Ship Motion Application

Commentary:

The formulation necessary to develop this app was based on the article Estimation of ship motions using closed-form expressions by Jensen and Mansour (2004). A validation of the results, comparing to model tests and linear strip theory are commented in the paper.

Example of theory validation for vertical acceleration ((m/s2)/m) at forward perpendicular as function of wave frequency for different headings, by Jensen and Mansour (2004).

The method is based on the wave response of a box shaped vessel with Length, Breadth and Draft as main dimensions. The responses are calculated for different wave frequencies and then plotted. To account for the hull shape in a real vessel, the waterline breadth, B₀, is multiplied by the Block Coefficient, C_b and the result, B, is used as the input for the Beam, as shown below:

B = B₀C_b

The article explains the motivation and the value of this approach:

In the design of ships, the wave-induced motions and accelerations are important to the assessment of the comfort of the crew and the passengers and to the scantlings of securing devices like lashing for container stacks. Usually, the design values are taken from the classification society rules where explicit formulas are given. However, since the formulas do not depend on the operational profile the naval architect cannot assess the influence of e.g. a weather routing system or speed reduction in heavy sea.
Direct calculation of the maximum wave-induced motions and accelerations a ship may encounter during its operational lifetime can be performed by taking into account the hull form, the mass distribution and the operational profile. A linear analysis is fairly straightforward using either two- or three-dimensional hydrodynamic procedures based on potential theory. However, such direct calculation procedures are not very useful in the conceptual design phase, because of lack of detailed data for the ship and because significant expertise and time are required to do the calculations.

The benefit of such simplified methodology comes from the fact that it efficiently allows the operational profile of a vessel to be used as initial parameter for the ship motion analysis at conceptual phase with reliable results. The method, however, has a few accuracy limitations and simplifications, which are listed as:

• Heave response is smaller than reality when λ/L > 1 (wave longer than the ship's length);
• Pitch response is larger than reality around λ/L = 1 for Froude numbers larger than 0.2;
• The simplified formulas predict zero pitch in beam sea (90 degrees);
• Pitch and heave movements are considered to have a phase difference of 90 degrees;
• The theory assumes very deep water waves;
• For the Bending Moment Calculation the minimum Cb allowed is 0.6

The app is coded in javascript and runs in a web based environment, allowing for a real time visualization of the results in different browsers and systems.

Preliminary Graphics' Explanantion

GRAPHIC 1 - Vertical motion (m/m) as function of wave frequency. This motion takes into consideration the combined movement from the pitch and heave at the desired location.

Length: Bigger length means reduced vertical motion;

Breadth: Bigger breadth means reduced vertical motion;

Cb: Bigger block coefficient means reduced vertical motion;

Draft: Increasing the draft value also increases the maximum vertical motion value and it occurs at a lower wave frequencies;

Speed: It affects the wave encounter frequency, which also affects the intensity of the motion. Higher speed translates into bigger motion;

Relative Position: It affects the movement caused by pitch, thus having bigger vertical motion;

Heading: It influences the wave frequency at which the maximum vertical motion values occur.

GRAPHIC 2 - Vertical acceleration (m/s²) as function of wave frequency. This motion takes into consideration the combined movement from the pitch and heave at the desired location. It is deriveted from the Vertical Motion calculated.

Length: Bigger length means reduced vertical acceleration;

Breadth: Increasing the breadth decreases the vertical acceleration to a point and then it increases again;

Cb: Increasing the block coefficient decreases the vertical acceleration to a point and then it increases again;

Draft: Increasing the draft decreases the vertical acceleration to a point, then the maximum value becomes constant, but the wave frequency at which it occurs starts to have higher values;

Speed: It affects the wave encounter frequency, which also affects the intensity of the vertical acceleration. Higher speed translates into bigger acceleration;

Relative Position: Bigger vertical acceleration happens at extremes;

Heading: It influences to the vertical acceleration varies greatly.

GRAPHIC 3 - Pitch motion divided by the wave number (k) is represented in this graphic as a function of wave frequency. It describes vertical motion caused by the pitch movement in meters at the desired location.

Length: Increasing the length increases the pitch motion to a point, then it decreases and become stable. However, the changes also affect when the resonance frequency occurs;

Breadth: Bigger breadth means increased pitch motion values;

Cb: Changes in the block coefficient causes the pitch motion intensity to decrease and increase again (Unrealistic values when Cb ranges from 0.75 to 0.85 at L/B = 96/40);

Draft: Increasing the draft value also increases the maximum pitch motion value and it occurs at a lower wave frequencies;

Speed: It affects the wave encounter frequency, which also affects the intensity of the pitch motion. Higher speed translates into bigger motion;

Relative Position: It peaks the vertical motion due to pitch at the extremes and is 0 at the Center of Gravity;

Heading: It varies a lot, with no pitch motion at 270 and 90 degrees (side waves).

GRAPHIC 4 - Heave motion (m/m) as function of wave frequency. It represents the heave induced vertical motion at the Center of Gravity of the vessel.

Length: Bigger length means reduced heave motion;

Breadth: Bigger breadth means reduced heave motion. Changing it might create a resonance spike (depends on the other parameters combination);

Cb: Block Coefficient variances do not affect the heave motion in this formulation;

Draft: Increasing the draft value also increases the maximum heave motion value and it occurs at a lower wave frequencies;

Speed: It affects the wave encounter frequency, which also affects the intensity of the heave motion. Higher speed translates into bigger motion;

Relative Position: Changes have no effect, since the heave motion is constant on the whole vessel length;

Heading: It influences the wave frequency at which the maximum heave motion values occur.

GRAPHIC 5 - Wave Induced Bending Moment (10⁶ N.m) as function of wave frequency.

Length: Bigger length means increased wave induced bending moment;

Breadth: Bigger breadth means increased wave induced bending moment;

Cb: Bigger block coefficient means increased wave induced bending moment;

Draft: Bigger draft means reduced wave induced bending moment;

Speed: Higher speed means increased wave induced bending moment;

Relative Position: Does not affect the formulation for wave induced bending moment. The maximum moment occurs at midship in ships;

Heading: The effect of changes in heading varies a lot.