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.

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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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High-speed milling is often modeled as a kind of highly interrupted machining, when the ratio of time spent cutting to not cutting can be considered as a small parameter. In these cases, the classical regenerative vibration model, playing an essential role in machine tool vibrations, breaks down to a simplified discrete mathematical model. The linear analysis of this discrete model leads to the recognition of the doubling of the so-called instability lobes in the stability charts of the machining parameters. This kind of lobe-doubling is related to the appearance of period doubling vibrations originated in a flip bifurcation. This is a new phenomenon occurring primarily in low-immersion high-speed milling along with the Neimark-Sacker bifurcations related to the classical self-excited vibrations or Hopf bifurcations. The present work investigates the nonlinear vibrations in the case of period doubling and compares this to the well-known subcritical nature of the Hopf bifurcations in turning processes. The identification of the global attractor in the case of unstable cutting leads to contradiction between experiments and theory. This contradiction draws the attention to the limitations of the small parameter approach related to the highly interrupted cutting condition.