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Stochastic error analysis in cam mechanisms
Mechanismand Machine Theory Vol. 13, pp. 631-641 © Pergamon Press Ltd., 1978. Printed in Great Bdtain
0094-114XI78/1201-06311502.0010
Stochastic Error Analysis in Cam Mechanisms H. R. Kimt and W. R. Newcombei" Abstract A probabilistic method is outlined for obtaining the variances in displacement, velocity and acceleration of followers in cam mechanisms for any given manufacturing tolerance on the cam profile. A follower motion equation is obtained by expansion in a Taylor series and considered in both linear and second-order terms. The equation is analyzed with random variables for the kinematic displacement, velocity and acceleration of the follower employing stochiastic techniques including finite differences and the maximum likelihood method. A general computer program allows the designer to obtain projected output errors for any commonly used disc cam type, and an example of one of these is included. Introduction STOCHASTICmethods and probabilistic techniques are being increasingly more widely used in the analysis and design of machines. In this investigation a probabilistic approach was employed to analyze the effect of manufacturing errors in cam profiles on the output or follower displacement, velocity and acceleration. Dhande and Chakraborty[l] have provided the necessary background, and their stochastic model is used. They employed a stochastic approach to analyze the three-sigma band of error in the follower displacement showing maximum and minimum errors, whereas, the method has been extended here to enable an analysis of the error in velocity and acceleration due to any given manufacturing tolerance on the cam profile. A general computer programme was written for this purpose. Several other investigations have been carried out on the error analysis of cam mechanisms. Rothbart [2] used a finite difference method to obtain the effects of surface roughness on follower acceleration and a relationship was derived for the allowable increment in cutter position for a maximum surface roughness error. Turtle [3] used vectorial addition to analyze errors in cam-operated devices and four-bar linkages. This method gave the approximation only of the error in a non-continuous system, and is not applicable to error analysis in the cam-follower mechanism itseff. Brittain and Horsnell[4] investigated the effects of grinding wheel wear and of machine setting errors on the surface of a production can specified for a required valve motion. Mischke[5] analyzed the effects of mechanical error in an eccentriccircular cam with a flat-faced translating follower without considering surface roughness. Filkin [6] as well as Sergeyev [7] investigated the effects of manufacturing and assembly errors on camoid mechanisms. The co-ordinates of discrete points on the camoid surface were considered instead of the surface geometry with approximate results. Mikhailov [8] investigated the relative effectiveness of various adjustment methods to reduce mechanical error in the output and Rumiantser[9] discussed the effects of assembly errors. Kawasaki[10] et al. presented methods for computing the cam profile lift error due to a change in grinding wheel diameter and Tesar[l l] has included a section on errors in cam mechanisms in his recent book. In the kinematic analysis and design of cam mechanisms a knowledge of the effect of manufacturing errors on the velocity and especially on the acceleration characteristics is often more important than the effect on displacement and reports on the investigation of this problem tDepartment of Mechanical Engineering, McMaster University, Hamilton, Ontario L8.S 4L7, Canada. 631
632
have not been seen. The variance of follower displacement due to mechanical and manufacturing errors can be obtained stochastically, but the variances of its first and second derivatives, i.e. velocity and acceleration are not easily obtained directly or indirectly. This paper will present a method of calculating the approximate value of these variances using a stochastic approach and the maximum likelihood estimate.
Cam-Follower System Model and Analysis Figure 1 shows the fundamental system used to model the cam-follower mechanism. St and $2 indicate the cam and follower surfaces fixed to the moving coordinates (x,.i, y,,i, ZmO and (xm2, Y,,2, z,,2) which move relatively to the fixed coordinates (xi~, Yit, zt0 and (xiz, Yt2, zt2). Two vectors /~l and /~2 are radius vectors of the contact point P between cam and follower expressed in the moving coordinate systems. YfZ
zu
$2
~
St '
"'~m,
z.
Figure 1. Fundamental coordinate system for cam follower mechanism. Let tt and t2 be the parameters of motion of cam and follower links and 0 and $ the parameters of the follower surface $2. Then, by vector and matrix notation, we have RI = [xrnl, Y,.l, Zmll r
/~2 = [Xm2,Y,.2, Z,.21T= / (0,¢)
(i)
6 = g (0,¢)
where ~2 is a unit normal vector to the surface $2 at contact point P. To satisfy the condition of contact, the dot product of the unit normal vector, ~ at contact point P and relative velocity vectors V'12at point P between the surface S~ and $2 must be equal to zero, i.e. ~7. I712= O.
(2)
Assuming tt is the input parameter, we can find the expression for output parameter t2 solving the above given eqns (1) and (2) for three unknown parameters 0, $ and t2 as t2 = h(x~ bYm l,Zm IXm2,ym2,Z~2,tO
(3)
in which the relationship between a coordinate fixed to the cam and the one fixed to follower may be given by introducing the transformation matrix. Equation (3) can be generalized denoting the variables as input random variables like t2 = h(rl,r2,r3 . . . . ri, tl).
(4)
Since it is extremely difficult to find the probability density function and distribution function of output variables t2 from eqn (4) theoretically, the mean and variance of t2 are obtained from those of the random variables, ri's by expanding eqn (4) in Taylor series about the mean value of ri's and taking the first two terms of the expanded equation. The former is called the moment generating function method complying with the theory of probability while the latter the partial derivative method for approximation.
633
They can be written as n
t~ = F + ~, Gi(ri - ~,)
(5)
i=1
where F = h(#l,p,z,#3 . . . . . P,n, tl) GI
=
(Ot2~ \7~i / ,,~
evaluated at the mean values of ri's; #i = mean values of r~'s. The mean value and the variance of output t2 are /~t2 = h(/~1,/~2,/.~3 . . . . /J,n, tl) [Oft2] = ~
Gi2[orri] 2 •
(6)
i=1
The mean value and variance of the follower displacement have been obtained from the stochastic model. It is now required to calculate the variances of velocity and acceleration of the follower to analyse the cam mechanism completely and generally. Thus is limited by the characteristics that the velocity and acceleration are the first and second derivatives of displacement respectively; the direct differentiation of displacement with respect to time for velocity makes no sense as the first derivative terms of both sides of the displacement equation eliminate each other. This is because the displacement equation was established by solving the parametric equations simultaneously which include displacement terms occurring from the known displacement relation like the basic motion curve. The same problem occurs in obtaining second derivatives from first derivatives even though the latter can be found. Therefore, a different condition or another method must be introduced. The Monte Carlo method, which is a powerful method for stochastic problems, may be employed by choosing random samples of displacement within the displacement error bound and using the finite differences method. However, several difficulties occur in determining sample size and incremental cam angle in the finite differences considering the contact condition between cam and follower. Moreover, the computer must be used with the Monte Carlo method, and the costs are high since an iteration should be done at every increment. The general equations of motion of a point in space using rotating and translating coordinate systems could be applied to this problem, but the same cancellation of derivatives as in the direct differentiation of displacement occurs for velocity and acceleration. On the other hand, the introduction of other conditions such as relative velocity or acceleration and radius of curvature of cam profile were considered, but this also failed to give the variance of velocity and acceleration as these conditions are not kinematically independent. The above reasons probably explain why the determination of velocity and acceleration errors has not been approached stochastically by previous researchers. To solve this problem finite differences were used to determine velocity and acceleration numerically. Then, the maximum likelihood technique of random variable analysis was introduced to obtain the desired stochastic results approximately.
Finite Dilferences The use of finite differences in cam-follower systems were introduced by R. C. Johnson[12] and developed by Rothbart[2] to find the effects of surface roughness on the follower acceleration. The follower characteristic curves can be obtained numerically by utilizing incremental tabulated values which may be estimated, taken from the actual cam shape measurements, or calculated from an analytical cam curve.
634
Let S = follower displacement (in.) V = follower velocity (in./sec.) A = follower acceleration (in./sec2.) 80 = incremental cam angle (rad.) 8t = time for cam to rotate through angle 80 (sec.) to = angular velocity of cam (rad./sec.) and assume that the cam displacement curve has three given points 1, 2 and 3 with small and equal time interval St, as shown in Fig. 2.
1 g / ~ i r n e ---.-
~St
Figure 2. Part of displacement curve and application of finite differences. The follower velocity at point 2 will be
V2=(-d-i):=limS3-- Sl=S~Tl'.,-~ dS
(7)
2~Jt
Knowing 8t = (801to) and substituting this into eqn (7), we get V2 ~
tO ~-~(S
3-
S 0.
(8)
For the acceleration of point 2, choose points 4 and 5 so as to be midpoints respectively between 1 and 2, and 2 and 3 as shown. Then, the follower acceleration will be d
dS
I14
(9)
8t Since V 5 = 53 - 52
8t
V 4 = 5 2 - SI a n d
'
8t
6t
= 80
a,
the acceleration is A2= (~0)2 (S1- 2S2 + $3).
(10)
This technique has shortcomings when applied to cams. It can be seen from (8) and (10) that the velocity and acceleration at dwell and intersecting points in the curves cannot be found, and as the increments of cam angle are taken larger and the jerk increases, the errors increase. The latter shortcomings, however, were somewhat reduced by using the computer to obtain a rough optimum value of the incremental angle.
The Maximum Ukelihood MethoO Three methods can be used in stochastic problems to compute moments of functional combinations of random variables; statistically, the moment of a function of random variables uniquely defines the distribution of the function, since a distribution is uniquely defined by its
635
moments if they are finite. These are: (i) Moment generating function, (ii) Partial derivative methods, (iii) Maximum likelihood methods. When the functions of density and distribution of random variable are known, the moment generating function is often employed. However, although this is a powerful method in statistics it has disadvantages when complicated and rigorous integrals that are hardly accessible must be handled since its principle follows the exact probabilistic theory. The second method, the partial derivative method which has already been used to obtain the variance of the displacement, has shortcomings when treating partial derivatives within Taylor series. In comparison, the maximum likelihood method is quite simple to apply to simple functions like eqns (8) and (10), and it can be employed to obtain the variances of velocity and acceleration from those of displacement. The maximum likelihood method[13] is based on a point estimate which is a single value that is used to estimate parameters such as mean values, variances and standard deviations. Assume normally distributed random variables X and Y as described by populations of values XI,X2 . . . . . . . . . . .
,X m
Yl,Y2 . . . . . . . . . . .
,Yn.
From these populations, the mean value estimator of te sum variate (~) can be calculated as ~=
1
m . f l [Xl + Yl) + (Xl + Y2) + . . . . + (Xl + Yn) + (X2 + Yl) + (X2 + Y2) + . . . . + (X2 + Yn)
+ . . . . . + (x., + yl) + (xm + y9 + . . . . . + (x. + yn)
~-±
(x,. + yj)
m.ti
(11)
Similarly, the standard deviation estimator of the sum variate is
D, = [ ~ . ~ l{e- (x, + y,)l ~ + { e - 0<, + sol ~ + . . . + { e - (x, + y.)}~ + {Z -- (X2 + yl)}2 + {Z - (X2 + y2)}2 + . . . . + {Z - (X2 + yn)}2
-F . . . . + {Z.(Xm + yl)} 2 + {Z. - (Xm + y2)}2 -F . . . . + {Z - (Xm "4-y2)}2]] 1/2
: [~ ~, (e- (x, + y,)~'l"2 =
~-_~
(t2)
j
m.n
Now, by arithmetic manipulation when X and Y are statistically independent these become, respectively ~ _. ( X l "["X 2 "l'.m_."" " + Xm ) .l. (.Y l "l- Y 2 + ~_ " " " "[" Y n )
(13)
=X+y Dz = {(xl - g)2 + (x2- g)2 + . . . . + (xm - ~)2 m + (Yl - y)2 + (Y2 - Y)2n+ . . . .
= (o,2 + o,2) "~.
+ (y" - y)2}t/2
(14)
The mean value and the standard deviation about the ditterences of two iiormally distributed statistically independent variates X and Y can be found likewise as
636
=__lm [(XI -- Yl) q- (Xl -- Y2)n+ . . . . + (X2 -- YO+ (x2 - Y2) + . . . .
+ (xi - Yn) + (x2 - Yn)
n
+ (Xm - Yl) + (Xm -- Y2)n+ . . . .
+ ....
+ (Xm - Yn)]
=i-y
(15)
and D~ =
{(xi - S) - (yj - •)}2] 1/2 •
i=1 "~
= (D~2 + Dy2)m
(16)
where D~ and Dy are, respectively, the standard deviation estimators of random variables X and Y. Considering the linear combination
W=aX+bY+cZ in which X, Y and Z are independent normal variates and a, b and c are the arbitrary constants; by a linear transformation of the normal variate, the mean of the linear combination becomes = aS+ by+ c~?
(17)
and the standard deviation will be D w = ( a 2 D x 2 + b2Dy 2 ''t- C2Dz2) I/2.
(18)
Now, noting that lim S =/zx and lira Dx = tr~ when the size of samples are large, eqns (13)--(18) m~
m---~
can be written as follows: For the sum tz~ =/~x + tzy
(13)'
cr~ = (Crx2 + ay2)1/2.
(14)'
For the differences tz~ = ~x -/zy
(15)'
orz = (o'x2 '1-o'y2)1/2.
(16)'
For a linear combination (17)' Orw _-- (a20rx 2 at- b20.y 2 -1- c20.z2)1/2"
(18)'
Now, with the mean value and standard deviation of displacement acquired from eqn(6), the variances of velocity and acceleration can be computed by combining eqns (8) and (9) and eqns (13)'-(18)'.
Example 1 Disc cam-translating roller follower-SHM The following specifications for a disc cam with an in-line translating roller follower have been taken from "CAMS", by Rothbart [2]:
637
Total Rise (T.R.) = 1.0 in. Radius of Base (R.B.) = 0.85 in. Circle Cam Angle for Rise (C.R.) = 150.0 deg. RPM = 180 Incremental Cam (I.A.) = 1.0 deg.
Angle Roller Radius (R.R.) Offset Distance (O.D.) Tolerance on Link (T.L.) Tolerance on Cam (T.P.) Profile Standard Motion Curve Maximum Pressure (@) Angle
= = = =
0.75 in. 0.0 in. -+0.0002 in. -+0.0001 in.
= SHM = 20 deg.
The computer programme developed for this project calculated the cam profile for exact contact conditions between cam and follower. The above design had been roughly minimized for a pressure angle of 20°. The exact value for the maximum pressure angle obtained from the programme was 16.4°. The results are shown in Figs. 3-5. Displacement, velocity and acceleration graphs are shown with their 3G bands, and the variance in each of these is plotted on the same graph to a different scale. The change in pressure angle is also plotted on each graph, and it is seen that the maximum values of the variances logically occur at maximum pressure angle for the in-line translating roller follower. TRANSLA]ING ROLLER FOLLOWER TOTAL R I S E RADIUS OF BASE CIRCLE CAM ANGLE FOR RISE RPH INCREKENTAL CAM ANGLE ROLLER RADIUS
= l,OO( TNCH ) ,85( INCH ) = 150,00( DEG, ) = 180,00 = 1,00( DEE, ) = ,75( INCH }
OFFSET DISTANCE O ,O0( I N C H } TOLERANCE OF L I N K PP~TS = TOLERANCE ON COMPONENT OF CAM P R O F I L E =
,0002( INCH } ,0001( INCH )
SIMPLE HARMONIC NOTION OR CRANK CURVE
1.000300
~
MEAN DISPLACEMENTp
i
( ~c
i
.?~o[15 • 1979BE-07 )
"
E--LJ ~J ZC: CJH
I
,/
. . . . .
36 8 A N D
-,,-,,-,,
VP~IANCEp
I
~''
-- ....
i
~''~',
I
PRESSURE ANGLE
i
90,0
17 s ~j o
~,~
../ ,500017 (,iS4S3£-07}
. . . . .
i
,%..
"
jw "'4.
./
E ¢1,o w
",.~
g .24ee6o
/
°'~
000iL% •o
1i,o
go,o
4i.o
$o.o
71,o
$o.o
1o6,o
1~o,o
lJllo
o ~lSO.o .o
CAM ANGLE CDEG, )
Rgure 3. Variance in displacement. Comparison tests were then carried out by varying cam specifications or characteristics one at a time. An arbitrary higher and a lower value of each characteristic was used while holding the other characteristics constant. The results of these tests are summarized in Table 1. The maximum values of the variances were selected from the computer print-outs, and these occurred at somewhat different cam-rotation angles than that of the base problem as indicated.
Example 2. Disk cam-oNset translating roller foUower-cycloidai The specifications for this cam mechanism were taken from Robert and Fenton[14]. This
-
-
HERN VELOCZTY~ I I
I
I/
.....
3[
-,,-,,-** .~
VRR~J:~NCEP ~.,~1
BANO
. . . . . I
PRESSURE RNGLE I I j
00.0
I.Yt0YIi ( .ll~41E-O;
.u H~
,/"
°.°00ZZ¢ (.°OSSSE-0Zl
""\
i,~
~Z
4 .... ~-:=
.. ~ .
~.$lti°? (.ili~7E-ot
.......
10. °
ll~0
to.0I
i°.oI
°o.oI
?°.0I
°0.0I
101.0I
it0 ! 0
110.0 0.o
lil,O
CRH RNGLE ( DEC. )
Figure 4. Variance in velocity translating NEPal ACCELER~TZON~
J.I].l * J.I J.II f
roller follower--SHM.
. . . . .
3ff B R N D
-,,-,,-,,
VRRII~ICE~
. . . . .
PRESSURE ;:iNGLE /
u~
( .~.l[lli*OI
Z HZ~ ~H
)
.' I..
•.
L~ LL O
]
/
zto011ttl
( .LttTIE*Oi I
45.0 ~j
•
-.,,.0,..0.~" " .~o
.rE
t
,,.o'
°0!o
,o.0' . . . . . ,,.0"T.... ,o.o~.... ~o,.0
,,oL.o
; .,.o ' ;-~,i o. 0.o
CAH ANGLE I DEG, )
Rgure 5. Variance in acceleration translating roller follower--SHM.
Table 1. In line translating roller follower Lower and Higher test Dimn's Basic Dimn. T.R. (l.0)
Radius of pitch circle C.A. (deg)
(in.)
Pressure angle C.A. (dell)
Max. (deg)
Variance of displacement C.A. (deg)
Max. (x 10=t)
64
1.2839
64
0.5
69
1.0853
69
9.264
68 69 79
1.5
59
1.4357
59
22.003
60
0.20697
R.B.
0.6
62
1.0329
62
18.617
0.20104
(0.85) C.R.
1.1 120
65 51
1.5322 1.3053
65 51
14.644 20.189
65 66 71 59
(150)
180
76
1.2633
76
13.774
73 74
RPM
100
64
!.2839
64
16.392
(180)
300
64
1.2839
64
16.392
16.392
R.R.
0.5
62
1.2519
62
18.617
(0.75) O.D.
1.0
65
i.3033
65
14.644
68 69 65 66 71
(0.0) C.A.: Cam rotation angle where maximum value occurred.
0.19795 0.19152
Variance of velocity C.A. (dell) 68 69 79 60 61
Max. (x 10-2) 1.1543 1.1168 1.2069
Variance of acceleration C.A. (deg) 68 69 79 6O 61
Max. (x 105) 1.6158 1.5633 1.6894
1.1723
0.19592 0.20237
65 66 71 59
0.19517
73 74
1.1381
65 66 71 58 73 74
0.19795
69
0.3563
69
0.1539
0.19795
69
3.2065
69
12.4680
65 66 71
1.5992
0.20104 0.19592
65 66 71
1.1425 1.1801
1.1723 1.1425
1.6410 1.5992 1.6519 1.5931
1.6410
639 cam was minimized for a 30° pressure angle and an offset was used to further decrease pressure on the rise stroke. Total Rise (T.R.) Radius of Base Circ. (R.B.) Cam Angle for Rise (C.R.) RPM Incremental Cam (I.A.) Angle Roller Radius (R.R.) Offset Distance (O.D.) Tolerance on Link (T.L.) Tolerance on Cam (T.P.) Profile Standard Motion Curve Maximum Pressure Angle
= = = = =
1.2 in. 0.71 in. 80.0 deg. 470 1.0 deg.
= = = =
0,5 in. 0.605 in. -+0.0002 in. •+0.0001 in.
= Cycloidal = 30 deg.
Here, the computer maximum value of the pressure angle was 30.08 ° . The results are shown in Figs. 6-8. The same type of comparison tests were performed as for Example 1, and the results are summarized in Table 2.
Table 2. Offset translating roller follower Lower and higher test Dimn's Basic Dimn. T.R. (1.2) R.B. (0.71) C.R. (90) RPM (470) R.R. (0.5) O.D. (0.605)
0.7 i.7 0.6 0.9 60 120 280 650 0.25 0.75 0.5 0.8
Radiusof pitch circle C.A. (deg)
(in.)
41 0 39 0 41
1.1645 0.7100 1.3272 0.6000 1.3644
0 41 41 0 42 41 0
0.7100 1.1645 1.1645 0,71 1,4576 1.4287 0,7100
Pressure angle
Varianceof displacement
C.A. M a x . C.A. ( d e g ) (deg) (deg) 41 0 39 0 41 23 0 41 41 0 42 41 0
30.085 30.000 41.200 33.367 26.956 45.250 30.000 30.085 30.085 39.065 26.109 32.027 41.388
40 3 37 39 41 4 40 40 1 41 40 2
Variance of velocity
Max.
C.A.
(X 10-7)
(deg)
0.31860 40 0.29538 3 0.38698 37 0.34942 39 0.28378 41 (not workable) 0.29628 4 0.31860 40 0.31860 40 0.42280 1 0.27573 41 0.30133 40 0.46832 2
Varianceof acceleration
Max. (× 10-t )
C.A. (deg)
Max. (x l0T)
1.2657 1.1733 1.5367 1.3889 1.1275
40 3 37 39 41
1.2086 1.1204 7.4677 1.3254 1.0765
1.1772 0.4920 2.4208 1.6777 1.0955 1.1971 1.8590
4 40 40 1 41 40 2
1.1239 0.1521 4.4211 1.6032 1.0460 1.1430 1.7760
C.A.: Cam rotation angle where maximum value occurred.
Summary The following observations can be made in general for both examples: 1. The maximum variances in displacement, velocity and acceleration occur at the point of maximum pressure angle. 2. Any change in cam dimension or specification which increases the pressure angle also increases the variances. 3. An increase in the angular velocity of the cam has no effect on the pressure angle, but it effects a large increase in variances of the velocity and acceleration. 4. An increase in roller radius effects a small decrease in the value of the variances.
Conclusions In general, errors in any machining or assembly process are statistical phenomena, and a probabilistic or statistical approach to the analysis of these is the only technique that can produce meaningful results.
640 TRANSLATING
ROLLER
FOLLO~ER
[[HAL R I S E = 1,20t I N C H ) AFII]IUS OF BASE CIRCLE , T i ( INCH ) CAN ANGLE FOR RISE = 90.00[ BEG. ) RPM = 470i00 [NCREMENTAL C A M ANGLE = i.00[ BEG, ] ROLLER R A D I U S = ,50{ I N C H ) OFFSET DISTANCE ,G I[ I N C H ) T O L E R A N C E OF L I N K P A R T S TOLERANCE ON COMPONENT OF CIqM P R O F I L E
CYCL01DAL
-
-
CURVE
OR S~NE ACCELERAT~0N
MEAN DISPLACEMENTe
.........
3C B~ND
-..-,.-,,
VARII~NCE+
o000~(INCH) ,0001(INCH)
=
PRESSURE ANQLE
. . . . .
90.0
( .IlII8E+0}' )
i4-'\ .59siTe
L,:CZ
[ .liS0iE-07 )
i
'.
~.
" \..
++.o
/
i
"'....
....
~
CT_ d ",,
-,~0~5~-10, o
'
0.0
,
11.0
zi~,ll
8~'.0
li.O
14,0
45.0
II.O
?~.0
90,0 o.o
11.0
CAM ANGLE (DEG, )
Rgure 8. Variance in displacement.
-
MEaN VELOCTTy~
-
.....
36" B~ND
-..-..-..
VARIANCE+
. . . . .
/?. .... . ~>+
l) ,6S~OIO
-
+ .+.o,,,,+++oo
PRESSURE
ANGLE
\
++ 41,0
\
lli&6~O.O
~
e.o
11.o
~?.O
15.0
lI.O
I+.O
61.0
7~.0
11.0
x
~I(~[o O o
CAM ANGLE (DEG,
Figure 7. Variance in velocity translating roller follower--cycloidal. 35+ B A N D
. . . . .
IBIGI,PIGI07
~
MEAN ACCELERATIONw I
~.
3.- - ~
I
- • -~.
-,,-,,-,, I
VARTANCEp I
. . . . .
l
t
PRESSURE ANGLE I
i
u)
PZII.I@IIII
(,l~O$$E'~OR)
~
/'"
" +
" "~"
~"
"~
+
/"
-+ -,IIPI@I
( ,10075E+01
•
)
I+. -L6517.470111
.
'
oo
I
,0
" ~ , - - - ~ ~.
•
./"\..
./
l ~
.,,0
••
"-
/
I
,,0
....
l
,.o
'
-
,
,,:o
+~
,~.0
~
67,I
+I,O
~
,,.'0
~0.0
~
../t
"'-~
zz ~-t ~,~
I
x
~
~/~-"
,,.io
,o~o oo
CAM ANGLE (DEG,)
Figure 8. Variance in acceleration translating roller follower--cycloidal.
('~ oc~
z (I
uJ
841
In this investigation the importance of keeping the pressure angle to a minimum is reaffirmed because maximum errors in follower output occur at points of maximum pressure angle. This is not a great revelation, but the techniques developed here now allow these errors to be quantized. The effect of the cam angular velocity is also emphasized. It will now be possible to extend this work to include all possible errors in cam mechanisms, to compare the relative performance of the standard motion curves and to introduce optimization techniques for minimizing output errors. Acknowledgements--The authors wish to thank the National Research Council of Canada for assisting with the financial support of this project.
References 1. S. G. Dhande and J. Chakraborty, Mechanical error analysis of cam-follower systems--a stochastic approach. I. Mech. E. p. 957-962 (1975). 2. H. A. Rothbart, Cams-Design, Dynamics and Accuracy. Wiley, New York (1956). 3. S. B. Tuttle, Error analysis of mechanism. Machine Design, 152-168 (June, 1960). 4. J. H. C. Brittain and R. Horsnell, A prediction of some causes and effects of cam profile errors. Proc. Inst. Mech. Engrs VoL 182, Pt3L (1967-68). 5. C. Mischke, Assessment of ultimate fidelity of an eccentric circular disk cam with translating fiat-faced follower as a function generator. Proc. Design Automation Conf. ASME Paper No. 73-DET-18 (1973). 6. V. P. Filkin, An investigation into the accuracy of a conoid mechanism. Problems of the Design and Accuracy of Complex Continuous-Action Devices and Computer Mechanisms, p. 137-178. Macmillan, New York (1964). 7. V. I. Sergeyev, The investigation of the accuracy of a non-automated friction devices. Problems of the Design and Accuracy of Computer Mechanisms, pp. 93-I I0. Macmillan, New York (1964). 8. E. A. Mikhailov, The accuracy and adjustment of mechanisms with a variable transfer ratio. Problems of the Design and Accuracy of Computer Mechanisms, pp. 191-232. Macmillan, New York (1964). 9. A. V. Rumiantsev, Some Problems of the Technology of Manufacture of Conoid Ovorongiz. (1953). I0. Y. Kawasaki, S. Yamamoto, Y. Yamamoto, K. Ito, H. Nagano, S. Inanobe and Y. Watanabe, Lift error on grinding cam profile. Ann. CIRP. 253-258 (1975). II. D. Tesar and G. K. Matthew, The dynamic syntheses, Analyses, and Design of Modeled Cam Systems. Lexington (1976). 12. R. C. Johnson, Method of finite differences for cam design. Mech. Design 27, 195 (Nov. 1955). 13. E. B. Haugen, Probabilistic Approaches to Design. Wiley, New York (1968). 14. R. G. Fenton, Cam design. Automobile Eng. 184--187 (1967). ANALYSE
H.R.
STOCHASTIQUE
Kim et W.R.
R~sum~
probabilistique
et de l ' a c c ~ l ~ r a t i o n
rance d ' u s i n a g e
sur le p r o f i l
par d ~ v e l o p p e m e n t
m i e r et du s e c o n d ordre. cement cin~matique, stochastiques, ordinateur
DANS UN M ~ C A N I S M E
du type ~ disque.
l'obtention
et l ' a c c ~ l ~ r a t i o n
la m ~ t h o d e
Un e x e m p l e
i'~quation
du m o u v e m e n t
est
de l ' e n t r a l n e m e n t ,
inclus.
maximale.
~ la s o r t i e pour
du d ~ p l a c e m e n t ,
~ cames pour route de l ' e n t r a i n e m e n t
~ la fois pour
avec des v a r i a b l e s
de la p r o b a b i l i t ~
les e r r e u r s
des v a r i a n t e s
des m ~ c a n i s m e s
Une ~ q u a t i o n
en s~rie de T a y l o r et a p p l i q u ~ e
la v i t e s s e
y compris
pour
de l ' e n t r a i n e m e n t
des cames.
On a n a l y s e
p e r m e t de p r ~ v o i r
M M T Vol, 13,No, 6=-F
A CAMES
Newcombe
- On d ~ c r i t une m ~ t h o d e
de la v i t e s s e
obtenue
DES E R R E U R S
les termes du pre-
al~atoires
pour
en e m p l o y a n t
Un p r o g r a m m e
tout m ~ c a n i s m e
tolLest
le d ~ p l a -
des m ~ t h o d e s
g~n~ral
sur
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0094-114XI78/1201-06311502.0010
Stochastic Error Analysis in Cam Mechanisms H. R. Kimt and W. R. Newcombei" Abstract A probabilistic method is outlined for obtaining the variances in displacement, velocity and acceleration of followers in cam mechanisms for any given manufacturing tolerance on the cam profile. A follower motion equation is obtained by expansion in a Taylor series and considered in both linear and second-order terms. The equation is analyzed with random variables for the kinematic displacement, velocity and acceleration of the follower employing stochiastic techniques including finite differences and the maximum likelihood method. A general computer program allows the designer to obtain projected output errors for any commonly used disc cam type, and an example of one of these is included. Introduction STOCHASTICmethods and probabilistic techniques are being increasingly more widely used in the analysis and design of machines. In this investigation a probabilistic approach was employed to analyze the effect of manufacturing errors in cam profiles on the output or follower displacement, velocity and acceleration. Dhande and Chakraborty[l] have provided the necessary background, and their stochastic model is used. They employed a stochastic approach to analyze the three-sigma band of error in the follower displacement showing maximum and minimum errors, whereas, the method has been extended here to enable an analysis of the error in velocity and acceleration due to any given manufacturing tolerance on the cam profile. A general computer programme was written for this purpose. Several other investigations have been carried out on the error analysis of cam mechanisms. Rothbart [2] used a finite difference method to obtain the effects of surface roughness on follower acceleration and a relationship was derived for the allowable increment in cutter position for a maximum surface roughness error. Turtle [3] used vectorial addition to analyze errors in cam-operated devices and four-bar linkages. This method gave the approximation only of the error in a non-continuous system, and is not applicable to error analysis in the cam-follower mechanism itseff. Brittain and Horsnell[4] investigated the effects of grinding wheel wear and of machine setting errors on the surface of a production can specified for a required valve motion. Mischke[5] analyzed the effects of mechanical error in an eccentriccircular cam with a flat-faced translating follower without considering surface roughness. Filkin [6] as well as Sergeyev [7] investigated the effects of manufacturing and assembly errors on camoid mechanisms. The co-ordinates of discrete points on the camoid surface were considered instead of the surface geometry with approximate results. Mikhailov [8] investigated the relative effectiveness of various adjustment methods to reduce mechanical error in the output and Rumiantser[9] discussed the effects of assembly errors. Kawasaki[10] et al. presented methods for computing the cam profile lift error due to a change in grinding wheel diameter and Tesar[l l] has included a section on errors in cam mechanisms in his recent book. In the kinematic analysis and design of cam mechanisms a knowledge of the effect of manufacturing errors on the velocity and especially on the acceleration characteristics is often more important than the effect on displacement and reports on the investigation of this problem tDepartment of Mechanical Engineering, McMaster University, Hamilton, Ontario L8.S 4L7, Canada. 631
632
have not been seen. The variance of follower displacement due to mechanical and manufacturing errors can be obtained stochastically, but the variances of its first and second derivatives, i.e. velocity and acceleration are not easily obtained directly or indirectly. This paper will present a method of calculating the approximate value of these variances using a stochastic approach and the maximum likelihood estimate.
Cam-Follower System Model and Analysis Figure 1 shows the fundamental system used to model the cam-follower mechanism. St and $2 indicate the cam and follower surfaces fixed to the moving coordinates (x,.i, y,,i, ZmO and (xm2, Y,,2, z,,2) which move relatively to the fixed coordinates (xi~, Yit, zt0 and (xiz, Yt2, zt2). Two vectors /~l and /~2 are radius vectors of the contact point P between cam and follower expressed in the moving coordinate systems. YfZ
zu
$2
~
St '
"'~m,
z.
Figure 1. Fundamental coordinate system for cam follower mechanism. Let tt and t2 be the parameters of motion of cam and follower links and 0 and $ the parameters of the follower surface $2. Then, by vector and matrix notation, we have RI = [xrnl, Y,.l, Zmll r
/~2 = [Xm2,Y,.2, Z,.21T= / (0,¢)
(i)
6 = g (0,¢)
where ~2 is a unit normal vector to the surface $2 at contact point P. To satisfy the condition of contact, the dot product of the unit normal vector, ~ at contact point P and relative velocity vectors V'12at point P between the surface S~ and $2 must be equal to zero, i.e. ~7. I712= O.
(2)
Assuming tt is the input parameter, we can find the expression for output parameter t2 solving the above given eqns (1) and (2) for three unknown parameters 0, $ and t2 as t2 = h(x~ bYm l,Zm IXm2,ym2,Z~2,tO
(3)
in which the relationship between a coordinate fixed to the cam and the one fixed to follower may be given by introducing the transformation matrix. Equation (3) can be generalized denoting the variables as input random variables like t2 = h(rl,r2,r3 . . . . ri, tl).
(4)
Since it is extremely difficult to find the probability density function and distribution function of output variables t2 from eqn (4) theoretically, the mean and variance of t2 are obtained from those of the random variables, ri's by expanding eqn (4) in Taylor series about the mean value of ri's and taking the first two terms of the expanded equation. The former is called the moment generating function method complying with the theory of probability while the latter the partial derivative method for approximation.
633
They can be written as n
t~ = F + ~, Gi(ri - ~,)
(5)
i=1
where F = h(#l,p,z,#3 . . . . . P,n, tl) GI
=
(Ot2~ \7~i / ,,~
evaluated at the mean values of ri's; #i = mean values of r~'s. The mean value and the variance of output t2 are /~t2 = h(/~1,/~2,/.~3 . . . . /J,n, tl) [Oft2] = ~
Gi2[orri] 2 •
(6)
i=1
The mean value and variance of the follower displacement have been obtained from the stochastic model. It is now required to calculate the variances of velocity and acceleration of the follower to analyse the cam mechanism completely and generally. Thus is limited by the characteristics that the velocity and acceleration are the first and second derivatives of displacement respectively; the direct differentiation of displacement with respect to time for velocity makes no sense as the first derivative terms of both sides of the displacement equation eliminate each other. This is because the displacement equation was established by solving the parametric equations simultaneously which include displacement terms occurring from the known displacement relation like the basic motion curve. The same problem occurs in obtaining second derivatives from first derivatives even though the latter can be found. Therefore, a different condition or another method must be introduced. The Monte Carlo method, which is a powerful method for stochastic problems, may be employed by choosing random samples of displacement within the displacement error bound and using the finite differences method. However, several difficulties occur in determining sample size and incremental cam angle in the finite differences considering the contact condition between cam and follower. Moreover, the computer must be used with the Monte Carlo method, and the costs are high since an iteration should be done at every increment. The general equations of motion of a point in space using rotating and translating coordinate systems could be applied to this problem, but the same cancellation of derivatives as in the direct differentiation of displacement occurs for velocity and acceleration. On the other hand, the introduction of other conditions such as relative velocity or acceleration and radius of curvature of cam profile were considered, but this also failed to give the variance of velocity and acceleration as these conditions are not kinematically independent. The above reasons probably explain why the determination of velocity and acceleration errors has not been approached stochastically by previous researchers. To solve this problem finite differences were used to determine velocity and acceleration numerically. Then, the maximum likelihood technique of random variable analysis was introduced to obtain the desired stochastic results approximately.
Finite Dilferences The use of finite differences in cam-follower systems were introduced by R. C. Johnson[12] and developed by Rothbart[2] to find the effects of surface roughness on the follower acceleration. The follower characteristic curves can be obtained numerically by utilizing incremental tabulated values which may be estimated, taken from the actual cam shape measurements, or calculated from an analytical cam curve.
634
Let S = follower displacement (in.) V = follower velocity (in./sec.) A = follower acceleration (in./sec2.) 80 = incremental cam angle (rad.) 8t = time for cam to rotate through angle 80 (sec.) to = angular velocity of cam (rad./sec.) and assume that the cam displacement curve has three given points 1, 2 and 3 with small and equal time interval St, as shown in Fig. 2.
1 g / ~ i r n e ---.-
~St
Figure 2. Part of displacement curve and application of finite differences. The follower velocity at point 2 will be
V2=(-d-i):=limS3-- Sl=S~Tl'.,-~ dS
(7)
2~Jt
Knowing 8t = (801to) and substituting this into eqn (7), we get V2 ~
tO ~-~(S
3-
S 0.
(8)
For the acceleration of point 2, choose points 4 and 5 so as to be midpoints respectively between 1 and 2, and 2 and 3 as shown. Then, the follower acceleration will be d
dS
I14
(9)
8t Since V 5 = 53 - 52
8t
V 4 = 5 2 - SI a n d
'
8t
6t
= 80
a,
the acceleration is A2= (~0)2 (S1- 2S2 + $3).
(10)
This technique has shortcomings when applied to cams. It can be seen from (8) and (10) that the velocity and acceleration at dwell and intersecting points in the curves cannot be found, and as the increments of cam angle are taken larger and the jerk increases, the errors increase. The latter shortcomings, however, were somewhat reduced by using the computer to obtain a rough optimum value of the incremental angle.
The Maximum Ukelihood MethoO Three methods can be used in stochastic problems to compute moments of functional combinations of random variables; statistically, the moment of a function of random variables uniquely defines the distribution of the function, since a distribution is uniquely defined by its
635
moments if they are finite. These are: (i) Moment generating function, (ii) Partial derivative methods, (iii) Maximum likelihood methods. When the functions of density and distribution of random variable are known, the moment generating function is often employed. However, although this is a powerful method in statistics it has disadvantages when complicated and rigorous integrals that are hardly accessible must be handled since its principle follows the exact probabilistic theory. The second method, the partial derivative method which has already been used to obtain the variance of the displacement, has shortcomings when treating partial derivatives within Taylor series. In comparison, the maximum likelihood method is quite simple to apply to simple functions like eqns (8) and (10), and it can be employed to obtain the variances of velocity and acceleration from those of displacement. The maximum likelihood method[13] is based on a point estimate which is a single value that is used to estimate parameters such as mean values, variances and standard deviations. Assume normally distributed random variables X and Y as described by populations of values XI,X2 . . . . . . . . . . .
,X m
Yl,Y2 . . . . . . . . . . .
,Yn.
From these populations, the mean value estimator of te sum variate (~) can be calculated as ~=
1
m . f l [Xl + Yl) + (Xl + Y2) + . . . . + (Xl + Yn) + (X2 + Yl) + (X2 + Y2) + . . . . + (X2 + Yn)
+ . . . . . + (x., + yl) + (xm + y9 + . . . . . + (x. + yn)
~-±
(x,. + yj)
m.ti
(11)
Similarly, the standard deviation estimator of the sum variate is
D, = [ ~ . ~ l{e- (x, + y,)l ~ + { e - 0<, + sol ~ + . . . + { e - (x, + y.)}~ + {Z -- (X2 + yl)}2 + {Z - (X2 + y2)}2 + . . . . + {Z - (X2 + yn)}2
-F . . . . + {Z.(Xm + yl)} 2 + {Z. - (Xm + y2)}2 -F . . . . + {Z - (Xm "4-y2)}2]] 1/2
: [~ ~, (e- (x, + y,)~'l"2 =
~-_~
(t2)
j
m.n
Now, by arithmetic manipulation when X and Y are statistically independent these become, respectively ~ _. ( X l "["X 2 "l'.m_."" " + Xm ) .l. (.Y l "l- Y 2 + ~_ " " " "[" Y n )
(13)
=X+y Dz = {(xl - g)2 + (x2- g)2 + . . . . + (xm - ~)2 m + (Yl - y)2 + (Y2 - Y)2n+ . . . .
= (o,2 + o,2) "~.
+ (y" - y)2}t/2
(14)
The mean value and the standard deviation about the ditterences of two iiormally distributed statistically independent variates X and Y can be found likewise as
636
=__lm [(XI -- Yl) q- (Xl -- Y2)n+ . . . . + (X2 -- YO+ (x2 - Y2) + . . . .
+ (xi - Yn) + (x2 - Yn)
n
+ (Xm - Yl) + (Xm -- Y2)n+ . . . .
+ ....
+ (Xm - Yn)]
=i-y
(15)
and D~ =
{(xi - S) - (yj - •)}2] 1/2 •
i=1 "~
= (D~2 + Dy2)m
(16)
where D~ and Dy are, respectively, the standard deviation estimators of random variables X and Y. Considering the linear combination
W=aX+bY+cZ in which X, Y and Z are independent normal variates and a, b and c are the arbitrary constants; by a linear transformation of the normal variate, the mean of the linear combination becomes = aS+ by+ c~?
(17)
and the standard deviation will be D w = ( a 2 D x 2 + b2Dy 2 ''t- C2Dz2) I/2.
(18)
Now, noting that lim S =/zx and lira Dx = tr~ when the size of samples are large, eqns (13)--(18) m~
m---~
can be written as follows: For the sum tz~ =/~x + tzy
(13)'
cr~ = (Crx2 + ay2)1/2.
(14)'
For the differences tz~ = ~x -/zy
(15)'
orz = (o'x2 '1-o'y2)1/2.
(16)'
For a linear combination (17)' Orw _-- (a20rx 2 at- b20.y 2 -1- c20.z2)1/2"
(18)'
Now, with the mean value and standard deviation of displacement acquired from eqn(6), the variances of velocity and acceleration can be computed by combining eqns (8) and (9) and eqns (13)'-(18)'.
Example 1 Disc cam-translating roller follower-SHM The following specifications for a disc cam with an in-line translating roller follower have been taken from "CAMS", by Rothbart [2]:
637
Total Rise (T.R.) = 1.0 in. Radius of Base (R.B.) = 0.85 in. Circle Cam Angle for Rise (C.R.) = 150.0 deg. RPM = 180 Incremental Cam (I.A.) = 1.0 deg.
Angle Roller Radius (R.R.) Offset Distance (O.D.) Tolerance on Link (T.L.) Tolerance on Cam (T.P.) Profile Standard Motion Curve Maximum Pressure (@) Angle
= = = =
0.75 in. 0.0 in. -+0.0002 in. -+0.0001 in.
= SHM = 20 deg.
The computer programme developed for this project calculated the cam profile for exact contact conditions between cam and follower. The above design had been roughly minimized for a pressure angle of 20°. The exact value for the maximum pressure angle obtained from the programme was 16.4°. The results are shown in Figs. 3-5. Displacement, velocity and acceleration graphs are shown with their 3G bands, and the variance in each of these is plotted on the same graph to a different scale. The change in pressure angle is also plotted on each graph, and it is seen that the maximum values of the variances logically occur at maximum pressure angle for the in-line translating roller follower. TRANSLA]ING ROLLER FOLLOWER TOTAL R I S E RADIUS OF BASE CIRCLE CAM ANGLE FOR RISE RPH INCREKENTAL CAM ANGLE ROLLER RADIUS
= l,OO( TNCH ) ,85( INCH ) = 150,00( DEG, ) = 180,00 = 1,00( DEE, ) = ,75( INCH }
OFFSET DISTANCE O ,O0( I N C H } TOLERANCE OF L I N K PP~TS = TOLERANCE ON COMPONENT OF CAM P R O F I L E =
,0002( INCH } ,0001( INCH )
SIMPLE HARMONIC NOTION OR CRANK CURVE
1.000300
~
MEAN DISPLACEMENTp
i
( ~c
i
.?~o[15 • 1979BE-07 )
"
E--LJ ~J ZC: CJH
I
,/
. . . . .
36 8 A N D
-,,-,,-,,
VP~IANCEp
I
~''
-- ....
i
~''~',
I
PRESSURE ANGLE
i
90,0
17 s ~j o
~,~
../ ,500017 (,iS4S3£-07}
. . . . .
i
,%..
"
jw "'4.
./
E ¢1,o w
",.~
g .24ee6o
/
°'~
000iL% •o
1i,o
go,o
4i.o
$o.o
71,o
$o.o
1o6,o
1~o,o
lJllo
o ~lSO.o .o
CAM ANGLE CDEG, )
Rgure 3. Variance in displacement. Comparison tests were then carried out by varying cam specifications or characteristics one at a time. An arbitrary higher and a lower value of each characteristic was used while holding the other characteristics constant. The results of these tests are summarized in Table 1. The maximum values of the variances were selected from the computer print-outs, and these occurred at somewhat different cam-rotation angles than that of the base problem as indicated.
Example 2. Disk cam-oNset translating roller foUower-cycloidai The specifications for this cam mechanism were taken from Robert and Fenton[14]. This
-
-
HERN VELOCZTY~ I I
I
I/
.....
3[
-,,-,,-** .~
VRR~J:~NCEP ~.,~1
BANO
. . . . . I
PRESSURE RNGLE I I j
00.0
I.Yt0YIi ( .ll~41E-O;
.u H~
,/"
°.°00ZZ¢ (.°OSSSE-0Zl
""\
i,~
~Z
4 .... ~-:=
.. ~ .
~.$lti°? (.ili~7E-ot
.......
10. °
ll~0
to.0I
i°.oI
°o.oI
?°.0I
°0.0I
101.0I
it0 ! 0
110.0 0.o
lil,O
CRH RNGLE ( DEC. )
Figure 4. Variance in velocity translating NEPal ACCELER~TZON~
J.I].l * J.I J.II f
roller follower--SHM.
. . . . .
3ff B R N D
-,,-,,-,,
VRRII~ICE~
. . . . .
PRESSURE ;:iNGLE /
u~
( .~.l[lli*OI
Z HZ~ ~H
)
.' I..
•.
L~ LL O
]
/
zto011ttl
( .LttTIE*Oi I
45.0 ~j
•
-.,,.0,..0.~" " .~o
.rE
t
,,.o'
°0!o
,o.0' . . . . . ,,.0"T.... ,o.o~.... ~o,.0
,,oL.o
; .,.o ' ;-~,i o. 0.o
CAH ANGLE I DEG, )
Rgure 5. Variance in acceleration translating roller follower--SHM.
Table 1. In line translating roller follower Lower and Higher test Dimn's Basic Dimn. T.R. (l.0)
Radius of pitch circle C.A. (deg)
(in.)
Pressure angle C.A. (dell)
Max. (deg)
Variance of displacement C.A. (deg)
Max. (x 10=t)
64
1.2839
64
0.5
69
1.0853
69
9.264
68 69 79
1.5
59
1.4357
59
22.003
60
0.20697
R.B.
0.6
62
1.0329
62
18.617
0.20104
(0.85) C.R.
1.1 120
65 51
1.5322 1.3053
65 51
14.644 20.189
65 66 71 59
(150)
180
76
1.2633
76
13.774
73 74
RPM
100
64
!.2839
64
16.392
(180)
300
64
1.2839
64
16.392
16.392
R.R.
0.5
62
1.2519
62
18.617
(0.75) O.D.
1.0
65
i.3033
65
14.644
68 69 65 66 71
(0.0) C.A.: Cam rotation angle where maximum value occurred.
0.19795 0.19152
Variance of velocity C.A. (dell) 68 69 79 60 61
Max. (x 10-2) 1.1543 1.1168 1.2069
Variance of acceleration C.A. (deg) 68 69 79 6O 61
Max. (x 105) 1.6158 1.5633 1.6894
1.1723
0.19592 0.20237
65 66 71 59
0.19517
73 74
1.1381
65 66 71 58 73 74
0.19795
69
0.3563
69
0.1539
0.19795
69
3.2065
69
12.4680
65 66 71
1.5992
0.20104 0.19592
65 66 71
1.1425 1.1801
1.1723 1.1425
1.6410 1.5992 1.6519 1.5931
1.6410
639 cam was minimized for a 30° pressure angle and an offset was used to further decrease pressure on the rise stroke. Total Rise (T.R.) Radius of Base Circ. (R.B.) Cam Angle for Rise (C.R.) RPM Incremental Cam (I.A.) Angle Roller Radius (R.R.) Offset Distance (O.D.) Tolerance on Link (T.L.) Tolerance on Cam (T.P.) Profile Standard Motion Curve Maximum Pressure Angle
= = = = =
1.2 in. 0.71 in. 80.0 deg. 470 1.0 deg.
= = = =
0,5 in. 0.605 in. -+0.0002 in. •+0.0001 in.
= Cycloidal = 30 deg.
Here, the computer maximum value of the pressure angle was 30.08 ° . The results are shown in Figs. 6-8. The same type of comparison tests were performed as for Example 1, and the results are summarized in Table 2.
Table 2. Offset translating roller follower Lower and higher test Dimn's Basic Dimn. T.R. (1.2) R.B. (0.71) C.R. (90) RPM (470) R.R. (0.5) O.D. (0.605)
0.7 i.7 0.6 0.9 60 120 280 650 0.25 0.75 0.5 0.8
Radiusof pitch circle C.A. (deg)
(in.)
41 0 39 0 41
1.1645 0.7100 1.3272 0.6000 1.3644
0 41 41 0 42 41 0
0.7100 1.1645 1.1645 0,71 1,4576 1.4287 0,7100
Pressure angle
Varianceof displacement
C.A. M a x . C.A. ( d e g ) (deg) (deg) 41 0 39 0 41 23 0 41 41 0 42 41 0
30.085 30.000 41.200 33.367 26.956 45.250 30.000 30.085 30.085 39.065 26.109 32.027 41.388
40 3 37 39 41 4 40 40 1 41 40 2
Variance of velocity
Max.
C.A.
(X 10-7)
(deg)
0.31860 40 0.29538 3 0.38698 37 0.34942 39 0.28378 41 (not workable) 0.29628 4 0.31860 40 0.31860 40 0.42280 1 0.27573 41 0.30133 40 0.46832 2
Varianceof acceleration
Max. (× 10-t )
C.A. (deg)
Max. (x l0T)
1.2657 1.1733 1.5367 1.3889 1.1275
40 3 37 39 41
1.2086 1.1204 7.4677 1.3254 1.0765
1.1772 0.4920 2.4208 1.6777 1.0955 1.1971 1.8590
4 40 40 1 41 40 2
1.1239 0.1521 4.4211 1.6032 1.0460 1.1430 1.7760
C.A.: Cam rotation angle where maximum value occurred.
Summary The following observations can be made in general for both examples: 1. The maximum variances in displacement, velocity and acceleration occur at the point of maximum pressure angle. 2. Any change in cam dimension or specification which increases the pressure angle also increases the variances. 3. An increase in the angular velocity of the cam has no effect on the pressure angle, but it effects a large increase in variances of the velocity and acceleration. 4. An increase in roller radius effects a small decrease in the value of the variances.
Conclusions In general, errors in any machining or assembly process are statistical phenomena, and a probabilistic or statistical approach to the analysis of these is the only technique that can produce meaningful results.
640 TRANSLATING
ROLLER
FOLLO~ER
[[HAL R I S E = 1,20t I N C H ) AFII]IUS OF BASE CIRCLE , T i ( INCH ) CAN ANGLE FOR RISE = 90.00[ BEG. ) RPM = 470i00 [NCREMENTAL C A M ANGLE = i.00[ BEG, ] ROLLER R A D I U S = ,50{ I N C H ) OFFSET DISTANCE ,G I[ I N C H ) T O L E R A N C E OF L I N K P A R T S TOLERANCE ON COMPONENT OF CIqM P R O F I L E
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Rgure 8. Variance in displacement.
-
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Figure 7. Variance in velocity translating roller follower--cycloidal. 35+ B A N D
. . . . .
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Figure 8. Variance in acceleration translating roller follower--cycloidal.
('~ oc~
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841
In this investigation the importance of keeping the pressure angle to a minimum is reaffirmed because maximum errors in follower output occur at points of maximum pressure angle. This is not a great revelation, but the techniques developed here now allow these errors to be quantized. The effect of the cam angular velocity is also emphasized. It will now be possible to extend this work to include all possible errors in cam mechanisms, to compare the relative performance of the standard motion curves and to introduce optimization techniques for minimizing output errors. Acknowledgements--The authors wish to thank the National Research Council of Canada for assisting with the financial support of this project.
References 1. S. G. Dhande and J. Chakraborty, Mechanical error analysis of cam-follower systems--a stochastic approach. I. Mech. E. p. 957-962 (1975). 2. H. A. Rothbart, Cams-Design, Dynamics and Accuracy. Wiley, New York (1956). 3. S. B. Tuttle, Error analysis of mechanism. Machine Design, 152-168 (June, 1960). 4. J. H. C. Brittain and R. Horsnell, A prediction of some causes and effects of cam profile errors. Proc. Inst. Mech. Engrs VoL 182, Pt3L (1967-68). 5. C. Mischke, Assessment of ultimate fidelity of an eccentric circular disk cam with translating fiat-faced follower as a function generator. Proc. Design Automation Conf. ASME Paper No. 73-DET-18 (1973). 6. V. P. Filkin, An investigation into the accuracy of a conoid mechanism. Problems of the Design and Accuracy of Complex Continuous-Action Devices and Computer Mechanisms, p. 137-178. Macmillan, New York (1964). 7. V. I. Sergeyev, The investigation of the accuracy of a non-automated friction devices. Problems of the Design and Accuracy of Computer Mechanisms, pp. 93-I I0. Macmillan, New York (1964). 8. E. A. Mikhailov, The accuracy and adjustment of mechanisms with a variable transfer ratio. Problems of the Design and Accuracy of Computer Mechanisms, pp. 191-232. Macmillan, New York (1964). 9. A. V. Rumiantsev, Some Problems of the Technology of Manufacture of Conoid Ovorongiz. (1953). I0. Y. Kawasaki, S. Yamamoto, Y. Yamamoto, K. Ito, H. Nagano, S. Inanobe and Y. Watanabe, Lift error on grinding cam profile. Ann. CIRP. 253-258 (1975). II. D. Tesar and G. K. Matthew, The dynamic syntheses, Analyses, and Design of Modeled Cam Systems. Lexington (1976). 12. R. C. Johnson, Method of finite differences for cam design. Mech. Design 27, 195 (Nov. 1955). 13. E. B. Haugen, Probabilistic Approaches to Design. Wiley, New York (1968). 14. R. G. Fenton, Cam design. Automobile Eng. 184--187 (1967). ANALYSE
H.R.
STOCHASTIQUE
Kim et W.R.
R~sum~
probabilistique
et de l ' a c c ~ l ~ r a t i o n
rance d ' u s i n a g e
sur le p r o f i l
par d ~ v e l o p p e m e n t
m i e r et du s e c o n d ordre. cement cin~matique, stochastiques, ordinateur
DANS UN M ~ C A N I S M E
du type ~ disque.
l'obtention
et l ' a c c ~ l ~ r a t i o n
la m ~ t h o d e
Un e x e m p l e
i'~quation
du m o u v e m e n t
est
de l ' e n t r a l n e m e n t ,
inclus.
maximale.
~ la s o r t i e pour
du d ~ p l a c e m e n t ,
~ cames pour route de l ' e n t r a i n e m e n t
~ la fois pour
avec des v a r i a b l e s
de la p r o b a b i l i t ~
les e r r e u r s
des v a r i a n t e s
des m ~ c a n i s m e s
Une ~ q u a t i o n
en s~rie de T a y l o r et a p p l i q u ~ e
la v i t e s s e
y compris
pour
de l ' e n t r a i n e m e n t
des cames.
On a n a l y s e
p e r m e t de p r ~ v o i r
M M T Vol, 13,No, 6=-F
A CAMES
Newcombe
- On d ~ c r i t une m ~ t h o d e
de la v i t e s s e
obtenue
DES E R R E U R S
les termes du pre-
al~atoires
pour
en e m p l o y a n t
Un p r o g r a m m e
tout m ~ c a n i s m e
tolLest
le d ~ p l a -
des m ~ t h o d e s
g~n~ral
sur
~ cames ordinaire