Page 161 - Mechatronics with Experiments

MECHANISMS FOR MOTION TRANSMISSION 147
The displacement, velocity, acceleration, and jerk (time derivative of acceleration) are
important since they directly affect the forces experienced in the mechanism. In cam design,
quite often we focus on the first, second, and third derivatives of the cam function with
respect to instead of t, time. In the final analysis, we are interested in the time derivative
values, since they determine the actual speed, acceleration, and jerk. The relationships
between the derivatives of cam function with respect to and t are as follows.

dx df( ) dx dx
= ; = ̇ (3.86)
d d dt d
2
2
2
2
d x d f( ) d x d x dx
̇ 2
= ; = + ̈ (3.87)
d 2 d 2 dt 2 d 2 d
3
3
3
2
3
d x d f( ) d x d x d x ̇ ̈ dx d ̈
̇ 3
= ; = + (3 ⋅ ⋅ ) + (3.88)
d 3 d 3 dt 3 d 3 d 2 d dt
̇
̈
̈
Notice that when = constant, then = 0, and d ∕dt = 0. When the input shaft of the cam
speed is constant, which is the case in most applications, then the relationship between the
time derivatives and derivatives with respect to of the cam function simplify as follows,
dx df( ) dx dx ̇
= ; = (3.89)
d d dt d
2
2
2
2
d x d f( ) d x d x
= ; = ̇ 2 (3.90)
d 2 d 2 dt 2 d 2
3
3
3
3
d x d f( ) d x d x ̇ 3
= ; = (3.91)
d 3 d 3 dt 3 d 3
Significant effort is made in selecting the f( ) cam function in order to shape the first,
second, and even the third derivative of it so that desired results (i.e., minimize vibrations)
are obtained from the cam motion. As a general rule in cam design, the cam function
should be chosen so that the cam function, first, and second derivatives (displacement,
speed, and acceleration functions) are continuous and the third derivative (jerk function)
discontinuities (if any) are finite. In general these continuity conditions are applied to
cam function derivatives with respect to . If the input cam speed is constant, the same
continuity conditions are then satisfied by the time derivatives as well. For instance, a cam
function with trapezoidal shape results in discontinuous velocity and infinite accelerations.
Therefore, pure trapezoidal cam profiles are almost never used. The alternatives for the
selection of functions that meet the continuity requirements in the cam function are many.
Some functions are defined in analytical form (parts of the cam function are portions of the
sinusoidal functions joined to form a continuous function) and some are custom developed
by experimentation and stored in numerical form, that is two column data of versus x .
i i
The four types of basic cam functions are discussed below (Figure 3.10).
1. Cycloidal displacement cam function has an acceleration curve for the rise portion
that is sinusoidal with single frequency,
2
d x( )
= C ⋅ sin( f ) (3.92)
1
1
d 2
where C is constant related to the displacement range of x and f is determined by
1
1
the portion of input shaft rotation to complete the rise portion of cam motion.For