Stochastic ship roll motion via path integral method

The stochastic chaotic motion of ship rolling is studied considering the static effect of water on deck. The wave excitation is treated as periodic excitation perturbed by white noise and the parameter region for chaotic motion is determined by Melnikov mean-square criterion. The Poincare’ maps are constructed and the probability density functions are calculated using path integral method. It is found that the chaotic motion can be restrained by increasing ship damping and ship rolling response converges to two attraction regions. In the non-chaotic parameter region, ship rolls within one attraction region. In the chaotic parameter region the probability density function has two connected peaks and ship rolling jumps randomly from one attraction region to another.

An intensity-modulated Gaussian process {x(t)} is defined as the product of a non-negative stationary random process {σx(t)} and a stationary Gaussian process {z(t)} that is statistically independent of {σx(t)}. When such an intensity-modulated process is the excitation to a linear time-invariant system, the response process {y(t)}, conditioned on the excitation-modulating function σx(t), is a generally nonstationary Gaussian process. This property is used to develop an approximation to the probability density function (PDF) of the system response by assuming that the time-varying conditional response variance σ2y(t), conditioned on the excitation-modulating function σx(t), is governed by the gamma PDF. The resulting response process PDF is characterized by two parameters: the mean and variance of the conditional response variance σ2y(t). An alternative approximation to the response PDF, characterized by these same two parameters, is developed by using a Taylor’s series representation in the variable σ2y of the response PDF conditioned on σ2y. This latter PDF approximation is shown to be identical to the Gram–Charlier series of type A through terms involving the Hermite polynomial of degree 6. The two response PDF approximations are shown to give essentially identical results for values of the coefficient of variation of σ2y of one-half or less. Simple, easily evaluated expressions for the mean and variance of σ2y are derived in terms of measurable quantities by utilizing the assumption that fluctuations in σx(t) occur slowly in comparison with those of z(t).

In this paper the nonstationary response of a class of nonlinear systems subject to broad-band stochastic excitations is examined. A version of the Path Integral (PI) approach is developed for determining the evolution of the response probability density function (PDF). Specifically, the PI approach, utilized for evaluating the response PDF in short time steps based on the Chapman–Kolmogorov equation, is here employed in conjunction with the Laplace’s method of integration. In this manner, an approximate analytical solution of the integral involved in this equation is obtained, thus circumventing the repetitive integrations generally required in the conventional numerical implementation of the procedure. Further, the method is extended to nonlinear oscillators, approximately modeling the amplitude of the system response as a one-dimensional Markovian process. Various nonlinear systems are considered in the numerical applications, including Duffing and Van der Pol oscillators. Appropriate comparisons with Monte Carlo simulation data are presented, demonstrating the efficiency and accuracy of the proposed approach.

Response probability structure of a structurally nonlinear fluttering airfoil in turbulent flow

We study the stability of small ships with water on deck in random beam waves from a probability perspective. Previous research has shown that this kind of ship motion is governed by two dynamical regions: Homoclinic and heteroclinic, where the heteroclinic model emulates symmetric vessel capsize and the homoclinic model represents a vessel with an initial bias caused by water on deck. We investigate the stability and capsizing of ships in the homoclinic region using the probability method and nonlinear stochastic dynamics theory. Simplifying the random wave excitation to a periodic force and white noise perturbation, the random Melnikov mean square criterion is used to determine the parameter domain for the ship's stochastic chaotic motion. The probability density function of the roll response is calculated by solving the stochastic differential equations using the path integral method. A mathematical example is presented. It is found that in the chaotic parameter region, the probability density function of the system has two peaks. The response of the system will jump from one peak to another for large amplitudes of periodic excitation. This will lead to instability and even capsizing.

The joint probability density function (PDF) of different structural responses is a very important topic in the stochastic response analysis of nonlinear structures. In this paper, the probability density evolution method, which is successfully developed to capture the instantaneous PDF of an arbitrary single response of interest, is extended to evaluate the joint PDF of any two responses. A two-dimensional partial differential equation in terms of the joint PDF is established. The strategy of selecting representative points via the number theoretical method and sieved by a hyper-ellipsoid is outlined. A two-dimensional difference scheme is developed. The free vibration of an SDOF system is examined to verify the proposed method, and a frame structure exhibiting hysteresis subjected to stochastic ground motion is investigated. It is pointed out that the correlation of different responses results from the fact that randomness of different responses comes from the same set of basic random parameters involved. In other words, the essence of the probabilistic correlation is a physical correlation.

Stochastic bifurcation and chaos study for nonlinear ship rolling motion with random excitation and delayed feedback controls

A generalized method for the stationary probabilistic response of nonlinear dynamical system

Path integrals offer a robust approach for simulating open quantum dynamics with advancements transcending initial system size limitations. However, accurately modeling systems governed by mechanisms that do not conserve the number of quantum particles, such as lossy cavity modes, remains a challenge. We present a method to incorporate such empirical source and drain mechanisms within a path integral framework using quantum master equations. This technique facilitates rigorous inclusion of bath degrees of freedom while accommodating empirical time scales via Lindbladian dynamics. Computational costs are primarily driven by the path integral method with minimal overhead from Lindbladian terms. We use it to study exciton transport in a four-site Fenna-Matthews-Olson model, examining the potential loss of the exciton to the reaction center. This path integral Lindblad method promises an enhanced ability to simulate dynamics and will be fundamental to simulation of spectra in diverse quantum processes in open systems.

An approximate analytical method is developed for calculating the probability density of the response of a Duffing oscillator subjected to narrow band random excitation when the stochastic jump phenomenon occurs. The stochastic jump phenomenon is characterized by the existence of multiple states of amplitude and sudden jumps between the states in the sample function of response. In this paper, the trigger for the stochastic jump is considered, and the probabilities of large and small amplitude states of the response are evaluated. The local probability density function of each state is separately assumed using the equivalent linearization method, and the probability density function of the overall response is derived by sum of these local probability density functions weighted with the probability of each state. In the illustrative example, the calculated probability density functions are compared with the Monte Carlo simulation results. It is shown that the present method well expresses the significant characteristic of the stochastic jump phenomenon.

New statistical theory was developed by the authors in the previous paper, which gives probability density function (p. d. f.) and extreme values of total second order responses of moored vessels in random seas. It assumes the response represented in the form of two term Volterra functional series and is based on the generalized Laguerre polynomial expansion of p. d. f. of which the first term is a gamma p. d. f. consisting of three parameters.This paper shows non-normality of surge responses of moored tanker and semi-submersible measured in a wave basin, and determines that the theory estimates it very well. Furthermore effect of time varying wave drift damping and other slow motion hydrodynamic force coefficients on p. d. f. and extreme values are discussed by time domain simulations. This effect cannot be represented in the form of two term Volterra functional series.

Parametric roll is a phenomenon in which there is a large rolling motion of a ship even when the ship is moving into head seas with no direct excitation. It is a nonlinear dynamic phenomenon of a ship rolling system with nonlinearities in the stiffness as well as the damping terms. Parametric roll of container ships in head seas is a relatively new problem, which has gained lot of importance after the catastrophic incidence of APL China in 1998. Analysis of parametric roll of container ships in regular head waves has been studied extensively. However, the ships do not encounter regular waves in the ocean. So, it is necessary to study how important parametric roll is in irregular seas. To study this, it is first important to model the variation of metacentric height in irregular waves, which is nonlinear as a result of the influence of underwater geometry and the motions of the ship in a seaway. In this work, the change of metacentric height (GM) in irregular waves has been modeled using a Volterra series approach. This transfer function for metacentric height (GM) is used to study parametric rolling of ships in irregular waves. Based on this study, roll motion sensitivity to the spectral peak period and significant wave height has been carried out.

The PDFs (probability density functions) and probability of a ship rolling under the random parametric and forced excitations were studied by a semi-analytical method. The rolling motion equation of the ship in random oblique waves was established. The righting arm obtained by the numerical simulation was approximately fitted by an analytical function. The irregular waves were decomposed into two Gauss stationary random processes, and the CARMA (2, 1) model was used to fit the spectral density function of parametric and forced excitations. The stochastic energy envelope averaging method was used to solve the PDFs and the probability. The validity of the semi-analytical method was verified by the Monte Carlo method. The C11 ship was taken as an example, and the influences of the system parameters on the PDFs and probability were analyzed. The results show that the probability of ship rolling is affected by the characteristic wave height, wave length, and the heading angle. In order to provide proper advice for the ship's manoeuvring, the parametric excitations should be considered appropriately when the ship navigates in the oblique seas.

The performance of offshore work from floating platforms dictates the desirability of "Minimum platforms dictates the desirability of "Minimum motions". One of the most troublesome motions, especially in the case of ship-shape or barge-shape platforms, is the roll motion because of the large platforms, is the roll motion because of the large amplitude and relatively large acceleration forces that could arise from such motion. Different devices have been employed to minimize The roll motion of ships bilge keels, gyroscopic stabilizers, solid moss transfer, stabilizing fins, U-tanks, flume tanks, active ballast transfer tanks, etc. The systems utilizing fins are effective only when the ship is under way and are not suitable for zero-speed offshore operations. The barge keels are generally very effective in damping the roll motions, and they prevent the angle of roll from becoming too large, but their reduction of the roll is limited to angles that are too great for the satisfactory performance of offshore operations, i.e., oil drilling. performance of offshore operations, i.e., oil drilling. The gyroscopic stabilizers generally are not used because of their high cost and complexity. The U-tanks, flume tanks, and active stabilizing tanks work by transferring ballast horizontally and by creating a stabilizing moment out of phase with the exciting force. This paper details use of the detuning tank. This system is different from others in the sense that it does not try to compensate the action of the forces imparted by the sea to the floating body with properly phased compensating forces, but tries to properly phased compensating forces, but tries to prevent the sea from imparting the forces to the prevent the sea from imparting the forces to the body. The effectiveness of the detuning tanks bas been experimentally verified in model basin motion tests and studies. Introduction Safety at sea is the first consideration of marine designers. Ships of conventional form or floating platforms for offshore operations must, under all platforms for offshore operations must, under all expected circumstances, float and be stable-hence, the application of suitable criteria for stability is one of the cornerstones of naval architectural design. The ability to maintain stability under design weather conditions, even after sustaining a certain amount of damage, has dictated maximum allowable heights of the center of gravity (KG) or minimum metacentric heights (GM). This limitation is certainly necessary in order to meet the demands of safety, but it is necessary only during certain extreme weather conditions or after flooding caused by damage. But this occurs, if it ever does, only during a very small percentage of the life of the ship, and to be prepared for this eventuality we may be forced to select some "safe parameter", like minimum GM, that may affect the parameter", like minimum GM, that may affect the motion performance of the ship during the major part of her useful working life. part of her useful working life. The motion performance may be of certain importance for ships engaged in ocean trade from the standpoint of crew comfort and the ability to maintain sea speed both of which have a certain economic value. But ships and platforms that are engaged An offshore work are more vitally affected by the motion performance, and their very effectiveness as offshore tools depends on how many days of the year they can perform their functions and the weather conditions that would force operations to be suspended. Roll motions have been one of the reasons for discontinuing offshore operations for ship-shape and barge-shape platforms not only because the amplitude of the motions but because of the high acceleration forces in the work area that can be originated by relative small amplitudes at short motion periods. The inclination due to wave action appears to be composed of two periodic functions, the period of one being the wave period T and the other period of one being the wave period T and the other the natural rolling period of the ship T . Rolling in still water is a free oscillation and the ship will roll in its own natural period. Among waves, the impulses producing the roll are periodic and tend to set up a forced oscillation of the ship in the period of the wave. If waves of constant period act for a sufficient time upon the ship, it will roll in the period of the waves, but if the period of the waves period of the waves, but if the period of the waves is not constant, the ship roll will not follow exactly that of the waves because of the tendency of the ship to revert to roll in its own natural period.

In this paper, the probability density function (PDF) and the mean up-crossing rate of nonlinear ship rolling in random beam seas are investigated. The excitation of stationary random sea waves is approximated as a second-order linear filtered white noise. The Fokker–Planck–Kolmogorov (FPK) equation governing the probability density function of ship rolling is a four-dimensional linear partial differential equation with varying coefficients, and obtaining its exact solution is much more sophisticated. The exponential-polynomial closure (EPC) method is applied to solve the corresponding FPK equation of the system. In numerical examples, linear-plus-cubic damping model and linear-plus-quadratic damping model with three different sea states are further examined. Comparison with the equivalent linearisation (EQL) method and Monte Carlo simulated results show that the proposed procedure is effective to obtain a satisfactory probability density function solution, especially in the tail region.