Page 166 - Mechatronics with Experiments

152 MECHATRONICS
the input shaft speed is constant, the time derivatives and derivatives with respect to input
shaft angle are related to each other as

3
2
dx d x d d x
̇ 2
̇ 3
̇
̇ x = ; ̈ x = ( ) ; ̈ x = ( ) (3.125)
d d 2 dt d 3
We will assume that the cam has rise and fall periods without any dwell period in between
them. For one half of a cycle (rise period), the cam profile is defined in three sections
1
as a function of input shaft displacement which are regions 0 ≤ ≤ , 1 ≤ ≤
8 rise 8 rise
7 , 7 ≤ ≤ .
8 rise 8 rise rise
It can be shown that the modified sine cam function in the rise region is as follows
(Cam ( )),
1
( )
2
d x( ) 4 1
= A ⋅ sin ;0 ≤ ≤ rise (3.126)
o
d 2 rise 8
( ) ( ( ))
dx( ) A 4
o rise
= ⋅ 1 − cos (3.127)
d 4 rise
( ( ))
A rise 4
o rise
x( ) = ⋅ − sin (3.128)
4 4
rise
Likewise, the cam functions for the other period of motion are defined as follows (Cam ( )),
2
( )
2
d x( ) 4 1 7
= A ⋅ sin + ; rise ≤ ≤ rise (3.129)
o
d 2 3 rise 3 8 8
( )( ( ))
dx( ) 3 rise 4
= V − A ⋅ cos + (3.130)
o
s1
d 4 3 rise 3
( ( ))
2
x( ) = X + V ⋅ − A ⋅ (3 ) ⋅ sin 4 + (3.131)
s1 s1 o rise
3 3
rise
7
and similarly, for the period of rise ≤ ≤ rise ,(Cam ( ))
3
8
2
d x( ) 7
= A ⋅ sin(4 ∕ rise − 2 ); rise ≤ < rise (3.132)
o
d 2 8
( )( ( ))
dx( ) rise 4
= V − A ⋅ cos − 2 (3.133)
d s2 o 4 rise
( ) 2 ( ( ))
rise 4
x( ) = X + V ⋅ − A ⋅ ⋅ sin − 2 (3.134)
s2 s2 o
4
rise
where the constants X , X , V , V s2 are determined to meet the continuity requirements
s1
s1
s2
at the boundaries of the cam function sections. Using the total travel range of the follower,
we can determine the constant A as a function of the total travel range specified. Below
o
are the five equations from which the above five constants can be calculated.
( 1 ) ( 1 )
Cam = Cam (3.135)
1 rise 2 rise
8 8
( 7 ) ( 7 )
Cam 2 rise = Cam 3 rise (3.136)
8 8
d ( ( 1 )) d ( ( 1 ))
Cam 1 rise = Cam 2 rise (3.137)
dt 8 dt 8
d ( ( 7 )) d ( ( 7 ))
Cam 2 rise = Cam 3 rise (3.138)
dt 8 dt 8
Cam ( rise ) = x rise (3.139)
3