Table 1

Identified stiffness and damping factor of the interface.

Interface Stiffness (N/m) Damping factor (N s/m)

Spring and ball retainer 1.13 × 106 24

Collet chuck 14.8 × 106 2

HSK 63 Infinite 0

2.1. Structural elements

The model for the spindle–SVDH–tool system is restricted to the rotating structure composed of the spindle shaft, the SVDH and the drill. This hypothesis was established by Gagnol et al. (2007a,b) through an experimental modal identification procedure carried out on spindle substructure elements. Dynamic equations were obtained using Lagrange formulation associated with a finite element method. Due to the size of the rotor sections, shear deformations had to be taken into account. Then the rotating sub-structure was built using Timoshenko beam theory. The relevant shape functions were cubic in order to avoid shear-locking. A spe-cial three-dimensional rotor-beam element with two nodes and six degrees of freedom per node was developed in the co-rotational ref-erence frame. The damping model used draws on Rayleigh viscous equivalent damping, which makes it possible to regard the damp-ing matrix D as a linear combination of the mass matrix M and the spindle rigidity matrix K:

D = ˛K + ˇM (1)

where ˛ and ˇ are damping coefficients. The set of differential equations are detailed in Gagnol (Gagnol et al., 2007a,b) and can be written as:

Mq¨N + (2˝G + D)q˙N + (K − ˝2N)qN  = F(t) (2)

where M and K are the mass and stiffness matrices. K results from

the assembly of the spindle rigidity matrix Kspindle and the rolling bearing rigidity matrix Kbearings. D is the viscous equivalent damping matrix, qN and F(t) are the nodal displacement and force vectors. G

and N are respectively representative of gyroscopic and spin soft-ening effects. ˝ is the rotor’s angular velocity.

2.2. Modelling of spindle–SVDH–tool interfaces

The dynamic behaviour of the interfaces represented by the HSK63 taper, spring and ball retainer, and collet chuck are taken into account (Fig. 2). The identification procedure of the interface models was carried out by Forestier et al. (2012) based on the receptance coupling method and then integrated into the model as illustrated in Fig. 2. The axial dynamic behaviour of the interface are modeled by a spring-damper element whose transfer function is

1 (3)

Hinterface(ω) = k + icω

The rigidity k and damping c values were determined by min-imizing the gap between the measured and the modeled tool tip node frequency response function for a non-rotating spindle, using an optimization routine and a least-squares type object function.

The identification of the interface model is carried out by com-paring the reconstructed assembly receptance with the measured receptance value. Identifications results for the spring and ball retainer, the collet chuck interface and the HSK 63 interfaces are given in Table 1. The experimental modal analyses carried out on the spindle/SVDH body system in the axial direction allow the HSK 63 interface to be considered as a rigid connection.

Fig. 1. The spindle–SVDH–tool system.

                             Fig. 2. The spindle–SVDH–tool system finite element model.