The mechanics of chip formation has been revisited in order to understand functional relationships between the process and the technological parameters. This has led to the necessity of considering the chip-formation process as highly nonlinear, with complex interrelations between its dynamics and thermodynamics. In this paper a critical review of the state of the art of modelling and the experimental investigations is outlined with a view to how the nonlinear dynamics perception can help to capture the major phenomena causing instabilities (chatter) in machining operations. The paper is concluded with a case study, where stability of a milling process is investigated in detail, using an analytical model which results in an explicit relation for the stability limit. The model is very practical for the generation of the stability lobe diagrams, which is time consuming when using numerical methods. The extension of the model to the stability analysis of variable pitch cutting tools is also given. The application and verification of the method are demonstrated by several examples.

A physical model to examine impact oscillators has been developed and analyzed. The model accounts for the viscoelastic impacts and is capable to mimic the dynamics of a bounded progressive motion (a drift), which is important in practical applications. The system moves forward in stick-slip phases, and its behavior may vary from periodic to chaotic motion. A nonlinear dynamic analysis reveals a complex behavior and that the largest drift is achieved when the responses switch from periodic to chaotic, after a cascade of subcritical bifurcations to period one. Based on this fact, a semianalytical solution is constructed to calculate the progression of the system for periodic regimes and to determine conditions when periodicity is lost.

A comprehensive study of the frictional chatter occurring during metal-cutting process is given. A general mathematical model of the machine-tool{cutting process is established, and then a high-accuracy numerical algorithm is developed. Next, a two-degree-of-freedom model of orthogonal metal cutting is examined. Then stochastic properties of the material being cut are introduced to re®ect variations in the workpiece properties, in particular, in the cutting resistance. Nonlinear dynamics techniques, such as constructing bifurcation diagrams and Poincaré maps, are employed to ascertain dynamics responses for both the deterministic and the stochastic model. Untypical routes to chaos and unusual topology of Poincaré cross-sections are observed. The conducted analysis has provided some practical design recommendations. Finally, occurrence of chatter was investigated analytically.

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