Experimental Assessments of the Added Mass of Flexible Cylinders in Water: The Role of Modal Shape Representation

Abstract

A flexible vertical cylinder model, fixed at both ends, is tested experimentally immersed in water and then in air. Galerkin’s decomposition is applied to obtain a Reduced Order Model (ROM) from a continuum one. Two closed-form trial modal shapes are chosen for the modal decomposition process. Then, modal added mass is assessed using classical Fourier and Hilbert transform (HT) signal analyses, comparing the model eigenvalues with the frequency evaluated from the experimental signals. The choice of modal shape is shown to alter significantly added mass experimental assessment. Similarity to classic results with rigid cylinders is achieved by taking a sufficiently proper modal representation. Moreover, the first mode added mass coefficient attains the same value of that previously determined for a cantilevered flexible circular cylinder, by Pesce and Fujarra in (Pesce and Fujarra, Int J Offshore Polar Eng 10:26–33, 2000) [1].

Appendices

Appendix 1—The HT Methodology as a Tool to Assess Optical Tracking Accuracy

During the experimental set up in air and water, the calibration of the optical tracking system revealed a measurement accuracy about a decimal of millimetre $(0.1\,{\text{mm}})$. On the other hand, by using the HT procedure to evaluate the instant damped frequency as function of vibration amplitude, resolution is clearly obtained, revealing a figure better than $0.005 \times 22.2\,{\text{mm}}\, \approx \,0.1\,{\text{mm}}$, as shows Fig. 14, for the decay test in air.

Fig. 14

Instant damped frequency from free-decay in air showing a clear bound for the optical tracking system accuracy

Appendix 2—Nomenclature

Latin Symbols

a: added mass coefficient, $a = m_{a} /m_{s}$	A: modal amplitude	$C_{a}$: added mass coefficient, $C_{a} = m_{a} /m_{d}$
$\widehat{C}_{a}$: ‘potential flow’ added mass coefficient, $\widehat{C}_{a} \, \approx \,1$	$C_{D}$: drag force coefficient	$C^{h}$: modal drag force coefficient
$C_{m}$: inertial coefficient, $C_{m} = 1 + C_{a}$	$C^{s}$: modal linear viscous damping coefficient	$c_{s}$: linear viscous damping coefficient
D: diameter	EA: axial stiffness	EI: bending stiffness
$f_{d}$: damped natural frequency (measured)	$\hat{f}_{n}$: natural frequency (calculated)	k: mode number
K, KC: Keulegan-Carpenter number	L: stretched length	$L_{0}$: unstretched length
$L_{i}$: immersed length	$L_{t}$: total length	M: modal mass
$m_{a}$: added mass per unit length	$m_{d}$: displaced mass per unit length	$m_{s}$: structural mass per unit length
$M_{a}$: modal added mass	$M_{d}$: modal displaced mass	$M_{s}$: modal structural mass
$m^{{ \star }}$: reduced mass parameter	$m_{1}^{{ \star }}$: first mode reduced mass	$N_{b}$: equivalent normal traction
$N_{b(0)}$: traction at the cylinder bottom	Re: Reynolds number	t: time
$T(z,t)$: tension	$\varvec{u}(z,t)$: displacement vector	U: mean velocity
x: cartesian coordinate	y: cartesian coordinate	z: cartesian coordinate

Greek Symbols

$\underline{\alpha }$: quasi-Bessel mode parameter	$\beta$: Sarpkaya’s $\beta$-parameter	$\underline{\beta }$: quasi-Bessel mode wave number
$\gamma$: linear weight	$\gamma_{i}$: immersed linear weight	$\eta$: modal rigidity
$\zeta$: linear viscous damping coefficient	$\nu$: kinematic viscosity	$\xi$: dimensionless modal amplitude
$\rho_{w}$: water specific mass	$\psi$: modal shape	$\omega$: angular frequency