## Abstract

A steady-state roll motion of ships with nonlinear damping and restoring moments for all times is modeled by a second-order nonlinear differential equation. Analytical expressions for the roll angle, velocity, acceleration, and damping and restoring moments are derived using a modified approach of homotopy perturbation method (HPM). Also, the operational matrix of derivatives of ultraspherical wavelets is used to obtain a numerical solution of the governing equation. Illustrative examples are provided to examine the applicability and accuracy of the proposed methods when compared with a highly accurate numerical scheme.

This is a preview of subscription content, access via your institution.

## References

Abualrub T, Abukhaled M (2015) Wavelets approach for optimal boundary control of cellular uptake in tissue engineering. Int J Comp Math 92(7):1402–1412. https://doi.org/10.1080/00207160.2014.941826

Abualrub T, Abukhaled M, Jamal B (2018) Wavelets approach for the optimal control of vibrating plates by piezoelectric patches. J Vibr Cont 24(6):1101–1108. https://doi.org/10.1177/1077546316657781

Abukhaled M (2013) Variational iteration method for nonlinear singular two-point boundary value problems arising in human physiology. J Math:ID:720134. https://doi.org/10.1155/2013/720134

Abukhaled M (2017) Green's function iterative approach for solving strongly nonlinear oscillators. J Comp Nonlinear Dyn 12(5):051021. https://doi.org/10.1115/1.4036813

Agarwal D (2015) A study on the feasibility of using fractional differential equations for roll damping models. Master thesis, Virginia Polytechnic Institute and State University. http://hdl.handle.net/10919/52959

Aloisio G, Felice F (2006) PIV analysis around the bilge keel of a ship model in a free roll decay. In: XIV Congresso Nazionale AI VE. LA., Rome, Italy

Bassler CC, Carneal JB, Atsavapranee P (2007) Experimental investigation of hydrodynamic coefficients of a wavepiercing tumble home hull form. In: Proceeding of the 26th International Conference on Offshore Mechanics and Arctic Engineering, San Diego, USA

Cardo A, Francescutto A, Nabergoj R (1981) Ultraharmonics and subharmonics in the rolling motion of a ship: steadystate solution. Int Ship building Progress 28:234–251

Cardo A, Francescutto A, Nabergoj R (1984) Nonlinear rolling response in a regular sea. Int Shipbuild Prog 31(360):204–206. https://doi.org/10.3233/ISP-1984-3136002

Chen CL, Liu YC (1998) Solution of two-point boundary-value problems using the differential transformation method. J Optimization Theory Appl 99:23–35

Comini G, Del Guidice S, Lewis RW, Zienkiewicz OC (1970) Finite element solution of non-linear heat conduction problems with special reference to phase change. Num Meth Engg 8:613–624

Demirel H, Alarcin F (2016) LMI-based H2 and H1 state-feedback controller design for fin stabilizer of nonlinear roll motion of a fishing boat. Brodogradnja: Teorija i praksa brodogradnje i pomorske tehnike 67(4):91–107. https://doi.org/10.21278/brod67407

Demirel H, Dogrul A, Sezen S, Alarcin F (2017) Backstepping control of nonlinear roll motion for a trawler with fin stabilizer. Transactions of the Royal Institution of Naval Architects Part A: International Journal of Maritime Engineering, Part A2 159, pp. 205-212. doi: 10.3940/rina.ijme.2017.a2.420

Doha EH, Abd-Elhameed WM, Youssri YH (2016) New ultraspherical wavelets collocation method for solving 2nth order initial and boundary value problems. J Egyp Math Soc 24:319–327. https://doi.org/10.1016/j.joems.2015.05.002

Froude W (1861) On the rolling of ships. Transactions of the Institution of Naval Architects 2, 1861, 180–229.

He JH (1999) Homotopy perturbation technique. Comp Meth Appl Mech Engg 178(4):257–262. https://doi.org/10.1016/S0045-7825(99)00018-3

Himeno Y (1981) Prediction of ship roll damping-a state of the art. DTIC Document, Ann Arbor.

Huang BG, Zou ZJ, Ding WW (2018) Online prediction of ship roll motion based on a coarse and fine tuning fixed grid wavelet network. Ocean Eng 160:425–437. https://doi.org/10.1016/j.oceaneng.2018.04.065

Ikeda Y, Ali B, Yoshida H (2004) A roll damping prediction method for a FPSO with steady drift motion. Proceeding of the 14th International Conference on Offshore and Polar Engineering Conference, Toulon, France 676–681.

Jang TS (2011) Non-parametric simultaneous identification of both the nonlinear damping and restoring characteristics of nonlinear systems whose dampings depend on velocity alone. Mech Syst Signal Process 25(4):1159–1173. https://doi.org/10.1016/j.ymssp.2010.11.002

Jang TS (2013) A method for simultaneous identification of the full nonlinear damping and the phase shift and amplitude of the external harmonic excitation in a forced nonlinear oscillator. Comput Struct 120:77–85. https://doi.org/10.1016/j.compstruc.2013.02.008

Jang TS, Choi HS, Han SL (2009) A new method for detecting non-linear damping and restoring forces in non-linear oscillation systems from transient data. Int J Non-Linear Mech 44(7):801–808. https://doi.org/10.1016/j.ijnonlinmec.2009.05.001

Kato H (1965) Effects of bilge keels on the rolling of ships. J. Soc. Nav. Archit. Jpn. 117:93–101

Khuri SA, Abukhaled M (2017) A semi-analytical solution of amperometric enzymatic reactions based on Green’s functions and fixed point iterative schemes. J Electroanalytical Chem 792:66–71. https://doi.org/10.1016/j.jelechem.2017.03.015

Kianejad SS, Enshaei H, Du_y J, Ansarifard N (2019) Prediction of a ship roll added mass moment of inertia using numerical simulation. Ocean Eng 173:77–89. https://doi.org/10.1016/j.oceaneng.2018.12.049

Lavrov A, Rodrigues JM, Gadelho JFM, GuedesSoares C (2017) Calculation of hydrodynamic coe_cients of ship sections in roll motion using Navier-Stokes equations. Ocean Eng 133:36–46. https://doi.org/10.1016/j.oceaneng.2017.01.027

Liao S (1997) Homotopy analysis method: a new analytical technique for nonlinear problems. Commun Nonlinear Sci Numer Simul 2(2):95–100. https://doi.org/10.1016/S1007-5704(97)90047-2

Liao S (2004) On the homotopy analysis method for nonlinear problems. Appl Math Comput 47(2):499–513. https://doi.org/10.1016/S0096-3003(02)00790-7

Liao SJ, Chwang AT (1998) Application of homotopy analysis method in nonlinar oscillations. J Appl Mech 65(4):914–922. https://doi.org/10.1115/1.2791935

Lihua L, Peng Z, Songtao Z, Ming J, Jia Y (2018) Simulation analysis of fin stabilizer on ship roll control during turning motion. Ocean Eng 164:733–748. https://doi.org/10.1016/j.oceaneng.2018.07.015

Lu J (2007) Variational iteration method for solving a nonlinear system of second-order boundary value problems. Comp Math Appl 54:1133–1138. https://doi.org/10.1016/j.camwa.2006.12.060

Oliveira AC, Fernandes AC (2012) An empirical nonlinear model to estimate FPSO with extended bilge keel roll linear equivalent damping in extreme seas. In: Proceeding of the 31st International Conference on Ocean, Offshore and Arctic Engineering, (OMAE). Paper OMAE2012-83360, Rio de Janeiro, Brazil.

Oliveira AC, Fernandes AC (2014) The nonlinear roll damping of a FPSO hull. J Off shore Mech Arct Eng 136:1–10. https://doi.org/10.1115/1.4025870

Rainville ED (1960) Special functions. The Macmillan Co., New York, 1960. MR 0107725.

Razzaghi M, Yousef S (2002) Legendre wavelets method for constrained optimal control problems. Math Met Appl Sci 25:529–539. https://doi.org/10.1002/mma.299

Sadek I, Abualrub T, Abukhaled M (2007) A computational method for solving optimal control of a system of parallel beams using Legendre wavelets. Math Comp Modell 45:1253–1264. https://doi.org/10.1016/j.mcm.2006.10.008

Salai Mathi Selvi M, Hariharan G (2016) Wavelet based analytical algorithm for solving steady-state concentration in immobilized isomerase of packed bed reactor model. J Memb Biol. 249(4):559–586

Salai Mathi Selvi M, Hariharan G (2017) An improved method based on Legendre computational matrix method for time dependent Michaelis-Menten enzymatic reaction model arising in mathematical chemistry. Proc Jangjeon Mathe Soc. 20(3):483–503. https://doi.org/10.17777/pjms2017.20.3.483

Salai Mathi Selvi M, Hariharan G, Kannan K (2017) A reliable spectral, method to reaction-diffusion equations in entrapped-cell photobioreaction packed with gel grnules using chebyshev wavelets. J Memb Biol 250:663–670

Tanaka N (1961) A study on the bilge keels. J Soc Nav Archit Jpn 109:205–212

Tavassoli Kajani M, Hadi Vencheh A (2004) Solving linear integro-differential equation with Legendre wavelets Int. J Comput Math 81(6):719–726. https://doi.org/10.1080/00207160310001650044

Yeung RW, Cermelli C, Liao SW (1997) Vorticity fields due to rolling bodies in a free surface - experiment and theory. Proceeding of the 21st symposium on naval hydrodynamics Trondheim Norway.

Zheng X, Wei Z (2016) Estimates of approximation error by Legendre wavelet. Appl Math 7:694–700. https://doi.org/10.4236/am.2016.77063

Salomi RJ, Sylvia SV, Rajendran L, Abukhaled M, (2020) Electric potential and surface oxygen ion density for planar, spherical and cylindrical metal oxide grains. Sensors and Actuators B: Chemical 321: 128576. https://doi.org/10.1016/j.snb.2020.128576

## Acknowledgements

The authors are thankful to Shri J. Ramachandran, Chancellor, Col. Dr. G. Thiruvasagam, Vice-Chancellor, Academy of Maritime Education and Training (AMET), Deemed to be University, Chennai, for their support .

## Author information

### Affiliations

### Corresponding author

## Additional information

### Article Highlights

• A mathematical model for rolling motion of ships in random beam seas is discussed.

• The second-order nonlinear differential equation is solved using a modified approach of the homotopy perturbation method.

• Analytical expression of the roll angle, velocity, acceleration, and damping and restoring moments is derived.

• Ultraspherical wavelet–based method was also employed to derive a numerical approximation of roll angle and restoring and damping moments.

## Appendices

### Appendix 1: Ultraspherical polynomials

The ultraspherical polynomials are special types of Jacobi polynomials that are associated with the real parameter \( \left(\alpha >\frac{-1}{2}\right) \). They are orthogonal polynomials on the interval (− 1, 1), with respect to the weight function \( w(x)={\left(1-{x}^2\right)}^{\alpha -\frac{1}{2}}, \) and defined by

where

are the eigenfunctions of the following singular Sturm-Liouville equation

The following integral formula and the theorem that follows are needed to establish the convergence of the expansion of the ultraspherical wavelets

Theorem 3 The following inequality holds for ultraspherical polynomials

### Appendix 2: Shifted ultraspherical polynomials

The shifted ultraspherical polynomials are defined on [0, 1] by

All properties of ultraspherical polynomials remain valid for the shifted polynomials.

The orthogonality relation for \( {\overset{\sim }{C}}_n^{\left(\alpha \right)}(x) \) with respect to the weight function \( \overset{\sim }{\omega }(x)={\left(x-{x}^2\right)}^{\alpha -\frac{1}{2}} \) is given by

For more properties of ultraspherical polynomials, see Rainville (1960).

### Appendix 3: Basic idea of HPM

Consider the nonlinear differential equation

with the boundary condition

where *A*, *B*, *f*(*r*), and Γ are a general differential operator, a boundary operator, a known analytical function, and the boundary of the domain Ω, respectively. Expressing *A*(*u*) as the sum of linear (*L*) and nonlinear (*N*) parts, Eq. (74) becomes

The homotopy technique begins by defining

*v*(*r*, *p*) : *Ω* × [0, 1] → *R*, such that

where *p* ∈ [0, 1] is an embedding parameter and *u*_{0} is an initial approximation of Eq. (74) that satisfies boundary conditions (Eq. 75). Evidently, Eq. (77) implies that

As *p* changes from 0 to 1, *v*(*r*, *p*) changes from *u*_{0} to *u*_{r}, a process known as a homotopy. The solution of Eq. (77) may be expressed in terms of a power series in the form:

An approximate solution to Eq. (77) is given by the following:

## Rights and permissions

## About this article

### Cite this article

Selvi, M.S.M., Rajendran, L. & Abukhaled, M. Estimation of Rolling Motion of Ship in Random Beam Seas by Efficient Analytical and Numerical Approaches.
*J. Marine. Sci. Appl.* (2021). https://doi.org/10.1007/s11804-020-00183-x

Received:

Accepted:

Published:

### Keywords

- Nonlinear damping
- Steady-state roll motion
- Ultraspherical wavelets
- Homotopy perturbation method
- Analytical solution