ManeuveringTopics Covered:1. Equations of Motion2. Stability Criteria

Coordinate Systems

• Inertial Coordinate System – Fixed relative to the Earth• Ship Coordinate System – moving with the ship

Orientation of Fixed Axis and Moving Axis

Newton’s Law in Inertial Reference Plane

Transformation of Coordinate System

Forces and Velocities Expressed in SCSThe motion of a ship is more conveniently expressed when referred to the (x, y) system of coordinates fixed with respect to the moving ship The ship fixed reference frame is always a right hand frame, with the x–axis pointing in the longitudinal direction, the y–axis positive starboard, and the z–axis positive down. We can transform coordinate systems using

Velocity and Acceleration Terms

where the dot above the symbol signifies the first derivative of the quantity with respect to time, and u and v are the components of V along x and y, respectively. Then

Equations of Motion• Differentiating and substituting into equations (1) we

obtain, using the fact that = r

Case 1 Roots are real, negative

• Roots are real• perturbations damp down to zero• motions are stable

ROUTH STABILITY CRITERIA

Case 2 Roots are real, positive

• Roots are real •Roots are positive• perturbations increase from zero• roll angle shall continue to increase and the ship shall capsize• motions are unstable

Case 3 Roots are complex, < 0

• Roots are complex • oscillatory response damps down to zero• motions are stable

Case 4 Roots are complex, > 0

• Roots are complex • oscillatory response increases and shall lead to ship’s capsize• motions are unstable