ting lip can be thought of as being composed of a number of small differential segments. The elementary tangential force dFt is per-pendicular to the cutting edge and the axial force dFz is in the drill axial direction. The cutting forces, axial forces and torque applied to the drill are evaluated by summing the elementary forces and torques acting on the basic elements of the edge of the drill and are expressed in Eq. (4) as a function of chip thickness h and width 

Ft  = dFt  = Kt bh

i

FZ  =dFZ  = KZ bh (4)

MZ  = 2 dFt · r(z) = Kt bhRav

where Rav is the average radius of the cutting force, h is the chip thickness and b is the radial depth of cut (drill diameter minus pilot hole diameter if existing). KZ and Kt are the average cutting pressures in the axial and tangential directions, respectively

3.1. Drill torsional–axial coupling

The development of a torsional–axial vibration model was a major milestone in understanding chatter in machining using a twist drill (Bayly and Metzler, 2001). The torsional–axial theory of Bayly is based on the fact that when a twist drill “untwists”, it extends in length (Fig. 6).

                           Fig. 6. (a) Dynamic model of drill bit. (b) Finite element model of the drill.

Twisting and axial deformations are coupled. Hence, the second

member of Eq. (2) becomes:.

F(t) · z = −Fz (t) − (  )MZ (t) (5)

The    term represents a torsional–axial coupling parameter,

which relates the applied torque to drill axial force excitation. It

is determined by FEM analysis (Fig. 6b). The coupling is negative,

since when the drill twists in −z resulting from the negative cutting

torque it also extends in +z which results in a negative axial force

contribution. According to Eq. (4), Eq. (5) can be rewritten

F(t) · z = h · b · Kt · ˛ with ˛ = KZ (6)

Kt + Rav ·

KZ  and Kt  represent the axial and tangential cutting pressures

and are identified through experimental drilling characterization(Guibert et al., 2009). The average tangential and thrust cutting pressures can be obtained from KZ = FZ /bf/2 and Kt = Ft /bf/2, where b is the radial width of cut, and f is the feed rate in m/rev. bf/2 represents the chip area. The average cutting pressures were obtained for the feed of 0.1 mm/rev and the speed of 6000 rpm. The tangential cutting pressures were estimated at Kt = 2.85 · 108 N/m2, and the thrust cutting pressures were found at KZ = 2.4 · 10 8 N/m 2 .

The coupling term Rav is estimated from the drill finite element model. It represents the amount of “unwinding”, expressed in terms of length of arch of rotation per unit of axial displacement. Fig. 7 shows this rotation at a radius ri  from the centre of the drill when a force Fz is applied at the tool tip in the axial direction.

As the arch of rotation ri · is variable along the cutting lip, an average of this parameter is used at the mid-point along the lip (Rav ·  ).