2.3. Model assembly and validation

As in a classic finite element procedure, dynamic equations of the overall system, composed of the drill, the SVDH vibrating subsystems, the SVDH body and the spindle, were obtained by assembling element matrices. Matrices and vectors for each indi-vidual element are formed first and then linked together into a set of system equations. The spring-damper connection param-eters between the drill and the SVDH vibrating subsystems and between the SVDH vibrating subsystems and SVDH body, identified by the receptance coupling method, enabled the rotor-beam mod-els of the components to be assembled. The rolling bearing model imposes the boundary conditions of the system. Angular contact ball bearings have a significant influence on spindle dynamics. In a high-speed spindle the bearing axial preload is kept essentially constant during thermally generated changes to spindle dimen-sions, by spring or hydraulic arrangements. However, the bearing properties are also speed-dependant. The rolling bearing stiffness matrices are calculated by in-house software developed on the basis of T.C Lim’s formulation (Lim and Singh, 1990). The bearing stiffness model represents the load-displacement relation com-bined with the Hertzian contact stress principle, and is calculated around a static function point characterised by the bearing preload. Based on Rantatalo’s prediction (Rantalalo et al., 2007), the initially calculated bearing stiffness is spindle speed-dependant because of the gyroscopic and centrifugal force Fc which acts on each ball (Fig. 3b). With increasing speed, the load conditions between the balls and the rings in the bearing change because of centrifugal force (Fig. 3a). Thus, speed-dependant bearing stiffness is integrated into the global spindle FEM and also influences the natural frequencies of the spindle–tool unit.

The spindle–SVDH–tool assembled model was validated by comparison between numerical and experimental FRF in the axial direction for a non-rotating state, as shown in Fig. 4.

                Fig. 4. Validation of the elements and the global model, in the axial direction, for a non-rotating spindle.

The frequency peaks governed by the spring and ball retainer at 60 Hz and the collet chuck at 4700 Hz, respectively, are in a good correlation with the numerical model. Parasitic frequency peaks can be observed on the measured FRF near 1200 Hz and 2400 Hz. These peaks are produced by the drill bending modes, which are also excited during an experimental axial impact. The simulations match the measurements well in the interesting frequency band below 1000 Hz.

Based on these results, it is possible to validate the use of the numerical model for further investigation, such as the prediction of adequate drilling conditions to ensure stable self-excited vibration.

Following Tlusty (Tlusty, 1985) and Stephenson (Stephenson and Agapiou, 1992), the net variations in the time-varying part of tangential and thrust forces at the drill tip are assumed to be pro-portional to the chip area. The influence of the ploughing effect and the effect of the chisel edge in zones 2 and 3 of Fig. 5 are not taken into account in the proposed cutting force model. Indeed, in this small region around the center of the chisel edge, the tool does not cut but rather extrudes the material (Guibert et al., 2009). Accord-ing to these assumptions, the thrust force, the axial force and the torque at the tool tip are respectively:

3.  Drilling force model

The energy required to maintain the self-excited vibration is provided by the cutting forces. These excite both the SVDH and the flexible low-diameter drill. The combination of rigid body motion and dynamic displacements of the drill induces mainly torsional and axial vibrations.

Several force models have been proposed in the literature for the primary cutting edge presented in Fig. 5 in zone 1. Both geo-metrical parameters and cutting pressures change greatly along the cutting lip of the twist drill (Roukema and Altintas, 2006). Each cut-