Thomas Industrial Library

Equation 9.12 (p. 241) gives the applied torque Tc in a camshaft. Equation 9.10b (p. 233) gives the follower force for a force-closed system. Combining them gives

The compensating cam needs to provide an equal and opposite torque Tb to cancel the applied torque

where the displacement of the compensating dummy follower is designated as y to distinguish it from the primary follower’s displacement x .

Equating the two torques gives

The terms on the left-hand side (LHS) of equation 9.21 are all known. The factors mb , cb , kb , and the initial displacement y 0 on the right-hand side (RHS) are all under the control of the designer. The goal is to solve for the displacement function y of the compensating cam. To do so requires iteration. Aviza[3] found that for realistic values of driving cam parameters, if the acceleration of the compensating cam follower train were included in equation 9.21, the numerical method failed to converge. The equation was successfully solved by assuming a zero mass for the compensating follower, thus eliminating the acceleration term on the RHS.

Figure 9-28 shows the s , v , and a diagrams of a test cam designed and built for this investigation. It is a simple, double-dwell design with a 1-in modified sine rise over 90 ° , dwell for 90 ° , 1-in modified sine fall over 90 ° , and dwell for 90 ° . Figure 9-29 shows the camshaft torque needed to drive a follower mass of 0.004 bl against a follower spring of 30.23 lb/in with a preload of 17.4 lb and 5% of critical damping.

Figure 9-30 shows the resulting s , v , and a diagrams of the required compensating cam to cancel the torque assuming zero follower mass. Figure 9-31a shows the original driving torque superposed on the countertorque from the compensating cam. Figure 9-31b shows their difference—the residual torque in the shaft—which is essentially zero.

However, this computation results from the unrealistic assumption of a zero-mass compensating cam follower train. When a realistic follower train mass for this system of 0.003 bl is applied to the compensating cam train, the resulting torque is as shown in Figure 9-32a, superposed on the original driving cam torque. Figure 9-32b shows their difference or residual torque in the shaft, now about 10% of the original, uncompensated torque. This technique, though approximate, has resulted in a 90% reduction of peak torque in the camshaft. These results are from a dynamic model of the system and were demonstrated experimentally using the apparatus shown in Figure 9-27.[3]

Note that it is possible in some cases for the contour of the calculated compensating cam to have unacceptable pressure angles and/or radii of curvature and thus be impractical to implement. In this study, the driving cam’s radii of curvature and pressure angles were of reasonable values and so resulted in a manufacturable compensating cam .