Partial cancellation of shaking force harmonics by cam modification

Abstract

An optimization procedure for minimizing the shaking force harmonics of machinery is described. Because of constraints imposed by the specification, optimization is unlikely to make any significant reduction in the fundamental, machine frequency, term of the shaking force. That term, normally the largest, must be substantially reduced by other means for the procedure described here to be cost effective. Earlier work describing methods of reducing the fundamental term are cited. An extrusion press machine incorporating two planar mechanisms, one crank driven and one cam driven, provides an example. Two designs of the cam-driven mechanism, referred to here as cams 1 and 2, are in use in installed extrusion press machines. Harmonic analysis of the shaking force as a function of time shows that cam 2 produces smaller amplitudes of shaking force harmonics. The ten variables used in the optimization process are the amplitudes and angles of the first five terms of a trigonometric series defining the acceleration of a reciprocating mass driven by a hypothetical third cam, cam 3, which is similar to cam 2 but stripped of all harmonics of order higher than the fifth. The objective function is the sum of the amplitudes of the horizontal components of the second to fifth shaking force harmonics on the machine as a consequence of accelerations of the moving parts of both mechanisms. Several equality and inequality constraints must be met. For the extrusion press machine the result is a reduction in the amplitudes that would be created if cam 3 were to be used by approximately 10, 75, 65 and 100 per cent respectively. The improvement over cam 1 is greater. The procedure is particularly suited as a retrofit measure where the number of installed machines is large in relation to the future rate of production of new machines: a cam that has been removed from one machine can be re-cut for use on another machine.