1. Reference problem#

1.1. Geometry and boundary conditions#

The figure below represents the reference problem: a hollow cylindrical beam with a length \(L=1m\), an outer radius \(R=0.5m\) and a thickness \(\mathit{ep}=0.01m\). It is immersed \(2.1m\) below the free surface in a water column of height \(h=30m\). The ends of the beam are supported by two identical springs, which are themselves embedded at their ends.

We note \({S}_{w}=\mathrm{\pi }{R}^{2}\) the hydraulic section (seen by the fluid) transverse to the beam and \(S=\mathrm{\pi }(2.R.\mathit{ep}-{\mathit{ep}}^{2})\) the real cross section.

1.2. Material properties#

The material has an isotropic elastic behavior:

Young’s module	\(200\mathit{GPa}\)
Poisson’s Ratio	\(\mathrm{0,3}\)
Density beam \(\mathrm{\rho }\)	\(7800\mathit{kg}/{m}^{3}\)
Water density \({\mathrm{\rho }}_{w}\)	\(1000\mathit{kg}/{m}^{3}\)
Spring stiffness according to z	\(50.0N/m\)
Drag coefficient \({C}_{d}\)	\(1.0\)
Added mass coefficient \({C}_{a}\)	\(0.9\)

1.3. Wave kinematics#

We consider a regular monodirectional swell directed along the +y axis, of order 1 (Airy swell), of height H, period T and wavelength \(\mathrm{\lambda }\), see diagram below.

The free area \(\mathrm{\eta }\) as a function of space \(y\) and time \(t\) is expressed:

(1.1)#\[ \ mathrm {\ eta} (y, t) =\ frac {H} {2}\ mathrm {cos} (ky-\ mathrm {\ omega} t)\]

with \(k=\frac{2\mathrm{\pi }}{\mathrm{\lambda }}\) the wave number and \(\mathrm{\omega }=\frac{2\mathrm{\pi }}{T}\) the pulsation.

The speed of the fluid \({V}_{w}=(u,v,w)\) is as follows:

(1.2)#\[\begin{split} \ begin {array} {c} u (y, z, t) =0\\ v (y, z, t) =\ frac {H} {2}\ mathrm {\ omega}\ frac {\ mathrm {sinh} (k, z, t) =0\\ v (y, z, t) =\ frac {array} {c}}\ mathrm {sinh} (k (h+z))} {\ mathrm {sinh} (k (h+z))} {\ mathrm {sinh} (k.y))}\ mathrm {sin} (k.y-\ mathrm {\ omega} t)\\ w (y, z, t) =\ frac {H} {2}\ mathrm {\ omega}\ frac {\ mathrm {sinh} (k (h+z))}} {\ mathrm {sinh})} {\ mathrm {sinh} (kh)}\ mathrm {sinh} (kh)}\ mathrm {sinh} (kh)}\ mathrm {sinh} (kh)} end {array}\end{split}\]

Fluid acceleration is noted \({a}_{w}=\frac{\partial {v}_{w}}{\partial t}\).

1.4. Loads#

The initial equilibrium position of the problem consists, starting from the reference position described in Géométrie, in applying a translational displacement of the \(1m\) beam in the +z direction. This imposed displacement is then relaxed leading the beam to adopt a dynamic oscillatory movement.

Note that Archimedes” weight and thrust are not taken into account in this problem.

Modeling A

The oscillating beam is subjected to Morison forces (drag term only) in a fluid at rest.

(1.3)#\[ {F} _ {\ text {mor}} =\ frac {1} {2} {2} {2} {C} _ {d} {\ mathrm {\ rho}}} _ {w} _ {w} ({v} _ {w}} ^ {v} _ {w} ^ {v}} ^ {\ text {ortho}}}) ({v} _ {w}} ^ {v} _ {w} ^ {w} ^ {\ text {ortho}} - {v} _ {s} ^ {\ text {ortho}})\]

where \({v}_{w}\) and \({v}_{s}\) respectively designate the fluid speed (zero here) and the solid speed (mechanical load input parameter FORCE_POUTRE). The symbol \(\text{ortho}\) represents the projection in the direction normal to the neutral fiber of the beam [1] _ . It should be noted that these are linear forces that are applied in a distributed manner on the beam.

B Modeling

The oscillating beam is subjected to Morison forces (drag term and inertia terms) in a fluid at rest

\[\]

: label: eq-4

{F} _ {text {mor}} =frac {1} {2} {2} {2} {C} _ {d} {mathrm {rho}}} _ {w} _ {w} ({v} _ {w}} ^ {v} _ {w} ^ {v}} ^ {text {ortho}}}) ({v} _ {w}} ^ {v} _ {w} ^ {w} ^ {text {ortho}} - {v} _ {s}} ^ {text {ortho}}}) + {mathrm {rho}} _ {w} {a} _ {a} _ {w} _ {w} _ {w}} ^ {w} ^ {w} ^ {w} ^ {w} ^ {w} ^ {w} ^ {w} ^ {w} ^ {w} ^ {w} ^ {w} ^ {text {ortho}} ^ {text {ortho}} ^ {text {ortho}} ^ {text {ortho}}} ^ {text {ortho}} ^ {text {ortho}} ^ {text {ortho}}} ^ {text {ortho}} ^ {text {ortho}} ({a} _ {w} ^ {text {ortho}}} - {a} _ {s} ^ {text {ortho}})

The first term (corresponding to the drag) is identical to the A model, the second and the third correspond to the inertia terms. Since the acceleration of the fluid is zero, these are reduced to the term of solid added mass \({-C}_{a}{\rho }_{w}{S}_{w}{a}_{s}^{\text{ortho}}\). This is taken into account in the mass matrix of the system by modifying the density of the beam:

\[\]

: label: eq-5

{rho} ^ {text {*}}} =rho + {C}} _ {a} {rho} _ {w}frac {{S} _ {w} _ {w}} {w}} {S}

Note: note that this change in density must be accompanied by an appropriate change in gravity in order to correctly account for the weight of the beam. However, we remind you that we are not taking this force into account in our problem.

C Modeling

The oscillating beam is subjected to Morison forces (see modeling B) in a fluid subjected to a swell of height \(H=3m\) and period \(T=12s\). Velocity and smooth acceleration are defined in Cinématique de houle.

The term solid added mass is taken into account in the mass matrix in the same way as for modeling B by modifying the density of the beam.

General note: Morison’s efforts depend on the position of the beam, they should be mentioned as follower loadings in the calculations.