Dynamic characterization of hysteresis elements in mechanical systems. I. Theoretical analysis

The pre-sliding-pre-rolling phase of friction behavior is dominated by rate-independent hysteresis. Many machine elements in common engineering use exhibit, therefore, the characteristic of "hysteresis springs," for small displacements at least. Plain and rolling element bearings that are widely used in motion guidance of machine tools are typical examples. While the presence of a hysteresis element may mark the character of the resulting dynamics, little is to be found about this topic in the literature. The study of the nonlinear dynamics caused by such elements becomes imperative if we wish to achieve accurate control of such machines. In this Part I of the investigation, we examine a single-degree-of-freedom mass-hysteresis-spring system and show that, while the free response case is amenable to an exact solution, the more important case of forced response has no closed form solution and requires other methods of treatment. We consider harmonic-balance analysis methods (which are common analysis tools in engineering) suitable for frequency-domain treatment, in particular the approximate describing function (DF) method, and compare those results with "exact" numerical simulations. The DF method yields basically a linear equation with amplitude-dependent modal parameters. We find that agreement in the frequency response function, between DF and exact solution, is good for small excitation amplitudes and for very large amplitudes. Intermediate values, however, show high sensitivity to amplitude variations and, consequently, no regular solution is obtainable by either approach. This appears to be an inherent property of the system pointing to the need for developing further analysis methods. Experimental verification of the analysis outlined in this Part I is given in Part II of the paper.