88
C. Ozoegwu and P. Eberhard
(a) Up-milling
(b) Down-milling
Fig. 2 Start angle θ s and end angle θ e of milling
where z(t) ∈ R
d F is the state, M( p y ) ∈ R
d F ×d F is the mass matrix, D( p y ) ∈ R
d F ×d F
is the damping matrix, and K( p y ) ∈ R
d F ×d F is the stiffness matrix. The matrices
B H ( p y ) ∈ R
d F ×n d and C H ( p y ) ∈ R
n d ×d F are used to project the n d -dimensional tool
excitation H(t) to the d F -dimensional workpiece response. The normalized parameter p y is indicative of the dependence of the modal matrices on tool location along
the feed direction (y) where p y = 0 at the beginning of tool pass and p y = 1 at the
end of the pass. Such high-fidelity FE model as given in Eq. (9) has many DOF and
is thus computationally cumbersome. Based on modal truncation, Eq. (9) reduces,
according to [23], to
˜
M( p y ) ¨ ˜
z(t) + ˜
D( p y ) ˙ ˜
z(t) + ˜
K( p y )˜ z(t) = ˜
B H ( p y )H(t) ˜
C H ( p y )(˜ z(t) − ˜
z(t − τ )),
(10)
where ˜
z(t) ∈ R
d R is the reduced state, ˜
M( p y ) = V
T
( p y )M( p y )V( p y ) ∈ R
d R ×d R is
the reduced mass matrix, ˜
D( p y ) = V
T
( p y )D( p y )V( p y ) ∈ R
d R ×d R is the reduced
damping matrix, ˜
K( p y ) = V
T
( p y )K( p y )V( p y ) ∈ R
d R ×d R is the reduced stiffness
matrix, ˜
B H ( p y ) = V
T B H ( p y ) ∈ R
d R ×n d , ˜
C H ( p y ) = C H ( p y )V ∈ R
n d ×d R , and
V( p y ) ∈ R
d F ×d R is the projection matrix. The projection matrix V( p y ) is a concatenation of the first d R eigenvectors where d R d F . In [23], each p y -dependent
matrix is derived from cubic spline interpolation of the values at a finite number of
tool location zones under a consistent system of coordinates ˜
z(t). Making the substitution x 1 (t) = ˜
z(t) and x 2 (t) = ˙ ˜
z(t), the first-order form of the reduced model for
stability analysis using the FDM reads
˙
x(t) = Ax(t) + B(t)x(t) − B(t)x(t − τ ),
(11)
where
A =
0
I
−( ˜
M( p y ))
−1 ˜
K( p y ) −( ˜
M( p y ))
−1 ˜
C( p y )
,
and
B(t) =
0
0
( ˜
M( p y ))
−1 B H ( p y )H(t)C H ( p y ) 0
. Due to the time delay in Eq. (11), the
C. Ozoegwu and P. Eberhard
(a) Up-milling
(b) Down-milling
Fig. 2 Start angle θ s and end angle θ e of milling
where z(t) ∈ R
d F is the state, M( p y ) ∈ R
d F ×d F is the mass matrix, D( p y ) ∈ R
d F ×d F
is the damping matrix, and K( p y ) ∈ R
d F ×d F is the stiffness matrix. The matrices
B H ( p y ) ∈ R
d F ×n d and C H ( p y ) ∈ R
n d ×d F are used to project the n d -dimensional tool
excitation H(t) to the d F -dimensional workpiece response. The normalized parameter p y is indicative of the dependence of the modal matrices on tool location along
the feed direction (y) where p y = 0 at the beginning of tool pass and p y = 1 at the
end of the pass. Such high-fidelity FE model as given in Eq. (9) has many DOF and
is thus computationally cumbersome. Based on modal truncation, Eq. (9) reduces,
according to [23], to
˜
M( p y ) ¨ ˜
z(t) + ˜
D( p y ) ˙ ˜
z(t) + ˜
K( p y )˜ z(t) = ˜
B H ( p y )H(t) ˜
C H ( p y )(˜ z(t) − ˜
z(t − τ )),
(10)
where ˜
z(t) ∈ R
d R is the reduced state, ˜
M( p y ) = V
T
( p y )M( p y )V( p y ) ∈ R
d R ×d R is
the reduced mass matrix, ˜
D( p y ) = V
T
( p y )D( p y )V( p y ) ∈ R
d R ×d R is the reduced
damping matrix, ˜
K( p y ) = V
T
( p y )K( p y )V( p y ) ∈ R
d R ×d R is the reduced stiffness
matrix, ˜
B H ( p y ) = V
T B H ( p y ) ∈ R
d R ×n d , ˜
C H ( p y ) = C H ( p y )V ∈ R
n d ×d R , and
V( p y ) ∈ R
d F ×d R is the projection matrix. The projection matrix V( p y ) is a concatenation of the first d R eigenvectors where d R d F . In [23], each p y -dependent
matrix is derived from cubic spline interpolation of the values at a finite number of
tool location zones under a consistent system of coordinates ˜
z(t). Making the substitution x 1 (t) = ˜
z(t) and x 2 (t) = ˙ ˜
z(t), the first-order form of the reduced model for
stability analysis using the FDM reads
˙
x(t) = Ax(t) + B(t)x(t) − B(t)x(t − τ ),
(11)
where
A =
0
I
−( ˜
M( p y ))
−1 ˜
K( p y ) −( ˜
M( p y ))
−1 ˜
C( p y )
,
and
B(t) =
0
0
( ˜
M( p y ))
−1 B H ( p y )H(t)C H ( p y ) 0
. Due to the time delay in Eq. (11), the