Limit cycles, bifurcations, and accuracy of the milling process

Time finite element analysis (TFEA) is used to determine the accuracy, stability, and limit cycle behavior of the milling process. Predictions are compared to traditional Euler simulation and experiments. The TFEA method forms an approximate solution by dividing the time in the cut into a finite number of elements. The approximate solution is then matched with the exact solution for free vibration to obtain a discrete linear map. Stability is then determined from the characteristic multipliers of the map. Map fixed points correspond to stable periodic solutions which are used to evaluate surface location error. Bifurcations and limit cycle behavior are predicted from a non-linear TFEA formulation. Experimental cutting tests are used to confirm theoretical predictions.