ME 352 - Machine Design I
Fall Semester 2016
Name of Student:____________________________________
Lab Section Number:_______________________________
Homework No. 6.
Part I. (20 points). Due at the end of lab on Wednesday, October 12th, or Thursday, October 13th.
Draw the lift curve and the cam profile of Problem 6.28 on pages 328 and 329.
Templates to draw the lift curve and the cam profile are provided on pages 2 and 3 of this homework.
Part II. (20 points). Due at the beginning of lecture on Friday, October 14th.
For the cam-follower system of Problem 6.28, at the cam angle   225o , determine numerical
values for:
(i) The first, second, and third-order kinematic coefficients of the displacement diagram.
(ii) The radius of the curvature of the cam surface.
(iii) The unit tangent and normal vectors to the cam at the point of contact with the follower.
(iv) The coordinates of the point of contact between the cam and the follower. Express your answers in
the moving Cartesian coordinate reference frame attached to the cam.
(v) The pressure angle of the cam. Is your answer acceptable for this cam-follower system?
-1-
Solution to Homework Set 6.
Part I. 20 Points. (i) 8 Points. The displacement diagram (or lift curve) is plotted in Figure 1.
Figure 1. The Displacement Diagram (or Lift Curve).
-2-
(ii) 8 Points. The cam profile is plotted in Figure 2.
Figure 2. The Cam Profile.
-3-
(iii) 4 Points. Comments on the curves and answers to the design issues:
(i) The recommended value of the pressure angle for a cam-follower system, see page 309 in the text
book, is less that 30 or 35. The pressure angle defines the steepness of the cam profile and is a
measure of the efficiency of the cam. A plot of the pressure angle against the cam position is shown in
Figure 3.
50
45
 - Pressure Angle - [deg]
40
35
30
25
20
15
10
5
0
0
50
100
150
200
250
Cam - Cam Angle - [deg]
300
350
Figure 3. A plot of the pressure angle against the cam position.
In this design, the pressure angle is more than the accepted value at the cam angles
  16  64,
131  180,
216  256,
and
299  341
Therefore, this cam profile is not a good cam design.
The high values of the pressure angle may be due to the selection of the segments for the
displacement diagram, and/or the dimensions of the cam and the diameter of the follower.
(ii) Position discontinuities never occur. Discontinuities in the derivatives will only occur at transitions
between dwell segments and lifting/returning segments of motion. Discontinuities in the derivatives are
undesirable especially for a high speed cam follower system.
There is an acceleration discontinuity at the beginning and end of the simple harmonic motions, both
rise and return. There is a jerk discontinuity at the beginning and end of the cycloidal motions, both rise
and return.
(iii) The radius of curvature of a cam profile should always be negative for a good cam design. A
positive radius of curvature means that the cam has a concave surface and there is the possibility that the
follower may lose contact with the cam.
If the radius of curvature of the cam profile is positive then the radius of curvature of the cam must
be greater than the radius of the follower.
-4-
In the proposed design, the positive values of the radius of curvature of the cam are always greater
than 0.5″ (i.e., the radius of the follower).
A plot of the radius of curvature of the cam profile against the cam position is shown in Figure 4.
10
Cam - Cam Radius of Curvature - [in]
8
6
4
2
0
-2
-4
-6
-8
-10
0
50
100
150
200
250
Cam - Cam Angle - [deg]
300
350
Figure 4. A plot of the radius of curvature of the cam profile against the cam position.
The radius of curvature of the cam is positive for the following ranges of the cam angle:
  3  25 ,
170  192 ,
208  225 ,
and
329  345
The radius of curvature of the cam is positive, and smaller than the radius of the roller follower, for the
following ranges of the cam angle:
  9  13 ,
181  187 ,
and
215  216
Note that the radius of curvature of the cam is zero between the cam angles 214 and 215 degrees
meaning that pointing has occurred. Also, it could imply that undercutting has occurred.
Also, with the exception of where the radius of curvature of the cam goes to zero, there is an inflection
point at the boundary of each range of angles for which the radius of curvature is positive.
(iv) Possible design changes to the cam-follower system include:
(a) Increasing the radius of the prime circle (with the same lift curve) in general would reduce the
pressure angle.
(b) Change the profiles to match acceleration at the transition (blend) points to eliminate acceleration
discontinuities.
(c) Change both SHM profiles to cycloidal would make accelerations continuous but would also
increase accelerations (and, therefore, the values of the pressure angle) in the middle parts of the rise and
the return profiles.
(d) May want to increase the diameter of the roller follower if the contact stresses are high.
-5-
(e) Change the eccentricity (or the offset). The effects may not be obvious from observation, so a
spreadsheet may be required to explore this problem numerically.
Part II (20 Points). The base circle diameter of the cam is D = 3 inches and the diameter of the
reciprocating roller follower is d = 1 inch. Therefore, the radius of the prime circle of the cam is
R0 
D  d 31

 2 inches
2
2
(1)
Since the problem states that there is a radial follower then the follower offset (or eccentricity) is   0.
Therefore, the coordinates of the follower center on the prime circle (that is, when the roller follower is
touching the base circle of the cam) in the fixed reference frame are
X0    0
(2a)
and
Y0  R0 2   2  2 2  0 2  2 inches
(2b)
The displacement of the follower center for full-rise simple harmonic motion, see Equation (6.12a),
page 291 in the text book, can be written as
y  ybase 
L
 * 
1  cos

 
2
(3)
where ybase is the base lift (that is, the lift at the beginning of the full-rise simple harmonic motion).
Therefore, from Table P6.28, see page 329, in the problem statement, the base lift is
ybase  0
(4)
Also, the total lift during the full-rise simple harmonic motion is
L  2 inches
(5)
The range of the full-rise simple harmonic motion is
  270o  210o  60o 

3
rad
(6a)
and
 *    0
(6b)
where 0 is the starting cam angle for the full-rise simple harmonic motion. The cam angle is   225o ,
and the starting cam angle for the full-rise simple harmonic motion is 0  210o , therefore
 *  225o  210o  15o 

12
rad
(6c)
and
 *


-6-
   3  
     rad
 12     4
(7)
Substituting Equations (4), (5), and (7) into Equation (3), the displacement of the follower center is
y  0 in 
or
2 in
2


 1  cos 
4

y  1 in (1  0.7071)  0.2929 in
(8a)
(8b)
This result implies that the follower center has risen from 0 in to 0.2929 in after 15o of the full-rise
simple harmonic motion (i.e., when the angle  *  15o or the cam angle θ  225 ).
For example, note that after 60o of full-rise simple harmonic motion (i.e., when the angle  *  60o
or the cam angle θ  270 ) then the displacement of the follower center, from Equation (8a), is
y  0 in 
2 in
1  cos    2 in
2
(8c)
(i) 3 Points. At The first, second, and third-order kinematic coefficients of the displacement diagram (or
the lift curve), see Equations (6.12b), (6.12c), and (6.12d), page 291 in the text book, can be written as
y 
L
 *
sin
2

(9a)
y 
2 L
 *
cos

2 2
(9b)
and
and
y  
3 L
 *
sin
2 3

(9c)
Substituting Equations (4), (5), (6), and (7) into Equations (9), the first-order, second-order, and thirdorder kinematic coefficients of the displacement diagram (or the lift curve) are
y  3 sin

4
y  9 cos
  2.121 in/rad

4
and
y   27 sin
  6.364 in/rad 2

4
  19.092 in/rad 3
(10a)
(10b)
(10c)
Partial Check: Note that the slope of the displacement diagram (i.e., the velocity), see Equation (10a), is
positive. From intuition (for a curve with rise) this is known to be correct.
(ii) 8 Points. At the cam angle   225o , the coordinates of the follower center (in the fixed Cartesian
reference frame) are
X FC  X 0    0
(11a)
and
YFC  Y0  y  2 in  0.293 in   2.293 in
(11b)
-7-
The coordinates of the follower center in the moving reference frame can be written as
and
xFC  X FC cos   YFC sin 
(12a)
yFC   X FC sin   YFC cos 
(12b)
Substituting Equations (11a) and (11b), and the cam angle   225o , into Equations (12), the
coordinates of the follower center in the moving reference frame are
xFC  0 cos 225  2.293 sin 225   1.621 in
(13a)
yFC  0 sin 225  2.293 cos 225   1.621 in
(13b)
and
Differentiating Equations (12) with respect to the cam angle, the first-order kinematic coefficients of
the follower center in the moving reference frame are
and
xFC   X FC sin   YFC cos   y sin 
(14a)
yFC   X FC cos   YFC sin   y cos 
(14b)
Therefore, the first-order kinematic coefficients of the follower center in the moving reference frame are
and
xFC  0 sin 225  2.293 cos 225  (  2.121) sin 225   3.121 in/rad
(15a)
  0 cos 225  2.293 sin 225  ( 2.121) cos 225   0.122 in/rad
yFC
(15b)
Differentiating Equations (14) with respect to the cam angle, the second-order kinematic coefficients of
the follower center in the moving reference frame are
and
xFC   X FC cos   YFC sin   2 y cos   y sin 
(16a)
yFC  X FC sin   YFC cos   2 y sin   y cos 
(16b)
Therefore, the second-order kinematic coefficients of the follower center in the moving reference frame
are
  0 cos 225  2.293 sin 225  2   2.121 cos 225  (  6.364) sin 225   5.878 in/rad 2
xFC
(17a)
  0 sin 225  2.293cos 225  2   2.121 sin 225  (  6.364) cos 225   0.121 in/rad 2
y FC
(17b)
and
The coordinates of the point of contact between the cam and the follower can be written as
xcam  xFC 
and
-8-
 
d  yFC


 
2  RFC
(18a)
 xFC
 


 
 RFC
(18b)
   xFC 2  yFC 2
RFC
(19a)
ycam  yFC 
d
2
where
The positive sign is used here because the input cam angle can always be chosen as positive, that is,
counterclockwise. Substituting Equations (15a) and (15b) into Equation (19a) gives
 
RFC
  3.121
2
   0.122    3.123 in
2
(19b)
CHECK: Squaring and adding Equations (14) and substituting into Equation (19a) gives the symbolic
equation that was presented in lecture, that is
   ( y  X FC ) 2  YFC 2
RFC
(19c)
Substituting Equations (10a), (11a) and (11b) into Equation (19c) gives
 
RFC
  2.121  0 
2
   2.293   3.124 in
2
(19d)
Substituting Equation (19b) and the known values into Equations (18a) and (18b), the coordinates of the
point of contact between the cam and the follower are
xcam   1.621 
1   0.122 

   1.601 in
2   3.123 
(20a)
ycam   1.621 
1   3.121 

   1.121 in
2   3.123 
(20b)
and
(iii) 3 Points. The radius of curvature of the pitch curve can be written in terms of the first-order and the
second-order kinematic coefficients of the follower center, as
 PC 
 3
RFC
 yFC
  xFC
 yFC

xFC
(21a)
Substituting the known numerical values into this equation, the radius of curvature of the pitch curve is
  3.123
  89.724 in
 PC 
  3.121 0.121    5.878  0.122 
3
(21b)
The positive sign implies that the center of curvature is along the positive unit normal direction. This
means that the cam profile is concave and may not be a good cam profile.
Check: Substituting Equations (14), (16), and (19c) into Equation (21a), the radius of curvature of the
pitch curve can be written in symbolic form as
 PC 
2 3/2
 [( y  X FC ) 2  YFC
]
YFC ( y  YFC )  ( y  X FC ) (2 y  X FC )
-9-
(22a)
Then substituting the known numerical values into Equation (22a), the radius of curvature of the pitch
curve is
 [(2.121  0) 2  2.2932 ]3/2
 PC 
(22b)
2.293 ( 6.364  2.293)  (2.121  0) [2(  2.121)  0]
Therefore, the radius of curvature of the pitch curve is
 PC 
 3.1243
  90.202 in
9.335  8.997
(22c)
Note that Equation (22c) is in good agreement with Equation (21b). There is a rounding error on the
order of one-half of one percent of the full value.
Since the radius of curvature of the pitch curve is a positive value then the radius of curvature of the
of the cam profile can be written as
d
(23a)
 cam   PC 
2
Substituting Equation (21b) and the diameter of the roller follower into this equation, the radius of
curvature of the cam profile is
 cam   89.724 
1
  90.224 in
2
(23b)
Note that the radius of curvature of the cam profile is a larger positive value than the radius of curvature
of the pitch curve (which agrees with fact that the cam profile is concave).
(iv) 4 Points. The unit tangent vector to the pitch curve can be written as
ut 
xFC ˆ yFC ˆ
i
j


RFC
RFC
(24a)
Substituting the known values into Equation (24a), the unit tangent vector to the pitch curve is
ut 
  3.121 iˆ   0.122 
  3.123   3.123
ˆj
(24b)
that is
ut   0.999 iˆ  0.039 ˆj
(24c)
The unit normal vector to the pitch curve can be written as
uN  
yFC ˆ xFC ˆ
i
j


RFC
RFC
(25a)
Substituting the known values into this equation, the unit normal vector to the pitch curve is
uN  
  0.122  iˆ    3.121
  3.123   3.123
ˆj
(25b)
that is
u N   0.039 iˆ  0.999 ˆj
- 10 -
(25c)
(v) 2 Points. The pressure angle of the cam can be written as
cos  

xFC
y
cos   FC sin 


RFC
RFC
(26a)
Substituting Equations (15) and (19b), and the specified cam angle θ  225, into Equation (26a) gives
cos    0.999 cos 225  0.039 sin 225   0.734
(26b)
Therefore, the pressure angle of the cam (for the given cam angle) is
  42.779
(27)
Check: The pressure angle of the cam (for the given cam angle) can also be written from Equation
(6.32), page 310 in the text book, as



 R2   2  y 
 O

  tan 1 
y  
(28a)
Substituting Equations (15) and (19b) into Equation (28a) gives



 22  02 in  0.293 in 


 2.121 in  0 in
  tan 1 
(28b)
Therefore, the pressure angle of the cam can be written as
 2.121 in 
1
  tan 0.925
 2.293 in 
(28c)
  42.768o
(29)
  tan 1 
The pressure angle of the cam is
Note that Equation (29) is in very good agreement with Equation (27).
The value of the pressure angle is not an acceptable value for this cam-follower system. Recall that a
good cam design should have a pressure angle in the range 0    30 . Therefore, this cam design is
not satisfactory (at least for this cam position of   225o ).
- 11 -