Comparison and evaluation of describing functions of multi-valued nonlinearities

The object of the describing function technique is to represent the actual nonlinearity of the system in terms of an equivalent linear system with the assumption that the input to the nonlinearity is approximately sinusoidal. Various schemes of generating describing functions of single-valued nonlinearities have been evolved in recent years. The most priminent schemes include (1) conventional, (2) new RMS, and (3) modified or corrected conventional schemes, which can also be extended to multi-valued nonlinearities. The usefulness of these schemes depends on the accuracy with which the describing function they produce can predict the amplitude and frequency of self-sustained oscillations in a nonlinear system. This study contains a comprehensive evaluation of the various schemes as applied to multi-valued nonlinearities. The expressions for describing functions of on-off type multi-valued nonlinearities are derived. These expressions are used to predict the amplitude and frequency of selfsustained oscillations. A design problem is presented as an illustration to show the necessity of a more accurate describing function. It concludes with the recommendation that the describing function that takes only the third harmonic into account is the most accurate one.