular gears is derived. The proposed mathematical model is very flexible with sufficient accuracy and can also be used to process the reverse engineering problem of the existing noncircular gears. (~) 2005 Elsevier Ltd. All rights reserved.
Keywords--Noncircular
gears, Mathematical model, Fourier series.
NOMENCLATURE ASp
a~ ) b(j) Bp
arc length NI measured from angle zero to angle Cp as shown in Figure 2 the coefficients of Fourier cosine series, n = 0, 1,2 .... n the coefficients of Fourier sine series, n = 0, I, 2 . . . , n
n np
tooth number unit normal vector at the point on the pitch curve in coordinate system Sp
Op
gear rotation center in Sp
rp
centrode of gear in Sp
l~i)
position vector for curves i in Sp
tangential shift parameter of rack cutter
Sp
coordinate systems p
C
distance between gear rotation centers O1 and 02
Pc Tp
circular pitch
F (j)
the Fourier series
Gp
position vector of the pitch curves in coordinate system Sp
tp u(~)
unit tangent vector of Tp
hk
addendum
ra
module of gear
mlj
ratio of angular velocities wj to wz
V
tangent vector at the point on the pitch curve in Sp curve variable for curves i of the rack the relative velocity pressure angle of gear
The author would like to thank the National Science Council of R.O.C. for financially supporting this work under Contract No. NSC 86-2212-E-194-007. 0895-7177/05/$ - see front matter © 2005 Elsevier Ltd. All rights reserved. doi: 10.1016/j.mcm.2003.08.010
Typeset by J 4 j ~ - ~
M.-F. TSAYAND Z.-H. FONG
556
8(i) Cp wp q, p
curve variable of the root fillet of the rack rotation angles in Sp angular velocity in Sp angle between the tangent vector and the Xp direction
superscript i superscript j subscript p
radius of curvature of the pitch
curve i of rack 1 Case (a) 2 Case (b) 1 Gear 1 2 Gear 2 c rack f frame
calve
1. I N T R O D U C T I O N The kinematical behavior of noncircular gears is similar to the cam-follower mechanism except that the noncircular gear set is a positive transmission. Noncircular gear set can be used to transmit varied gear ratio with good accuracy, higher power density, ease for dynamic balance, and compact in size. However, the mathematical model for tooth profile of noncircular gears is rather difficult to derive. The most typical examples of application of noncircular gears are as follows, (i) as the deriving mechanism for a linkage to modify the displacement function of the velocity function, and (ii) for the generation of a prescribed function. The applications of noncircular gear include the Geneva mechanism with modified angular velocity, the Bopp and Reuter meters for the measurement of the discharge of liquid, and the automatic planting machine for agriculture. Litvin [1,2] proposed a methodology by extending from tooth evolute to generate tooth profile of noncircular gears. Kuczewski [3] used spur gears to locally mimic ellipse gears. Chang [4,5] developed a mathematical model for modified elliptical gears. Most mathematical models proposed are limited to certain type of pitch curve [6,7]. In this paper, a methodology is proposed to establish tooth profile of noncircular gears. The pitch curve or motion relationship between gears (a motion program) is approximated by applying Fourier series technique. The numerical accuracy of the pitch curve can be controlled by effective terms ofthe Fourier series in the cost of computation time. Based on the equation of approximated pitch curve, generalized equations of tooth profile and design limits of noncircular gears are derived. Computer-aided design software for noncircular gears is developed based on the proposed methodology. Similar methodology can be used to process the reverse engineering problem of the existing noncircular gears. The proposed mathematical model is very flexible and with sufficient accuracy as shown by the presented numerical examples.
2. A P P R O X I M A T E THE CENTRODE OF NONCIRCULAR GEAR BY FOURIER SERIES In order to improve the flexibility in the design of pitch curve of noncircular gear, the Fourier series is applied to approximate either the pitch curve or the instantaneous gear ratio program. Applied coordinate systems are shown in Figure 1. Coordinate systems Sl(xl,yl, zl) and $2(x2, Y2, z2) are rigidly attached to Gear 1 and Gear 2, respectively, while the coordinate system Sf(xf,yf, zf) is rigidly attached to the frame. There are two cases; one is for the known centrode curve and the other is for the known instantaneous gear ratio program as follows. (a) If the centrode of noncircular Gear 1 is known, the centrode of Gear 1 can be represented by the Fourier series F(1)(¢1) as
r~ (¢1) = F ~ (¢1).
(1)
The centrode of Gear 2, which is conjugated with the centrode of Gear 1, is represented in terms of ¢1 as
r2 (¢5 (¢1))
=
c
-
~
(¢1).
(2)
Mathematical Model of Noncircular Gears
557
I
t ~b2
t k
_
_
_
l
r,(~,)
'
C i
i
o, Figure 1. The coordinate system applied to the noncircular gear set.
The instantaneous ratio of angular velocities can be represented as wl d¢1 r2 m12 - w2 - d¢2 = rl"
(3)
The angular relationship between rotation angles ¢1 and ¢2 are shown as the following equation, ¢2 = f0 ¢1 d¢1 ,
(4)
m12
where C is the distance between gear rotation centers O1 and 02, and 42 is the rotation angle of Gear 2. (b) If the ratio of angular velocities m21(¢1) is known, gear ratio can be represented by the Fourier series F (2) (¢1) as m21 ~-
1
_ F(2)(¢1),
(5)
m12
and the pitch curve of Gears 1 and 2 as C
~1 = - - - - Z '
(8)
m12 + m12
~2 = ~ C .
(7)
m12 + 1
There are two approaches to obtain the coefficients of Fourier series as follows. (1) Assume the exact function F(J)(¢I) is known, the orthogonal trigonometric system is applied to obtain the coefficients as follows [8], OO
F (j) ( ¢ 1 ) = a ( J ) + ~
[a(nj) cos ( n ¢ 1 ) + b(nj) sin (n¢1)]
j=
1,2,
n=l
a~j) = ~1 fo 2'~ F(J) (¢1) d¢1,
(S)
a(nj) _-- 1¢r --/2, F (j) (41) cos (n¢l) d¢1, J0
b~ ) -= _1 f2~r F(j)(¢1) sin (n¢1) d¢l, 7r J0
n -- 1,2,3,4,...,
where a 0(j) , a (j), 1 . .. , a~ ) , b~j), ... , b~ ) E R are the coefficients of Fourier series.
558
M.-F. TSAY AND Z.-H. FONG
(2) If the coordinate of tack points ((at, F(J)(al)), n = 1, 2, 3, 4 , . . . , ) are known, where w is the number of tack points, numerical method [9] is applied to obtain the coefficients as follows, h
F (j) ( a t ) = a(J) + E
[a~) c°s (nat) + b(nJ)sin (nal)] '
j = 1, 2, k = 2h + 1,
[Q]~xl = [M]~xk [D]kxl,
(9) (10)
where h is the number of effective terms of Fourier series, k is the total numbers of the unknown coefficients, [M] is the matrix of orthogonal trigonometric functions cos (nat) and sin (naz), [Q] is the matrix of numerical values FJ (a~), and [D] is the matrix of Fourier series coefficients a 00) , a (J) 1 , . . . ~ u n _(J)~ b~), . . . ~ n ~(J) • [Q]T = [F j (al)
[DiT=[4 Ii
F j (a2)
a(
"-"
FJ (a~)].
(ii)
at b2 . .
(12)
c°sal sinai cos a2 sin a2
c°s2al sin2al cos 2a 2 sin 2a2
"
"
cos aw sin aw
cos 2aw sin 2aw
• .. • ..
cos kal sin kal cos ka2 sin ka2
•
[M] =
o
(13)
• ..
coska,~ sinka~
If w = k, then coefficient matrix can be obtained by [D] = [M] -1 [Q].
(14)
If w > k, the least square method shall be used to obtain the approximated results,
[D1 = ([M] T [M]) -1 [M] T [Q].
(15)
3. B A S I C G E O M E T R I C A L C O N S I D E R A T I O N S OF T H E N O N C I R C U L A R P I T C H C U R V E The position vector Gp(¢p) of the pitch curve rp is represented in the Cartesian coordinate system Sp as shown in Figure 2.
Y
tp
" Xp
Figure 2. Geometrical properties of the noncirculax pitch curve•
Mathematical Model of Noncircular Gears
559
GP(¢P):{xP}={rpc°sCp} " y prpsinCp
(16)
The tangent vector Wp at the point I on the pitch curve can be obtained by differentiation,
dxp } Tp =
Tpy
(17)
dyp "
=
The unit tangent vector tp can be derived as follows,
(18)
(19)
n P = { - t PtYp }= As shown in Figure 2, the unit tangent vector at point I can be simplified as
sin ~/
tpu
(20)
'
where 7 can be obtained from equation (20) as follows, 7 = tan-1
(tp~,tpy).
(21)
The arc length A¢, measured from point N to point I on the pitch curve can be calculated by the following equation,
=
+
\ dep ]
de,.
(22)
The circumference arc length A2~r should equal to integer multiples to the pitch of rack cutter so as to keep all tooth profile complete, therefore,
A2~r= ~rrnn,
(23)
where m is the module of gear, and n is the tooth number. 4. M A T H E M A T I C A L
MODEL
OF
THE
RACK
CUTTER
Assume the noncircular gear is generated by a rack cutter. There are some limitations on the concave portion of noncircular gear by using the rack cutter. However, the working part of tooth profile (the involute part) is virtually the same whether the gear is generated by a rack cutter or a shaper. A standard rack cutter composes five curves as shown in Figure 3. The parameters of the rack cutter include pressure angle ~, tip radius re, circular pitch Pc, and addendum hk. As shown in Figure 3, Curves 1 and 5 of the rack cutter are straight lines that generate the involute working part on tooth surface. Curves 2 and 4 are circular arcs that generate the root fillet of tooth surface. Curve 3 is straight line that generates the bottom land of noncircular gear. Tangential
560
M.-F. TSAYAND Z.-H. FONG
l~
Xc
II ~,1I
Bp
r
.....
.......
i ........
(2~ -
13)
7----
.....
Figure 3. The coordinate system applied to the rack cutter.
shift parameter Bp is the distance between the rack center and the origin Oc of coordinate system Sc.
Bp=
0,
if p = 1,
0.5Pc,
if p = 2 ,
(24)
where subscript p indicate which gear is being generated, p = 1 for Gear 1 and p = 2 for Gear 2. Tangential shift parameter is applied to synchronize the initial generating point of Gear 1 and Gear 2.
Bp
4.1. T h e M a t h e m a t i c a l
Equations for Curves 1 and 5 of the Rack Cutter
Curves 1 and 5 are used to generate opposite sides of the working tooth profiles of noncircular gears. Parameter u (i) is the curve variable for Curves 1 and 5. The position vector for Curves 1 and 5 of rack cutter can be obtained by the following equations, =
=
Pc+u
- h k sec ~0 _< u (i) < hk sec ~
(1) s i n ~
'
(i = 1, 5),
and the corresponding unit normal vectors are
{~(1)}
n(1)
{ sin~
,oxc
:
~(1)
=
-- C O S ~0
} '
(26) = 4.2. T h e M a t h e m a t i c a l
(5) 1~yc
=
-- COS~
"
:Equations for Curves 2 and 4 of the Rack Cutter
Curves 2 and 4 are used to generate opposite sides of root fillets of generated t o o t h profile.
Parameter 0 (i) is the curve variable of the root fillet. The position vectors and unit vectors for Curves 2 and 4 of rack cutter can be obtained by the following equations,
yc(2)
R(4) -=--
y(4)
-Bp + ~
=-
- hk tan ~0 -- rc cos ~ + rc sin 0 (2)
'
Pc + hk tan ~o + re cos ~ -- rc sin 0 (4) - B p -- -~-
'
Mathematical Model of Noncircular Gears 0 _> 0 (i) _> ~~ - ~
561
(i = 2, 4 ) ,
(27)
and the corresponding unit normal vectors are
n (2)
n(2) =
-- cos 0 (2)
,(2) po y c
=
sin 8 (2)
'
(28) n~4)=
n(y~)
(sinO(4)
"
4.3. T h e M a t h e m a t i c a l E q u a t i o n for C u r v e 3 o f t h e R a c k
Cutter
Curve 3 is used to generate the bottom land of the generated tooth profile of noncircular gears. Parameter u (3) is the curve variable of the Curve 3. The position vector and unit normal vector for Curve 3 of the rack cutter surface can be obtained by
n(3) yc
=
.
5. T H E M A T H E M A T I C M O D E L O F GENERATED NONCIRCULAR GEARS Assume the noncircular gear is generated by a rack cutter. As shown in Figure 4, the pitch line of the rack cutter performs pure rolling on the pitch curve of noncircular gears during the generating process. The coordinate system Sc(X,, Yc), S f ( X f , Y f ) , and Sp(Xp, Yp) are rigidly attached to the rack cutter, frame, and the generated gear, respectively. The generated gear rotates about a fixed center Op while the rack cutter translates without rotation in the plane. The relative velocity is zero at the instantaneous center I and the parameter Acp is the translational distance from the rack cutter origin point O~ to instantaneous center of rotation I. The coordinate transformation matrix from coordinate system S¢ to Sp is shown as follows,
[Mpc]=
sin 7
cos 7
rp cos Cp + Acp COS7 ]
-cos7
sin7
-rpsin¢p+A¢
0
0
sin7
.
(31)
1
Therefore, the locus of the rack cutter represented in coordinate system of the generated gear is R(~ ) = [Mpc] R (1)
(i = 1 , 2 , . . . , 5 ) .
(32)
According to the theory of gearing, the unit normal at the contact point must pass through the instantaneous center of rotation I. Equation of meshing may be represented in coordinate
system S~ as follows,
n(O
,(i)
(i -- 1 , 2 , . . . , 5 ) ,
(33)
where (X (i) , y(O) ___(0, -Acp) is the position vector of the instantaneous center I while (x (0, y(i)) and (n ~c,,oy¢ (i) ~(i)~j are the position vector and unit normal vector of the contact point, respectively.
562
M.-F. TSAY AND Z.-H. FONG
Figure 4. The coordinate systems applied to the gear generation procedure.
The equation for the tooth profile of generated gear can be obtained by solving equations (32) and (33), simultaneously. The simplified equations for complete tooth profile of noncircular gears are listed as follows. According to equations (32), (33), (25), and (26), the working tooth profiles of generated noncircular gear can be represented by x (~) = u (1) cos~vsin~ +
( -Bp + -~Pc+u(1)sin~)c°sT+rpC°SCp+A~ c°sT'
( Pc y(1) =_u(1) cos~ocosT÷ - B p + ~ - +
A¢~ = ( Bp Pc4
u(1)) sin~0 s i n ~ - r v s i n C v + A ~ sinT,
u(Ocsc ~o),
(34)
x(pS)=u(S)cos~osinT+(-Bp-(-~+u(5)sin~o))cosT+rpcosCp+A¢ cosT, y(pS)=u(5)cos~cosv+(-Bp-(-~+u(5)sin~v))sinv-rvsinCp+A, sinv,
-hksec~o < u (i) < hk sec~
(i ----1,5).
Mathematical Model of Noncircular Gears
563
According to equations (32), (33), (27), and (28), the root fillet of generated tooth profile can be represented by X(p2) = ( - h k + r c s i n ~ - rccos0 (2)) sin 7
+ (-Bp + (-~ - hk tan~- rccos~) + rcsinO(2)) cosT + rpCOS¢, + A¢, cosT, y(2)
=--(-hk
+ rc sin~ - rceos0(2)) cos~'
+ (-Bp + ( ~ - hk tan ~ - re eos ~) + rc sin O(2)) sin ~ - rp sin Cp + Acp sin T, Acp = Bp - ( ~ - hk tan~- rcCOS~+ (-hk + r~sin~)cot O(2)) ,
(36)
X(p4) = (--hk + r e s i n s - - r ~ c o s 0 (4)) sin7
-Bp-(~-hktan~-recos~-resinO(4))cos~+rpCOSCp+A¢pcos% yp(4) = _
(--hk +
+ (-B,-
rc s i n ~ - rc cos0 (4)) cosq'
(-~ - hk tan~- r~cos~) - rcsinO(4)) sinT- rpsin¢, + A¢,sin%
A¢ = B, + (-~ - hk tan~- rccos~) + (-hk + r~sin~)cot O(4), 0 _< 0 (~) _< ~~r-
~
(37)
(i = 2, 4 ) .
According to equations (32), (33), (29), and (30), the bottom land of generated tooth profile of noncircular gears can be represented by X(p3) =
(-hk + r~ sin ~
- re) sin 3, + (-Bp
yp(3) = ( h k - r~ sin ~ + r~) cos~/+
A¢ = Bp - u (s),
(-Bp u(3)
+ u (3)) cos ~/+ rp cos Cp +
ACp cos %
+ u (3)) sin~/- rp sin Cp + A¢~ sin%
<_ -~-hktan~--rcCOS~].
(38)
6. E Q U A T I O N OF U N D E R C U T T I N G A N D D I S C U S S I O N ON THE D E S I G N LIMITS According to the theory of gearing [1], undercutting occur when singular point exists on generated tooth profile. The mathematical definition of singularity of the tooth profile is represented by the following equation, V(2) --- V(1) Jr- V (12) ~-- 0. (39) Equation (39) implies the following two determinants are equal to zero,
of
of&,
Ou
0~/ dt
=
of
Ofd~
Ou
07 dt
= o.
(40)
Suppose that the undercutting points on the generated tooth profile of noncircular gears are generated by the working Curve 1 of the rack cutter. The relative velocity between the gear blank and the rack cutter, represented in coordinate system S¢, is shown as follows, V(12)
= wp (-Bp + -~Pe+u(1)sin~+ A¢,)ic+W,(-uO)cos~)jc.
(41)
564
M.-F. TSAY AND Z.-H. FONG
Equation of meshing f , derived from equation (33), corresponding to Curve 1 is simplified as Pc
f = u + ~- sin q0 - Bp sin qo + Acp sin ~.
(42)
The angular velocity of noncircular gears is assumed to be Wp = dd-~t . Substitute equations (41) and (42) into equation (40), the condition of undercutting is simplified as OA4~ 2 u cos 9~ + - ~ sin ~ = 0.
(43)
Parameters u = ~=hk sec ~0 at the lowest/highest end points on Curve 1 of the rack cutter. Assume the addendum parameter hk = 1.0 m. Equation (43) implies the limiting values of the gear module m as '/nma x ~
OAcp sin2 ~ = [ppsin 2 ~[
(44)
~
The minimum number of teeth thus is derived from the pitch curve circumference A2~ = ~rmn. n m i n _~ ~
S•Tr
nrain E N.
,
(45)
71"?TI,m a x
This result is substantially the same as that obtained by Wu [10] and Chang [5] with Pv(¢p) is the radius of curvature of the pitch curve [1].
PP = - dnp~
--
dnpy
--
II o
r~ + 2r~ - rprp
(46)
The maximum gear module of the rack cutter is limited by the minimum curvature radius [Pp[min of pitch curve. The minimum or maximum pitch radius (]Pp[min or [Pp]max) can be located when equation (47) is observed. dpp = O.
d¢v 7. E X A M P L E S
(47)
AND DISCUSSIONS
7.1. E x a m p l e 1 The basic data of this example is the same as provided by Chang [5]. As shown in Chang's paper [6], an elliptical gear with module m = 5.0 mm, number of teeth n -= 45, pressure angle ~c = 20 °, radius of circular arc rc = 0.3m, and eccentricity e -= 0.608, major semi-axis a -- 125 mm, and the elliptical gear pitch curve, rl = (al (1 - e12))/(1 + el cos ¢1) is explicitly defined. Fourier series is used to simulate the elliptical gear's pitch curve. Since the function of elliptical pitch curve is an even function, the function can be simplified as r l = ~n=o[a~ cos(n¢l)]. Fourteen items of above-mentioned series are used to approximate the pitch curve. The coefficients a3o,a~ . . . . , a~3 are calculated by equation (9) and the calculated values are listed in Table 1. The maximum deviation of the pitch curves between elliptical gear and approximated gear is less than 10 -a mm at ¢1 = zr as shown in Figure 5. The tooth profile based on approximated pitch curve is shown in Figure 6. The normal deviation of tooth surface between approximated gear and exact gear obtained by Chang's method is below 2 x 10 -5 mm as shown in Figure 7 in three typical teeth located at the major axis, minor axis, and ¢1 = 0.25r. The numerical results suggest that
Mathematical Model of Noncircular Gears
565
T a b l e 1. T h e Fourier series coefficients of E x a m p l e 1.
Coefficmnt a0 al a2
Data Coefficient 99.2608483 a7 -67.2555863 as 22.7849850 a9
Data -0.1016835 0.0344486 --0.0116706
a3
--7.7191439
al0
0.0039538
a4 a5
2.6115107 --0.8859514
all a12
--0.0013395 0.0004538
a6
0.3001444
a13
0.0001576
0.000075 0.00005 0.000025
,.AA.AA,
-0.000025 -0.00005 -0.000075
Ra~us Devia~on(mm)
Figure 5. The radius deviation of approximatedpRch curve of Example 1.
rl
" X1
Figure 6. The elliptical gear generated based on approximated pitch curve. approximated pitch curve and corresponding tooth profiles are fairly close to the theoretical ones and the deviations are controllable. The undercutting of this example is checked by equations (43)-(46). Figure 8 shows the radius of elliptical pitch curve and the corresponding curvature approximated by Fourier series. The minimum curvature [Pt Imin occurred at both end points of major axis of the ellipse and [Pl Imin = 78.8218 ram. The limiting value mmax of the gear module calculated by equation (44) is 9.22 ram, which is identical to the result provided by Chang [5]. Since the addendum of the rack cutter is just 5.0 mm, which is smaller than the limiting value of 9.22 mm. Therefore, undercutting will not occur on the tooth profile of the generated gear. 7.2. E x a m p l e 2
Assume a pitch curve (Gear 1) of cam with seven tack points are given as listed in Table 2. Based on the given pitch curve of Gear 1, we designed two conjugated gears with total gear ratios ~ equal to 1 and 2. Gear 1 rotates ~ revolutions per revolution of Gear 2. Assume that pressure angle ~ = 20 °, tip radius rc = 0.25m. The center distance C of the gear pair is obtained by equation (4).
566
M.-F. TSAY AND Z.-H. FONG mr~
0 . 00002
/
0.00001
-
-
!
the region 1
?/lm
-2
o. o-~oor----"--
workingsurfacen (ram) the region 5
--0 . 00002
(a). The normal deviation of the tooth surface at the major semi-axis. 0.00002
i'TII
the region I 0.00001
2
\
-0.00001
4
workingsurfaceu (ram)
theregion5
-0.00002 (b). The normal deviation of the tooth surface at the minor semi-axis.
0.00002
ml~
theregion1
F
t
\ the region 5
working surfaceu (ram)
(c). The normal deviation of the tooth surface at ¢1 = 45 °. Figure 7. The normal deviation on tooth surface generated by Curves 1 and 5 of rack cutter of Example 1.
(mm)
14o1~°2°°18° ~curve
r (¢0
lOO 8o 2
3
4
5 ~
el(Degree)
Figure 8. The radius of approximated elliptical pitch curve and curvature of Example 1.
Mathematical Model of Noncircular Gears
567
Table 2. The tack points of cam. Radian
Radius (ram)
Radian
Radius (mm)
0 0.9425 1.8401 2.7378
4o 48 52 57
3.6353 4.5329
54 47
5.43O5
43
20(
15(
IOC
5C rain 1
2
3
4
5
6
Figure 9. Curvatures of noncircular gears of Example 2.
Y,
- r , , t v v
"
"
'
'
V
:
|
,
Figure 10. Noneireular gears of Example 2.
Apply Fourier series to approximate the pitch curve of Gear 1 and use the approximated pitch curve r l to generate the pitch curve r2 of Gear 2. The curvature radius pp calculated by equation (46) is shown in Figure 9. [P2[min is 33.7457mm on Gear 2 when total gear ratio v = 1. T h e m a x i m u m module m is 3.93116mm as calculated by equation (23). Thus, the minimum teeth number of gears 1 and two corresponding conjugate gears (v = 1 and 2) are 25, 25, and 50, respectively. The center distances C = 98.7228mm when v -= 1, while C = 147.158mm when v -- 2. Figure 10 shows the approximated Gear 1 and the conjugated noncircular gears with total gear ratios v equal to 1 or 2.
7.3. E x a m p l e 3 Design a gear set with center distance C -- 1 5 0 m m and Gear 2 rotates at a predetermined angular velocity ratio, d¢2 d¢1 W2 ~-- dt
=m21
dt '
and a predetermined angular acceleration,
dm21 d¢1 ~2---
d2¢1 d--~ d~ ÷ m 2 1 dt 2 ,
568
M.-F. TsAY AND Z.-H. FONG
1.5 1.25 1 0.75
(2.) 1
0.25 f 0 -0.25
1
2 ~ 4
5
~/
Figure 11. The modified trapezoidal function.
\ Figure 12. The noncircular gear set of Example 3.
with the modified trapezoidal function [11] as shown in Figure 11. Apply the Fourier series to simulate the modified trapezoidal function and calculate the instantaneous ratio of angular velocities m21 by equation (5). The design limits can be obtained by equations (44)-(46). Figure 12 shows the tooth profiles of the noncircular gear pair with n = 40 and module m -- 3.7135 mm. 7.4. E x a m p l e 4 This example is aimed to check whether the proposed m a t h e m a t i c a l model can be applied to simulate the concave pitch curve or not. Figure 13 shows the concave noncircular gear pair with pressure angle ~ -- 20 °, the module m = 0.9489mm, the teeth number of gears n = 100, the radius of circular arc rc = 0.15 m, and the center distance C = 90.921 m m . The numerical results show t h a t m a t h e m a t i c a l model of working part of the tooth profile is acceptable while tooth profile in the root fillet m a y have some troubles.
Figure 13. The noncircular gear pair of Example 4.
Mathematical Model of Noncircular Gears
569
8. C O N C L U S I O N S Fourier series and numerical techniques are applied to approximate the pitch curves or instantaneous gear ratio. Based on the approximated pitch curves, the mathematical model of the tooth profiles of noncircular gear is proposed. Equations for the undercutting are used to determine the design limitations of noncircular gears. According to the presented numerical results, following conclusions are drawn. 1. The proposed pitch curve approximation method is effective and can be used to design the tooth profile of noncircular gear with sufficient accuracy. 2. There are two possible ways to design noncircular gear set with discrete data. Either pitch curve or instantaneous gear ratio can be approximated by the discrete data set. According to the approximated pitch curve or instantaneous gear ratio, the tooth profiles of the noncircular gear can be easily derived. The proposed mathematical model is very flexible. 3. Undercutting equation is very helpful to determine the design parameters. 4. It is possible to design noncircular gear without exact equations for the tooth profile. This is very important to do the reverse engineering for existing noncircular gear set.
REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.
F.L. Litivin, Theory of Gearing, NASA Publication, Washington, DC, (1989). F.L. Litivin, Gear Geometry and Applied Theory, Prentice-Hall, New Jersey, (1994). M. Kuczwski, Designing elliptical gears, Machine Design, 166-168, (1988). S.-L. Chang, Designing Involute Elliptical Gears, Master's Thesis, National Tsing Hua University, Taiwan, (1992). S.-L. Chang and C.-B. Tsay, Mathematical model and undercutting analysis of elliptical gears generated by rack cutters, Mech. Ma~h. Theory 31 (7), 879-890, (1996). T. Bernett, Elliptical gears for irregular motion, Mechanical Engineering 89 (6), 33-39, (1967). S.H. Tong and C.H. Yang, Generation of identical noncircular pitch curves, ASME, Journal of Mechanical Design 120 (2), 337-341, (1998). E. Kreyszig, Advanced Engineering Mathematics, Seventh Edition, John Wiley and Sons, Inc., (1993). B. Faires, Numerical Analysis, Sixth Edition, Brooks/Cole Publishing Company, (1997). X. Wu, S. Wang, G. Wang and A. Yang, Theory Mach. Prague, In Proe. 8~h Wld Congr., pp. 391-394, Czechoslovakia, (1991). J. John, Design of Machinery, McGraw-Hill International Editions, (1992).