Nonlinearly Coupled Pitch and Roll Motions in the Presence of Internal Resonance; Part I, Theory

Article type: Research Article

Authors: Nayfeh, A.H. | Oh, I.G.

Affiliations: Department of Engineering Science and Mechanics, Virginia Polytechnic Institute and State University, Blacksburg, Virginia 24061 USA

Abstract: The dynamic stability and complicated motions of a vessel in following or head regular waves are investigated when the frequency in pitch is nearly twice the frequency in roll. The damping in the pitch mode is modeled by a linear viscoins damping term, whereas that of the roll mode is modeled by the sum of a linear viscous part and a quadratic viscous part. The method of multiple scales is used to determine a system of four nonlinear first-order equations governing the modulation of the amplitudes and phases of the pitch and roll. Force-response and frequency-response curves are generated. Coexistence of multiple solutions is found. The jump phenomenon continues to exist, whereas the saturation phenomenon ceases in the presence of quadratic damping. Hopf bifurcations are found. Near these bifurcations, the modulation equations possess limit-cycle solutions and hence the steady-state motion is a periodically modulated pitch and roll motion. Numerical simulations are used to investigate the bifurcations of these limit cycles and how they lead to chaos and hence chaotically modulated pitch and roll motions.

DOI: 10.3233/ISP-1995-4243202

Journal: International Shipbuilding Progress, vol. 42, no. 432, pp. 295-324, 1995