Model for the prediction of whirling vibrations in drilling processes through semi-discretization of the drill motion equation

摘要

In this work, a model for the prediction of drilling stability against low-frequency lateral vibrations, named as whirling in the literature, is proposed. These vibrations are lateral displacements of the tool that arise at frequencies near multiples of the rotation frequency of the drill. The appearance of whirling vibrations leads to the generation of lobe-shaped holes. In order to predict whirling vibrations, the motion equation of the drill is deduced taking into account the modal characteristics of the drill and the cutting and process damping forces that act on it. In this paper, forces that arise in two different regions of the drill are considered: (1) forces on the main cutting edges and (2) forces on the chisel edge. Different force models are presented for each region that include both the regenerative effect of the vibration on the cutting area and the process damping. An oblique cutting model and an orthogonal model are employed for the calculation of cutting forces acting on the main cutting edges and on the chisel edge, respectively. The cutting force model for the main cutting edges takes into account the cutting angle (inclination angle, rake angle, and chip flow angle) variation along the main cutting edges. For the chisel edge region, where the feed speed is no longer negligible with respect to the cutting speed, the dynamic cutting angles are employed for the force model development. Concerning the process damping force model, previous works in the literature consider a constant value of the clearance angle for the calculation of the process damping coefficient. However, in this work, the variation of the normal clearance angle along the main cutting edges is considered. It is shown that, depending on the clearance face grinding parameters employed, the clearance angle can double its value along the main cutting edges. Considering the force models and through the semi-discretization of the motion equation of the drill, the appearance of low-frequency lateral vibrations is predicted regarding the drill geometry and cutting conditions such as drill rotation speed and feed. In addition, given cutting conditions at which whirling vibrations are expected to occur, the model is able to predict the vibration frequencies that are excited. The drilling model and the stability predictions are experimentally validated by means of drilling tests with different drill diameters and cutting conditions. In comparing the experimentally obtained results and the predictions obtained by the model, it is concluded that the model can reasonably predict the appearance of whirling vibrations as a function of drill geometry and cutting conditions. Generated hole shape is also analyzed through the measurement of hole roundness and bottom surface geometry. It is observed that, when drilling in the presence of whirling vibrations, holes with lobed shape and polygonal bottom surface are generated. It is also noticed that both the number of lobes and the number of sides of the polygonal bottom surface are directly related to the vibration frequencies that arise.