Twenty-First Symposium on Naval Hydrodynamics (1997) / Chapter Skim
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Nonlinearly Interacting Responses of the Two Rotational Modes of Motion-Roll and Pitch Motions
Nonlinearly Interacting Responses of the Two Rotational Modes of Motion-Roll and Pitch Motions
Pages 205-219
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From page 205... ...
Also to take into account the damping effects, a linear-plusquadratic damping model is introduced in the equatio for the roll mode while a linear damping model is kept in the equation for the pitch mode. The method of multiple scales is used to determine the influence of the quadratic nonlinear roll damping on the periodic responses of both modes and their stability.
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From page 206... ...
Nayfeh ~ considered the nonlinearly coupled roll and pitch motions of a ship in regular Y' 206 head waves in which the couplings are primarily in the hydrostatic terms when the pitch frequency is approximately twice the roll frequency and the encounter frequency is near either the pitch or roll natural frequency. He demonstrated He saturation phenomenon when the encounter frequency is near the pitch natural frequency.
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From page 207... ...
( 16) where the -an and ·~3n are the amplitudes and phases of the roll and pitch modes.
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From page 208... ...
4. PRIMARY RESONANCE OF THE PITCH MODE To express the nearness of the resonances, we introduce two detuning parameters, cry and ~2' defined as w2=2~+~ and Q=CU2+~2 (23a)
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From page 209... ...
where c; ~ i = 0,1,2, and 3) are constant coefficients determined from control parameters in use, such as By, ~2, At,, p2, ~3, f2, and lo,; they are Even by Co -(~2+~2)
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From page 210... ...
A Hopf bifurcation point is defined as a critical value of a control parameter at which a pair of complex-conjugate eigenvalues of the Jacobian matrix cross the imaginary axis into the right half of the complex plane with nonzero speed. This type of loss of stability fits the Hopf bifurcation theorem, according to which the modulation equations possess limit-cycle solutions near the bifurcation points.
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From page 211... ...
= (2 When F~ > 0, equation (53) has two real roots when 1~21 < f2 <(if +~2)
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From page 212... ...
Moreover, for a given f2, the amplitude a2 of the pitch mode increases svith increasing quadratic roll damping coefficient p3 . Figures 2 show typical force-response curves when it, > 0 ~ F~ = 0.2484)
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From page 213... ...
Then, when this happens, the response of the roll mode would no longer exhibit the coexistence of two stable motions and the subsequent jump phenomenon for large values of the quadratic roll damping coefficient ~3 Figure 2 (c) shows the force-response curves of the pitch mode corresponding to Figure 2 (b)
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From page 214... ...
In Figure 3 (c) , we show the frequency-response curves for the pitch mode corresponding to Figure 3 (b)
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From page 215... ...
Figure 4 shows that the Hopf bifurcation curves move upward and approach each other as the quadratic roll damping coefficient p3 increases. This implies that increasing p3 causes the disappearance of aperiodic responses.
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From page 216... ...
If a pair of complex-conjugate eigenvalues crosses the imaginary axis with nonzero speed, then we have a Hopf bifurcation. Near these bifurcation points, the response is an amplitude- and phase-modulated combined pitch and roll motion, with the energy being continuously exchanged between the two modes.
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From page 217... ...
Moreover, the Hoff bifurcation points approach each over as p3 incr~. For example, the Hopf bifurcation interval ~ 09995 ~ CT2 ~ 0.09995 for p3 = 0 shrinks gradually to ~.055 ~ ~2 ~ 0.055 for p3 = 0.9 .
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From page 218... ...
When the encounter frequency is near the roll natural frequency, the amplitudes of both the roll mode and the pitch mode decrease as the magnitude of the quadratic roll damping coefficient p3 increases. Again, Hopf bifurcations occur as either the encounter frequency or excitation amplitude is varied.
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From page 219... ...
H., "Nonlinear Rolling Motions of Ships in Longitudinal Waves," International Shipbuilding Progress, Vol.
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