Calculating the damping c based on an assumed value of  requires specifying a value for the overall system k and for its effective mass. The choice of k will affect both the natural frequency of the system for a given mass and the available force to keep the joint closed. Some iteration will probably be needed to find a good compromise. A selection of data for commercially available helical coil springs is provided in Appendix D. Note in equations 9.10 that the terms involving acceleration and velocity can be either positive or negative. The terms involving the spring parameters k and Fpl are the only ones that are always positive. To keep the overall function always positive requires that the spring force terms be large enough to counteract any negative values in the other terms. Typically, the acceleration is larger numerically than the velocity, so the negative acceleration usually is the principal cause of a negative force Fc.

The principal concern in this analysis is to keep the cam force always positive in sign as its direction is defined in Figure 9-1. The cam force is shown as positive in that figure. In a force-closed system the cam can only push on the follower. It cannot pull. The follower spring is responsible for providing the force needed to keep the joint closed during the negative acceleration portions of the follower motion. The damping force has an effect, but the spring must supply the bulk of the force to maintain contact between the cam and follower. If the force Fc goes negative at any time in the cycle, the follower and cam will part company, a condition called follower jump . When they meet again, it will be with large and potentially damaging impact forces. The follower jump, if any, will usually occur near the point of maximum negative acceleration. Thus, we must select the spring constant and preload to guarantee a positive force at all points in the cycle. In automotive engine valve cam applications, follower jump is also called valve float , because the valve (follower) “floats” above the cam, also periodically impacting the cam surface (sometimes called valve crash ). This will occur if the cam rpm is increased to the point that the larger negative acceleration makes the follower force negative. The “redline” maximum engine rpm often indicated on its tachometer is to warn of impending valve float above that speed that will damage the cam and follower.

Program DYNACAM allows the iteration of equation 9.10 to be done quickly for any cam whose kinematics have been defined in that program. The program's Dynamics button will solve equation 9.10 for all values of camshaft angle, using the displacement, velocity, and acceleration functions previously calculated for that cam design in the program. The program requires values for the effective system mass m , effective spring constant k , preload Fpl , and the assumed value of the damping ratio . These values need to be determined for the model by the designer using the methods described in Sections 8.9 and 8.10 (pp. 185-197). The calculated force at the cam-follower interface can then be plotted or its values printed in tabular form. The system’s first natural frequency, based on the model of Figure 9-1, is also reported when the tabular force data are printed.

EXAMPLE 9-2

Kinetostatic Force Analysis of a Force-Closed (Spring-Loaded) Cam-Follower

System.

Given: A translating roller follower as shown in Figure 9-1 is driven by a forceclosed

radial plate cam which has the following program:

Segment 1: Rise 1 inch in 50 ° with modified sine acceleration

Segment 2: Dwell for 40 °

Segment 3: Fall 1 inch in 50 ° with cycloidal displacement

Segment 4: Dwell for 40 °

Segment 5: Rise 1 inch in 50 ° with 3-4-5 polynomial displacement

Segment 6: Dwell for 40 °

Segment 7: Fall 1 inch in 50 ° with 4-5-6-7 polynomial displacement

Segment 8: Dwell for 40 °

Camshaft angular velocity is 18.85 rad/sec

Follower effective mass is 0.0738 lb-sec2/in (blobs)

Damping is 10% of critical ( = 0.10)

Problem: Compute the necessary spring constant and spring preload to maintain contact between cam and follower, and calculate the dynamic force function for the cam. Calculate the system natural frequency with the selected spring. Keep the pressure angle under 30 ° .