In our case, using a carbide drill of 5 mm diameter, the helix angle is 30◦ and the material to be machined is 35MnV7. The identification of terms Rav ·   and ˛ is respectively −0.45 and 0.39.

4.  Chatter vibration prediction

  In this section, the proposed SVDH dynamic model is used for the purpose of stability analysis. The aim is to use natural axial chatter to fragment the chips. The challenge is to keep the system stabilized at a suitable frequency and magnitude for a good cutting quality. The cutting parameters are therefore chosen to be in the unstable domain.

4.1.  Stability model and numerical predictions torsional–axial coupling in the drill and SVDH vibration provides a mechanism for torsional–axial chatter. The torsional vibrations. lead to the shortening and lengthening of the drill, which results in a wavy surface of the bottom of the hole. The main difficulty of vibratory drilling is to foresee the cutting conditions that will generate regular vibrations able to induce interrupted cutting. In this study, the drilling operation is considered as having one degree of freedom in thrust force, taking into account the torsion effect in this direction. The stability of the system is investigated analytically from the study of the chip thickness (Fig. 8a). The regenerative chatter of system can be presented by the block diagram shown in Fig. 8b.The cutting forces and coupling term, Eq. (6), are substituted into Eq. (2) for motion in the drill’s axial direction:

(Mq¨N + (2˝G + D)q˙N + (K − ˝2N)qN ) · z = ˛Kt b(Z(t) − Z(t −   ))         （7）

The chip thickness resulting from the regenerative displacement

is expressed in the Fourier domain as:

h(jω) = fZ + Z(jω) − Z (jω)e−jωT (8)

where fz is the feed rate per tooth. T is the tooth period and Z(j  ) is the axial tool tip displacement. The stability diagram is obtained by integrating the numerical-predicted axial tool tip frequency response into the chatter stability approaches. The axial transfer function H(j  ) representing the ratio between the Fourier transform of the axial displacement Z(j ) at the tool tip and the axial dynamic cutting force F(j ) is expressed as:

Fig. 9. Stability lobes with and without torsional–axial coupling.

H(jω) = x(jω) = inv(−Mω2 + (2˝G + D)ω + (K − ˝2N)) (9)

F (jω)

H(jω) = R(ω) + jI(ω) where R and I are, respectively the real part and the imaginary part of the transfer function. They depend on the adjustable parameter of the SVDH, i.e. the axial spring rigidity and the additional SVDH mass.

For a given axial SVDH rigidity or SVDH mass, the resulting sta-bility relationships are established as:

N = 60ωc with : ϕ = tan−1 I(ωc ) (10)

Z(2ϕ + 3  + 2p )

R(ωc )

where ωc is the chatter pulsation. N is the spindle speed in rpm, p is an integer that corresponds to the inpidual lobe number. The linear stability analysis of the modeled drilling system enables the identification of the stable and unstable domains and can be illustrated in a traditional stability lobes diagram (Fig. 8) accord-ing to the cutting parameters of depth of cut and spindle speed. The stability limit, integrating drill torsional–axial vibration, can be established: