Back Lash and Describing Function for second order system

Cuthbert Nyack
Backlash can occur in geared positional systems where there is some play between the teeth of the gears. This effect adds a phase delay to the system and can make a second order system oscillatory. Describing function analysis can be used to analyse the behaviour of this system. More information on describing function analysis can be found in Modern Control Engineering by K Ogata, published by Prentice Hall. In describing function analysis a Fourier series analysis is done of the backlash transfer characteristics to derive a function relating the magnitude and phase of the fundamental component in the fourier series. The result can be plotted as shown in the red curve in the applet below. This is plotted with the magnitude vs phase of the linear transfer function(yellow plot below).

The frequency of the point of intersection is determined, in this case w = 1.57. The amplitude of the backlash vs phase is also plotted with the same axes(green curve). The intersection of the w line with this curve give the amplitude of the backlash which is also the amplitude of the resulting oscillations(in this case ~ 2).

Gif image below shows how applet should appear.

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COPYRIGHT 2006 Cuthbert Nyack.