A long stepped shaft fixed at the centers is considered. When processing it at a lathe, nonlinear effects (soft and rigid modes of self-oscillations) can be observed. To investigate these effects and the reasons of their occurrence, a mathematical model is constructed. It considers elastic properties of a shaft and the dynamic characteristics of cutting obtained experimentally, By means of a power balance method, expressions for amplitudes of self-oscillations and conditions, occurrence of soft and rigid modes of excitation of self-oscillations are derived
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