Higher-order estimation of limit cycle amplitude in metal cutting

Abstract eng:
Global dynamics of an orthogonal cutting model is discussed, which is governed by a nonsmooth delay differential equation (DDE), namely, the delayed term intermittently disappears if the amplitude of the vibrations exceeds some critical value. Along the linear stability boundaries of the fixed point of the DDE, subcritical Hopf bifurcation occurs and consequently, a parameter domain of bistability can be identified where the stable fixed point, an unstable periodic orbit, and a large-amplitude stable bounded motion (chatter) coexist. The first approximation of the amplitude of the unstable periodic orbit is the standard square-root function of the bifurcation parameter that is the chip width. A non-standard higher-order approximation is proposed that takes into account that the periodic solution disappears as the chip width approaches zero. This allows the construction of a more accurate still simple formula for the size of the bistable region, which is highly important for the real-world application of regenerative cutting theory.

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