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

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 visco ins 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 andfrequency-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.

[1]  A. Nayfeh,et al.  NONLINEAR COUPLING OF PITCH AND ROLL MODES IN SHIP MOTIONS , 1973 .

[2]  A. Nayfeh Introduction To Perturbation Techniques , 1981 .

[3]  H. Hatwal,et al.  Non-linear vibrations of a harmonically excited autoparametric system , 1982 .

[4]  J. Dugundji,et al.  Resonance oscillations in mechanical systems , 1976 .

[5]  A. H. Nayfeh,et al.  The response of two-degree-of-freedom systems with quadratic non-linearities to a combination parametric resonance , 1986 .

[6]  Ali H. Nayfeh,et al.  ON THE UNDESIRABLE ROLL CHARACTERISTICS OF SHIPS IN REGULAR SEAS , 1988 .

[7]  Ali H. Nayfeh,et al.  NONLINEAR ROLLING MOTIONS OF SHIPS IN LONGITUDINAL WAVES , 1990 .

[8]  Vimal Singh,et al.  Perturbation methods , 1991 .

[9]  J. Tukey,et al.  An algorithm for the machine calculation of complex Fourier series , 1965 .

[10]  J. E. Kerwin,et al.  Notes on Rolling in Longitudinal Waves , 1955 .

[11]  R. Singleton An algorithm for computing the mixed radix fast Fourier transform , 1969 .

[12]  W. Błocki Ship safety in connection with parametric resonance of the roll , 1980 .

[13]  T. Aprille,et al.  A computer algorithm to determine the steady-state response of nonlinear oscillators , 1972 .

[14]  Dean T. Mook,et al.  Theoretical and experimental study of modal interaction in a two-degree-of-freedom structure , 1984 .

[15]  Ali H. Nayfeh,et al.  Subharmonic and Superharmonic Resonances in the Pitch and Roll Modes of Ship Motions , 1974 .

[16]  A. H. Nayfeh,et al.  PERTURBATION-ENERGY APPROACH FOR THE DEVELOPMENT OF THE NONLINEAR EQUATIONS OF SHIP MOTION , 1974 .