Nonlinear Rolling Motion of a Statically Biased Ship Under the Effect of External and Parametric Excitation

1993 ◽  
Author(s):  
Isaac Esparza ◽  
Jeffrey Falzarano
Keyword(s):  
Steady State ◽  
Parametric Excitation ◽  
Poincaré Map ◽  
Initial Conditions ◽  
Global Analysis ◽  
Wave Train ◽  
Periodic Wave ◽  
Rolling Motion ◽  
Poincare Map ◽  
Safe Basin

Abstract In this work, global analysis of ship rolling motion as effected by parametric excitation is studied. The parametric excitation results from the roll restoring moment variation as a wave train passes. In addition to the parametric excitation, an external periodic wave excitation and steady wind bias are also included in the analysis. The roll motion is the most critical motion for a ship because of the possibility of capsizing. The boundaries in the Poincaré map which separate initial conditions which eventually evolve to bounded steady state solutions and those which lead to unbounded capsizing motion are studied. The changes in these boundaries or manifolds as effected by changes in the ship and environmental conditions are analyzed. The region in the Poincaré map which lead to bounded steady state motions is called the safe basin. The size of this safe basin is a measure of the vessel’s resistance to capsizing.

A PHYSICAL INTERPRETATION OF MELNIKOV’S METHOD

1992 ◽  
Vol 02 (01) ◽  
pp. 1-9 ◽  
Author(s):  
YOHANNES KETEMA
Keyword(s):  
Physical Interpretation ◽  
Poincaré Map ◽  
Initial Conditions ◽  
Homoclinic Orbits ◽  
Poincare Map ◽  
Stable And Unstable Manifolds ◽  
Melnikov's Method ◽  
Melnikov’S Method ◽  
Sensitive Dependence ◽  
The Stability

This paper is concerned with analyzing Melnikov’s method in terms of the flow generated by a vector field in contrast to the approach based on the Poincare map and giving a physical interpretation of the method. It is shown that the direct implication of a transverse crossing between the stable and unstable manifolds to a saddle point of the Poincare map is the existence of two distinct preserved homoclinic orbits of the continuous time system. The stability of these orbits and their role in the phenomenon of sensitive dependence on initial conditions is discussed and a physical example is given.

EFFECTIVE COMPUTATION OF THE POINCARÉ MAP FOR THE ANALYSIS OF NONLINEAR DYNAMIC CIRCUITS/SYSTEMS USING RUNGE-KUTTA TRIPLES

1994 ◽  
Vol 04 (01) ◽  
pp. 93-98 ◽  
Author(s):  
L. FINGER ◽  
H. UHLMANN
Keyword(s):  
Numerical Simulations ◽  
Nonlinear Dynamic ◽  
Poincaré Map ◽  
Global Analysis ◽  
Interpolation Polynomial ◽  
Poincare Map ◽  
Dynamic Circuits ◽  
Dense Output ◽  
Computer Aided ◽  
Runge Kutta

An enhancement of the classical Runge—Kutta technique for numerical simulations is presented for the computer-aided global analysis of nonlinear dynamic circuits/systems. With Runge—Kutta triples a remarkable saving of calculation time can be achieved by using an interpolation polynomial for dense output. The Runge—Kutta triples are applied to calculate the Poincaré map for autonomous models/systems.

APPLICATION OF GLOBAL METHODS FOR ANALYZING DYNAMICAL SYSTEMS TO SHIP ROLLING MOTION AND CAPSIZING

1992 ◽  
Vol 02 (01) ◽  
pp. 101-115 ◽  
Author(s):  
JEFFREY M. FALZARANO ◽  
STEVEN W. SHAW ◽  
ARMIN W. TROESCH
Keyword(s):  
Nonlinear Dynamic ◽  
Invariant Manifolds ◽  
Initial Conditions ◽  
Global Analysis ◽  
Relative Size ◽  
Nonlinear Dynamic Analysis ◽  
Periodic Wave ◽  
Industry Standards ◽  
Highly Nonlinear ◽  
Lobe Dynamics

Ship capsizing is a highly nonlinear dynamic phenomenon where global system behavior is dominant. However the industry standards for analysis are limited to linear dynamics or nonlinear statics. Until recently, most nonlinear dynamic analysis relied upon perturbation methods which are severely restricted both with respect to the relative size of the nonlinearity and the region of consideration in the phase space (i.e., they are usually restricted to a small local region about a single equilibrium), or on numerical studies of idealized system models. In this work, recently developed global analysis techniques (e.g., those found in Guckenheimer and Holmes [1986], and Wiggins [1988, 1990]) are used to study transient rolling motions of a small ship which is subjected to a periodic wave excitation. This analysis is based on determining criteria which can predict the qualitative nature of the invariant manifolds which represent the boundary between safe and unsafe initial conditions, and how these depend on system parameters for a specific ship model. Of particular interest is the transition which this boundary makes from regular to fractal, implying a loss in predictability of the ship’s eventual state. In this paper, actual ship data is used in the development of the model and the effects of various ship and wave parameters on this transition are investigated. Finally, lobe dynamics are used to demonstrate how unpredictable capsizing can occur.

Analytical Study and Experimental Confirmation of SNA Through Poincaré Maps in a Quasiperiodically Forced Electronic Circuit

2015 ◽  
Vol 25 (08) ◽  
pp. 1530020 ◽  
Author(s):  
A. Arulgnanam ◽  
Awadesh Prasad ◽  
K. Thamilmaran ◽  
M. Daniel
Keyword(s):  
Analytical Study ◽  
Poincaré Map ◽  
Initial Conditions ◽  
Nonlinear Element ◽  
Phase Portraits ◽  
Poincare Map ◽  
Explicit Analytical Solution ◽  
Strange Nonchaotic Attractors ◽  
First Time ◽  
Bifurcation Process

Quasiperiodically forced series LCR circuit with simple nonlinear element is studied analytically and experimentally. To the best of our knowledge, this is the first time that strange nonchaotic attractors (SNAs) are studied analytically. From the explicit analytical solution, the bifurcation process is shown. With a single negative conduction region of the nonlinear element two routes namely, Heagy–Hammel and fractalization routes to the birth of SNA are identified. The analytical analysis are confirmed by laboratory hardware experiments. In addition, for the first time, a detailed stroboscopic Poincaré map is generated experimentally for two different frequencies, for the above two routes, which clearly confirm the presence of SNAs in these two routes. Also, from the experimental data of the corresponding attractors, we quantitatively confirm the presence of SNAs through singular-continuous spectrum analysis. The analytical results as well as experimental observations are characterized qualitatively in terms of phase portraits, Poincaré map, power spectrum, and sensitivity dependance on initial conditions.

Second Order Poincaré Map to Study Dynamical Behavior of Nonsymmetrical Gyrostat Satellite

2000 ◽  
Author(s):  
K. H. Shirazi ◽  
M. H. Ghaffari-saadat
Keyword(s):  
Phase Space ◽  
Poincaré Map ◽  
Initial Conditions ◽  
Equations Of Motion ◽  
Dynamical Behavior ◽  
Two Dimensions ◽  
Second Order ◽  
Poincare Map ◽  
Runge Kutta Method ◽  
Final Space

Abstract The second order poincare’ map is described and used for investigation of the dynamical behavior of a gyrostat satellite. The normalized attitudinal equations of motion for a typical non-symmetric gyrostat satellite are considered. For different sets of initial conditions the equations simulated by Runge-Kutta method. The poincare’ section is used to dimension reduction of system phase space. By this map the dimension reduced from six to five. Using secondary map the dimension of phase space can be reduced to four and considering symmetry of phase space the final space has two dimensions that is presentable at the plane. Bifurcation in the attitudinal behavior can be demonstrated easily by the derived map.

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