Sinusoidal input describing function for hysteresis followed by elementary backlash

The author proposes a new sinusoidal input describing function which accounts for the serial combination of hysteresis followed by elementary backlash in a single nonlinear element. The output of the hysteresis element drives the elementary backlash element. Various analytical forms of the describing function are given, depending on the a/A ratio, where a is the half width of the hysteresis band or backlash gap, and A is the amplitude of the assumed input sinusoid, and on the value of the parameter representing the fraction of a attributed to the backlash characteristic. The negative inverse describing function is plotted on a gain-phase plot, and it is seen that a relatively small amount of backlash leads to domination of the backlash character in the describing function. The extent of the region of the gain-phase plane covered by the describing function is such as to guarantee some form of limit cycle behavior in most closed-loop systems.