Page 257 - Cam Design Handbook

THB8 9/19/03 7:25 PM Page 245

CAM MECHANISM FORCES 245

mission index can take throughout the motion of the mechanism. The overall size of the
mechanism is not important as this is a function generation problem. Consequently, the
nondimensional ratios of (r/d) and (d/a) are used. The value of d determines the size of
the mechanism. The procedure consists of the following steps.
Step 1: For the specified minimum value of (r/d) and a chosen value of (d/a), using
Eq. (8.27), the two segments of n vs q curve are drawn. This determines the permissible
zone as the region enclosed between the two curves for specifying the motion in the first
half of the cycle. Drawing the (-dh/dq) vs q curve in the same graph aids the visualiza-
tion of the motion in the remaining half of the cycle.
Step 2: Depending on the nature of the desired function f(q), the n vs q can be speci-
fied within the permissible region. The (d/a) ratio can be changed if necessary. Second-
order (acceleration) and third-order (jerk or shock) derivatives of the function f(q) can be
controlled in this process. Continuity of the function f(q) and its derivatives can also be
ensured. Harmonic, cycloidal, polynomial, spline, and other types of curves can be used
to specify the n vs q curve to satisfy all of the design objectives in the design of the cam.
Then, the motion in the second half of the cycle is easily computed using Eq. (8.17). The
(-dh/dq) vs q curve should be appropriately used to control the motion in the second half
of the cycle by specifying a curve in the first half.
Step 3: The rotation of the cam in the entire 360° range can be readily obtained by
computing the cumulative area under the n vs q curve from 0 to 360°. This can be done
either analytically or numerically depending on how the n vs q curve is specified. Note
that the analytical expression for (-dh/dq) is readily available from Eq. (8.19).
Step 4: Using the kinematic inversion technique the endpoint of the two-link open serial
chain can be made to trace the pitch curve of the cam profile as per the function f(q)
obtained in step 3. The finite radius of the roller is then used to obtain the actual profile
of the cam as an envelope of all the positions of the rollers. This can be done by com-
puting the equidistant offset curve of the pitch profile of the cam using the numerically
determined normal at the point of contact. In practice, by choosing a milling cutter of the
same radius as the roller, the pitch profile is sufficient.
All of the above steps are implemented in Matlab (2000) and the entire design proce-
dure automated. The Matlab scripts generate the G-code for the computer numerically con-
trolled Fadal vertical machining center to manufacture the cam.

8.12.4 Force Closure
Form closure and force closure are two ways in which the cam and roller can be kept in
constant contact even at high speeds. Force closure with a helical extension spring is
disscussed here.
Figure 8.23 shows the arrangement of the spring. Three parameters—s 1 , s 2 , and y—
are identified to locate the spring on the roller crank and the cam. These three parameters
are determined to meet the following objectives. As the mechanism is in motion, the length
of the spring should not change excessively as this will impose additional loads on the
mechanism. The direction of the spring force should be such that it counteracts the sepa-
rative forces acting on the cam and roller. An optimization problem was solved for the
single dwell cam to minimize the maximum variation of the spring length with s 1 , s 2 , and
y as the design variables. The Nelder-Mead simplex algorithm was used for this purpose.

8.12.5 Quasikinetostatic Analysis

In this section, the expressions for computing the input torque required at the roller crank
are presented. If the mechanism operates at high speeds, inertia forces should also be taken