The literature on the modelling and analysis of spindle systems shows that the tool tip FRF is also greatly influenced by the contact dynamics of the spindle–holder–tool interfaces (Erturk et al., 2006; Schmitz et al., 2007). The flexibility of the aforementioned inter-faces plays an important role in spindle dynamics (Ahmadian and Nourmohammadi, 2010). The identification of contact dynamics in spindle–SVDH–tool assemblies has been carried out by Forestier(Forestier et al., 2011) using the receptance coupling method on the basis of experimental substructure characterization. The iden-tified interface models are then integrated into the global model. The global model is then validated by comparing the numerical and experimental Frequency response Functions.

In Section 3, a generic accurate drilling force model is developed by taking into account the drill geometry, cutting parameters and effect of torsion on the thrust force.

Section 4 is dedicated to the prediction of adequate drilling con-ditions based on controlled self-excited drill vibration. A specific instability lobes diagram is elaborated by integrating into an ana-lytical stability analysis the overall structural model-based tool tip FRF of the system associated with the proposed drilling force model. The torsional–axial coupling of the twist drill is investigated on the basis of Bayly’s model (Bayly and Metzler, 2001) and consequences on drilling instability are established.

In Section 5, the stability lobes are compared to experimen-tally determined stability boundaries for validation purposes. By taking into account the drill torsional–axial coupling, the model-based stability lobes are modified and correspond better with the experimentally determined stability limits.

Finally, a conclusion is presented.

Nomenclature

Hmodel interface model receptance

k interface stiffness

c interface damping factor

D damping matrix

M mass matrix

K stiffness matrix

G gyroscopic matrix

N spin softening effects matrix

˝ rotor angular velocity

qN nodal displacement

F(t) force vector

FZ axial cutting forces

KZ axial cutting pressure

Kt tangential cutting pressure

D drill diameter

h chip thickness

B chip width

Z axial tool tip displacement

fz feed rate per tooth

c chatter pulsation

N spindle speed

2.  Model building

The vibratory drilling system is composed of a SVDH body clamped to the spindle by a standard HSK63A tool-holder interface. A SVDH vibrating subsystem is jointed to the SVDH body using a specific spring, and axially guided by a ball retainer. Finally, a long drill is held in the SVDH vibrating subsystem with a standard ER25 collet chuck. The SVDH system is mounted on a spindle capable of speeds up to 15,500 rpm. The spindle has four angular bearings in overall back-to-back configuration (Fig. 1).

The spindle–SVDH–tool system is composed of four structural subsystems: the drill, the SVDH vibrating subsystem, the SVDH body and the spindle.