Civil Engineering Reference
In-Depth Information
ª
S
ˆ
S
ˆ
º ª
n
1
º
ª
M
M
0
a
Sn
ˆ
1
º
f
ij
i
T
j
i
«
» «
»
«
»
«
» «
»
«
S
ˆ
S
ˆ
T
ˆ
T
ˆ
n
1
S
ˆ
n
1
T
ˆ
n
1
»
x
[5.7]
MMMM a
f
f
G
j
GT
GT
G
J
T
G
G
«
» «
»
«
»
«
» «
n
1
»
«
T
ˆ
n
1
T
ˆ
T
ˆ
a
f
»
0
M
M
J
I
«
» «
»
¬
¼
I
T
IJ
¬
¼ ¬
¼
where the indexes
,
ij
refer to the dofs inside the numerical structure,
GT
the dofs
common to both parts and
I J
the dofs inside the tested structure. Now if we
condense the accelerations
a
, which are inside the numerical structure, we obtain
the following control equation, for the whole dofs present in the lab:
S
ˆ
S
ˆ
S
ˆ
1
S
ˆ
T
ˆ
T
ˆ
ª
º
ª
n
1
º
(
MMMM MM
a
)
GT
G
j
ji
i
T
GT
G
J
T
«
»
«
»
x
n
1
«
T
ˆ
T
ˆ
»
«
»
a
M
M
¬
¼
¬
¼
J
I
T
IJ
[5.8]
Sn
ˆ
1
S
ˆ
S
ˆ
1
Sn
ˆ
1
Tn
ˆ
1
ª
º
(
f
MMf
)
f
G
G
j
ji
i
G
«
»
«
T
I
ˆ
1
»
f
¬
¼
and the numerical structure is ruled by:
S
ˆ
n
1
S
ˆ
n
1
S
ˆ
n
1
M
a
f
M
TT
[5.9]
ij
j
i
i
Equation [5.8] allows us to stress the differences between a simple PSD test and
a test with sub-structuration. As far as sub-structuration is concerned, an additional
mass matrix related to the connection nodes (term between brackets in the first
member of [5.8]) should be added to the experimental pseudo-mass. In the same
way, at each time step, the pseudo-force vector should include a contribution (term
between brackets in the second member of [5.8]) that will become the numerical
sub-structure. It is interesting to note that during the time-step loop, the data flow
between the digital process and the experimental process is balanced. If the
experimental part needs the contribution of forces on the connection nodes, the
numerical process will require acceleration on the same connection nodes to be able
to develop according to equation [5.9]. Let us also note that such an implementation
is mainly asynchronous: each process has to wait for the other to be able to go
ahead. That approach is natural within the frame of the conventional PSD method,
where the implementation length of a loading step in displacement is not set
a
priori
. As we will see, this is no longer the case with the continuous PSD method
(see section 5.2).
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