Nonlinear Dynamics 35: 123–146, 2004. © 2004 Kluwer Academic Publishers. Printed in the Netherlands. Polyharmonic Balance Analysis of Nonlinear Ship Roll Response J. C. PEYTON JONES 1,∗ and I. ÇANKAYA 2 1 Center for Nonlinear Dynamics & Control, Villanova University, 800 Lancaster Ave, Villanova, PA 19085, U.S.A.; 2 Department ofElectronics and Computer, Technical Education Faculty, Sakarya University, 54187 Esentepe Campus, Adapazarı, Turkey; ∗ Author for correspondence (e-mail: jamespj@ece.vill.edu; fax: +1-610-519-4436) (Received: 13 August 2003; accepted: 7 November 2003) Abstract. A recently developed analytical algorithm for generating the polyharmonic balance equations of a class of nonlinear differential equations has been used to perform a detailed four-component analysis of a single- degree-of-freedom ship roll model with cubic damping and quintic stiffness terms subject to a biased sinusoidal forcing function. The tradeoff between accuracy and complexity has then been systematically investigated across a range of input sinusoidal frequency, amplitude, and input bias, by investigating the effect of neglecting harmonic or model terms in the more detailed analysis and then comparing the results to independent solutions obtained through direct numerical integration of the governing equation. Keywords: Harmonic balance, jump resonance, ship roll, nonlinear systems. 1. Introduction The roll response of a ship in regular beam waves may be modeled using a single degree of freedom differential equation. Typically such models include nonlinear stiffness terms in order to characterize the shape of the static stability diagram, and nonlinear damping terms which represent effects due to frictional resistance or to eddies behind bilge keels. As wave amplitude increases, the effect of these terms becomes increasingly significant, so studies of roll behavior in conditions that might cause capsize necessitate the use of some form of non- linear analysis. Historically, such analysis has generally used perturbation-based methods to predict the frequency response of the vessel in the region of the main, sub- or super-harmonic resonances [1–3]. The use of harmonic balance techniques is much less common, despite their potential benefit and their widespread application in other areas. A possible reason for this lies in the high degree of nonlinearity of the system, coupled with the sharply resonant nature of ship roll models where higher harmonics can make a significant contribution to the response. The presence of a dc or bias term in the forcing function also excites the even harmonics in the reponse. Such factors significantly increase the complexity (or if ignored, reduce the accuracy) of the harmonic balance method. The R-harmonic expansion of a system containing a single nth order nonlinearity, for example, contains (2R) n terms, which must then be combined and simplified in order to determine the frequency, amplitude, and phase of each distinct output component. For low values of R and n, this process is not particularly onerous, but complexity increases rapidly as these values are incremented. A simple cubic nonlinearity subject to a single sinusoid, for example, generates only 8 terms in the expansion or two distinct output sinusoids, but a small change to a quartic