Md. Shamim Mahmud

Naval Architecture and Marine Engineering, Bangladesh University of Engineering and Technology, Dhaka 1000, Bangladesh

1 Introduction1

Ship stability depends upon the ship’s control system,which have been used for over 100 years. Various hydrodynamic characteristics have been studied to reduce ship capsizing and to improve control system designs (Zaheret al., 2007; Kreuzer and Wendt, 2000; Janget al., 2007;Beaver and Zseleczky, 2009). However, if hydrofoil technology is used instead, it provides greater maneuvering stability, particularly with respect to changes in wave elevation.

A mathematical model for the theoretical maneuvering of ships in deep and confined waters was first proposed by Norrbin (1970). The most important factor affecting the dynamic stability of ships in realistic seas was found to be the rolling motion. Thus, the ship’s roll is the most crucial degree of freedom in a ship’s motion dynamics. Ship’s roll can be predicted through numerical methods (Janget al.,2010).

After the last addition of rudder roll-damping systems in the late 1970s and early 1980s, the majority of stabilityrelated research shifted toward the development of better control system design, as opposed to the development of new stabilization concepts (Whiteet al., 2007; White, 2013;Hatzakis and Sclavounos, 2006). Recently, there has been a surge in the revitalization of roll gyrostabilizers, as lift forces, due to the angle of the flow around their fins,produce anti-roll moment (Ghassemiet al., 2010). Designs for zero-speed stabilizers and a cycloidal propeller have also been proposed (Moaleji and Greig, 2005; Haroet al., 2011;Jurgens and Moltrecht, 2001).

These recent developments have been pushed by the luxury yacht industry. Additionally, several navies are again pursuing the use of rudder roll-damping systems. Previous research did not delve into the potential of a movable hydrofoil, only fixed-supported hydrofoils were investigated.

This paper first considers the subject of ship stability. The source of roll stability in three types of ships, a displacement ship, a surface piercing hydrofoil, and a submerged-foil hydrofoil, is considered. The roll disturbance for each of the ships is determined using both the ship’s displacement and the surface-piercing hydrofoil. The righting moment is produced entirely by the change in the vessel’s attitude relative to the water surface. In the case of the ship’s displacement, the righting moment is due to a shift in the center of buoyancy, while for the surface-piercing hydrofoil,the righting moment is caused by a shift in the hydrodynamic pressure center of the foils (Lloyd, 1975;Matusiak, 2007). However, in a submerged-foil hydrofoil,no righting moment is produced by the change of the ship’s relative position on the water’s surface. The righting moment for this ship must be produced by underwater control surfaces, in response to the change of ship’s attitude relative to inertial space, as sensed by a vertical gyro.

The improvement of computer technology has led to the development of different algorithms for automatic control systems (ACS), so now a ship’s attitude can be sensed using a nonlinear mathematical model (Renand and Yang, 2004).The application of these artificial control systems has been discussed by Roskillyet al. (2002), Surendran and Kiren(2007), Renet al. (2005), and Zheng (2011). ACS functional configurations were built using a highly-developed digital craft simulation, control hardware, and sea conditions. This simulation includes three control algorithms, a proportionalintegral-derivative (PID), a linear quadratic regulator (LQR),and sliding mode controls that are applied to regular and irregular wave conditions (Stark, 1974; Bai and Kim, 2010).It is this transfer of stability reference, from the water surface to an inertial reference that leads to the superior sea keeping capabilities of the submerged-foil hydrofoil.

The mathematical model presented in this paper represents the multipurpose usage of hydrofoil in vessels, to reduce the heeling and trimming moment, which aids in maintaining an upright position and lessening the water resistance. The more prevalent adoption of hydrofoil technology could lead to increased ship’s stability under sea conditions that challenge conventional water-tank ballast control systems, as well as overall vessel maneuverability.

2 Hydrofoil proposition and positioning

2.1 Proposition

A multipurpose hydrofoil is recommended, as it could lessen the trimming and heeling moments of the craft by maintaining the vessel in an upright position and by lessening the resistance, thus stabilizing the ship. The design of the hydrofoil was inspired from observations of dolphin movement, particularly how dolphin’s balance their body using their fins in different wave conditions. Inspiration also came from observing the relationship between the rudder force and center of gravity force on the ship.

2.2 Positioning the hydrofoil

A pair of hydrofoil fins are attached to the craft, a National Advisory Committee for Aeronautics (NACA)profile 4412 (4% Max camber, maximum thickness at the midpoint >amp; maximum thickness 12% of the chord). These two fins are accommodated along the side wall of the boat,as shown in Figs. 1 and 2. The center pressure of these two hydrofoils must be in line with the boat’s center of gravity.

For each hydrofoil, two supports are provided (Figs. 3 and 4). The first support has a rotating axis and is denoted byAandA’, while the second axis is movable and is denoted byBandB’. The movable axis is used to create an angle of attack for the foils.

Fig. 1 Top view of hydrofoil position

Fig. 2 Side view of hydrofoil position

Fig. 3 Center of floatation and gravity position

Fig. 4 Center of pressure position of hydrofoil

3 Mathematical model: turning, heel, and trim

3.1 Removing capsizing moment during turning

The abbreviations for the mathematical model are provided as follows:ARis surface area of the hydrofoil and is equal to half the area of the rudder,Vis the speed of the ship in m/s,δRis the angle of attack,Cfis the center of floatation,Cgis the center of gravity,?represents the mass of the ship,Ris the turning radius,Gis the gravitational acceleration,Mhis the moment-causing heel,Mtis the moment-causing trim,Ffis the force exerted on each foil,andMfis the moment provided by the foil.

The positioning of a right-turning boat is depicted in Fig.5 and the cross section of this boat is shown in Fig. 6, whereFris the rudder force,Fhis the body force, andRis the turning radius.

Fig. 5 Turning position of ship

Fig. 6 Cross-section of ship

Therefore, the moment-causing heel is

whereEis the center of lateral resistance.EandHare very close, so the moment-causing heel can be simplified as:

Usually, the value of E is closer to half of the draft, so, the moment becomes a function of the turning radius R and the ship speed V. This moment should be neutralized by the moment created by the hydrofoils, as the capsizing moment is anti-clockwise. In this case, there must be a clockwise moment produced by the hydrofoils. Let us assume the angle of attack is δR. The left-hand hydrofoil should be rotated positively and the right-hand hydrofoil should be rotated negatively, by direction. In other words, these two hydrofoils should provide force in the vertically upward and downward directions, respectively, and provide a clockwise moment, see Fig. 7.

Fig. 7 Clockwise and anti-clockwise moment

For the foil area AR, according to the Baker and Bottomley formula, the force on each foil is

So, the moment produced would be where b is the molded breadth and t is the distance (Cgof the foil, determined from the ship’s side).

This moment should equal the capsizing moment of the boat. From Eqs. (3) and (5), we get

This angle will be positive for the left-hand foil and be negative for the right-hand one.

3.2 Control of heel

This mathematical system can be used when the boat rotates to a considerable degree (the high angle of movement but not beyond the stability angle) due to either wave or wind forces, called heeling, Fig. 8. Excessive heeling, i.e., tilt, can be disconcerting for passengers. For this purpose, an automated control system could be introduced to the both sides of the ship to measure the amount of heel of the boat. If there is a change in the draft on both sides, let it be ?T.

The force on each portion can be defined as:

with moment-causing heel being

whereWA is the water plane area. The moment is acting in an anti-clockwise direction. To reduce the heeling, a clockwise moment should be provided to correct the ship’s position, as stated by Eq. (5), from Eqs. (6) and (7).

Therefore, δRHis

In this case, the angles of attack on the left- and right-hand foils should be positive and negative, respectively.

Fig. 8 Displacement due to heel

3.3 To correct unwanted trim

Due to non-uniform loading or any external force, the boat may not be in the correct trim position.

Fig. 9 Top view in trim condition

Fig. 10 Unwanted trim

If he boat is trimmed via the bow, the boat will rotate around the center of floatation (Cf), Fig. 9. Therefore, a coordinate system can be constructed starting at aft perpendicular and positive rightward, Fig. 10. The force caused in this forward rotation is:

Therefore the forward rotation force is

The distance from theCgfor the two portions would be approximately.

Therefore, the trimming moment is

In this case, the force will be provided by both the foils in the upwards direction. The force should be equal to

The anti-trimming moment provided by foil is

Thus, from Eqs. (11) and (13)

Fig. 11 shows the possible control system setup. The heave, heave velocity, pitch and pitch angular velocity are filtered and used to determine the flap angle in the control system. Also, the wave elevation is estimated by using relative bow height and filtered ship motion. Finally, the pulse for the linear motor is calculated by using filtered ship motion to operate the fore and aft foil flap.

Fig. 11 Hydrofoil control system

Hydrofoil ships use the dynamic lift of submerged lifting surfaces to support their weight, which in conventional ships is supported by the buoyancy of the hull. In this way, the ship’s hull can be lifted clear of the water, eliminating hull drag and the forces imposed on the hull by the sea. However,the advantages of hydrofoil use are exact a price. As with any dynamic lift vehicle, hydrofoil ships are weight sensitive and must operate at relatively high speeds in order to generate the dynamic lift required to support their weight with a reasonable size foil system (Spyrou, 1996). They also have dynamic instability in quartering seas, described by the behavior of a hydrofoil ship when broaching. Weight sensitivity and speed, coupled with the problems associated with operating in the worst of the marine environments place special demands of the ships subsystems. It may be expected that the hydrofoil will spend the major of its operating lifetime in the hull-borne mode. If the foils are extended during hull borne operation, there is a significant effect on craft motion, particularly in the roll mode, which is normally not heavily damped. The foil system gives hydrofoil crafts the hull-borne motion characteristics of ships that have much larger displacements.

The ship’s control system comprise those components necessary to control the ship’s speed, attitude, and direction,and if necessary, the dynamic stabilization. As with any dynamic lift vehicle, the control system of a hydrofoil can be divided into five functional areas: sensor, computer,actuator, force producer, and the vehicle itself. The vehicle and control system react to two inputs: the command and the external disturbance, which are shown in a typical block diagram in Fig. 12.

The foils themselves act both as sensors and as control devices, by virtue of the changes in force and moments that occur, which correspond with the depth of foil submergence.As already noted, hydrofoils provide persuasive advantages of extreme simplicity and high reliability, although this simplicity is bought at the cost of rough water seaworthiness.From Eq. (14) we have obtained the angle of attackδRof the foil. Now, how can we control the foil? Twentieth century aerospace technology introduced us to the flapping/movable airfoil. We can use this same technology to create a flapping or movable hydrofoil. The determining factor for this transfer of technology is the air–water interface drag(Mortezazadehet al., 2014; Giesing and Smith, 1967;Forbes, 1985), which is higher than the aerodynamic drag.The characteristics of these moveable hydrofoils in different sea condition can be approximated through the application of numerical methods (Pascarelliet al., 2002; Kim and Yamato, 2004; Surendran and Venkata Ramana Reddy, 2003;Jang and Kinoshita, 2000). Before reaching the stall angle,in many cases the lift force dramatically decreases, due to the free surface effect. Consequently, theh/cratio is an important factor for attaining the desirable lift force using movable hydrofoils, wherehis the free surface height andcthe chord length. Our research discerned the moveable hydrofoil drag and lift characteristics from our (NACA 4412)model when theh/cratio is 1, as displayed in Figs. 13 and 14, respectively.

Fig. 12 Motion control

Fig. 13 Drag coefficient versus Froude number graph

Fig. 14 Lift coefficient versus Froude number graph

Figs. 13 and 14 represent the hydrofoil characteristics.This hydrodynamic characteristic is important for designing the control system. We have already discussed that positioning of hydrofoil is an important matter for automatic control system. From this graph we will be able to know which position (h/cration) is appropriate for getting the maximum efficiency of the hydrofoil. To get the higher efficiency at minimum angle of attack in this perspective,the importance of this graph is undeniable.

4 Conclusions

Eqs. (6), (8), and (14) express how ships can be controlled by their parametric values. These mathematical equations can be used in the mechanical programming for maneuvering the gear systems of hydrofoils. In an automatic controlling system, the captain of the ship will input these parameter values and then the computer program will determine the angle of attack to optimizing the vessel’s heel and trim. Consequently, the gear system will be automatically activated by the computer signal. Future work would be a matter of design, and would entail a more exact representation of the mechanical elements necessary for the implementation of the drive systems. Simulations using a more precise model of the hydrodynamics (Figs. 13 and 14),as well as an investigation of other controllers would also be required to confirm these findings. However, these initial results show promise, and merit a more detailed investigation. We can conclude that our mathematical model,which is dependent on the sea conditions, may help in algorithm and control system design of future automotive craft systems.

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