Neeraj UAbin Devasia

K.Vijayan

Nonlinear Rolling of Ships in Ocean

Contents

1.Introduction

2.Motivation

2.Formulation of rolling equation

3.Analysis Methodology

4.Results

5.Conclusion & Future works

6..References

Introduction

● The ships moving in the oceans are continuously subjected to waves

subjecting it to high amplitude ● Extreme rolling and - ultimately - capsizing in critical wave and operational

conditions may occur according to the following mechanisms: ! ○ Static loss of stability !○ Dynamic loss of stability ! ○ Broaching ! ○ Combined modes with additional factors !

● Rolling is the rigid body oscillation of a ship about its longitudinal axis

Motivation

● Roll motion is the most important phenomenon for ships ,coupled with a few

others,which may lead to capsizing.

● Large amplitude rolling motion makes large container ships extremely unstable and

vulnerable to crew and cargo accidents, machinery failure, structural damage and

possibly capsizing.

● Roll motion affects the performance of seagoing surface vessels by limiting the

effectiveness.

● These necessitate a thorough study of ship roll motion.

Problem Formulation - Geometry

Length L 100 cm

Breadth 32 cm

Density 1000 kg/m3

Height 30 cm

Draft 4.88 cm

Weight 23.44 Kg

Geometric properties of the Barge

The Considered Rectangular Barge has following assumptions .• The Block Coefficient if approximately taken as

1 .• Shear deformations are ignored .• Added mass of slender bodies is calculated by

strip theory .

Problem Formulation A typical non-linear roll motion of equation can be expressed as

where i is mass moment of Inertia B,C are functions of response .

Analysis Methodology

The governing differential equation of the non linear rolling motion of a ship is analyzed as

The solution procedure is carried out by Duffing’s method.It can be rewritten as a special case of Duffing’s equation with quintic restoring terms as follows:

Analysis MethodologySubstitute Using method of harmonic balance and neglecting terms with higher harmonics

Left hand side of the equation can be written as Coparing the amplitude of RHS and LHS

Analysis Methodology

● The amplitudes obtained from the 10th order equation was plotted.The imaginary

roots are ignored and the three real roots are shown in different colour. The

minimum amplitude against the frequency is plotted.

Analysis Methodology

● Further, Numerical analysis was carried out in the time domain by solving the

nonlinear second order differential equation using ode45 solver in MATLAB.

● The analysis was done for a range of forcing frequencies.

● Maximum amplitude was obtained for each time series .

● The transients in the time series as a result of limited cycle oscillations were

neglected to get the steady state maximum amplitude.

● There were plotted against the corresponding excitation frequencies.

Results

Roll motion

● The plot obtained from the solution of roll equation through harmonic balance

indicates a jump behaviour with the amplitude suddenly decreasing to a lower

value after a certain frequency.

● The plot obtained through numerical methods is almost similar to the one before

showing a jump behaviour.

ResultsControlled Roll motion

● Generally, to reduce roll motion a counteracting moment is provided by various

methods such as anti-roll tank, active fins [6].

● The effect of this on the roots obtained from the harmonic balance were analyzed

with a reduced net excited moment.

● The result was a shift of the peak amplitude value to the left side and the jump

phenomena observed previously disappeared.

● The response shifted to the linear form as indicated by the roots

Results

Amplitude vs Frequency plot showing

the real roots obtained from the

analytical solution of duffing

equation

Amplitude vs Frequency showing all roots from solving the 10th order equation

Results

Amplitude vs frequency plot obtained from plotting the maximum amplitude from time series analysis using ode45 solver

Conclusion and Future Work

● The amplitude was observed to have a jump at a particular frequency when the

response was analysed analytically and numerically using time domain results.

● This jump was avoided as the value of forcing term was reduced. The response

shifted to a linear form on the change.

● Further study will explore the effect of control methods on the roots of the roll

motion.

● Coupling with other degrees of motion are possible in the real life situations. But

these are beyond the scope of the paper and are hence not considered in the study.

This could be studied in detail in future works.

References

1. M.Taylan (2000), The effect of nonlinear damping and restoring in ship rolling. Ocean

Engineering 27 921–932

2. Taylan M. 1999. Solution of the nonlinear roll model by a generalized asymptotic method. Ocean

Engineering 26:1169–1181.

3. Taylan M. 2000. The effect of nonlinear damping and restoring in ship rolling. Ocean

Engineering, 27:921–932

4. Dalzel JF. 1978. A note on the form of ship roll damping. Journal of Ship Research, 22:178–185

5. S. Surendran & V. Kiran (2006), Technical note Studies on the feasibilities of control of ship roll

using fins, Ships and Offshore Structures, 1:4, 357-365.

6. S. Surendran & V. Kiran (2007), Control of ship roll motion by active fins using fuzzy logic, Ships

and Offshore Structures, 2:1, 11-20.

7. Subrata Chakrabarti (2001), Empirical calculation of roll damping for ships and barges, Ocean

Engineering 28915– 932.

8. Bhattacharyya R. 1978. Dynamics of marine vehicles, Wiley, New York.

9. Edward V. Lewis, Principles of naval architecture, Volume 1-Stability and strength.10. A. Nayfeh, D. Mook, L. Marshall, Non-linear coupling of pitch and roll modes in ship motion, J. Hydronaut., 7 (1973), pp. 145–15211. A. Nayfeh, D. Mook, Nonlinear Oscillations, Wiley, New York (1995)12. Nayfeh, A.H. and Khdeir, A.A., “Nonlinear Rolling of Biased Ships in Regular Beam Seas,” Int. Shipbuild. Prog., Vol. 33, pp. 89-93 (1986)13. Virgin, L.N., “The Nonlinear Rolling Response of a Vessel Including Chaotic Motions Leading to Capsize in Regular Seas,” Appl. Ocean Res., Vol. 9, pp. 89-95 (1987).14. A. Cardo, A. Francescutto, R. NabergojNonlinear rolling response in a regular seaInt. Shipbuilding Prog., 31 (1984), p. 20415. Denise, J.-P.F., 1983. On the roll motion of barges. Transactions RINA 125, 255–268.16. Yoji Himeno, Prediction of Ship Rolling-A State of Art

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