Investigations of Nonlinear Forces in Metal Cutting

During chatter in metal cutting the tool vibration would cause a variation in "effective" rake angle of the cutter, generating a force variation that depends on penetration rate: a kind of process damping. This effect is examined for forces computed both from a theoretical Merchant-type model, and from a numerical database of forces for metal cutting constructed from a suite of AdvantEdge simulations. Since the tool can potential vibrate at any angle relative to the workpiece, the forces for varying angle of vibration were computed, and the dynamic stability consequences considered. It is found that the two models lead to similar forces for varying vibration angle, at least through first order. Depending on the vibration angle the force will either increase or decrease with both chip load and penetration rate, reflecting the difference in the effect on the chipload and cutting speed with varying vibration direction. Second order terms in penetration rate were different in the two formulations, possibly a result of approximations used in the calculations involved in the Merchant formulation. Dynamically this means that the linear stability of each vibration angle is the similar for the two models, while the differences in nonlinear terms results in differences in the type Hopf bifurcation observed upon loss of stability.