study on ship maneuverability of turning in regular ...study on ship maneuverability of turning in...

Study on Ship Maneuverability of Turning in Regular Transverse Wave

1Linjia YANG, 2Yihan TAO 1,First AuthorDalian maritime university, ,Dalian, China, [email protected]

*2,CorrespondingAuthor Dalian maritime university,Dalian, China, [email protected]

Abstract As is known to all, it is very dangerous when a ship turns in transverse waves. So it is of great

significance to find out the effect of regular transverse wave and eliminate the effects. Using computer simulation method could grasp ship’s motion law in regular transverse waves. The simulation was based on MMG (usually called Maneuvering Model Group) ship mathematical model and a wave force mathematical model. During the simulation, let the ship sail both in regular waves and calm sea with the rudder angle being 0°and 20°, then recorded the tracks. To verify the authenticity of simulation result, a ship trail was made. Based on the analysis of the simulation results, a discussion will be made in order to design a rudder controller, and the effect of waves can be reduced. In conclusion, the effect regular transverse wave acting on ship turning maneuverability can be obtained, and it is a practicable plan to design a rudder controller.

Keywords: Turning Maneuverability, Regular Transverse Wave, MMG, Rudder Controller

1. Introduction

In recent years, the research of ship controller has greatly developed [1]. Many autopilots can be used

to reduce the effect of the waves while ship is sailing straight [2, 3]. But when it comes to turning, especially with a large rudder angel, the autopilot is not so effective any more. Hence, the research about the effect of waves on ship’s turning maneuverability and the control of ship’s turning is significant. Ship’s tuning maneuverability in regular transvers wave is a typical example [4-7].

Computer simulation is a very important research method on the field of ship design and ship research. It is faster, cheaper, and more flexible than many other methods, such as real ship test and ship model experiment. The establishment of mathematical model is one of the most important parts of the computer simulation. Generally, the methods to establish mathematical model can be divided into two groups. One is Europe and the United States group which is widely used by research institution equipped with advanced equipment. The other one is Japan group MMG, which is not widely used but can give a very accurate result with little investment. The research about how to establish MMG mathematical ship model is of great significance.

2. Turning maneuverability

The ability that a ship enters turning movement, by steering a rudder angle and keeping it when the

ship sailing straight at a certain speed, is called turning maneuverability, which is extremely important to ship. Turning movement can be divided into three states, according to the change of external forces and the difference of movement states.

The first state is called side’s way and introversion section. After steering, the ship still keeps its forward speed, and sails almost along the origin course, the stern moves outsides. Besides, the ship will introvert, because the acting point of rudder force is lower than gravity center.

The second state is called transition section [7]. The ship’s side way speed and drift angle gradually increase. In the beginning, the ship’s angular acceleration is large [8, 9]. With the continuous improvement of ship angular velocity, angular acceleration is gradually reduced. Accompanied by the disappearance of introversion, the camber angle gradually increases [10-14]. The ship gradually enters steady turning movement.

The third state is called steady turning section [15-17]. The ship’s angular acceleration reduces to zero. The camber angle, sidesway speed, drift angle and the line speed of the ship are all tend to be steady [18-20].

Study on Ship Maneuverability of Turning in Regular Transverse Wave Linjia YANG, Yihan TAO

Journal of Convergence Information Technology(JCIT) Volume8, Number8, April 2013 doi:10.4156/jcit.vol8.issue8.114

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3. Mathematical model There are two kinds of common coordinate system are used for ship maneuvering motion analysis:

inertial coordinate system and body-fitted coordinate system. As Figure 1 shows, oxyz are inertial coordinate system fixed on the surface of the Earth, taken as the baseline reference system. Set axis Ox pointing due north, axisOypointing due east, and axis Ozpointing the center of the earth. For the body-fitted coordinate systemoxyz, in order to simplify the equations, we choose ship’s center of gravity as the point of origin. The axisOx points to bow, axisOy points to starboard, and axis Oz points to the keel. The heading angle is ψ.

The wave coordinate systemζηzis introduced, in order to establish the mathematical model of wave forces. The axis ζ point the direction that regular waves come from, and ψ is the encounter angle.

Figure 1.Coordinate system

In the research ship’s turning maneuverability, the main motion variables are forward speedu,

traverse speedv, and turning angular velocityr. Talking about ship planer motion, Heaving, rolling and pitching motions can be ignored which means w = p = q = 0 . Considering the mass of the ship distributes symmetrically to planxz, we can conclude that the moment of inertia about axisOx and axisOy is zero, and the position of gravity center is zero in axisOy which means I = I = 0, andy = 0. Because the origin point of the ship located at the gravity center, the position of the gravity center is zero in axisOx which meansx = 0. So the ship motion equation is simplified to a well-known format. (m + m)(u − rv) = X + X + X + X （1）

m +m(v − ur) = Y + Y + Y + Y （2）

I r = N + N + N + N （3）

In the above formula:m is the mass of the ship.mandmare additional mass caused by inertia of hull.uandvare ship’s speeds along axisOxand axisOy in body-fitted coordinate system.ris turning angular velocity. X, YandN are viscous hydrodynamic forces and moment on the hull.X, YandNare thrust forces and moment provided by propeller.X, YandNare rudder forces and moment.X, YandNare wave forces and moment acting on the hull [1].

The additional mass of the ship can be calculated with multiple regression formulas proposed by Zhaoming Zhou [2]. The viscous hydrodynamic forces and moment on the hull can be calculated with approximate estimation model proposed by Kijima [3]. Because ship discussed in this article uses a controllable pitch propeller, the thrust force and moment are calculated by method proposed by DMU (Dalian Maritime University) [4]. The rudder forces and moment can be calculated with a conventional rudder force calculation model [5]. The wave force and moment calculation is the focus of this paper, which will be detailed below.


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When discuss the effect of the regular wave on ship maneuverability, the wave force can be divided into two kinds: First-order disturbance wave force and second-order drift wave force.

In calculation of disturbance wave force, the angular between wave direction of propagation and heading is called wave angularχ, shown in figure 2. And there is: χ = π − (α −Ψ) (4) χ = 0means following sea, χ = π means head sea, χ = ± means transvers wave (+ means wave from starboard); ship encounter frequency against wave is [6]: ω = ω − ku cos χ + kv sinΨ (5)

Figure 2.Wave angular χ

In the formula (5), ω is circular frequency of regular wave, k is wave number, there is: k = πω

= ω = πω (6)

In the formula, Lω is wave length, Tω is wave period, which is related to wind velocity. The relationship may be different in different area on the sea.

Kallstrom gave the formula to calculate Tω and hω with least square regression method [7]. Tω = −0.0014V + 0.042V + 5.6 (7) hω = 0.015V + 1.5 (8) In the formula, V is wind velocity. And we should be attention that, formula (7-8) can only be used

whenV ≤ 20m/s . WhenV > 20m/s , extrapolation should be cautiously made. When V = 0 , Tω = 5.6sandhω = 1.5m. In the Atlantic Ocean, the situation may be appropriate. It is very complicated to use the well-known Froude-krylov assumption. Hence, under the premise of

ensuring a certain precision, we come up with a simplified method to solve the problem. Assume that the shape of the ship is a regular hexahedron, draftd(x)and beam B(x)do not vary with x , and the cross-sectional areaA(x) will also be a constant. The simplified formulas are as follows. X = −2aB ∙ s(t) （9） Y = −2aL ∙ s(t) （10） N = ak B sin b ∙ − L sin c ∙ ζ(t) （11）

In the formulas，a = ρg1 − e/k；b = kL/2 ∙ cos χ；c = kB/2 ∙ sin χ； s(t) = (kh/2) sin(ωt)；ζ(t) = (h/2) cos(ωt)；h is the wave height. Attention: When b = 0 (transvers wave), there is… X = 0 （12） Y = −2aL sin c ∙ s(t) （13） N = 0 （14）

O

Wave propagation direction

χ


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When c = 0 (head sea or following sea), there is… X = −2aB sinb ∙ s(t) （15） Y = 0 （16） N = 0 （17） In order to calculate the drift wave force, Daidola [8] proposed following wave drift force and

moment formulas, considering the effect of wave on ship maneuverability [9-11]. X = ρLa cos χ C (λ) （18） Y = ρLa sin χ C (λ) （19） N = ρLa sin χ C (λ) （20）

In the formulas，C ，C ，C are test coefficient. According to English ship model test [12] Daidola regressed: C (λ) = 0.05 − 0.2 λ + 0.75 λ − 0.51 λ （21） C (λ) = 0.46 + 6.83 λ − 15.65 λ + 8.44 λ （22） C (λ) = −0.11 − 0.68 λ − 0.79 λ + 0.21 λ （23）

4. Motion simulation

We take the training ship YUKUN of Dalian Maritime University as test model, use Matlab to

establish ship mathematical model for planer motion, and make turning simulations in calm seas and in regular transverse waves. Choose the simulation environment as follows, in order to ensure that the simulation data as authentic as possible.

Table1.Ship Parameters

Initial speed 16.7kn; initial heading 000; engine speed 173r/min; no current; wavelength 120m; wave

from the angle 090; wave height 2m.Ignoring the ship's rolling and pitching. Besides, the wind force was ignored in the simulations, in order to make the effect of regular waves

more obvious. We did a circle-turning ship trail on MV/YUKUN, and recorded positions of the ship; the result is

shown in Figure 3and Figure 4 [13]. By comparing the simulation track and the real positions, we can estimate whether the mathematical model is precise enough to reflect the real motion law of the ship. It is obvious that the simulation result is close to real ship position. So the MMG mathematical model is precise enough to reflect the real motion of the ship.

Length between perpendiculars[m]

Molded breadth[m] Design draft / Depth[m]

105 18m 5.4/8.35 number of Paddle Propeller speed[r/min] Design Speed[kn] 4 173 16.7 Block coefficient Displacement[m3] Position of the center of

gravity[m] 0.561 5878.8 -0.51 Height of rudder[m] Rudder area[m2] Propeller diameter[m] 4.8 11.8 3.8


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Figure 3.Ship trail and simulation comparison (with portside rudder)

Figure 4.Ship trail and simulation comparison (with starboard rudder)

The simulation of sailing in regular transverse waves was a comparative test. Comparing the

simulation results of turning against waves with 20° rudder angle with no rudder angle, the tracks are

shown in Figure 5.Comparing the simulation results of turning along waves with 20° rudder angle with no rudder angle, the tracks are shown in Figure 6.


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Figure 5. Turning against waves Figure 6. Turning along waves

5. Discussion

Comparing the curve and the points in Figure 3 and Figure 4, the errors of the simulation results of

ship’s mathematical model are very small. The ship’s mathematical model can reflect the real situation. Because it is extremely dangerous sailing in transverse waves, we can’t conduct test with real ship. Hence, we make conclusion by simulation results.

Compare the turning circles of ship’s sailing in calm seas and in regular transverse waves as Figure 7 and Figure 8 and results of Figure 5 and Figure 6.

Figure 7.Turning circle comparison of starboard rudder


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Figure 8.Turning circle comparison of portside rudder

It can be concluded that: In Fig.4, the ship will experience a long period of transverse waves, when steering towards waves.

Because of the waves force, the turning angular velocity extremely declined. And the diameter of turning circle will decrease because of the falling of forward speed u. The track of the ship and the center of the turning circle have a sideway along the waves. The rolling amplitude will increase, and the ship’s stability is deeply decreased.

In Fig.5, the heading of the ship are changed extremely quickly, when steering along waves. Rolling amplitude will extremely increase with large turning angular velocity. The ship can easily capsize, in this period. And it is almost impossible to control the ship. Because the ship follows wave in a long period, the forward speed u is increased, and the diameter of the turning circle is also increased.

In a conclusion, the turning maneuverability is greatly decreased in regular transverse waves. We should never steer along wave in regular transverse waves which is extremely dangerous. And steering towards waves is also very dangerous for ship to capsize, if the GM is not very big [14, 15].

To control the ship, we should apply a controlling forceY provided by rudder to balance the wave force [16, 17]. If the ship is turning, the rudder must provide a centripetal forceY. The rudder forceY must be the resultant force of controlling force and centripetal force, which meansY = (Y + Y). The magnitude of controlling force should be equal to the wave force, and the direction should be opposite to the wave force, which meansY = −Y. And the wave force can be calculated by the mathematical model of wave force with the data observed by sensors. If we get the data, including the wavelength, wave height and encounter angle, the rudder force that we need can be calculated [18-20]. Considering the effect of steering delay, inertia, and error of wave model, the work done by rudder force are usually smaller than the work done by wave force. Therefore the rudder force should be multiplied with a gain K, so that the total work done by resultant force gradually converges to 0 and wave effect is gradually reduced, until be eliminated. We can get the K value by making many actually tests. Knowing the rudder forceY = KY = (Y + Y), we can takeY to the rudder model to get the rudder angle by making the inverse solution. By adjusting the rudder angle, the controller can reduce the effect of the regular transverse wave.

6. Conclusion

To study ship motion law in regular transverse waves, it is feasible to use simulation method based on

MMG ship mathematical model. Regular transverse wave has a significant effect on ship turning maneuverability. In regular transverse wave, it takes a long time for ship to start turning, and ship experiences severe rolling. The shape and size of ship turning circle changes irregularly. It is extremely dangerous for ships to turn in transverse waves. The effect of regular transverse wave can be reduced by a rudder controller, which can provide a transverse force to balance wave force.


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