Page 164 - Mechatronics with Experiments

150 MECHATRONICS
dx( ) C 1 1
=− + C 1 ⋅ cos( f (2 rise + dwell − )) (3.111)
1
d f 1 f 1
2
d x( )
= C ⋅ sin( f (2 rise + dwell − )) (3.112)
1
1
d 2
2. Modified sine function is a modified version of the first function, cycloidal displace-
ment function. Compared to the original version, this version results in lower peak
acceleration and speed values while maintaining a similar cam displacement function.
This cam function uses at least two frequencies of the sinusoidal profile. Pieces of
the two sine functions are combined to form a smooth cam function.
2
d x( )
= C ⋅ sin( f ) + C ⋅ sin( f ) (3.113)
2
2
1
1
d 2
where C , C constants would have non-zero values in some segments and zero values
1
2
in other segments of the cam. In other words, in some segments of the cam function
one of the sections of the sinusoidal function is used, in other segments the other
sinusoidal function is used with its appropriate segment connecting to the preceeding
and the following curve. Let rise be the period of input shaft rotation for the rise
period. f , f are two constant frequencies.
2
1
2 4
f = = (3.114)
1
∕2
rise rise
2 4
f = = (3.115)
2
3 ∕2 3
rise rise
The acceleration function is constructed from the segments of two sinusoidal func-
tions as follows,
( )
2
d x( ) 2 1
= A ⋅ sin ; for 0 ≤ ≤ rise (3.116)
o
d 2 ( rise ∕2) 8
( )
2
d x( ) 2 1 7
= A ⋅ sin + ( ∕3) ; for rise ≤ ≤ rise (3.117)
o
d 2 (3 rise ∕2) 8 8
2
d x( ) ( 2 ) 7
= A ⋅ sin − 2 ; for rise ≤ ≤ rise (3.118)
o
d 2 ( rise ∕2) 8
The acceleration function like this describes the movement of the follower for the rise
period. If the follower cycle is made of rise and fall without any dwell periods, the
complete displacement cycle is performed over an input shaft rotation of = 2 rise .
Displacement and speed curves are determined by directly integrating the acceleration
curve with zero initial condition for speed and zero initial condition for displacement
at the beginning of integration.
The fall motion curves are simply obtained using a mirror image of the rise motion
curves. During the dwell portion (if used in the cam design), the cam follower
displacement stays constant, and speed and acceleration are zero.
3. Modified trapezoidal acceleration function modifies the standard trapezoidal acceler-
ation function around the points where the acceleration changes slope. The accelera-
tion function is smoothed out with a smooth function, such as a sinusoidal function,
hence eliminating the jerk discontinuity. If purely trapezoidal acceleration profiles
are used, there would be discontinuity in the jerk function when the acceleration
function slope changes from a finite value to zero value (constant acceleration).
In order to reduce the effect of this in terms of vibrations in the mechanism, the