DESIGN OF MACHINERY SOLUTION MANUAL 8-8-1

PROBLEM 8-8
Statement: Design a double-dwell cam to move a follower from 0 to 1.5 in in 45 deg, dwell for 150 deg, fall 1.5 in
in 90 deg and dwell for the remainder. The total cycle must take 6 sec. Choose suitable programs
for rise and fall to minimize velocities. Plot the SVAJ diagrams.
Given: RISE DWELL FALL DWELL
|
1
45 deg := |
2
150 deg := |
3
90 deg := |
4
75 deg :=
h
1
1.5 in := h
2
0 in := h
3
1.5 in := h
4
0 in :=
Cycle time: t 6 sec :=
Solution: See Mathcad file P0808.
1. The camshaft turns 2t rad during the time for one cycle. Thus, its speed is
e
2 t rad
t
:= e 1.047
rad
sec
=
2. The modified sinusoidal motion is defined in local coordinates by equations 8.15 through 8.19. The numerical
constants in these SCCA equations are given in Table 8-2.
b 0.25 := c 0.00 := d 0.75 :=
C
a
5.5280 :=
3. The SCCA equations for the rise or fall interval (|) are divided into 5 subintervals. These are:
for 0 <= x <= b/2 where, for these equations, x is a local coordinate that ranges from 0 to 1,
y
1
x ( ) C
a
x
b
t

b
t
|

\
|
|
.
2
sin
t
b
x
|

\
|
|
.

(
(

:= y'
1
x ( ) C
a
b
t
1 cos
t
b
x
|

\
|
|
.

\
|
|
.
:=
y''
1
x ( ) C
a
sin
t
b
x
|

\
|
|
.
:= y'''
1
x ( ) C
a
t
b
cos
t
b
x
|

\
|
|
.
:=
for b/2 <= x <= (1 - d)/2
y
2
x ( ) C
a
x
2
2
b
1
t
1
2

\
|
|
.
x + b
2 1
8
1
t
2

\
|
|
.
+

(
(
(

:= y'
2
x ( ) C
a
x b
1
t
1
2

\
|
|
.
+

(
(

:=
y''
2
x ( ) C
a
:= y'''
2
x ( ) 0 :=
for (1 - d)/2 <= x <= (1 + d)/2
y
3
x ( ) C
a
b
t
c
2
+
|

\
|
|
.
x
d
t
|

\
|
|
.
2
+ b
2 1
8
1
t
2

\
|
|
.
+
1 d ( )
2
8

d
t
|

\
|
|
.
2
cos
t
d
x
1 d
2

\
|
|
.

(
(

(
(
(

:=
y'
3
x ( ) C
a
b
t
c
2
+
d
t
sin
t
d
x
1 d
2

\
|
|
.

(
(

(
(

:=
y''
3
x ( ) C
a
cos
t
d
x
1 d
2

\
|
|
.

(
(

:= y'''
3
x ( ) C
a

t
d
sin
t
d
x
1 d
2

\
|
|
.

(
(

:=
DESIGN OF MACHINERY SOLUTION MANUAL 8-8-2
for (1 + d)/2 <= x <= 1 - b/2
y
4
x ( ) C
a
x
2
2

b
t
1 +
b
2

\
|
|
.
x + 2 d
2
b
2
( )
1
t
2
1
8

\
|
|
.
+
1
4

(
(
(

:=
y'
4
x ( ) C
a
x
b
t
+ 1 +
b
2

\
|
|
.
:= y''
4
x ( ) C
a
:= y'''
4
x ( ) 0 :=
for 1 - b/2 <= x <= 1
y
5
x ( ) C
a
b
t
x
2 d
2
b
2
( )
t
2
+
1 b ( )
2
d
2

4
+
b
t
|

\
|
|
.
2
sin
t
b
x 1 ( )

(
(

(
(
(

:=
y'
5
x ( ) C
a
b
t
1 cos
t
b
x 1 ( )

(
(

(
(

:=
y''
5
x ( ) C
a
sin
t
b
x 1 ( )

(
(

:= y'''
5
x ( ) C
a
t
b
cos
t
b
x 1 ( )

(
(

:=
4. The above equations can be used for a rise or fall by using the proper values of u, |, and h. To plot the SVAJ
curves, first define a range function that has a value of one between the values of x1 and x2 and zero elsewhere.
R x x1 , x2 , ( ) if x x1 > ( ) x x2 s ( ) . 1 , 0 , [ ] :=
5. The global SVAJ equations are composed of four intervals (rise, dwell, fall, and dwell). The local equations
above must be assembled into a single equation each for S, V, A, and J that applies over the range 0 <= u <= 360
deg.
6. Write the local svaj equations for the first interval, 0 <= u <= |
1
. Note that each subinterval function is multiplied
by the range function so that it will have nonzero values only over its subinterval.
For 0 <= u <= |
1
(Rise)
s
1
x ( ) h
1
R x 0 ,
b
2
,
|

\
|
|
.
y
1
x ( ) R x
b
2
,
1 d
2
,
|

\
|
|
.
y
2
x ( ) + R x
1 d
2
,
1 d +
2
,
|

\
|
|
.
y
3
x ( ) +
R x
1 d +
2
, 1
b
2
,
|

\
|
|
.
y
4
x ( ) R x 1
b
2
, 1 ,
|

\
|
|
.
y
5
x ( ) + +
...
|

\
|
|
|
|
.
:=
v
1
x ( )
h
1
|
1
R x 0 ,
b
2
,
|

\
|
|
.
y'
1
x ( ) R x
b
2
,
1 d
2
,
|

\
|
|
.
y'
2
x ( ) + R x
1 d
2
,
1 d +
2
,
|

\
|
|
.
y'
3
x ( ) +
R x
1 d +
2
, 1
b
2
,
|

\
|
|
.
y'
4
x ( ) R x 1
b
2
, 1 ,
|

\
|
|
.
y'
5
x ( ) + +
...
|

\
|
|
|
|
.
:=
a
1
x ( )
h
1
|
1
2
R x 0 ,
b
2
,
|

\
|
|
.
y''
1
x ( ) R x
b
2
,
1 d
2
,
|

\
|
|
.
y''
2
x ( ) + R x
1 d
2
,
1 d +
2
,
|

\
|
|
.
y''
3
x ( ) +
R x
1 d +
2
, 1
b
2
,
|

\
|
|
.
y''
4
x ( ) R x 1
b
2
, 1 ,
|

\
|
|
.
y''
5
x ( ) + +
...
|

\
|
|
|
|
.
:=
j
1
x ( )
h
1
|
1
3
R x 0 ,
b
2
,
|

\
|
|
.
y'''
1
x ( ) R x
b
2
,
1 d
2
,
|

\
|
|
.
y'''
2
x ( ) + R x
1 d
2
,
1 d +
2
,
|

\
|
|
.
y'''
3
x ( ) +
R x
1 d +
2
, 1
b
2
,
|

\
|
|
.
y'''
4
x ( ) R x 1
b
2
, 1 ,
|

\
|
|
.
y'''
5
x ( ) + +
...
|

\
|
|
|
|
.
:=
DESIGN OF MACHINERY SOLUTION MANUAL 8-8-3
7. Write the local svaj equations for the second interval, |
1
<= u <= |
1
+ |
2
. For this interval, the value of S is the
value of S at the end of the previous interval and the values of V, A, and J are zero because of the dwell.
For |
1
<= u <= |
1
+ |
2
s
2
x ( ) h
1
:= v
2
x ( ) 0 := a
2
x ( ) 0 := j
2
x ( ) 0 :=
8. Write the local svaj equations for the third interval, |
1
+ |
2
<= u <= |
1
.+ |
2
+ |
3
.
For |
1
+ |
2
<= u <= |
1
.+ |
2
+ |
3
s
3
x ( ) h
3
1 R x 0 ,
b
2
,
|

\
|
|
.
y
1
x ( ) R x
b
2
,
1 d
2
,
|

\
|
|
.
y
2
x ( ) + R x
1 d
2
,
1 d +
2
,
|

\
|
|
.
y
3
x ( ) +
R x
1 d +
2
, 1
b
2
,
|

\
|
|
.
y
4
x ( ) R x 1
b
2
, 1 ,
|

\
|
|
.
y
5
x ( ) + +
...
|

\
|
|
|
|
.

(
(
(
(

:=
v
3
x ( )
h
3
|
3
R x 0 ,
b
2
,
|

\
|
|
.
y'
1
x ( ) R x
b
2
,
1 d
2
,
|

\
|
|
.
y'
2
x ( ) + R x
1 d
2
,
1 d +
2
,
|

\
|
|
.
y'
3
x ( ) +
R x
1 d +
2
, 1
b
2
,
|

\
|
|
.
y'
4
x ( ) R x 1
b
2
, 1 ,
|

\
|
|
.
y'
5
x ( ) + +
...
|

\
|
|
|
|
.
:=
a
3
x ( )
h
3
|
3
2
R x 0 ,
b
2
,
|

\
|
|
.
y''
1
x ( ) R x
b
2
,
1 d
2
,
|

\
|
|
.
y''
2
x ( ) + R x
1 d
2
,
1 d +
2
,
|

\
|
|
.
y''
3
x ( ) +
R x
1 d +
2
, 1
b
2
,
|

\
|
|
.
y''
4
x ( ) R x 1
b
2
, 1 ,
|

\
|
|
.
y''
5
x ( ) + +
...
|

\
|
|
|
|
.
:=
j
3
x ( )
h
3
|
3
3
R x 0 ,
b
2
,
|

\
|
|
.
y'''
1
x ( ) R x
b
2
,
1 d
2
,
|

\
|
|
.
y'''
2
x ( ) + R x
1 d
2
,
1 d +
2
,
|

\
|
|
.
y'''
3
x ( ) +
R x
1 d +
2
, 1
b
2
,
|

\
|
|
.
y'''
4
x ( ) R x 1
b
2
, 1 ,
|

\
|
|
.
y'''
5
x ( ) + +
...
|

\
|
|
|
|
.
:=
9. Write the local svaj equations for the fourth interval, |
1
+ |
2
+ |
3
<= u <= |
1
.+ |
2
+ |
3
+ |
4
. For this interval, the
values of S, V, A, and J are zero because of the dwell.
For |
1
+ |
2
+ |
3
<= u <= |
1
.+ |
2
+ |
3
+ |
4
s
4
x ( ) 0 := v
4
x ( ) 0 := a
4
x ( ) 0 := j
4
x ( ) 0 :=
10. Write the complete global equation for the displacement and plot it over one rotation of the cam, which is the
sum of the four intervals defined above.
Let u
1
|
1
:= u
2
u
1
|
2
+ := u
3
u
2
|
3
+ := u
4
u
3
|
4
+ :=
S u ( ) s
1
u
u
1
|

\
|
|
.
R u u
1
, u
2
, ( ) s
2
u u
1

u
2
u
1

\
|
|
.
+
R u u
2
, u
3
, ( ) s
3
u u
2

u
3
u
2

\
|
|
.
R u u
3
, u
4
, ( ) s
4
u u
3

u
4
u
3

\
|
|
.
+ +
... :=
u 0 deg 1 deg , 360 deg .. :=
DESIGN OF MACHINERY SOLUTION MANUAL 8-8-4
0 60 120 180 240 300 360
0
0.5
1
1.5
2
DISPLACEMENT, S
Cam Rotation Angle, deg
D
i
s
p
l
a
c
e
m
e
n
t
,
i
n
S u ( )
in
u
deg
11. Write the complete global equation for the velocity and plot it over one rotation of the cam, which is the sum of
the four intervals defined above.
V u ( ) v
1
u
u
1
|

\
|
|
.
R u u
1
, u
2
, ( ) v
2
u u
1

u
2
u
1

\
|
|
.
+
R u u
2
, u
3
, ( ) v
3
u u
2

u
3
u
2

\
|
|
.
R u u
3
, u
4
, ( ) v
4
u u
3

u
4
u
3

\
|
|
.
+ +
... :=
0 60 120 180 240 300 360
2
0
2
4
VELOCITY, V
Cam Rotation Angle, deg
V
e
l
o
c
i
t
y
,
i
n
V u ( )
in
u
deg
12. Write the complete global equation for the acceleration and plot it over one rotation of the cam, which is the sum
of the four intervals defined above.
A u ( ) a
1
u
u
1
|

\
|
|
.
R u u
1
, u
2
, ( ) a
2
u u
1

u
2
u
1

\
|
|
.
+
R u u
2
, u
3
, ( ) a
3
u u
2

u
3
u
2

\
|
|
.
R u u
3
, u
4
, ( ) a
4
u u
3

u
4
u
3

\
|
|
.
+ +
... :=
DESIGN OF MACHINERY SOLUTION MANUAL 8-8-5
0 60 120 180 240 300 360
20
10
0
10
20
ACCELERATION, A
Cam Rotation Angle, deg
A
c
c
e
l
e
r
a
t
i
o
n
,
i
n
A u ( )
in
u
deg
13. Write the complete global equation for the jerk and plot it over one rotation of the cam, which is the sum of the
four intervals defined above.
J u ( ) j
1
u
u
1
|

\
|
|
.
R u u
1
, u
2
, ( ) j
2
u u
1

u
2
u
1

\
|
|
.
+
R u u
2
, u
3
, ( ) j
3
u u
2

u
3
u
2

\
|
|
.
R u u
3
, u
4
, ( ) j
4
u u
3

u
4
u
3

\
|
|
.
+ +
... :=
0 60 120 180 240 300 360
100
0
100
200
300
JERK, J
Cam Rotation Angle, deg
J
e
r
k
,
i
n
J u ( )
in
u
deg