31_add_Added Masses of Ship Structures.pdf - 1.5 Ellipsoids of Added Masses and Ellipsoids of Added Moments of Inertia 19 The main axes of the ellipsoid

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1.5 Ellipsoids of Added Masses and Ellipsoids of Added Moments of Inertia 19 The main axes of the ellipsoid have the following property: if the body moves with constant velocity in an ideal fluid without rotation along the main axis, then the hydrodynamic forces and torques are equal to zero. Indeed, the added masses λ 12 13 23 vanish in the coordinate system, whose axes coincide with the main axes. The vanishing of forces and torques follows from expressions ( 1.22 )–( 1.27 ) since in the new coordinate system R x = R y = R z = L x = L y = L z = 0. The mo- tion with constant velocity along one of main axes is stable; along the two other main axes such motion is unstable. Let the main axes be chosen such that λ 11 < λ 22 < λ 33 . Suppose that a body moving with constant velocity along the axis Ox 1 receives a deviation—a rotation around the axis Oz 1 in the negative (clockwise) direction (Fig. 1.2 ) by an angle ϕ . Then the velocity gets the second component u 2 in the direction of the main axis Oy 1 attached to the body. Equation ( 1.27 ) shows that in this situation the body is

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