locmot
DESCRIPTION
lTRANSCRIPT
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1. General
The following quantities can be calculated locally on a structure by aNySIM:
- Motions.
- Velocities.
- Accelerations.
2. Local position
The location is defined in the local system of coordinates of the structure. Let’s call P the
point of interest attached to the structure. The user should define the position of P as
reference position in the input to have access to output signals at this location. Let’s consider
the case of a semi-sub with two cranes at the stern on each side. Let’s assume that the user
is interested in the motion at the crane at portside.
Ship-fixed coordinates of the point of interest
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3. Pure rotations
The motions (velocities, accelerations) are integrated at the centre of gravity G in aNySIM.
The following convention is used for the definition of the rotation angles.
Rotation convention
Ф is used for the roll angle.
Θ is used for the pitch angle.
Ψ is used for the yaw angle.
Let’s consider a semi-sub with no heel, no trim in its initial orientation. We called {SFt0} the
system of axes of the semi-sub as defined by its ship-fixed axes at t= 0s. This system of axes
is fixed. At time t, the system of axes attached to the semi-sub is called {SFct} for ‘ship-fixed
at current time’. {SFct} is moving with the semi-sub.
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If the semi-sub has been purely rotated (no translation), then the point G has not moved
whereas the point P has moved. The new position of P is determined by the total rotation of
the vector GP.
By convention, we pass from {SFt0} to {SFct} through three elementary rotations in the
following order:
- The rotation around the axis Gz, yaw.
- The rotation around the axis Gy, pitch.
- The rotation around the axis Gx, roll.
A rotation matrix can be defined for each rotation.
For yaw:
−=
100
0cossin
0sincos
ψψ
ψψ
ψR
For pitch:
−
=
θθ
θθ
θ
cos0sin
010
sin0cos
R
For roll:
−
=
φφ
φφφ
cossin0
sincos0
001
R
A vector V is transformed from {SFt0} to {SFct} by:
[ ] 0SFtSFct VRRRV ⋅⋅⋅= ψθφ
In the other way, a vector V is transformed from {SFct} to {SFt0} by:
[ ] SFctSFt VRRRV ⋅⋅⋅=−1
0 ψθφ
Which can be written in the following form:
[ ] SFctT
SFt VRRRV ⋅⋅⋅= ψθφ0
Or:
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[ ] SFctTTT
SFt VRRRV ⋅⋅⋅= φθψ0
The new position of P at time t in {SFt0} is calculated by:
[ ] SFTTT
t GPRRRGP ⋅⋅⋅= φθψ
The motions at P induced by the total rotation is defined as:
==
P
P
P
tt
z
y
x
PP0
The motions in {SFt0} can be written in the following way:
[ ] SFTTT
tttt GPRRRGPGPPP ⋅
−⋅⋅=−= ==
100
010
001
00 φθψ
The motion at P in {SFt0} can be formulated as:
[ ] SFTTT
P
P
P
GPRRR
z
y
x
⋅
−⋅⋅=
100
010
001
φθψ
4. Pure translation
If the semi-sub has been submitted to a translation only (no rotation) then the motion at P is
identical to the motion at G.
=
G
G
G
P
P
P
z
y
x
z
y
x
Where:
- xG is the surge at CoG
- yG is the sway at CoG
- zG is the heave at CoG
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5. Translation + rotation
The new position of P in {SFt0} results from the translation of the structure plus its rotation(s)
around the point G.
[ ] SFTTT
G
G
G
tGPRRR
z
y
x
PG ⋅⋅⋅+
= φθψ0
The position of P in the earth-fixed global system of coordinates with origin O is given by:
{ } { }EFtEFEF PGOGOP
00 +=
We pass from the earth-fixed global system of axes {EF} to {SFto} by a yaw rotation of the
initial heading Ψ0.
−=
100
0cossin
0sincos
00
00
0
ψψ
ψψ
ψR
Finally the coordinates of P in the global referential {EF} are given by:
{ } [ ]tEFEF PGROGOP
000
⋅+= ψ
6. Local motion
The motion of P is defined by the difference between the current position P and its initial
position.
{ } { }EFEFtEF
EF
OPOPPP
dZ
dY
dX
00−==
This motion vector can be projected in 3 different system of axes:
0) The earth-fixed {EF} (global) SOA.
1) The {SFt0}, global SOA of which the direction of the (xx’) axis is given by the heading
of the structure at t= 0s.
2) The {SFct}, local (or relative) SOA which is moving with the structure.
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The norm of the motion vector gives the radius of the sphere that includes all motions of the
point P.