Chapter 3.1 Newton-Euler about the CG

Newton’s second law

$$m\dot{\vec{v}}={\vec{f}}_g$$

subscript g denotes the center of gravity (CG)

Euler’s first and second axioms

Conservation of linear momentum:

$${\vec{p}}_g$$

Conservation of angular momentum:

$${\vec{h}}_g$$

f_g and m_g are forces/moments about the body’s CG w_ib is the angular velocity of frame b relative to frame i I_g is the intertia dyadic about the body’s CG

Translational motion about the CG

When deriving the equations of motion, it will be assumed that:

1. The vessel is rigid
2. The NED frame is inertial

The first assumption eliminates considerations of forces acting between elements in the mass.

The second assumption eliminates forces due to earth’s movement relative to a star-fixed inertial system.

Translational motion about the CG expressed in {b}

Rotational motion about the CG

Equations of motion about the CG