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Research Papers: Ocean Engineering

# A Comparative Assessment of Simplified Models for Simulating Parametric Roll

[+] Author and Article Information
Abhilash S. Somayajula

Marine Dynamics Laboratory,
Department of Ocean Engineering,
Texas A&M University,
College Station, TX 77843
e-mail: s.abhilash89@gmail.com

Jeffrey Falzarano

Professor
Marine Dynamics Laboratory,
Department of Ocean Engineering,
Texas A&M University,
College Station, TX 77843
e-mail: jfalzarano@civil.tamu.edu

1Corresponding author.

Contributed by the Ocean, Offshore, and Arctic Engineering Division of ASME for publication in the JOURNAL OF OFFSHORE MECHANICS AND ARCTIC ENGINEERING. Manuscript received April 5, 2016; final manuscript received September 25, 2016; published online January 31, 2017. Assoc. Editor: Robert Seah.

J. Offshore Mech. Arct. Eng 139(2), 021103 (Jan 31, 2017) (11 pages) Paper No: OMAE-16-1035; doi: 10.1115/1.4034921 History: Received April 05, 2016; Revised September 25, 2016

## Abstract

The motion of a ship/offshore platform at sea is governed by a coupled set of nonlinear differential equations. In general, analytical solutions for such systems do not exist and recourse is taken to time-domain simulations to obtain numerical solutions. Each simulation is not only time consuming but also captures only a single realization of the many possible responses. In a design spiral when the concept design of a ship/platform is being iteratively changed, simulating multiple realizations for each interim design is impractical. An analytical approach is preferable as it provides the answer almost instantaneously and does not suffer from the drawback of requiring multiple realizations for statistical confidence. Analytical solutions only exist for simple systems, and hence, there is a need to simplify the nonlinear coupled differential equations into a simplified one degree-of-freedom (DOF) system. While simplified methods make the problem tenable, it is important to check that the system still reflects the dynamics of the complicated system. This paper systematically describes two of the popular simplified parametric roll models in the literature: Volterra GM and improved Grim effective wave (IGEW) roll models. A correction to the existing Volterra GM model described in current literature is proposed to more accurately capture the restoring forces. The simulated roll motion from each model is compared against a corresponding simulation from a nonlinear coupled time-domain simulation tool to check its veracity. Finally, the extent to which each of the models captures the nonlinear phenomenon accurately is discussed in detail.

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Topics: Simulation , Waves , Ships , Hull

## References

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## Figures

Fig. 1

Body plan of the C11 hull form

Fig. 2

Separation of C11 hull form into 200 two-dimensional strips

Fig. 3

Variation of G1(x) and G2(x) along the length for C11 hull form

Fig. 4

First-order transfer function for GM variation for C11 hull form in 5 kn forward speed

Fig. 5

Plot of the second-order transfer function g1(ωm, ωn) for C11 hull form in 5 kn forward speed: (a) real part of g1(ωm, ωn) and (b) imaginary part of g1(ωm, ωn)

Fig. 6

Plot of the second-order transfer function g2(ωm, ωn) for C11 hull form in 5 kn forward speed: (a) real part of g2(ωm, ωn) and (b) imaginary part of g2(ωm, ωn)

Fig. 7

Comparison of Volterra first-order GM variation with simulated GM variation for C11 hull form in 5 kn forward speed subjected to an input wave realized from Bretschneider spectrum with significant wave height Hs = 3 m and modal period Tp = 14.1 s: (a) GM time series comparison and (b) power spectral density comparison of δGM

Fig. 8

Square of IGEW transfer functions

Fig. 9

Comparison of IGEW and Grim effective wave with actual wave elevation from Bretschneider spectrum with Hs = 3 m and Tp = 14.1 s

Fig. 10

Comparison of GM variation using IGEW with simulated GM variation for C11 hull form in 5 kn forward speed subjected to an input wave realized from Bretschneider spectrum with significant wave height Hs = 3 m and modal period Tp = 14.1 s: (a) GM time series comparison and (b) power spectral density comparison of δGM

Fig. 11

GZ variation calculation using IGEW: (a) GZ curve for a 4 m high regular wave, (b) quasi-static trim for a 4 m high regular wave, and (c) comparison of quasi-static trim with dynamic pitch for C11 hull form in 5 kn forward speed subjected to an input wave realized from Bretschneider spectrum with significant wave height Hs = 3 m and modal period Tp = 14.1 s

Fig. 12

Comparison of parametric roll amplitudes from SIMDYN simulations with experiments [33]: (a) H = 6 m, (b) H = 8 m, and (c) H = 10 m

Fig. 13

Comparison of roll time histories from simplified models with SIMDYN for C11 hull form in 5 kn forward speed subjected to the same input wave realized from Bretschneider spectrum with significant wave height Hs = 3 m and modal period Tp = 14.1 s: (a) wave elevation used in SIMDYN and 1DOF models, (b) Volterra model, (c) GM variation model using IGEW, and (d) GZ variation model using IGEW

Fig. 14

SIMDYN simulation of parametric roll for C11 hull form in 5 kn forward speed subjected to regular wave (T = 14.1 s) with and without nonlinear Froude–Krylov force: (a) H = 2 m and (b) H = 3 m

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