The effects of backlash and of speed-dependent friction on the stability of closed-cycle control systems

The possibility of hunting in closed-cycle control systems is often contributed to by: (a) backlash in gearing which is part of the main sequence of the control. (b) friction that falls with increase in speed over some range of low speeds. It is shown how, to an approximation sufficiently close for practical purposes, the effect of each of these features may be introduced into the usual treatment of stability by the loop response-vector locus or Nyquist diagram. For this purpose, vector loci are provided that give the additional phase-shift and amplitude ratio due to backlash and speed-dependent friction respectively, for sinusoidal inputs of any frequency or amplitude. The effects of the harmonics that are also produced because of the essentially non-linear nature of these phenomena are disregarded. The general method is based on unpublished work of the late Professor P.J. Daniell, who provided an analytical treatment of the effect of backlash of which the present paper is essentially an interpretation in geometrical terms.