A note on the describing function of an element with coulomb, static, and viscous friction

IT IS NOW a standard procedure to use the sinusoidal-analysis, or describing-function, method in the investigation of quasilinear servomechanisms. The Nyquist criterion is applied to determine the stability conditions, and the Nyquist diagram provides some information as to the type of equalization that should be employed.1 These techniques may be used even when the describing-function of the nonlinear element is both frequency and amplitude sensitive, but this requires the graphing of a family of describing functions.2 This complication may be avoided by isolating the frequency-sensitive and amplitude-sensitive portions of the describing function into separate terms. In the trivial case, the describing function Ho´ may be represented as $H_o{\prime} = H_{n}H_{o}\eqno{\hbox{(1)}}$ where Ho is amphtude dependent and frequency independent, and Hn is amplitude independent and frequency dependent. Unfortunately, this easy separation of terms canot be achieved for the describing function of an element with coulomb, static, and viscous friction, a type of element present in many servo systems. This paper with show, however, that an equivalent result is achieved by means of an equivalent, block-diagram representation of Ho´.