by Sthefano L. Andrade and Henrique M. Gaspar

(sthefano.lande@gmail.com; hega @ hials.no), v0.1, Oct 2015.

A Ship Design and Operations Lab App.

Graphical representation of estimated motion responses for ships. Heave, pitch, roll, vertical motion and vertical acceleration responses are calculated as function of length, breadth, draft, block coefficient, waterline breadth and operational profile. Bending moment is also estimated. Values can be changed by clicking and dragging the sliders. Based on the article **Estimation of ship motions using closed-form expressions by Jensen and Mansour (2004)**.

Length:

Waterline Breadth:

Block Coefficient:

Draft:

Speed:

Relative Position to CG (%) :

Heading:

Wave Amplitude:

Froude Number:

Waterplane Area Coefficient:

Transverse Metacentric Height:

Roll Natural Period:

Estimated Tn Value =

Empirical damping ratio (%):

Prismatic Length Ratio:

(for old browsers)

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

GRAPHIC 2 - Vertical acceleration (m/s^{2}) as function of wave frequency. Combined movement from the pitch and heave at the desired location. Deriveted from the Vertical Motion calculated.

RAPHIC 3 - Pitch motion is represented in this graphic as a function of wave frequency. Describes vertical motion caused by the pitch movement in meters at the desired location.

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

GRAPHIC 5 - Wave Induced Bending Moment (10^{6} N.m) as function of wave frequency.

**Length (m):** Ship's waterline length.

**Waterline Breadth (m):** Ship's maximum breadth at the waterline.

**Draft (m):** Distance between the bottom of the vessel and the waterline.

**Block Coefficient:** Ratio between the ship's displacement volume and the volume of a prism with dimensions defined by Length, Waterline Breadth and Draft.

**Speed (kts):** Speed of the vessel, in knots.

** Relative position to CG (%):** Longitudinal position on the vessel where the user wishes to study the motion. Defined as a percentage of the ship length in relation to its center of gravity (in this case considered to be located at *L*/2). Positive to the fore and negative to the aft.

** Heading (degrees):** Direction of the waves in relation to the ship, where at 180 degrees they are head on waves.

** Wave Amplitude(m):** Amplitude of the waves.

**Waterplane Area Coefficient:** Ratio between the ship's waterplane area at desired draft and its waterline length times waterline breadth.

**Transverse Metacentric Height (m):** Distance between the ship's transverse center of gravity and its transverse meta-center.

**Roll Natural Period (s):** Natural period of roll of the ship. When unknown by the user, the program suggests a value based on the following rule of thumb: *Tn* = 0.85**Breadth*/*GM*^{2}

**Critical Damping (%):** This parameter is used to take into account the water viscosity. The viscose roll damping is approximately accounted for by adding a percentage of the critical damping to the inviscid wave damping.

**Prismatic Length Ratio:** Length ratio of two prismatic bodies that represent the ship. The ship is assumed to consist of two prismatic elements with the same draft, T, but different breadths B0 and B1 and cross-sectional areas A0 and A1. As shown:

*Model of the simplified ship used in roll motion analysis (Jensen and Mansour, 2004)*

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).*

*B* = *B _{0}*

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.

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.

GRAPHIC 2 - Vertical acceleration (m/s^{2}) 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.

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.

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.

GRAPHIC 5 - Wave Induced Bending Moment (10^{6} N.m) as function of wave frequency.