Chapter 3 - Dynamics of Marine Vessels
Chapter 3 - Dynamics of Marine Vessels Chapter 3 - Dynamics of Marine Vessels
16 3.3.1 Nonlinear Equations of Motion Properties of the NED Vector Representation M ∗ ̈ C ∗ , ̇ D ∗ , ̇ g ∗ J − g o w (1) M ∗ M ∗ 0 ∀ ∈ 6 (2) s ̇ M ∗ − 2C ∗ , s 0 ∀ s ∈ 6 , ∈ 6 , ∈ 6 (3) D ∗ , 0 ∀ ∈ 6 , ∈ 6 if M M 0and ̇ M 0. It should be noted that C ∗ , will not be skew-symmetrical although C is skew-symmetrical. Ivar Ihle – TTK4190 Spring 2006
17 3.3.1 Nonlinear Equations of Motion Property (System Inertia Matrix) For a rigid body the system inertia matrix is strictly positive if and only if M A >0, that is: If the body is at rest (or at most is moving at low speed) under the assumption of an ideal fluid, the zero-frequency system inertia matrix is always positive definite, that is M M 0 where: M M MRB MA 0 m − Xu̇ −Xv̇ −Xẇ −Xv̇ m − Yv̇ −Yẇ −Xẇ −Yẇ m − Zẇ −Xṗ −mzg−Yṗ my g −Zṗ mzg−Xq̇ −Yq̇ −mxg−Zq̇ −my g −Xṙ mxg−Yṙ −Zṙ M ≠ M −Xṗ mzg−Xq̇ −my g −Xṙ −mzg−Yṗ −Yq̇ mxg−Yṙ my g −Zṗ −mxg−Zq̇ −Zṙ Ix−Kṗ −Ixy−Kq̇ −Izx−Kṙ −Ixy−Kq̇ Iy−Mq̇ −Iyz−Mṙ −Izx−Kṙ −Iyz−Mṙ Iz−Nṙ Ivar Ihle – TTK4190 Spring 2006
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16<br />
3.3.1 Nonlinear Equations <strong>of</strong> Motion<br />
Properties <strong>of</strong> the NED Vector Representation<br />
M ∗ ̈ C ∗ , ̇ D ∗ , ̇ g ∗ J − g o w<br />
(1) M ∗ M ∗ 0 ∀ ∈ 6<br />
(2) s ̇ M ∗ − 2C ∗ , s 0 ∀ s ∈ 6 , ∈ 6 , ∈ 6<br />
(3) D ∗ , 0 ∀ ∈ 6 , ∈ 6<br />
if M M 0and ̇ M 0.<br />
It should be noted that C ∗ , will not be skew-symmetrical although C is skew-symmetrical.<br />
Ivar Ihle – TTK4190 Spring 2006
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