Analysis of time series from stochastic processes governed by a Langevin equation is discussed. Several applications for the analysis are proposed based on estimates of drift and diffusion coefficients of the Fokker-Planck equation. The coefficients are estimated directly from a time series. The applications are illustrated by examples employing various synthetic time series and experimental time series from metal cutting.
A two degree of freedom model of the milling process is investigated. The governing equation of motion is decomposed into two parts: an ordinary differential equation describing the periodic chatter-free motion of the tool and a delay-differential equation describing chatter. The stability chart is derived by using the semi-discretization method for the delay-differential equation corresponding to the chatter motion. The periodic chatter-free motion of the tool and the associated surface location error (SLE) are obtained by a conventional solution technique of ordinary differential equations. It is shown that the SLE is large at the spindle speeds where the ratio of the dominant frequency of the tool and the tooth passing frequency is an integer. This phenomenon is explained by the large amplitude of the periodic chatter-free motion of the tool at these resonant spindle speeds. It is shown that large stable depths of cut with a small SLE can still be attained close to the resonant spindle speeds by using the SLE diagrams associated with stability charts. The results are confirmed experimentally on a high-speed milling center.
Two methods for automatic chatter detection in outer diameter plunge feed grinding are proposed. The methods employ entropy and coarse-grained information rate (CIR) as indicators of chatter. Entropy is calculated from a power spectrum, while CIR is calculated directly from fluctuations of a recorded signal. The methods are verified using signals of the normal grinding force and RMS acoustic emission. The results show that entropy and CIR perform equally well as chatter indicators. Based on the normal grinding force, they detect chatter in its early stage, while only cases of strong chatter are detected based on RMS acoustic emission.
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