1. Use the attached sheets for dynamic (marine) systems (*);

2. Make assumptions based on relevant theories related to the marine systems in order to derive equations governing characteristics of the marine systems.

4. Determine which numerical integration methods can be used in order to have acceptable accuracy;

5. Make simulation programs for the system under various working/operational conditions using both MATLAB/GUI and Simulink/GUI;

6. Use the simulation programs to learn the characteristics of the marine systems via various simulated scenarios, and compare simulated results by MATLAB and Simulink (if applicable); and

7. Write an individual report and submit it and all simulation programs on the due date.

## Answer:

r yaw rate [rad/sec]

ψ yaw angle [rad]

The equation for the trajectory can be represented using a state space model which is

gn: justify;">Here, u is the velocity of surge and v is the sway velocity. The co-ordinate of hull is represented by x and y axis.

Now, given that the inertia moment about the z axis = 22.5 kgm^2.

G to rudder distance is 1085 mm

The distance between the twin propellers = 118 mm.

The distance from the bow thruster to G = 852 mm.

The distance from G to stern thruster = 338 mm.

Vessel total length = 2570 mm

Vessel mass = 63.4 kgs.

Propeller maximum speed = 1000 RPM.

The maximum amount of generated drag force = 50 N.

The torque coefficient of water resistance between the hull and water = 6.75 Nm/rad/s.

The moment of rudder constant = 2.5 Nm/rad.

The maximum ladder thrust force = 25 N.

Now, the Yaw rate

Now, the equation of rudder angle is

The pitch angle is given by,

Where,

= Commanded rudder angle

= Commanded pitch angle

= time constant of the rudder

= time constant of the pitch

a = constant

Now, the steering equation of the hull is

Surge

Sway

Yaw

Here, m = mass of the vessel = 63.4 kg.

Xr, Yr = hydrodynamic forces acting on ship’s rudder

Xh, Yh = hydrodynamic forces acting on the ship’s hull

Izz = moment of inertia about z axis = 22.5 kgm^2

, is the acceleration in the x and y direction.

Yt = hydrodynamic force by thruster = 25 N.

Nt = hydrodynamic moment by the thruster

**Simulink model of the steering mechanism and ship hull:**** **

**Rudder angle plot:**

**Yawrate plot:**

**Yaw position plot:**

**Trajectory plot (XY co-ordinate):**

Now, the MMG model of Hoorn vessel is constructed in Simulink from the numerical values of the variables as given above. The values of the different variables are calculated given in the file modelparam.m file.

**Simulink model:**

Initially the rudder angle is taken as 60 degrees and then the angle is gradually reduced in large steps. The model is simulated for 100 secs and the hull position in XY co-ordinate is obtained as given below.

Hence, it is seen that when the magnitude of the rudder angle is large then the hull displacement is large from its initial position as the locus is spiral but with low magnitude of the rudder angle the locus tends to complete circle and hence the displacement of the hull from its initial position is small.

## This problem has been solved.

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