On the manifestation of coexisting nontrivial equilibria leading to potential well escapes in an inhomogeneous floating body
Introduction
A classic engineering rule of thumb states that a floating object can be made stable in the upright position if its center of mass is located below the centroid of the displaced fluid, or the center of buoyancy. This fact can be proven by showing that the center of buoyancy shifts to the right for clockwise rotation about the center of mass and to the left for counter-clockwise rotation which results in a restoring torque for both cases. More sophisticated analysis has been conducted in the field of Naval Architecture to show that this upright position can remain stable even if the center of gravity is located above the center of buoyancy by introducing the concept of a metacenter [1]. The metacenter is defined as the point of intersection between a vertical line drawn through the center of buoyancy when the body is tilted and a vertical line drawn through the original center of buoyancy at equilibrium. It has been shown that a floating object will remain stable even if the center of mass is located above the center of buoyancy as long as the metacenter is located above the center of mass [2]. The analysis to prove this linearizes the system about its equilibrium point to determine whether small perturbations will result in an instability commonly called capsizing. This criterion has been well documented and is extremely useful for Naval Engineers in designing hull shapes, prescribing maximum payloads, and constructing ballasts [3].
While this work can be applied to inform design decisions, scientists and engineers have examined the stability of floating bodies on a more fundamental level in attempt to gain deeper phenomenological insights. These studies have looked to investigate the equilibrium and stability behavior of various symmetric floating objects and have been particularly interested in the emergence of non-trivial tilted equilibrium positions [[4], [5], [6]]. Gilbert investigated the equilibria of homogeneous cylinders and tetrahedrons for various specific density and geometric ratios to illustrate the emergence of tilted equilibrium positions [7]. Rorres expanded on this work by conducting similar studies on the more complicated paraboloid [8]. Erdos et al. then progressed the field even further by showing how the same non-trivial equilibria states could be realized for inhomogeneous shapes by simply shifting the location of the object’s center of mass [9].
These studies offer useful insights into the behavior of floating bodies, however little investigation has been done to examine scenarios in which multiple non-trivial equilibrium solutions coexist simultaneously. Intuition suggests that tilted equilibrium solutions should be possible for scenarios in which the center of mass is located off the prism’s centerline, however, it is not intuitive whether this is possible for a symmetric mass distribution and, if so, which of these solutions would be stable. Furthermore, if situations can be identified where coexisting stable equilibrium positions exist, would it be possible to transition back and forth between stable orientations through dynamic wave excitation? If such conditions could be determined, this behavior could be manipulated for a host of hydrodynamic engineering applications [10]. This paper seeks to answer these questions by examining the bifurcation characteristics of a floating rectangular prism (hereafter referred to as a buoy) to identify and study coexisting tilted equilibrium positions in floating bodies. In particular, this paper develops a nonlinear mathematical model to determine the static stability of the upright and tilted equilibria positions as a function of the vertical position of the center of mass within the buoy, validates the numerical results through experimental studies, and then applies this knowledge to examine the buoy’s behavior under dynamic wave excitation.
The remainder of this paper is organized as follows. Section 2 defines the geometry and fundamental variables used to describe the problem. In doing so, six distinct regions that correspond to scenarios where different combinations of corners are submerged below the waterline are identified [6]. Section 3 uses this geometry to derive the governing equations of motion and explain the key parameters involved. Section 4 then describes how these equations are used to determine equilibrium positions and stability through numerical methods. Results are presented for four particular cases with qualitatively different bifurcation behaviors. Finally, Section 5 describes experimental tests that successfully validate numerical equilibrium results for the static case and examine responses where the buoy oscillations hop back and forth between potential wells under dynamic wave conditions.
Section snippets
Important geometry
Fig. 1 shows a schematic of an upright and tilted rectangular prism along with its important geometry. As illustrated, five fundamental geometric parameters are used to describe the prism: its height , width , length , mass , and vertical distance from the base to its center of mass . Analyzing what happens when is varied for a prism with a fixed set of parameters , , , and is this paper’s primary focus. From these parameters, the submerged depth , can be calculated as where
Equations of motion
Equations of motions for this system are described by restricting the problem to planar motion: rotation about the axis coming out of the page, translation in the vertical direction, and translation in the horizontal direction. Under these constraints, the following three general equations of motion are obtained:
where is the gravitational force, is defined as the system’s moment of inertia, and
Equilibrium and stability
This section describes how the equations of motion derived above are used to generate equilibrium solutions and determine their stability. It outlines the computational methods used to step through these solutions for varying values of the parameter and discusses some of the results that are obtained. Specifically, it focuses on four Cases to highlight their qualitatively different bifurcation diagrams and basins of attraction.
Experimental studies
This section describes experimental testing that was conducted for situations in which multiple stable equilibrium orientations were found to coexist simultaneously. Static experiments were undertaken in still water to verify the numerical results obtained above and then dynamic experiments were conducted in a waveflume to examine responses where the buoy hopped back and forth between coexisting stable orientations.
Conclusions
The intent of this paper was to examine coexisting equilibrium positions for a floating rectangular prism and to investigate how this phenomenon might manifest itself in nature. In doing so, a nonlinear model was developed to determine equilibrium positions and evaluate their stability as a function of the vertical position of the center of mass within the prism, . The resulting bifurcation diagrams and basins of attraction reveal a critical value where the upright equilibrium becomes
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