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How comes it to large roll angles in intact vessel case?

Roll motion is a periodic motion. High amplitudes of periodic motion are observed, usually when the excitation and the vibrating system are in resonance.


What is the case in beam sea?

w_e = w
encounter frequency = wave frequency

The wave and the ship have the same frequency so the ship moves with the waves (transverse)


What is the maximum value of the roll angle dependent on?

The damping in the system. Radiated waves, the viscosity and hydrodynamic lift forces plays a role for the damping of the roll motion.


What is important to consider for the damping in beam sea and with no forward speed?

Damping is strongly dependent on the size and arrangement of the bilge keels. Vessels without bilge keels can achieve high roll motions.


What is the case with forward speed?

Roll damping for the hull and appendages (bilge keel etc.) is strong and high roll angles are unlikely to occur.


Explain pure loss of stability

Ships have typically slender frames in the forward and the aft region. The waterline surface/area is decreased on a wave crest and increased in a wave trough. The most dangerous situation is in following seas due to the time span where the ship is at the wave crest (GM small). In this case the initial stability can be negative and the righting lever arm may be so low that the ship takes on very large roll angles or even capsizes.


Explain broaching/surf-riding

When a ship is moving in following seas, the waves are equal to the ship length, but faster than the ship. The ship is therefore accelerated significantly from the wave in the longitudinal direction, so that the relative velocity will be small. The stability is lost or reduced significantly when the bow of the ship dips into the forward-running wave (fex). Relative velocity between the flow and the rudder is reduced by the orbital velocity of the follower waves, rudder effectiveness decreases. This leads to an uncontrolled strong turning manouver of the ship leading to a large heel angle due to centrifugal accelerations. Consequently, the ship will lay transverse to the wave with a large heel angle and will be overrun transversely by breaking crest of the incoming wave.


Explain parametric rolling

New container ships have large bow and stern flare and wide beam above the load waterline. As the wave travels along the hull, the waterline changes (region bow and stern) to the crests and troughs accordingly. This creates a periodic oscillation of the GM-value and the to the excitation of roll motion. In the trough phase, more stability and a larger righting moment.

Parametric rolling pitches twice while completing one roll cycle. Te = 0.5*To
- The encounter period is the same as the half of the natural rolling period

When the ship is located in the wave troughs the rolling speed is increased due to the stability, and on wave crests the GM is decreased leading to an increase in the roll angle. This means that the roll motion is excited for every roll period twice which is very dangerous. Typically the roll amplitudes on both sides are approximately the same once the system is excited.


How can parametric rolling occur when Te =To?

The excitation takes place only once during a roll period, from the same side. The duration of the transient oscillation is longer, and the rolling angle to one side is always greater than the other.


How can parametric rolling be investigated?

- Model tests
- Simple simulation, in which roll is not treated linearly
- Simulations in which all the DOFs are treated non-linearly, but the flow forces are determined by using simple method (SIMBEL, Uthlande)
- Simulation with RANSE solvers, where the fluid dynamics problem is solved in the time domain


Explain non-linear kinematic coupling

A further reason for occurrence of high roll motion is the non-linear kinematic coupling between the roll motion and the pitch motion of a ship.

Autoparametric resonance - the pitch natural frequency is equal to a multiple of roll natural period


Mathieu equation - What does it say?

It explains the resonance conditions for parametric excited roll which can help us with making an overview of the unstable and stable regions (periodic solutions). The value of the amplitudes occurring depends on the shape of the righting lever arm.
Damping affects the resonance and in areas with high increasing damping, the instability regions get smaller.
epsilon (y-axis) =
delta (x-axis) =