Sunao Murashige

Nonlinear analyses of roll motion of a flooded ship in waves

This paper investigates nonlinear responses of a flooded ship in regular waves. In previous experimental work, we found that the roll motion of a flooded ship can exhibit complicated irregular behaviour even in waves of a moderate height. First, we analyse the fractal dimension and the Lyapunov exponents of the experimental data and show that they have chaotic characteristics. We also show that a radial basis function network obtained directly from the data can reproduce a geometrical structure of the reconstructed attractor and provide good short-term prediction on the dynamical motion. Next, in order to understand this nonlinear phenomenon, we derive a simple mathematical model for the nonlinearly coupled motion of roll and flooded water in regular waves. This model has a form of coupled Duffings equations with a bistable restoring term and a nonlinear inertial coefficient matrix. We obtain bifurcation diagrams of periodic solutions of this model and examine the intricate structure of this nonlinear system. Chaotic responses are found in wide regions of the parameter space, even if the wave-exciting moment is not large. Furthermore, the attractor structure of the chaotic solution is similar to that of the measured chaotic motion in the experiments. The results suggest that bifurcation analyses in this work help us understand the complex dynamics of nonlinear motion of a flooded ship in waves.

Coexistence of periodic roll motion and chaotic one in a forced flooded ship

This letter describes the coexistence of periodic and chaotic roll motion of a flooded ship in waves. We found experimentally, both with a flooded ferry model and with a simplified box-shaped model, that the two types of roll motion can coexist under the same wave condition. A trajectory reconstructed in a delay-coordinate state space from the time series data of the measured roll angle looks like a low-dimensional strange attractor. Moreover, a mathematical model for the simplified box-shaped ship shows the coexistence of a periodic solution and a chaotic one with a positive maximum Liapunov exponent.

Boundary element simulation of large amplitude standing waves in vessels

Abstract In this paper, a fluid in a vessel is considered and large amplitude standing waves (LASW) of the fluid are simulated directly using the boundary element method (BEM). In the simulation, two problems come out. The first problem is that the energy of the LASW increases gradually when using double nodes at corners in the BEM. The second problem is that projection-like profiles appear near the point where the free surface meets the vessel wall when regridding is not used at each time step. These projection-like profiles are not physical and indicate numerical error, and cause the simulation to break down. We found that the use of discontinuous elements solves the first problem, and the use of the ‘half shift technique’ solves the second problem. In addition, a method called RIG for highly accurate simulation using regridding is proposed and verified.

BIFURCATION STRUCTURES OF PERIOD-ADDING PHENOMENA IN AN OCEAN INTERNAL WAVE MODEL

In this paper, we study bifurcation structures of period-adding phenomena in an internal wave model that is a mathematical model for ocean internal waves. It has been suggested that chaotic solutions observed in the internal wave model may be related to the universal property of the energy spectra of ocean internal waves. In numerical bifurcation analyses of the internal wave model, we illustrate bifurcation routes to chaos and parameter regions where chaotic behavior is observed. Furthermore, it is found that the chaotic solutions are related to the period-adding sequence, that is, successive generations of periodic solutions with longer periods as a control parameter is changed. Considering the period-adding sequence as successive local bifurcations, we discuss a mechanism of the phenomena from the viewpoint of bifurcation analysis. We also consider similarity between period-adding phenomena in the internal wave model and ones in the Lorenz model.

Analysis of Flagellar Bending in Hamster Spermatozoa: Characterization of an Effective Stroke

Abstract The mechanism by which flagella generate the propulsive force for movement of hamster spermatozoa was analyzed quantitatively. Tracing points positioned 30, 60, 90, and 120 μm from the head-midpiece junction on the flagellum revealed that they all had zigzag trajectories. These points departed from and returned to the line that crossed the direction of progression. They moved along the concave side (but not the convex side) of the flagellar envelope that was drawn by tracing the trajectory of the entire flagellum. To clarify this asymmetry, the bending rate was analyzed by measuring the curvatures of points 30, 60, 90, and 120 μm from the head-midpiece junction. The bending rate was not constant through the cycle of flagellar bending. The rate was higher when bending was in the direction described by the curve of the hook-shaped head (defined as a principal bend [P-bend]) to the opposite side (R-bend). We measured a lower bending rate in the principal direction (R-bend to P-bend). To identify the point at which the propulsive force is generated efficiently within the cycle of flagellar bending, we calculated the propulsive force generated at each point on the flagellum. The value of the propulsive force was positive whenever the flagellum bent from an R-bend to a P-bend (when the bending rate was lowest). By contrast, the propulsive force value was zero or negative when the flagellum bent in the other direction (when the bending rate was higher). These results indicate that flagellar bending in hamster spermatozoa produces alternate effective and ineffective strokes during propulsion.

On necessary and sufficient conditions for numerical verification of double turning points

Summary.This paper describes numerical verification of a double turning point of a nonlinear system using an extended system. To verify the existence of a double turning point, we need to prove that one of the solutions of the extended system corresponds to the double turning point. For that, we propose an extended system with an additional condition. As an example, for a finite dimensional problem, we verify the existence and local uniqueness of a double turning point numerically using the extended system and a verification method based on the Banach fixed point theorem.

Numerical verification of solutions of periodic integral equations with a singular kernel

This paper proposes the method of numerical verification of solutions of a periodic integral equation with a logarithmic singular kernel, which is typically found in some two-dimensional potential problems. The verification method utilizes a property of the singular integral for trigonometric polynomials, the periodic Sobolev space and Schauders fixed point theorem.

AN IDEAL OCEAN WAVE FOCUSING LENS AND ITS SHAPE

Abstract A hydrodynamic singularity distribution for an ideal wave focusing lens is derived on the assumption of its slenderness and high frequency of incident waves. The result indicates that its sectional shape should satisfy necessary conditions which are equivalent to geometrical optics. Some experiments and computations show that an array of submerged circular cylinders almost satisfy the above necessary conditions for a wide band of wave frequencies. Furthermore, using the slender ship theory, it is shown that a convex circular cylinder type gives better wave focusing performance than a flat plate type, which has been used in previous work, particularly in irregular waves.

Dwarf solitary waves and low tsunamis

This work applies the regularized solitary wave theory to develop accurate computational method for evaluating the dwarf solitary waves, with amplitude-to-water depth ratio α ≤ 10−2, as a useful model of one-dimensional tsunamis propagating in the open ocean. The algebraic branch singularities of these solitary waves magnifying with diminishing wave amplitude, making their computations insurmountable by existing methods, are removed by the regularized coordinates given by this new theory. Numerical examples show that this new method can produce accurate results even for α ≅ 10−3 or less.

Boundary Conditions for Numerical Stability Analysis of Periodic Solutions of Ordinary Differential Equations

This paper considers numerical methods for stability analyses of periodic solutions of ordinary differential equations. Stability of a periodic solution can be determined by the corresponding monodromy matrix and its eigenvalues. Some commonly used numerical methods can produce inaccurate results of them in some cases, for example, near bifurcation points or when one of the eigenvalues is very large or very small. This work proposes a numerical method using a periodic boundary condition for vector fields, which preserves a critical property of the monodromy matrix. Numerical examples demonstrate effectiveness and a drawback of this method.