# Numerical Modeling of Wave Disturbances in the Process of Ship Movement Control

## Abstract

**:**

## 1. Introduction

## 2. Wave Disturbances Generated by Wind

- $S\left(\omega \right)$—spectral density of wave ordinate;
- $\omega $—angular frequency of the wave (pulsation);
- D—dispersion of wave coordinates $\left(D=0.143{\left({h}_{3\%}\right)}^{2}\right)$;
- ${h}_{3\%}$—wave height with 3% probability of exceedance (Table 1);
- $\alpha $—parameter dependent on parameter $\beta $$\left(\alpha =0.21\beta \right)$.

- Waves coming at a specific angle from the direction opposite to ship movement (diagonal waves);
- waves coming at an angle, but in the direction of ship movement (quartering waves);
- waves coming from the bow (head waves);
- waves perpendicular to the ship side (beam waves);
- waves striking the stern (following waves).

- ${\alpha}_{k}$—parameter dependent on parameter ${\beta}_{k}$$\left({\alpha}_{k}=0.21{\beta}_{k}\right)$;
- ${\beta}_{k}$—parameter calculated from the formula $\left|\beta +V{g}^{-1}\mathrm{cos}\left(\xi \right){\beta}^{2}\right|$;
- V—ship’s speed;
- G—gravitational acceleration;
- $\xi $—wave angle (Figure 2).

- ${S}_{r}\left(\omega \right)$—spectral density of wave angular velocity;
- ${x}_{r},{x}_{T}$—dimensionless reducing parameters (Figure 3) dependent on the length of the ship (L), wavelength ($\lambda $), and the ship’s maximum draught marked by the waterplane (T).

_{w}should be added to—or, for positive wave angles, subtracted from—the ship’s angular speed caused by rudder deflection.

- $polyval\left(p,{x}_{i}\right)$—a function returning the value of the polynomial with coefficients written in a table p, in a set point x
_{i}; - $cutf\left(f(x),a,b\right)$—function returning the value $f\left(x\right)$ for $x\in \langle a,b\rangle $, value $f\left(a\right)$ for $x<a$ and value $f\left(b\right)$ for $x>b$.

Algorithm 1 Using Lagrange interpolation |

% x = [x0, x1, …, xN], y = [y0, y1, …, yN] |

% p—coefficients of Lagrange polynomial |

function p = lagranp(x,y) |

N = length(x) − 1; |

p = 0; |

for m = 1:N + 1 |

P = 1; |

for k = 1:N + 1 |

if k ~= m |

P = conv(P,[1 − x(k)])/(x(m) − x(k)); |

End |

End |

p = p + y(m) * P; |

end |

end |

- Parameters associated with the wave:
- -
- Wave angle $\xi $;
- -
- wave height ${h}_{3\%}$;
- -
- wave length $\lambda $;

- parameters of the ship:
- -
- Maximum draft T;
- -
- length L;
- -
- speed V.

## 3. Computing Experiments

- $({x}_{1},{x}_{2})=(x,y)$—Cartesian coordinates (ship’s position);
- ${x}_{3}=\psi $—deviation from the course;
- ${x}_{4}=r$—angular velocity;
- ${x}_{5}$—longitudinal speed;
- ${x}_{6}$—transverse speed;
- $u=\delta $—rudder angle;
- ${u}_{z}={\delta}_{z}$—rudder angle setting;
- ${\delta}_{\mathrm{max}}$—maximum rudder deflection;
- ${\dot{\delta}}_{\mathrm{max}}$—maximum rate of turn of the rudder;
- S
_{t}—propeller thrust; - ${a}_{1},{a}_{2},{a}_{3},f,W,{r}_{1},{r}_{3}$—coefficients determined from model tests (different for different types of vessels).

^{2}], ${a}_{3}=0.001$ [1/s

^{2}], $f=0.014$ [1/s], $w=124$ [m/rad

^{2}], ${s}_{t}=0.11$ [m/s

^{2}], ${r}_{1}=-69.5$ [m/rad], ${r}_{3}=0$ [m/s

^{2}/rad

^{3}]. This particular ship has the following characteristics: 9214 gross registered tons [t], 13,498 DWT, single screw, length L = 172 [m], maximum draft indicated by the waterplane T = 8 [m], maximum speed 20 [w], maximum (minimum) angular velocity ${r}_{\mathrm{max}}=0.0191$ [rad/s] (${r}_{\mathrm{min}}=-0.0191$ [rad/s]), maximum (minimum) rudder angle ${\delta}_{\mathrm{max}}=0.6$ [rad] (${\delta}_{\mathrm{max}}=-0.6$ [rad]), maximum (minimum) rudder rate of turn ${\delta}_{\mathrm{max}}=0.066$ [rad/s] (${\delta}_{\mathrm{max}}=-0.066$[rad/s]).

## 4. Conclusions

## Funding

## Conflicts of Interest

## References

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**Figure 4.**The trajectory of ship movement and signal of angular speed disturbance r

_{w}for sea state 4 and the wave angle −π/4.

**Figure 5.**The trajectory of ship movement and signal of angular speed disturbance r

_{w}for sea state 5 and the wave angle 5π/12.

**Figure 6.**The trajectory of ship movement and signal of angular speed disturbance r

_{w}for sea state 5 and the wave angle 0.

**Figure 7.**The trajectory of ship movement and signal of angular speed disturbance r

_{w}for sea state 6 and the wave angle −π/5.

**Table 1.**Summary of selected parameters of waves depending on the sea state and the Beaufort wind scale.

Sea State | Sea State Designation | Probability of Occurrence [%] | ${\mathbf{h}}_{3\%\left[\mathbf{m}\right]}$ | Maximum Wave Length [m] | Beaufort Wind Scale (Approx.) |
---|---|---|---|---|---|

0 | calm–glassy | 11.2486 | 0 | – | 0 |

1 | calm–rippled | 0.00–0.25 | 5 | 1 | |

2 | smooth wavelets | 0.25–0.75 | 25 | 2–3 | |

3 | slight | 31.6851 | 0.75–1.25 | 50 | 4 |

4 | moderate | 40.1944 | 1.25–2.00 | 75 | 5 |

5 | rough | 12.8005 | 2.00–3.50 | 100 | 6 |

6 | very rough | 3.0253 | 3.50–6.00 | 135 | 7 |

7 | high | 0.9263 | 6.00–8.50 | 200 | 8 |

8 | very high | 0.1190 | 8.50–11.0 | 250 | 9–10 |

9 | phenomenal | 0.0009 | 11.0–… | >250 | 11–12 |

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Borkowski, P.
Numerical Modeling of Wave Disturbances in the Process of Ship Movement Control. *Algorithms* **2018**, *11*, 130.
https://doi.org/10.3390/a11090130

**AMA Style**

Borkowski P.
Numerical Modeling of Wave Disturbances in the Process of Ship Movement Control. *Algorithms*. 2018; 11(9):130.
https://doi.org/10.3390/a11090130

**Chicago/Turabian Style**

Borkowski, Piotr.
2018. "Numerical Modeling of Wave Disturbances in the Process of Ship Movement Control" *Algorithms* 11, no. 9: 130.
https://doi.org/10.3390/a11090130