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where the summation is over all nodes in the rigid block,
m
k
and
r
k
are the mass
and position of node
k
, and
M
= Σ
k
m
k
and
r
CM
are the mass of the block and
position of its center of mass. Note that Eq. (7.24) holds not only for velocities,
but also for positions and higher time derivatives. The matrix elements
projecting from the full 3
N
-dimensional space to translations in the block space
are
ɺ
∂
(
r
)
m
M
CM
µ
k
=
δ
,
(7.25)
µν
ɺ
∂
(
r
)
k
ν
where µ and ν indicate
x
-,
y
- or
z
-components of the vector (or tensor) enclosed
in parentheses, and δ
µν
is the Kronecker delta function. Similarly, angular
momentum conservation gives
ɺ
∑
1/ 2
I
θ=
r
×
r
ɺ
,
m
(
)
(7.26)
k
k
k
k
where
I
and θ are the block's moment of inertia tensor (the elements of which are
given by I
µα
≡ Σ
k
m
k
(r
k
2
δ
µα
- (
r
k
)
µ
(
r
k
)
α
) and angular displacement vector. The
components of θ
ɺ
are
ɺ
∑ ∑
−
1/ 2
ɺ
θ
=
m
(
I
)
(
r
×
r
)
(7.27)
µ
k
µα
k
k
α
k
α
=
∑ ∑
∑
−
1/ 2
ɺ
m
(
I
)
(
r
)
(r
)
ε
,
(7.28)
k
µα
k
β
k
ν αβν
k
α
βν
where ε
αβν
is the permutation symbol (ε
123
= ε
231
= ε
312
= 1; ε
213
= ε
321
= ε
132
=
-1; otherwise ε
αβν
= 0), also known as Levi-Civita symbol. Differentiating the
components of θ
ɺ
with respect to the components of
r
ɺ
the matrix elements
projecting from the full 3
N
-dimensional space to rotations in the block space are
,
∂
θ
µ
∑
−
1/ 2
=
m
(
I
)
(
r
)
ε
.
(7.29)
k
µα
k
β αβν
ɺ
∂
(
r
)
αβ
k
ν
Using Eqs. (25) and (29) and the notation of Li and Cui (2002), the elements of
the 3
N
× 6
n
b
projection matrix
P
are
m
/
M
δ
for
µ
=
1, 2, 3
j
J
µν
µ
P
=
(7.30)
J
,
j
ν
∑
−
1/ 2
0
(
I
)
m
(
r
−
r
)
ε
for
µ
=
4, 5, 6
(
µ
−
3),
α
j
j
J
β αβν
αβ
where
J
is the rigid block index,
r
J
0
is the position vector of the mass center of
block
J
,
j
is the node index, µ designates the rigid block translation (1 ≤ µ ≤ 3) or
rotation (4 ≤ µ ≤ 6), and ν is the node displacement component (1 ≤ ν ≤ 3). The
