The Ideal Locations for the Contra-Rotating Shafts of Generalized Lanchester Balancers

Abstract

The inertia of the moving parts of high-speed planar mechanisms creates a shaking force that is variable in magnitude, direction and line of action throughout a cycle. Alternatively, the instantaneous shaking force vector can be represented by an equal parallel force acting through an arbitrarily chosen origin, together with a moment about that origin. The two components of the force and the moment can be computed by kinetostatic analysis at discrete intervals of a cycle to create three arrays.The discrete Fourier transform converts these arrays from the time to the frequency domain. The lowest frequency term of order k = 1 is of cycle frequency. Terms of higher order k > 1 have frequencies that are k times cycle frequency.The shaking force frequency term of order k can be represented by two contra-rotating force vectors of constant, but generally unequal, magnitude rotating at k times cycle frequency. The forces can be eliminated by equal and opposite forces exerted by counterweights mounted on contra-rotating shafts rotating at the same speed. However, if the locations of these shafts are both arbitrarily chosen, a shaking couple of order k will generally remain unbalanced.Fast Fourier transform (FFT) analysis of the array of moments as well as the force components enables suitable shaft locations to be determined in such a way that this couple vanishes. The couple vanishes if the balancing shafts are coaxial with a centre at an invariant location in the plane. The couple can also be eliminated even though the location of either shaft, but not both, is chosen arbitrarily. The location of the second shaft is then determinate and two procedures are explained to determine that location.