Chapter 3 - Dynamics of Marine Vessels
Chapter 3 - Dynamics of Marine Vessels Chapter 3 - Dynamics of Marine Vessels
22 3.3.2 Linearized Equations of Motion The gravitational and buoyancy forces can also be expressed in terms of VP coordinates. For small roll and pitch angles: g Notice that this formula confirms that the restoring forces of a leveled vessel ( ) is independent of the yaw angle . 0 ≈ PG PGP p Gp G 0 For a neutrally buoyant submersible (W=B) with x g =x b and y g =y b we have: G diag0, 0, 0, 0, zg − zbW, zg − zbW,0 For a surface vessel G is defined as: G 022 032 023 G r 0 0 0 0 0 0 0 0 0 0 , G r −Zz 0 −Z 0 −K 0 −Mz 0 −M P Notice that: GP ≡ G Ivar Ihle – TTK4190 Spring 2006
23 3.3.2 Linearized Equations of Motion Low-Speed Maneuvering and DP: ≈ 0 implies that the nonlinear Coriolis, centripetal, damping, restoring, and buoyancy forces and moments can be linearized about 0 and 0. Since C(0)=0 and Dn (0)=0 it makes sense to: approximate: Ṁ C D Dn 0 D The resulting state-space model becomes: A ̇ p Ṁ D G p w 0 I −M−1G −M−1D P p , B g Gp 0 M−1 g o w ̇x Ax Bu Ew x p , , u , E 0 M−1 which is the linear time invariant (LTI) state-space model used in DP. Ivar Ihle – TTK4190 Spring 2006
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22<br />
3.3.2 Linearized Equations <strong>of</strong> Motion<br />
The gravitational and buoyancy forces can also be expressed in terms <strong>of</strong> VP<br />
coordinates. For small roll and pitch angles:<br />
g<br />
Notice that this formula confirms that the restoring forces <strong>of</strong> a leveled vessel<br />
( ) is independent <strong>of</strong> the yaw angle .<br />
0<br />
≈ PG PGP p Gp G<br />
0 <br />
For a neutrally buoyant submersible (W=B) with x g =x b and y g =y b we have:<br />
G diag0, 0, 0, 0, zg − zbW, zg − zbW,0<br />
For a surface vessel G is defined as:<br />
G <br />
022<br />
032<br />
023<br />
G r<br />
0 0 0 0 0 0<br />
0<br />
0<br />
0<br />
0<br />
, G r <br />
−Zz 0 −Z<br />
0 −K 0<br />
−Mz 0 −M<br />
P Notice that:<br />
GP ≡ G<br />
Ivar Ihle – TTK4190 Spring 2006
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