6 Degrees Of Freedom

1.

CAUSES OF THE SIX DEGREES OF FREEDOM. a.

Pitching

-

b.

Heaving

-

c.

Surge

-

Swaying along the lateral Y-axis. Motion about the vessel’s transverse axis. To have a constant trim (trim being the difference of the forward and after drafts), stability in the motion of pitch must be acquired. Governing stability – Longitudinal stability The yawing along the vertical Z-axis. The vertical bodily motion of the vessel. To keep the vessel on the surface at a relatively constant mean draft, stability in heave is necessary. Governing stability – Positional motion stability Rolling along the longitudinal X-axis. Longitudinal bodily motion. There is a desire for a vessel to maintain a constant speed, this would require that the vessel have stability along the surge axis. Governing stability – Stability in motion ahead & astern

d.

Rolling -

Motion about the vessel’s longitudinal axis. One of the most undesirable characteristic of ships. Setting up stresses in the structure, causing discomfort to both passengers and crew. Generates risk of cargo shifting and increasing the cost of operation. Rolling is sometimes a direct cause of speed reduction but more often a change in course, which in turn, may result in speed reduction. It is a concern to keep the vessel from capsizing, stability in rolling motion must be sufficient. Governing stability – Transverse stability

e.

Yawing-

motion about the vessel’s vertical axis. Rotation of a ship about a vertical axis approximately through its center of gravity. This is undesirable because its correction requires the use of a rudder with increase in resistance to propulsion and because it produces yaw-heel. It is desirable for a vessel to be able to stay on course and not swing wildly from it. This can be construed to mean that the vessel is stable in yaw motion or heading. Governing stability – Directional stability

f.

Sway

-

Bodily transfer of the ship in a lateral direction due to orbital motion of the water in a wave. Lateral, side to side, bodily motion. It is important to minimize a vessel’s sideways or lateral motion. This requires a high degree of stability in sway. Governing stability – Lateral motion stability

2.

COMPUTATION OF ROLLING PERIOD. -

For ships of ordinary form at moderate angles of roll, the axis is not far from the center of gravity; and, where the simplification assists in the solution of a problem, the axis of roll is assumed to pass through G. Under this assumptions, the equation of motion of the ship is: IdØ+M=0 dt

- equation no. 1

where: I = mass moment of inertia of the ship about a long’l axis through the center of gravity M = is the righting moment Ø = angle of inclination of the ship from the vertical

I= Δ k g -

- equation no. 2

For small angles of inclination: M = ΔGZ = ΔGM sinØ = ΔGMØ

-

2πk √gGM

- equation no. 5

Therefore the rolling period of a ship is: TØ = 1.108k √GM

-

- equation no. 4

Equation no. 1 is the equation for simple harmonic motion having the period: TØ =

-

- equation no. 3

Substituting these values, we arrive: d Ø + g GM Ø = 0 dt k

-

where: k = radius of gyration of mass of a ship about a long’l axis through G

- equation no. 5

For induced rolling (sallying), rolling period is: Period = 1.108k √GM

2.

DRAW THE SIX DEGREES OF FREEDOM.

HEAVE YAW

SWAY PITCH SURGE ROLL