Discussion on Analytical Approaches to the Study of Vessel Dynamics—Outcomes of the Two-Part Minisymposium at the 2007 SIAM Conference on Applications of Dynamical Systems

2008 ◽  
Vol 45 (04) ◽  
pp. 211-220
Author(s):  
Laura Alford ◽  
Atul Banik ◽  
Vadim Belenky ◽  
Katrin Ellermann ◽  
Hirotada Hashimoto ◽  
...  
Keyword(s):  
Dynamical Systems ◽  
High Speed ◽  
Chaotic Behavior ◽  
Applied Mathematics ◽  
Ship Motions ◽  
Parametric Rolling ◽  
Strongly Nonlinear ◽  
Current Trends ◽  
Military Applications ◽  
Analytical Approaches

While the study of ship stability dates back to Archimedes, modern research on vessel dynamics is at the forefront of applied mathematics. Large-amplitude ship motions result in strongly nonlinear, even chaotic behavior. The current trends toward high-speed and unique hullform vessels in commercial and military applications have broadened the need for robust mathematical approaches to studying the dynamics of these innovative ships. The presentations in this minisymposium focus on analytical formulations to model and understand the complicated dynamics leading to vessel phenomena such as capsizing, broaching, and parametric rolling.

A New Numerical Tool for Fast Ships in Following Seas

2010 ◽  
Vol 132 (3) ◽  
Author(s):  
Ray-Qing Lin ◽  
John G. Hoyt
Keyword(s):  
High Speed ◽  
Transfer Functions ◽  
Ship Motion ◽  
New Method ◽  
Frequency Resolution ◽  
Ship Motions ◽  
Following Seas ◽  
Strongly Nonlinear ◽  
Singularity Point ◽  
Incident Waves

The six-degrees-of-freedom ship motions of a ship at speeds other than zero are always measured in terms of encounter frequency, and often, the incident waves in experimental data are also measured only in the encounter frequency domain. Using these measured data to obtain transfer functions from irregular following sea ship motions is complicated by the combined effects of very low encounter frequencies and the “folding” of the sea spectra. This results in having both overtaking and encountered waves of the same encounter frequency but different wavelengths. Computing transfer functions becomes untenable when the ship speed approaches the wave phase velocity, where the encounter spectrum has a mathematical singularity. St. Denis and Pierson (1953, “On the Motions of Ships in Confused Seas,” Soc. Nav. Archit. Mar. Eng., Trans., 61, pp. 280–357) suggested the basic relationships between response ship motions or moments that can be developed in the wave frequency domain at the outset. The St. Denis–Pierson method is based on a linear theory and works well when the ship response regime is linear or weakly nonlinear. However, for high-speed craft operating at different headings where the problems are nonlinear, especially strongly nonlinear, the St. Denis–Pierson assumptions will break down inducing error (1953, “On the Motions of Ships in Confused Seas,” Soc. Nav. Archit. Mar. Eng., Trans., 61, pp. 280–357). Furthermore, using the frequency resolution method to remove the singularity point may also induce errors, especially when the singularity point is located near the peak of stationary frequency. How to obtain the correct frequency resolution in the local region of singularity point is still an unsolved problem. In this study, we will propose a new method capable of predicting ship response motions for crafts with nonlinear or strongly nonlinear behaviors quantitatively. For example, using this method, one can use measured ship motion data in head seas to predict the motions of the ship at high speed in following seas. The new method has six steps, including using a filter to eliminate those unexpected modes that are not from incident waves, inertial motions, or nonlinear interactions, and applying a higher-order Taylor expansion to eliminate the singularity point. We refer to the new method as the Lin–Hoyt method, which agrees reasonably well with computations of the nonlinear “digital, self-consistent, ship experimental laboratory ship motion model,” also known as DiSSEL (2006, “Numerical Modeling of Nonlinear Interactions Between Ships and Surface Gravity Waves II: Ship Boundary Condition,” J. Ship Res., 50(2), pp. 181–186). We also use experimental head sea data to validate the simulations of DiSSEL. The Lin–Hoyt method is fast and inexpensive. The differences in the results of the numerical simulations obtained by the Lin–Hoyt method and other linear methods diverge rapidly with increased forward ship speed due to the nonlinearity of ship motion responses.

A DYNAMICAL SYSTEM WITH CHAOTIC BEHAVIOR

1989 ◽  
Vol 03 (15) ◽  
pp. 1185-1188 ◽  
Author(s):  
J. SEIMENIS
Keyword(s):  
Dynamical System ◽  
Dynamical Systems ◽  
Infinite Number ◽  
Chaotic Behavior ◽  
Equations Of Motion ◽  
Hamiltonian Dynamical Systems

We develop a method to find solutions of the equations of motion in Hamiltonian Dynamical Systems. We apply this method to the system [Formula: see text] We study the case a → 0 and we find that in this case the system has an infinite number of period dubling bifurcations.

scholarly journals A computational framework for finding parameter sets associated with chaotic dynamics

2021 ◽  
pp. 1-11
Author(s):  
S. Koshy-Chenthittayil ◽  
E. Dimitrova ◽  
E.W. Jenkins ◽  
B.C. Dean
Keyword(s):  
Experimental Data ◽  
Dynamical Systems ◽  
Lyapunov Exponents ◽  
Parameter Space ◽  
Chaotic Dynamics ◽  
Chaotic Behavior ◽  
Computational Framework ◽  
Independent Variables ◽  
Systems Model ◽  
Small Support

Many biological ecosystems exhibit chaotic behavior, demonstrated either analytically using parameter choices in an associated dynamical systems model or empirically through analysis of experimental data. In this paper, we use existing software tools (COPASI, R) to explore dynamical systems and uncover regions with positive Lyapunov exponents where thus chaos exists. We evaluate the ability of the software’s optimization algorithms to find these positive values with several dynamical systems used to model biological populations. The algorithms have been able to identify parameter sets which lead to positive Lyapunov exponents, even when those exponents lie in regions with small support. For one of the examined systems, we observed that positive Lyapunov exponents were not uncovered when executing a search over the parameter space with small spacings between values of the independent variables.

Operation of Ride Control Systems to Minimise Slamming of Wave-Piercing Catamarans

2021 ◽  
Vol 163 (A1) ◽  
pp. 29-40
Author(s):  
M R Davis
Keyword(s):  
Control Systems ◽  
High Speed ◽  
Vertical Motion ◽  
Ship Motions ◽  
Wave Surface ◽  
Active Motion ◽  
Ride Control ◽  
Heave Motion ◽  
Control Feedback ◽  
Active Controls

Wave slam produces dynamic loads on the centre bow of wave piercing catamarans that are related to the relative vertical motion of the bow to the encountered wave surface. Rapid slam forces arise when the arch sections between centre bow and main hulls fill with rising water. In this paper time domain solutions for high speed ship motion in waves, including the action of active motion controls, are used to compute the slam forces. Slamming occurs at specific immersions of the bow whilst the peak slam force is characterised by the maximum relative vertical velocity of the bow during bow entry. Vertical motions of bow and encountered wave are in antiphase at encounter frequencies where slamming is most severe. The range of encounter frequencies where slamming occurs increases with wave height. Wave slam loads reduce ship motions, the heave motion being most reduced. Deployment of a fixed, inactive T-foil can reduce slamming loads by up to 65 %. With active controls peak slamming loads on the bow can be reduced by up to 73% and 79% in 4 m and 3 m seas, local control feedback being marginally the most effective mode of control for reduction of slamming.

Some Observations of Chaotic Vibration Phenomena in High-Speed Rotordynamics

1991 ◽  
Vol 113 (1) ◽  
pp. 50-57 ◽  
Author(s):  
F. F. Ehrich
Keyword(s):  
Numerical Model ◽  
High Speed ◽  
Chaotic Behavior ◽  
Narrow Region ◽  
System A ◽  
Vibratory Response ◽  
Local Contact ◽  
Jeffcott Rotor ◽  
Subharmonic Response ◽  
Static Part

Subharmonic response in rotordynamics may be encountered when a rotor is operated with its rotational centerline eccentric to that of a close clearance static part, so that local contact can take place during each orbit when the rotor is excited by residual unbalance. The rotor will tend to bounce at or near its fundamental frequency when the rotor is operated at or near a speed which is a whole number [n] times that frequency. Using a simple numerical model of a Jeffcott rotor mounted on a nonlinear spring, it is found that the vibratory response in the transition zone midway between adjacent zones of subharmonic response has all the characteristics of chaotic behavior. The transition from subharmonic to chaotic response has a complex substructure which involves a sequence of bifurcations of the orbit with variations in speed. This class of rotordynamic behavior was confirmed and illustrated by experimental observations of the vibratory response of a high-speed turbomachine, operating at a speed between 8 and 9 times its fundamental rotor frequency when in local contact across a clearance in the support system. A narrow region between zones of 8th order and 9th order subharmonic response was identified where the response had all the characteristics of the chaotic motion identified in the numerical model.

A case study in bifurcation theory

2017 ◽  
Vol 28 (08) ◽  
pp. 1750104 ◽  
Author(s):  
Youssef Khmou
Keyword(s):  
Dynamical Systems ◽  
Bifurcation Theory ◽  
Chaotic Behavior ◽  
Numerical Study ◽  
Logistic Map ◽  
Logistic Function ◽  
Second Order ◽  
Short Paper ◽  
Chaotic Region

This short paper is focused on the bifurcation theory found in map functions called evolution functions that are used in dynamical systems. The most well-known example of discrete iterative function is the logistic map that puts into evidence bifurcation and chaotic behavior of the topology of the logistic function. We propose a new iterative function based on Lorentizan function and its generalized versions, based on numerical study, it is found that the bifurcation of the Lorentzian function is of second-order where it is characterized by the absence of chaotic region.

scholarly journals Hyperfine Interactions in Organic Fragments

2021 ◽  
Author(s):  
◽  
John Patrick Macarthur Bailey
Keyword(s):  
High Speed ◽  
Organic Molecules ◽  
Electronic Computer ◽  
Applied Mathematics ◽  
Hyperfine Interactions ◽  
Industrial Research ◽  
Molecular Orbital Calculations ◽  
Master Of Science ◽  
Alternative Approaches ◽  
Division Department

<p>This thesis, the first thesis in theoretical chemistry submitted for the degree of Master of Science at Victoria University of Wellington, has been designed to illustrate two alternative approaches to theoretical studies. The first five chapters illustrate the modern use of operator methods; the last two are concerned mainly with molecular orbital calculations for large organic molecules, using a giant high speed electronic computer. I am deeply indebted to Mr Keith Morris, of the Applied Mathematics Division, Department of Scientific and Industrial Research, for his generous and highly competent help in writing computing programs, and operating computers, at all odd hours of the day and night, for the calculations in this thesis. I would also like to thank Dr R.M. Golding, for useful discussions, and the Director, Applied Mathematics Division, Department of Scientific and Industrial Research, for making computing facilities available.</p>

Development and Optimization of Mathematical Model of High Speed Planing Dynamics

2015 ◽  
Author(s):  
Prin Kanyoo ◽  
Dominic J. Taunton ◽  
James I. R. Blake
Keyword(s):  
Relative Velocity ◽  
High Speed ◽  
Lift Force ◽  
Hydrodynamic Pressure ◽  
Physical Nature ◽  
Ship Motions ◽  
Additional Contribution ◽  
Forward Speed ◽  
Planing Craft ◽  
Hydrostatic Force

The primary difference between a planing craft and a displacement ship is that the predominant force to support the conventional or displacement craft is hydrostatic force or buoyancy. While in the case of planing craft, the buoyancy cedes this role to hydrodynamic lift force caused by flow and pressure characteristics occurring when it is travelling at high forward speed. However, the magnitude of hydrostatic force is still significant that cannot be completely neglected. Due to the high forward speed and trim angle, the flow around and under the planing hull experiences change of momentum and leads to the appearance of lift force according to the 2ndlaw of Newton. In other words, there is a relative velocity between the craft hull and the wave orbital motion that causes hydrodynamic pressure generating hydrodynamic lift force act on the hull surface. Then, in case of behaviors in waves, an additional contribution of ship motions is necessary to be considered in the relative velocity, resulting in nonlinear characteristic of its physical nature.

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