Investigations into the Influence of Tank Position and Nonlinear Damping Effects on the Performance of an Anti-Rolling Tank

This is an English translation of the original German document: “Untersuchungen über den Einfluß der Tanklage und nichtlinearer Dämpfungseffekte auf die Wirkung eines Schlingertanks” by Yoshifumi Takaishi, Institut für Schiffbau der Universität Hamburg, Report No. 155, August 1965.

Rolling Motion of a Ship with Anti-Rolling Tank

The equations of motion for a ship with an anti-rolling tank according to Chadwick and Klotter [1] can be supplemented by additional terms for the coupling between rolling and lateral (sway) motion (A.1). These additional terms, like the others caused by hydrodynamic forces, are expressed according to Grim [2] (A.2).

Figure 1 shows a schematic drawing of the ship and the anti-rolling tank of the Frahm type. H_t is negative when the tank is located above the center of gravity, and the equations of motion are formulated with respect to this point.

Rolling motion

Motion of the tank water

Lateral (sway) motion (Equation 1)

where:

J_s = I + I″
K_s = ρg V_t · MG
M_Q = m · OG + m_H · h_sr
M_s = m + m″
B_Q = –ρ·ω·g·(A_s²/h_w)
G_r = m · OG + I″
C_r = ρg/2 · (A_s²/h_w)

The dimensionless forms of the equations are (Equation 3):

(Terms involving φ̈, φ̇, φ, q̈, q̇, q with coupling coefficients Ks, At, ψs, ψt)

The asterisked terms (·) denote the additional coupling terms between rolling and lateral motion.

The coefficients in the equations are described by the hydrodynamic coefficients of the ship hull and the properties of the anti-rolling tank as follows:

K_s: damping coefficient of the ship
A_t: tank effectiveness
ψ_s: influence of water inertia
ψ_t: influence of tank position

where N = number of strips, δ = block coefficient of the ship, α_w = waterplane coefficient, H = B/2T; B_s, T_s = breadth and draft at the midship section.

It is assumed that the hydrodynamic forces excited by a wave on a cross-sectional strip element of the ship hull are identical to the forces that would act on a two-dimensional body of the same cross-section in a transverse wave of the same wave obliquity and circular frequency, and that the strip method is applicable for integrating the forces.

The tank-water-related coefficients in the equations have the following meanings:

ψ_t: the static stability loss due to the free surface of the tank water
μ_t: the natural frequency of the free motion of the tank water divided by the natural frequency of the ship
K_t: the damping coefficient of the tank water
ψ_s: the influence of the water inertia
ψ_t: the influence of the tank position

While the linear equations of motion can be solved directly, for studying nonlinear effects (especially nonlinear damping) it is recommended to investigate using an analog computer. The analog computer also displays the influence of parameter variations in a clear and intuitive form.

In this paper, the influences of the coefficients of the motion of the tank water, as well as the influence of tank position and lateral motion, are investigated using an analog computer. Figure 2 shows the circuit diagram of the analog computer for simulating the equations of motion, with a nonlinear element included.

1. Influence of the Coefficients of Tank Water Motion

First investigated was the influence of the coefficients K_t and μ_t for the following simplified equations of motion:

φ″ + K_s·φ′ + φ + ψ_s·φ″ + A_t·φ_r″ = 0

φ_r″ + K_t·μ_t·A_t·φ_r′ + μ_t²·φ_r + μ_t²·φ = 0

The coefficients K_s, A_t, ψ_s, and ψ_t were taken from an example in [1].

Figure 3a shows the influence of the damping of the water in the anti-rolling tank, and Figure 3b shows the case in which the damping is nonlinear, where K_t = 0.6 + a·φ′² was used.

Figure 4 shows the influence of the natural frequency of the water in the anti-rolling tank, where K_t = 0.2 was assumed and μ_t was varied with μ_t.

Figures 5a and 5b show the influence of water damping when the tank effectiveness A_t has been doubled. Figure 5c shows the influence of nonlinear damping.

2. Influence of Tank Position Relative to the Ship’s Center of Gravity

It is noteworthy that the coefficients ψ_s and ψ_t become smaller and even negative when the tank is positioned higher, because both coefficients depend on T_t and H_t, and H_t is negative when the tank is above the center of gravity.

Figure 6 shows the rolling motion of the ship with the tank positioned so high that ψ_s and ψ_t are negative. Comparison of this figure with Figure 1a reveals that the higher tank position is more effective. Chadwick and Klotter already noted that the anti-rolling tank becomes particularly effective for imaginary ψ_st in applied examples. The imaginary ψ_st in [1] corresponds to negative ψ_s and ψ_t in the equations of motion (3).

Gerritsma has also shown that the measured anti-rolling moment is greater when the tank is installed higher [3].

3. Influence of Lateral Motion Coupling

The coupling between lateral (sway) motion and the motion of the tank water has already been treated by Chadwick and Klotter [1]. The equations of motion are (Equation 6).

Figure 7 shows the influence of lateral motion coupling for negative ψ_s and ψ_t: the rolling motion becomes smaller in the lower frequency range and larger in the higher frequency range.

Investigation with a Two-Dimensional Ship Hull (Lewis Form)

First investigated was the influence of tank position as well as the coefficients related to tank water for a two-dimensional ship hull whose cross-section is a Lewis form (H = 1.2, δ = 0.9). The hydrodynamic coefficients of the Lewis form for rolling and lateral motion are sufficiently computed and available. μ_t = 0.25 was assumed. The equations of motion (1) were computed.

Figure 8 shows the roll amplitude for three different tank positions, from which it can be seen that the highest tank position would be most favorable. Figures 9a, 9b, 10, and 11 show the influences of tank water damping, the influences of the tank water natural frequency, and the influences of ship hull damping, respectively.

Comparison of the calculated results between Figures 1–7 and Figures 8–11 reveals that the optimal choice of tank parameters depends substantially on the ship form.

Influence of Ship Heading Angle Relative to the Wave

Finally, the influence of the ship’s heading angle relative to the wave was investigated. For this purpose, a mathematical ship form was chosen whose hydrodynamic coefficients were calculated as a function of heading angle [4].

The following values were used in the calculation as coefficients for tank water motion:

μ_t	ψ_t	K_t	ψ_s	ψ_t
0.25	1.0	0.5	–0.08	–0.8

These values had proven in previous calculations with the analog computer to be the more favorable values for the ship in beam seas.

In this case, the calculations were carried out with a digital computer. The calculated results are shown in Figure 12, from which an increase in roll amplitude in a certain frequency range for oblique waves can be read.

Since the ratio of the transverse force excited by the waves to the excited moment, as well as the phase shift between transverse force and moment, depends on very many parameters and in particular on the heading angle, it cannot be expected that the effect of the passive anti-rolling tank will be equally good in all cases. It is quite conceivable that, for example, the effect in beam seas is good, but in oblique seas it is substantially worse. A comprehensive judgment about the effectiveness of the anti-rolling tank is only possible when very comprehensive investigations are carried out for the various possible operating conditions.

References

[1] Chadwick, I.H. and K. Klotter: On the Dynamics of Anti-Rolling Tanks. Schiffstechnik, Vol. 2, No. 8, February 1955.

[2] Grim, O.: Die Schwingungen von schwimmenden zweidimensionalen Körpern. USVA-Bericht Nr. 1090, Nr. 1117. [Oscillations of Floating Two-Dimensional Bodies.]

[3] van den Bosch, J.J. and J.U. Vugts: Some Notes on the Performance of Free Surface Tanks as Passive Anti-rolling Devices. Netherlands Shipbuilding Laboratory, Report No. 119, 1964.

[4] Grim, O. and Y. Takaishi: Das Rollmoment in schrägläufender Welle. Institut für Schiffbau der Universität Hamburg, Bericht Nr. 148. [The Rolling Moment in Oblique Waves.]

[Translation covers the first 27 pages (the complete document); the original contains figures (oscilloscope traces and response amplitude curves) that are not reproduced in this translation.]