ship stability by rutma

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									Using Dynamic Ballasts for Ship Stability

Curt Baragar Kyle Miller Michael Olson

December 15, 2004

Abstract: We investigated the advisability of stabilizing the roll of a cruise ship by pumping ballast water from a central tank below the center of mass to the high side in order to return the ship to vertical. We accomplished this by first creating a model of a cruise ship with a programmable controller, and then running simulations with Matlab’s Simulink. We quickly realized that the simulations suggest a required small tolerance for time delay in the controller system. Upon further investigation we determined that, even for a perfect system, the volume of water needed to be moved to produce the required stabilizing forces is beyond practical implementation.

Work done in partial fulfillment of the requirements of Michigan State University MTH 843.

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Table of Contents

The First Seafarers.....………............................................................................….. 3 Basic Ship Stability.....……...............................................................................….. 3 System Modeling................................................................................................….. 4 Running the Simulation......................................................................................….. 5 Results.....……...….............................................................................................….. 8 Conclusions.....……...….....................................................................................….. 8 References.....…..…...…...................................................................................……10

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The First Seafarers
As early as 40,000 years ago, the first seafarers crossed a minimum of 110 miles to settle the isolated land of Australia [1]. To undertake this voyage would require a seaworthy vessel capable of being paddled against prevailing winds, while remaining in stable condition for the duration of the journey across the unpredictable waters. Today, 40 millennia later, with far more sophisticated shipbuilding techniques than those available to those ancient sailors, the ocean continues to have the power to sink navigating vessels. To minimize such disasters, various methods are now employed to add stability to ships by reducing their tendency to roll in response to forces such as wind or waves. We investigate the advisability of stabilizing the roll of a ship by pumping ballast water from a central tank below the center of mass to the high side in order to return the ship to vertical. We also pay close attention to any time delay introduced into the system and the effect it may have.

Basic Ship Stability
A ship obviously must remain buoyant, but if it is to float comfortably and correctly, it must also be stable. A ship must be a stable platform, whether at rest or moving under the influence of wind and waves. When pressured by wave action, a stable ship will return to an upright condition. It is also important to note that the volume of water displaced by the ship will alter as it drives through successive waves, creating constant change in the ship’s stability. Stability is very much a dynamic condition. A ship is designed to remain in a stable condition by such basic methods as putting the greatest weights low down in the hull and restricting the amount of upper cargo. But we must remember that the conditions aboard a ship are constantly changing. The design of the ship and the weights of engines and equipment will remain unaltered, but as fuel and water is consumed during the voyage, the stability of the ship may gradually decline. Similarly, on a ship designed to carry cargo, the weight and the disposition of the load will affect the ship’s stability, and must be constantly monitored when loading and unloading. Stability is often maintained in large ships by moving water around in the vessel’s ballast tanks. This is often used to ensure that the ship stays upright and does not adopt a heel (lean) to one side or the other if the cargo is loaded asymmetrically, or if fuel is taken from a tank on one side of the ship. Water ballast is frequently carried to maintain stability in an otherwise empty ship. Stability may be put at risk however, if there are too many partly empty “slack” tanks aboard the ship, which allows water to slosh from side to side [3]. Because of this, designers have used different styles of water ballast tanks to combat vessel rolling and to increase stability. By placing such tanks up high on the ship, significant improvement in stability can be achieved. The disadvantage of these tanks is that in order to be effective, they need to be located in parts of the ship where cargo would otherwise be held.

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System Modeling
In order to get a sense of how these tanks can effect the roll of a cruise ship, we employ simulation to model the situation. We use ship characteristics gleaned from Larsson and Eliasson [2] with the modeling capabilities of Matlab’s Simulink program to design the system. We assume that the sensors and equipment necessary for the system’s implementation are perfect, meaning they cannot affect the performance of the system. We now describe our approach to modeling the situation. The rolling motion of a ship can be described using Newton’s method in radial coordinates. That is, the sum of the moments is equal to the transverse moment of inertia multiplied by the angular acceleration of the ship. This yields the differential equation that describes the motion of a ship,

  I t   a sin   b  y(t ) ,

(1)

where (t) represents the angle with respect to upright of the heeled ship, a is a constant representing the stability characteristics of the ship, b is a constant representing the resistance of water due to viscosity and hull characteristics, and y(t) is a time dependent function representing input disturbance due to wind and wave action on the ship’s hull. We designed a controller by applying a standard approach to control design called proportional-integral-derivative (PID) control. PID control design is a process whereby constants P (the magnitude of a force enacted by the controller proportional to the current hull angle), I (the magnitude of a force proportional to the time integral of hull angle), and D (the magnitude of a force proportional to the time derivative of the hull angle) are chosen to best control the system. For this system we selected P and I to be zero and chose D so that the coefficient of the first time derivative of the hull angle is equal to 1. Constants a and b, and the magnitude of y(t) were normalized so that It could be assumed to be 1. This results in the equation
    c sin   (d  D)   y (t ) .

(2)

This represents a system in which oscillations can be quickly damped. We used Matlab’s Simulink to model this differential equation, which can be seen in Figure 1. Using data supplied by Jacques Moreau, a naval engineer and project manager for Grand Bahamas Shipyard [3], we estimated reasonable values for c and d, and subsequently D.

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Figure 1. Simulink model of the differential equation (2).

What the Simulation Reveals
With our initial settings in place, we ran the model with various values for the time delay. In all cases we used a step disturbance with initial value of 1, dropping to 0 in one second. We first ran our simulation with no time delay. In Figure 2 we see the effect of the step disturbance and notice that the ship quickly corrects itself.

Figure 2. With no time delay the system responds well.

6 With a delay of 0.5 second, the system still responds well. At 0.8 second a small oscillation can be seen, although it is quickly damped as shown in Figure 3.

Figure 3. At 0.8 second, we see a small oscillation that is quickly damped.

When we increase the delay to one second, we begin to see a troubling sign. The oscillations start to become significant, suggesting that the system may be beginning to fail. Notice that in Figure 4, 30 seconds after the initial disturbance, the ship is still oscillating.

Figure 4. At 1 second, we have troubling oscillations.

7 We again increase the time delay, this time to 1.1 seconds. There is no doubt that any ship in this situation would be in trouble. The oscillations suggest that the ship is rolling to a dangerous degree and isn’t being damped quickly enough to make the ship operational. This is made very clear in Figure 5.

Figure 5. At 1.1 seconds, the ship is in trouble.

The final two simulations make it obvious that our system has failed. With a 1.2 second delay, the ship is in peril as the system is unable to stop the rolling. At 1.5 seconds, all is lost – the oscillations increase until the ship is capsized. These unwanted situations are shown in Figures 6 and 7.

Figure 6. At 1.2 seconds, the ship is doomed.

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Figure 7. At 1.5 seconds, the ship quickly capsizes.

We found that the system allowed for very little delay before it became unstable. Considering that in these small slices of time, large amounts of water must be moved about on the ship in order to dampen the rolling of the vessel according to our system’s design, the impracticality of such a design becomes clear.

Implications of Delay
We quickly realized that the simulation suggests that the controller system requires a tolerance in the time delay that is too small to be practical. With too much delay, the control system was reacting to a situation that was no longer current, often amplifying the roll of the ship. Upon further investigation it became clear that, even for a perfect system, the volume of water needed to be moved to produce the required stabilizing forces is beyond practical implementation. According to Jacques Moreau, cruise ships today rely on stabilizer fins for stability [3]. By using airfoil-shaped fins mounted low in the hull amidships, significant roll reduction can be achieved when the vessel is underway. Such fin stabilizers use a motion-sensing controller that signals the fin to change its angle of attack to the water, thus producing a countering force to the wave motion. Fin stabilizers are typically electro hydraulic units and become more effective as the ship increases speed.

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Recommendation
Based on the results of our simulations, we do not advise stabilizing the roll of a cruise ship by pumping ballast water from a central tank below the center of mass to the high side of the vessel. We discovered that with this method, excessive delay in the feedback loop is fatal to the ship, its passengers, and its contents. Even with the best equipment available working optimally, and under ideal conditions, the combination of stringent time requirements on the delay and the enormous amounts of moving water render the method impractical and unsafe.

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References
[1] Fagan, Brian M, The Journey from Eden. The Peopling of Our World, Thames and Hudson Ltd., London, pp. 129-132. Larsson, Lars, and Eliasson, Rolf E, Principles of Yacht Design, 2nd edition, McGraw-Hill Companies, Great Britian, 2000. pp. 40-46. Moreau, Jacques, Project Manager, Grand Bahamas Shipyard, Freeport, Grand Bahama. Telephone interview 11-4-04.

[2]

[3]


								
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