det I — 2 Ktc apð1 — e—

ÞA0 GðiocÞ

¼ 0, (3)

relationship between the cutting force and vibrations at the tool tip become:

GðioÞ ftðioÞ ¼ utðioÞ。 (1)

In general, GðioÞ would be a 3 ~ 3 matrix, as displace- ments and the force vector are defined in the three- dimensional Cartesian system。 However in the milling process, the axial direction (Z) is typically much stiffer than the feed force direction (X) and the cross feed force direction (Y) (see Fig。 1)。  Therefore

" GXX ðioÞ    GXY ðioÞ #

where ap is the nominal depth of cut, Ktc is the tangential cutting coefficient, oc is the chatter frequency, T  is  the tooth passage period and A0 is the immersion dependent matrix which is a function of the cutting   coefficients。

The cutting experiments were done for estimating the cutting coefficients and to verify the analytical stability lobes by verifying the regions of stability and instability。 The cutting coefficients are constant for a given tool insert–workpiece combination and not affected by the change in machine structure。 The axial cutting coefficients are not necessary for finding the stability lobes because  the

structure has been assumed rigid along the axial   direction。

Feed and cross-feed forces were measured while cutting in full  immersion  with  two  inserts  only。  The   mechanistic

Here, GXX  and GYY  are FRFs that were experimentally

determined by an impact hammer test。 The off-diagonal cross coupling terms GXY, GYX in Eq。 (2) are  relatively small and are neglected。

The vibrations of the machine affect the quality of machining due to chatter (i。e。, regenerative self excited vibrations)。 Chatter occurs due to the interaction of the workpiece and tool, which leads to vibrations near one of the structural modes。 At some combinations of spindle speed and depth of cut, the cutting forces can become unstable and cause chatter。 The analytical chatter predic- tion model presented by Altintas and Budak [22,23]   gives

cutting force model [22] for the milling cutting process is described as

FX ¼ —Ktc ap f t sin f sin f — Kte ap sin f

— Krc ap f t sin f cos f — Kre ap cos f,

FY ¼ Ktc ap f t sin f cos f þ Kte ap cos f

— Krc ap f t sin f sin f — Kre ap cos f。 ð4Þ

Knowing FX, FY for different values of instantaneous cutter angular locations f, the cutting coefficients Ktc, Kte, Krc and Kre can be estimated using the least-squares approach。 The forces FX and FY are the components of ft in

T

the  characteristic  equation  of  the  milling  process     and

Eq。 (1), i。e。, ft  ¼ ½FX ; FY ]

。 The constants Ktc and Krc arise

equates it to zero to find the stability limit。 The characteristic  equation  for  finding  the  stability  limit  for

due to the shearing action in the tangential and radial directions  respectively。  Kte  and  Kre  are  the corresponding

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

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