Page 41 - Cam Design Handbook

THB2 8/15/03 12:48 PM Page 29

BASIC CURVES 29

The instantaneous angular rate of change of velocity
dy
2
y ¢¢ = = follower acceleration. (2.6)
dq 2
The instantaneous cam angle rate of change of acceleration
3
dy
y ¢¢¢ = = follower jerk. (2.7)
dq 3
By the use of Eq. (2.2) the follower characteristics can be expressed as direct time
dependent as follows:

The follower velocity can be written as
)(
w
˙ y = dy dt = ( dq dt dy d ) = ( dy d ) =q w y¢ (2.8)
q
the follower acceleration
d Ê dy ˆ d d Ê dy ˆÊ dq ˆ
˙˙ y = d y dt = w = w = w
2
2
dt Ë d ¯ dq d Ë q dt ¯
d ¯Ë
q
q
q
= w 2 ( dy d ) = w 2 y¢¢ (2.9)
2
2
and the follower jerk
3
Ê dy ˆ
˙˙˙ y = w 3 = w 3 y¢¢¢ . (2.10)
Ë dq 3 ¯
These equations facilitate converting from one family of dimensional units to another.
By having the cam profiles in the mathematical form of Eq. (2.1) or Eq. (2.2) we also can
easily find the other characteristics, by differentiation.
From the foregoing equations, we see that each successive derivative can be deter-
mined from the slopes of any point on the previous curve. Thus, at any point the slope of
the displacement curve yields the velocity of the follower. The slope of the velocity curve
is the acceleration of the follower, and the slope of the acceleration curve is the jerk of
the follower. This procedure is called the graphical-slope differentiating method. It is pre-
sented for a quick appraisal of the displacement, velocity, acceleration, and jerk curves.
Utilizing the graphical-slope method, the velocity curve is reasonably accurate and the
acceleration curve is at best only an approximation.
With the graphical-slope method, the determination of the signs (positive or negative)
is essential for a fundamental understanding of the follower motion and dynamics. Let us
establish as positive (+) the displacement of the follower above the lowest point in the dis-
placement diagram (Fig. 2.1a). Therefore, the velocity is positive if the follower moves
in the direction of positive displacement and negative if it moves in the opposite direc-
tion. That is, velocity is positive on the rise and negative on the fall. The acceleration
follows a similar convention. Therefore the acceleration is positive when its direction is
that of the positive displacement. Throughout this book, the word deceleration is not used
because it does not indicate the direction of acceleration and would therefore cause con-
fusion regarding the direction of inertia load. The slope of the curves at any point gives
us another method for determining signs. Sloping upward to the top right the slope is pos-
itive, and sloping upward to the top left the slope is negative. Also, a vertical slope at a
point in any of the curves has an infinite value and is called a discontinuity because it has
two values at the point being studied.
Some concepts discussed in this chapter are: