# A Parametric Study and Optimization of Ship-Stabilization Systems

Osama A. Marzouk, Ali H. Nayfeh

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Osama A. Marzouk, Ali H. Nayfeh. A Parametric Study and Optimization of Ship-Stabilization Systems. 1st WSEAS International Conference on Maritime and Naval Science and Engineering (MN'08), World Scientific and Engineering Academy and Society (WSEAS), Sep 2008, Malta, Malta. pp.169-174. [hal-04924840](https://hal.science/hal-04924840v1)

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# A Parametric Study and Optimization of Ship-Stabilization Systems

OSAMA A. MARZOUK and ALI H. NAYFEH

Virginia Polytechnic Institute and State University
Department of Engineering Science and Mechanics
Campus Mail Code 0219, Virginia 24061
USA

omarzouk@vt.edu

*Abstract:* We conduct a parametric study on the performance of passive and active anti-roll tank systems that stabilize the motion of ships to identify the influence of continuous and discrete design variables; such as the number of tanks, their locations, and their overall mass (relative to the mass of the ship). We carried out simulations for the motion of a typical cargo ship with and without control. We modified the original geometry such that pitching can excessively excite rolling through autoparametric resonance. We found interesting phenomena, such as sharp loss-of-effectiveness and saturation. We suggest a two-tank system (either passive or active); its mass is determined from the desired roll limit.

*Key-Words:* Ship, Stabilization, Roll, Tanks, Active, Passive

## 1 Introduction

Based on long-term research work [1, 2, 3, 4] on roll motion of ships and its nonlinear and large-amplitude behavior due to resonance, it was shown that anti-roll tanks can achieve a high level of roll reduction. Anti-roll tank systems have proved to be a very efficient technique at certain operation conditions, such as low forward speeds and moored vessels, when other roll stabilization techniques, such as rudders, become much less efficient [5, 6, 7, 8]. Active tanks are equipped with a pumping device to force the motion of the working liquid to induce moments that continuously counteract those moments induced on the vessel during its maneuver. On the other hand, passive tanks are not equipped with such pumping devices, and the working liquid moves entirely due to the sloshing effects. While active tanks can achieve same level of roll reduction as passive tanks but with a significantly smaller size and mass, passive ones have the advantages that they do not require a power supply and they need less maintenance. Thus, both types are demanded and we consider them in the present study.

We consider the design of high-performance tank systems for a typical cargo ship whose pitch frequency is near twice its roll frequency. We carried out simulations for the motion of a modified geometry of a cargo ship (series 60) such that the natural frequency of pitch is near twice the natural frequency of roll. Therefore, low-amplitude pitching motion can excite large-amplitude roll motion autoparametric (or inter-

nal) resonance. This is because once the pitch amplitude reaches a certain limit, additional energy transferred to the ship due to the exciting sea waves is transferred primarily to the rolling mode [9]. This makes the problem more challenging for the control systems.

We consider one-, two-, three-, and four-tank systems, with masses that vary by two orders of magnitude. This work is repeated for several wave angles at representative rough sea conditions. We show that as the mass ratio (defined as overall mass of the tank system relative to the mass of the ship) decreases, all tank systems exhibit a critical mass at which they lose their effectiveness steeply, resulting in large-amplitude roll motion. This critical value depends on the heading and the number of tanks. On the other hand, as the mass ratio increases, performance of the active systems saturates and becomes independent of the mass. We also examine the responsiveness of both passive and active tanks to the change of their locations and number of tanks, and how it varies with the heading. We finally identify a preferred or optimum tank design and eliminate the ones that are susceptible to decreased masses and demonstrate their deficiencies.

## 2 Analysis

We use a modified version of the three-dimensional simulation program LAMP (Large-Amplitude-Motion Program) [10] in conducting our study, which we incorporated appropriate routines in it to introduce the
control system and to couple it with the ship motion in the time domain. The analysis is based on the incompressible potential-flow theory, but uses empirical formulas to account for viscous effects, and predicts the linear (small-amplitude) or nonlinear (large-amplitude) ship motion. The forces and moments induced by the motion of the sloshing liquid in the tanks, which is fresh water, are added to the hydrodynamic ones that are computed by solving the flow field around the ship. More details about the equation of motion of the ship can be found in Ref. [11], whereas derivation and details about the equations describing the induced forces and moments by the tanks can be found in Ref. [3]. Finally, the equation of motion of the liquid inside the tank is described in Refs. [3] and [4].

The geometric idealization system of the ship hull consists of 16 water lines and 32 station stations. The bow view of the ship is illustrated in Fig. 1. The length overall (LOL) is 128 m and the mass is 11,700 tons. The ship moves with a representative forward speed of 13.4 Knots in rough conditions (sea state 5). This is a reasonable speed for our problem knowing that practicing engineers and planners usually classify a vessel whose maximum operating speed is 30 knots and above as a “high-speed vessel” [12]. It was selected such that the Froude number is 0.2. We utilized the original ship geometry, but modified the idealized mass and inertia information that accompanies the geometrical data.

The anti-roll tank system is composed of one or more identical tanks, each of which has a U-tube shape. In case of a single-tank system, it is installed on the ship such that its center lies vertically under the center of gravity of the ship. In case of multi-tank system, the tanks are placed symmetrically about the center of gravity in the longitudinal direction. A sketch of one tank is shown in Fig. 2. In this figure, $Z(t)$ is the time-dependent displacement of the liquid (we use fresh water as the working liquid) inside the tank, being positive when the free surface in the port column is above the equilibrium position. As an arbitrary design choice, we fix $H_1$ (the distance from the equilibrium level to the centerline of the horizontal pipe) at 7 m and $B$ (the semi-length of the horizontal pipe) at 5 m. We also use a constant coefficient of minor losses in the tank, which is 0.2. These values are very reasonable and practical for typical operating conditions of the tanks. The diameter of the two identical vertical columns ($D_1$) and the horizontal connecting pipe ($D_2$) will be design variables that we determine to satisfy a desired mass ratio and natural frequency. After fixing $H_1$ and $B$, the tank frequency becomes a function of the ratio $D_1/D_2$. For a fixed natural frequency of 0.4 Hz, this ratio becomes 3.3. Specifying a mass ra-

tio for the tank system determines its mass (the liquid only), which enables us to find the values of $D_1$ and $D_2$. As mentioned before, in the passive-control case, the pump is switched off, whereas in the active-control case it is switched on. This is reflected in the solver of the motion of the tank liquid since the pump power appears in this equation as a driving term for the liquid acceleration. The pump power is found at each time step of the simulation through a feedback law that calculates this power using the current values of the roll angle and the roll rate.

## 3 Results

To illustrate the remarkable effect of the control system (passive or active) on damping the undesirable roll motion, we compare in Fig. 3 the roll motion (in degrees) for the uncontrolled ship (no tanks are installed) and a passively-controlled at oblique (135°) sea state 5 waves using a typical 3-tank passive system with a mass ratio of 2.1%. The amplitude of the encountered roll is reduced from 17.9° to 1.96°. In Fig. 4, we demonstrate how the active control overpowers the passive one in the performance, where the amplitude of the encountered roll is reduced further from 1.96° to 0.42° although the mass is 1/5 the mass of the passive system shown in Fig. 3. We define the amplitude here as $(\max(\phi) - \min(\phi))/2$, where $\phi(t)$ is the time-dependent roll angle.

We considered two wave angles and examined at each one the effect of varying the mass ratio and the number and locations of the tanks on their performance. These wave angles are 135° (oblique waves) and 180° (head waves). These cases were carefully selected after extensive simulation runs. The head-waves case corresponds to the most excessive roll motion for the uncontrolled ship (amplitude 28.5°). We wanted to examine a less-critical case also to see how the performance of the control systems is different from the head-waves case. For this purpose, we chose the oblique-waves case. We define the wave angle such that 90° waves come toward the ship from the port side when the ship is moving toward the global longitudinal axis.

For each wave angle, we considered two ranges of the mass ratio: from 0.7% to 5.0% for the passive case, and from 0.02% to 0.56% for the active case. We chose different ranges because active systems can be much lighter than the passive ones and achieve comparable roll damping. In Fig. 5, the values of $D_1$ for the considered ranges of mass ratio for the passive and active cases are given.

We start with the passive-control case and show in Fig. 6 how the peak roll angle is reduced as the
mass ratio increases for each wave angle. The peak here is defined as $max(|\phi|)$, which is more important to damp than the amplitude. All tank arrangements show improved performance as the mass ratio increases. However, for head waves, the one-tank passive system shows lower rate of improvement in its performance (i.e., rate of decreasing the roll peak) as the mass ratio increases.

We then consider the active-control cases. In Fig. 7, the peak roll also decreases as the mass ratio increases. However, we found a critical mass ratio of 0.03% where the performance of the one-tank active control system deteriorates substantially at and below this value. The peak roll angle for the oblique wave with this one-tank active system at mass ratio 0.03° is 15.8°, whereas its 28.2° for head waves. These values are close to those obtained for the uncontrolled roll, which indicates a big loss of functionality of the control system. For oblique waves, other multi-tank systems undergo the same process but at lower critical mass of 0.02%, so they were able to maintain their performance for lower mass ratios than in the case of one-tank system.

For all wave angles, the one-tank active system achieves larger roll (i.e., lower performance) than the multi-tank active systems. The difference is slight for beam and oblique waves but relatively large for head waves. All these results indicate that multi-tank active systems have comparable performance which surpasses the performance of the one-tank active system. The roll is reduced to very low values for head waves using active systems. In these cases (regardless of the number of tanks), the performance of the active systems is quite independent of their mass. Increasing the mass does not yield further reduction of the roll, but leads to very slight increases or decreases with the roll level being essentially the same. Thus, the system exhibits saturation in the performance.

After exploring the effects of varying the number of tanks and the mass ratios of passive and active systems, we want to investigate the influence of changing the locations of the tanks (for multi-tank systems only) while keeping their mass ratio constant. For this purpose, we limit ourselves to mass ratio 2.1% for passive systems and 0.42% for active ones. The tanks of each of the two-, three-, and four-tank systems were placed symmetrically about the ship CG. Again, we start with the passive-control case, and show in Fig. 8 the influence of the longitudinal tanks distance (distance between the most-forward tank and most-backward tank) on the peak roll for the three wave angles. We did not find a clear trend and the peak angle changes weakly even when the tanks distance increases by a factor of six. This finding is not only common among all wave angles for the passive control,

but also for the active control.

Finally, we compare in the time-domain the roll angle with single- and multi-tank active systems at a selected mass ratio of 0.42% for oblique and head waves in Figs. 9 and 10, respectively. This step aims at giving more details about the effect of changing the number of tanks while keeping the mass constant (i.e., the effect of distributing the mass of the tank liquid), which cannot be seen in Figs. 9 and 10. The difference is insignificant among the multi-tank cases. However, for head waves the single-tank system results in one-order-of-magnitude larger roll compared to the multi-tank systems.

# 4 Conclusion

The design space of anti-roll tanks has many dimensions. In this numerical study, we focused on some of them (discrete and continuous), including the number of tanks, the overall mass of the system, and the locations of the tanks. We explored the effect of these design variables, for critical (head) and non-critical (135° oblique) sea waves, on the roll reduction attained by passive- and active-control systems. We used modified data of a typical cargo ship to have excessive roll due to sea waves.

We studied the reduction of the peak roll angle as the mass ratio (fraction of the ship mass) of one-, two-, three-, and four-tank passive and active control systems decreases and found that at a very small values (critical value 0.03%), the one-tank system loses its effectiveness sharply. For oblique waves, all active systems lose their effectiveness at very low critical mass ratios. However, the multi-tank systems extended their excellent performance slightly beyond the one-tank system, which has the largest (worst) critical mass. In addition, this one-tank system gives higher roll angles than its multi-tank counterparts that have same mass, especially for critical waves. So, the active one-tank system should be ruled out when looking for optimum tank design. For the multi-tank systems (passive or active), we found that distributing the tanks away from the CG of the ship has weak and non-monotonic effect on the tank performance. So, we leave this design variable to be chosen based on installation constraints or practical preferences.

Although we found that multi-tank systems have comparable performance (versus mass ratio or tank placement). It is preferred to reduce the number of tanks since this reduces the manufacture, installation, and maintenance costs. So, we propose a two-tank control system. The size of the tanks (i.e., the diameters of the columns and connecting pipe) is correlated to its mass ratio, which is in turn related to
the peak encountered roll angle. Since different wave angles correspond to different mass-roll relationship. So, depending on the allowed roll angle, a mass ratio is found from either Fig. 6 or Fig. 7, and the corresponding tank dimensions are then found from Fig. 5 and from the $D_1/D_2$ value.

Although the study considered a particular vessel, its findings can be useful and applicable to a wider range of configurations. The illustrated loss-of-effectiveness phenomenon that occurs at a critically low mass, for example, can be used to find a lower limit on the mass of the system during the design phase. One should ensure the mass of an active system is far enough from the critical mass that correspond to design and off-design operating conditions.

References:

[1] A.-H. Nayfeh, On the Undesirable Roll Characteristics of Ships in Regular Seas, *J. Ship Res.* 32, 1988, pp. 92–100.

[2] I.-G. Oh, A.-H. Nayfeh, and D.-T. Mook, A Theoretical and Experimental Investigation of Indirectly Excited Roll Motion in Ships, *Philos. T. of the Roy. Soc. A.* 358, 2000, pp. 1853–1881.

[3] K.-S. Youssef, D.-T. Mook, A.-H. Nayfeh, and S.-A. Ragab, Roll Stabilization by Passive Anti-Roll Tanks Using an Improved Model of the Tank-Liquid Motion, *J. Vib. Control* 9, 2003, pp. 832–862.

[4] O.-A. Marzouk, and A.-H. Nayfeh, Mitigation of Ship Motion Using Passive and Active Anti-Roll Tanks, *Proc. 21st Conf. Mech. Vib. Noise*, 2007.

[5] J.-J. Van den Bosch and J.-H. Vugts, Some Notes on the Performance of Free Surface Tanks as Passive Anti-Rolling Devices, *The Report of the Ship-building Laboratory*, Tech. Uni. Delft 119, 1964, pp. 3–52.

[6] T.-W. Treakle, A Time Domain Numerical Study of Passive and Active Anti-Roll Tanks to Reduce Ship Motion, MS Thesis, Department of Aerospace and Ocean Engineering, Virginia Polytechnic Institute and State University, 1998.

[7] T. Perez, *Ship Motion Control: Course Keeping and Roll Stabilisation Using Rudder and Fins*, Springer –London 2005.

[8] R. Moaleji and A.-R. Greig, On the Development of Ship Anti-Roll Tanks, *Ocean Eng.* 34, 2007, pp. 103–121.

[9] A.-H. Nayfeh, *Introduction to Perturbation Techniques*, Wiley, New York 1981.

[10] Science Applications International Corporation, Large Amplitude Motions Program (LAMP) for Ship Motions and Wave Loads Predictions – User’s Guide, 1998.

[11] T.-I. Fossen, *Guidance and Control of Ocean Vehicles*, Wiley, New York 1994.

[12] Z. Demirbilek, Review of Hydrodynamics of High-Speed Marine Vehicles by Odd M. Faltinsen, *J. Waterw. Port. C-ASCE* 132, 2006, pp. 427–428.

<figure><img src="image_2.png" /><figcaption>Figure 1: Bow view of the ship (dimensions in m).</figcaption></figure>

<figure><img src="image_3.png" /><figcaption>Figure 2: A sketch of one tank.</figcaption></figure>
<figure><img src="image_4.png" /><figcaption>Figure 3: Roll angle for oblique waves, without anti-roll control and with 3-tank passive system of 2.1% mass ratio (longitudinal tanks distance = 10 m).</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>Figure 4: Roll angle for oblique waves, with 3-tank passive system of 2.1% mass ratio and active system of 0.42% mass ratio (longitudinal tanks distance = 10 m).</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>Figure 5: Diameter of the vertical column ($D_1$) versus the mass ratio for passive systems (up to 3%) and active systems.</figcaption></figure>

<figure><img src="image_7.png" /></figure>

<figure><img src="image_8.png" /><figcaption>Figure 6: Peak encountered roll angle, with a passive system, versus the mass ratio at different wave angles.</figcaption></figure>
<figure><img src="image_9.png" /></figure>

<figure><img src="image_10.png" /><figcaption>Figure 7: Peak encountered roll angle, with an active system, versus the mass ratio at different wave angles.</figcaption></figure>

<figure><img src="image_11.png" /></figure>

<figure><img src="image_12.png" /><figcaption>Figure 9: Roll angle for oblique waves, with active system (mass ratio 0.42%, different num of tanks).</figcaption></figure>

<figure><img src="image_13.png" /></figure>

<figure><img src="image_14.png" /><figcaption>Figure 8: Peak encountered roll angle, with a passive system, versus the longitudinal tanks distance at different wave angles.</figcaption></figure>

<figure><img src="image_15.png" /></figure>

<figure><img src="image_16.png" /><figcaption>Figure 10: Roll angle for head waves, with active system (mass ratio 0.42%, different num of tanks).</figcaption></figure>