D lim = −1 (11)

˛ · Kt · 2 · G(ω)

where G(ω) is the real part of the global system (Spindle–SVDH–drill) transfer function. ˛ is the torsional–axial coupling term (Eq. (6)). As presented in reference (Gagnol et al., 2007a,b), the stability lobe diagram is developed in a two-step method. First a preliminary lobe diagram is computed on the basis of the tool tip speed-dependant transfer function in the axial direction. One valid point can be extracted from this simulation at the considered spindle speed. Finally the new dynamic stability lobe diagram is elaborated by assembling the single valid point (Dlim , n) resulting from the appropriate spindle speed tool point FRF (Fig. 9). However, this procedure is used primarily to take into account variations in bearing stiffness as gyroscopic effects described in Eq. (2) do not affect the spindle axial dynamic’s properties.

Fig. 9 represents stability predictions in the plane of spindle speed and drill diameter. It is established for SVDH rigidity of up to 1.2e7 N m, where the operating domain representative of self-excited vibrations is increased. For a drilling operation with a 5 mm diameter drill, the maximum spindle speed is increased from 16,800 to 19,500 rpm by taking into account the torsional–axial coupling of the drill. The zone outside the limits corresponds to conventional drilling because it does not provide sufficient energy for the chatter. The zone inside the limits covers the vibratory field of instability, which is of interest in the context of this study.

Fig. 10. Stability lobes with and without torsional–axial coupling respectively in the stiffness-spindle speed planes (a) and mobile mass-spindle speed plane (b)

For a given drill diameter, the width of cut is imposed and it is more convenient to establish stability lobes diagram according to the adjustable parameters of the SVDH: i.e. the spring stiffness and the adjustable mass according to spindle speed. Fig. 10 illus-trates the stability diagrams of the SVDH system according to the two considered planes, respectively represented by SVDH spring stiffness according to spindle speed (a) and SVDH additional mass according to spindle speed (b).

It can be noticed that the torsional–axial coupling effect influ-ences the instability of the process. For a given drill diameter, it corresponds to a reducing of the application zone of vibratory drilling by decreasing the maximum spindle speed.

4.2. Experimental validation

The cutting stability boundaries predicted in the above sections were experimentally verified by a machining test (Forestier et al.,2012). As shown in Fig. 11, the drilling tests were conducted on a high-speed machining centre (Hermle C800U 3-axis equipped with Weiss 15500 rpm spindle), on a 35 MnV7 steel workpiece, with a drill of 116 mm length and 5 mm diameter. The work piece was fixed on a Kistler dynamometer (9257A) in order to monitor the cutting force during drilling. Captured signals were processed using LMS® Pimento software and enabled drilling oscillation frequency to be measured. The lobe number is determined on this basis.

The experimental results were compared with the modeled sta-bility lobes (Fig. 12).

The stability frontiers without torsional–axial coupling are rep-resented by black curves in Fig. 12. It can be noticed a shifted to the left of the stability lobes by taking into account the drill torsional phenomenon.

Experimental results show that new stability lobes consider-ing drill twisting effect are more accurate than the stability lobes established without this effect. For example, the two non-vibrating experimental points are well-positioned in the stable zone in the new stability lobes. It can be observed that the experimental points and the lobe number information are in good agreement with the numerical stability predictions. All experimental points are plotted near the correct numerical area.