Environmental Engineering Reference
In-Depth Information
two-bladed teetering rotor will transmit all loads to the drive train except for the
teetering moment. Also the displacements and the two rotations at the end of the
drive train will be communicated to the blades. Since the teetering moment is not
transmitted it can only be zero and this corresponds to the dynamic equation that
determines the teetering angle.
Certain connections, involve more than two bodies. In such cases note that while
displacements and rotations should not be added, loads must. So if body 1 pro-
vides a displacement then it must receive the sum of the corresponding loads from
all other connected bodies.
4.3 Implementation issues
In order to facilitate the code implementation for systems with several connec-
tions, the set of all kinematic DOF involved in the connections is introduced as
an additional unknown denoted as q. Therefore extra equations must be defi ned.
Note that q will contain not only elastic DOF but also all rigid body DOF like
yaw, teeter or pitch. If a specifi c q i is indeed an elastic degree of freedom u p then
we simply set q i
u p Otherwise an additional equation is needed. The condition
of zero moment for the teeter angle is such an example. Another example is the
pitch angle which is specifi ed by the control system. In this case the extra equation
would correspond to the controller equation (or equations).
The introduction of q , specifi es the form of R and T k Starting from the local
system of component k , a series of system displacements and rotations will bring
us to the global system:
r
= +
R
T
(
+
T
(
Rr
+ =+ =+ +
)
)
Rr
T
Rr
T
T Su
( 13 )
G
,
k
,
k
1,
k
1,
k
k
k
k k
k
k
0
k
In the above relation, each T m ,k may contain several consecutive rotations and
therefore appear as a product of elementary rotation matrices of the type: T * (
ϕ
)
defi ned for a given direction * = x , y , z and a given angle
ϕ
. As an example consider
the case of the drive train.
q
1
r
=
Hq
+
+
TTT
(
q
)
(
q
)
(
q
+
j
)
T
( )(
r
+
Su
)
G
T2
z
4
x
5
y
6
yaw
z
0
q
where H T denotes the tower height, q 1 -q 6 denote the elastic displacements and
rotations at the tower top and f yaw denotes the yaw angle. The 90 o rotation is here
added so that the axis of the drive train is in the y local direction.
Since
3
R
and T k depend on q ,
2
R
=∂
q
,
R
=∂
q q
+∂
q
k
jkj
k
jkij
jkj
( 14 )
2
TTTT
=∂
q
,
=∂
q q
+∂
T
q
k
jkj
k
jkij
jkj
2
2
∂=∂∂ ∂=∂∂∂ and repeated indexes indicate summation. It is
clear that all components of q are not always needed. Nevertheless eqn (14) reveals
/
q
,
/
q
q
in which
j
j
ij
i
j
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