Nonlinear differential equations with delay as models for vibrations in the machining of metals

Traditional manufacturing processes, such as turning, milling, and drilling, involve relative motion between the bulk material, the workpiece, and a harder material which accomplishes the removal, the tool. The interaction of the workpiece and the tool often causes a vibration of either the tool or the workpiece. A large amplitude manifestation of this vibration is known as chatter. Tools from the field of nonlinear dynamics are used to interpret tool vibrations measured in the turning of aluminum. The method of false nearest neighbors shows that the attractor governing these vibrations can be embedded in a four to six dimensional space. Experimental Poincare sections show that the attractor governing tool vibration has interesting structure when embedded in three dimensions. This structure has the same overall shape seen in mathematical models, but lacks fine details.; The notion that tool chatter is one possibility in a range of tool vibrations is advanced. Experiments show that tool vibrations occur well before the onset of chatter. These smaller amplitude vibrations share many qualitative features with the larger amplitude chatter. A decrease in the magnitude of cutting forces with increasing cutting speed, similar to that seen in dry friction, and subcritical Hopf bifurcations are advanced as potential mechanisms for these smaller amplitude vibrations. A theory proposed by Doi and Kato (1956) suggests there is a time lag between the tool motion and the cutting forces. The existence of this time lag is also sufficient to explain small amplitude vibrations before the onset of chatter.; The mathematical models given primary emphasis are differential difference equations, or differential equations with delay. These models are capable of predicting periodic, quasi-periodic, and aperiodic tool vibrations. The models can also undergo subcritical Hopf bifurcations similar to those in the turning experiments of Hooke and Tobias (1963). Despite their infinite dimensional character, the steady state behavior of these models can be visualized in finite dimensional spaces. The Fourier power spectra and Poincare sections for these models with delay are similar to those seen in turning experiments. The false nearest neighbors method shows the attractors for these models can also be embedded in four to six dimensional spaces.