2。2。 Dynamics of machine tools

This paper describes the FRFs and the stability lobe diagrams for chatter for the arch-type RMT at the y ¼ 01 and 451 reconfiguration position,  to investigate   variations

Fig。 1。  Final design for RMT with its finite element   model。

328 J。 Dhupia et al。 / International Journal of Machine Tools & Manufacture 47 (2007) 326–334

Table 1

Modal data for final RMT design

Mode # Frequency (Hz) Comments

1 36 Mainly due to weak connection between the ram and the arch。

2 53 Mainly due to column/base   structure

3 60 Mainly due to a combination of weak connection between the ram and the arch,

4

69 and stiffness of Y-axis

Mainly due to weak connection between the ram and the arch

5 72 Mainly due to a combination of weak connection between the ram and the arch,

and stiffness of Z-axis

These data are valid only for a position of the machine in which: (a) spindle is half the way extended, i。e。 250 m, (b) ram is positioned at 451 angle, and (c) arch is positioned at the middle of its 775 mm   stroke。

Fig。 2。  Different pivot-type RMT design  concepts。

Fig。 3。  Arch-type  RMT。

in arch-type RMT dynamic performance。 While the FRF gives the description of structural vibrations and stiffness of the machine tool [21], the stability lobes for chatter can be used as a direct assessment of the expected machining quality  and productivity。

The FRF matrix, also known as the receptance matrix for the relative workpiece-machine tool vibration,GðioÞ,  is

the transfer function from the cutting force to the rela- tive workpiece–machine displacement。 GðioÞ ¼ GtðioÞ— GwðioÞ, where GtðioÞ and  GwðioÞ  are  the  FRF matrices of tool and workpiece, respectively。 Let the displacement of the tool be ut and that of the workpiece be uw。 The cutting forces acting on the tool ðftÞ and the workpiece ðfwÞ have the same magnitude but opposing directions (i。e。 ft ¼ —fw)。

J。 Dhupia et al。 / International Journal of Machine Tools & Manufacture 47 (2007)   326–334 329

10。15 10。2 10。25 10。3 10。35 10。4 10。45

Time, t [sec]

Fig。 4。 Experimental and estimated cutting forces at ap ¼ 3:3 mm, N ¼ 1700 RPM and f t ¼ 7:2 mm=s。

In  this  paper  the  workpiece  is  assumed  rigid,  and    the this system is

assumption  was  verified  by  measuring  the  FRF  of   the

workpiece  and  measuring  the  vibrations  displacement of

the  workpiece  during  the  cutting  experiments。  Thus, the