Environmental Engineering Reference
In-Depth Information
the complex non-linear character of multi-body systems and this is independent
of the elastic modelling. Equation (14) is introduced in (10) and subsequently into
(12). The resulting equations are discretized using FEM approximations. For each
component of the system the dynamic equations will then take the following form:
ˆ
ˆ
G
ˆ
M u +C qqu
(, )
+
(
K
(,, )
qqq
+
K u
)
+
R qqq
(,, )
+
BL
=
Q
(15)
kk
k
k
k
k
k
k
k
Note in (15) the appearance of a damping like term defi ned by C k , an extra stiff-
ness term defi ned by
G
k
K
and a non-linear term R k depending on q and its time
derivatives.
The boundary loads BL are determined by the virtual work done by the reacting
forces and moments F and M
at the connection points:
⎧⎫
F
M
T
T
T
T
(
d
uI
+
d
u I
)
=
d
u K u
[
+
K u
′′
+
(
K u K u
)
′′
]
d
u K u
′′
⎩⎭
F
M
11
12
21
22
22
100000
00000 1
010000
000000
I
=
,
001000
I
=
F
M
000100
(16)
000010
000000
after eliminating the virtual displacements and rotations. The boundary loads will
introduce stiffness which will however depend on the DOF of the bodies connected
to k. The fi nal step is to assemble all component equations into one fi nal system.
In view of obtaining a more manageable set of equations, linearization is usu-
ally carried out based on formal Taylor's expansions with respect to a reference
state. The reference state can be either fi xed as in the case of linear stability analy-
sis, or represent the previous approximation within an iterative process towards the
non-linear solution. By collecting all unknowns into one vector x , the following
form is obtained:
ˆ
TTT
Mx + Cx + Kx = Q
,
x
=
(
u
,
q
)
(17)
In eqn (17) although there is no dependence indicated, M , C , K and Q depend on
the reference state x 0 and its time derivatives. The structure of the mass, damp-
ing and stiffness matrices is given in Fig. 6 in the case of a three-bladed wind
turbine. The contribution of the local equations for each component are block
diagonal. The kinematic conditions at the connection points appear in the out
right column. The static connection conditions appear as isolated rows denoted
as “dynamic coupling terms” and correspond to the terms appearing in eqn (16).
Finally the extra equations for q appear last.
4.4 Eigenvalue analysis and linear stability
Eigenvalue analysis is a useful tool in structural analysis because it provides a concise
dynamic characterization of the system considered. For a linear system without
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