Page 165 - Mechatronics with Experiments

MECHANISMS FOR MOTION TRANSMISSION 151
acceleration profile function includes sinusoidal functions at the corners of the curve
and constant acceleration functions in other segments of the cam. Let rise be the
period of input shaft rotation for the rise period. The modified trapezoidal accelera-
tion profile is formed by constructing a function where the pieces of it are obtained
from a sinusoidal function with period rise ∕2 and constant acceleration function.
Let us consider a modified trapezoidal acceleration function. The acceleration
function has positive and negative (symmetric to the positive) portions. The variable
portions of the acceleration function are implemented with portions of the sinu-
soidal, and constant acceleration portions are implemented with a constant value
(Figure 3.16),

2
d x
= A ⋅ sin(2 ∕( ∕2)); for 0 ≤ ≤ ∕8 (3.119)
d 2 o rise rise
2
d x 1 3
= A ; for rise ≤ ≤ rise (3.120)
o
d 2 8 8
2
d x 3 5
= A ⋅ sin(2 ∕( ∕2) − ); for ≤ ≤ (3.121)
d 2 o rise 8 rise 8 rise
2
d x 5 7
=−A ; for rise ≤ ≤ rise (3.122)
o
d 2 8 8
2
d x 7
= A ⋅ sin(2 ∕( ∕2) − 2 ); for ≤ ≤ (3.123)
d 2 o rise 8 rise rise
Again, the acceleration functions describe half of the displacement cycle of the
follower. For a symmetric return function, the mirror image of the same acceleration
function is implemented in the cam profile.
4. Polynominal functions of many different types, that is including third, fourth, fifth,
sixth, or seventh order polynomials, are used for cam functions.
3
2
x( ) = C + C + C + C + ... + C n (3.124)
1
3
0
2
n
Typically, the input shaft is driven at a constant speed, and output follower motion is
obtained as a cyclic function of input shaft rotation. Before the development of computer
controlled programmable machines, almost all of the automated machines had one large
power source driving a shaft. Many different cams generated desired motion profiles at
individual stations from the same shaft. Hence, all the stations are synchronized to a master
shaft. The synchronization is fixed as a result of the machined profiles of cams. If a different
automated control is needed, cam profiles had to be physically changed. As programmable
computer controlled machines replace the fixed automation machines, the use of mechanical
cams has decreased. Instead, cam functions are implemented in software to synchronize
independently controlled motion axes. If a different type of synchronization is required, all
we need to do is to change the cam software. In mechanical cam synchronized systems, we
would have to physically change the cam.

Example Consider a cam with its follower as shown in Figure 3.9. Let us consider
the displacement, speed, acceleration, and jerk profiles for modified sine cam functions.
̈
̈
d
Assume that the input shaft of the cam runs at a constant speed, hence = 0, dt = 0.0.
̇
̇
Assume that nominal operating speed of input shaft of the cam is = 2 ⋅ 10 rad∕s( =
10 rev∕s = 600 rev∕min ), the total displacement of the cam follower is 2.0 in, and the period
of cam motion for a complete up–down cycle is once per revolution of the input shaft. Since