The transient motion of a floating body

Abstract

An analytical method of calculating the body motion was given in an earlier paper. Viscosity and surface tension were neglected, and the equations of motion were linearized. It was found that, for a half-immersed horizontal circular cylinder of radius a, the vertical motion at time τ(a/g)½ is described by the functions h1(τ) (for an initial velocity) and h2(τ) (for an initial displacement) where \begin{eqnarray*} h_1(\tau) &=& \frac{1}{2\pi}\int_{-{\infty}}^{\infty}\frac{e^{-iu\tau}du}{1-\frac{1}{4} \pi u^2(1+\Lambda(u))}\\ {\rm and}\qquad\qquad\qquad h_2(\tau) &=& -\frac{1}{8}i\int_{-\infty}^{\infty}\frac{u(1+\Lambda(u))e^{-iu\tau}du}{1-\frac{1}{4}\pi u^2(1+\Lambda (u))}. \end{eqnarray*} The function ∧(u) in these integrals is the force coefficient which describes the action of the fluid on the body in a forced periodic motion of angular frequency u(g/a)½. To determine ∧(u) for any one value of u an infinite system of linear equations must be solved.In the present paper a numerical study is made of the functions h1(τ) and h2(τ). The integrals defining h1(τ) and h2(τ) are not immediately suitable for numerical integration, for small τ because the integrands decrease slowly as u increases, for large τ because of the oscillatory factor e−iur. It is shown how these difficulties can be overcome by using the properties of ∧(u) in the complex u-plane. It is found that after an initial stage the motion of the body is closely approximated by a damped harmonic oscillatory motion, except during a final stage of decay when the motion is non-oscillatory and the amplitude is very small. It is noteworthy that the motion of the body can be found accurately, although little can be said about the wave motion in the fluid.

Simulation of floating structure dynamics in waves by implicit coupling of a fully non-linear potential flow model and a rigid body motion approach

“…For freely floating bodies, the convergence of the algorithm is studied for various amplitudes of motion in free decay tests. We find that the implicit method of van Daalen (1993) and Tanizawa (2000) gives a satisfactory agreement with the analytical results of Maskell and Ursell (1970). A larger discrepancy, however, is found for large amplitude free decay motions, which indicates the need for developing a way to include viscous effects in the present FNPF model.…”

“…In the linear limit, Maskell and Ursell (1970) derived a semianalytical solution of the vertical displacement of the center of mass, x G3 (t) for: (i) a zero initial velocity condition…”

“…11, we compare the computed time evolution of the cylinder elevation, for x G3 (0) = R/8, using the direct method or the implicit one, to the semi-analytical method of Maskell and Ursell (1970);and (ii) to the experimental values of Ito (1977). The experiments of Ito were performed with various initial displacements: 1 in, 1.5 in and 2 in and for a diameter of the cylinder equal to 2R = 6 in (in the foot-Implicit, Taylor 2nd order Implicit, symplectic-like 1st order Ito R/3 Navier Stokes, Ghasemi et al (2014) pound system).…”

“…In Sect. 5, we study the model convergence properties for free decay tests of a floating circular cylinder, for which a semi-analytical solution due to Maskell and Ursell (1970) is available. Many numerical and physical parameters will be varied, to assess the sensitivity of the model solution.…”

Natural Frequency of 2-Dimensional Heaving Circular Cylinder: Time-Domain Analysis

“…Using our numerical and experimental results, the natural frequency was defined as its average value given by the motion data excluding those of the initial stage. Our results were then compared with those of the existing investigations such as Maskell and Ursell (1970), Ito (1977) and Yeung (1982) as well as the newly obtained results of Lee (2012). Comparison showed that the natural frequency obtained here agrees well with that of Lee (2012), which was found through the frequency domain analysis.…”