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Ch32-I044963.fm Page 155 Monday, August 7, 2006 11:28 AM
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11:28 AM
Ch32-I044963.fm
Monday, August 7,2006
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Estimation of Cutting Depth using Existence Evaluation of Workpiece Volume
In order to estimate the cutting depth without repetition of workpiece shape estimation, we introduce
some estimation models proposed in former studies (Takata S. 1989). In these models, it is assumed
that cutting edge is regarded as a set of finite flutes and cutting force can be estimated as the sum
total of the force loaded on each finite flute as Figure 3. Furthermore, if both finite flute and tool feed
is sufficiently small, the cutting depth at can be calculated from tool feed in each cutting edge passing
/and the result of existence evaluation EE, as shown Equation 1 and 2.

at = EE \r + Jb-J' fijr -l)+r ) ! (1)
b — sin 6sin gcos0 + cos q sin (p (2)
v
v v Tool feed φ
Cutting edge Tool feed u
u
u
θ q EE 0
at
wt EE 1
Cutting area of one finite flute : atwt
Cutting area of one finite flute : at wt
Figure 3: Cutting depth estimation based on the idea of finite flute and existence evaluation
This equation means that cutting force estimation can be realized by referring results of the existence
evaluation for each finite flute. So, following section, we propose a new method of existence
evaluation without the explicit information about workpiece shape in machining.

Efficient solution of Existence Evaluation based on the Idea of Tool Swept Volumes
Existence evaluation requires only judgment of workpiece volume existence where the finite flute is
located. It does not surely need the explicit information of workpiece shape. So, we introduce the
idea of tool swept volume and set operation between volumes. As shown in Equation 3, workpiece
volume in machining of «th tool moving step MWV n'can be described by volume of initial workpiece
MWVo, /th tool swept volume TSVi, a part of «th tool swept volume SubTSV n and set operations.

MWV, =\fWV,r\C\TSV \c\SubTSV, (3)
I | |
Then, we define function of existence evaluation EE(p, V). If point/) is located in the inside of volume
V, the value of EE(p, V) is 1. In the case of others, EE(p, V) is 0. Applying this function to Equation 3,
the existence evaluation for volumes performed set operations can be achieved by multiplication of
result about the existence evaluation for each volume, as illustrated in Figure 4.

MWV
MWV MWV 0
TSV 1
n ⎛ −1 ⎞ EE c ,MWV ) ' = EE c ,MWV )
MWV n ' ⎜ ⎜ ⊃ TSV ⎟ ⎟ i ∩ SubTSV n ( ijm n ( ijm 0
c
EE ( ijm ,MWV n ) ' i ⎝ =1 ⎠ ⋅ EE ( ijm ,TSV 1 )
c
c
c
EE ( ijm ,MWV ' n ) = EE ( ijm ,MWV 0 ) TSV i …
⎛ n− 1 ⎞ ⎛ n− 1 ⎞ ⋅ EE ( ijm ,TSV 1 - n )
c
,MWV ∩
SubTSV
EE
c ij
⎟
= = EE(c ( ijm ,MWV 0 ⎜ ⎜ ⊃ TSV ⎟ ⎟ i ∩ SubTSV ) × ×EE(c ( ijm ⎜ ⊃ ,⎜ TSV ⎟ i ∩ SubTSVn )
EE
c ijm
0 ⎠ n i ⎝ ⎠ n
EE
i ⎝ = 1 1 = ⋅ EE(c ( ijm ,SubTSV n ) )
,SubTSV n
c ijm
SubTSV n
Figure 4: Decomposition of existence evaluation process for workpiece volume in machining
By using this relation, we can judge whether workpiece volume exists on the finite flute c ijm with the
result of existence evaluation between c,y m and each tool swept volume. Because the tool swept