2. Benchmark solution#
2.1. Calculation method used for the reference solution#
To calculate the added coefficients:
we show [bib1] that the added mass and damping coefficients depend on the permanent potential of the fluid speeds \(\stackrel{ˉ}{\phi }\) as well as on two fluctuating potentials \({\phi }_{1}\) and \({\phi }_{2}\): these potentials can be written in the case of the rotational movement of the external cylinder around the pivot C [bib1]:
However, the added modal coefficients projected onto this mode of rotation can be written as:
either:
For the system with an equivalent degree of freedom:
It is a mass-spring-shock absorber system representing the rotational movement of the cylinder around the downstream pivot \(C\).
The inertia of the mechanical system subjected to the flow is written as:
where
is the inertia of the pivoting external cylinder with respect to the axis \(\mathrm{Cx}\) (see figure below) in air.
We show [bib2] that this inertia is equal to:
where
is the mass of the cylinder:
where
is the thickness of the cylinder,
its total length.
is the density of the cylinder.
thus
the damping of the mechanical system subjected to the flow is written as:
where
is the damping of the mechanical system in air. Usually,
is equal to a few% of the critical damping of the system:
.
where
is the inertia of the air cylinder calculated above and
spring stiffness at the pivot point
. We take the reduced depreciation
equal to 1%.
Thus, the total damping of the system under flow is written as:
the stiffness of the mechanical system subjected to flow is written as:
where
is the stiffness of the spring in air.
is the stiffness added by the flow. We show [bib1] that it is zero in this rotation mode.
Thus the total rigidity of the system is independent of the flow speed.
We then calculate the complex modes of this mechanical system under flow (damped free vibrations):
The complex natural frequencies of this system are written as [bib3]:
with
: reduced system damping
: natural pulsation.
Digital applications:
Three calculations of added damping were made corresponding to three flow velocities that cause three vibratory behaviors of the structure:
Speed to \(4m/s\)
Speed to \(4.24m/s\)
Speed to \(6m/s\)
The values of the mechanical system are:
The added weight and damping provided by the flow are equal to:
(independent of the flow speed value)
Depending on the fluid entry speed, we have:
The amortization of the fluid/structure system is written as:
to
The flow does not amplify the vibrations. The internal structural damping is significant enough to dissipate the energy provided by the flow to the structure. The system is still depreciated.
to
The depreciation of the system is cancelled out.
to
The damping of the system at this last speed is negative: the system then enters floating instability.
The corresponding reduced amortization is written as:
The natural pulsation remains unchanged:
.
2.2. Benchmark results#
Analytical result.
2.3. Bibliographical references#
ROUSSEAU G., LUU H.T.: Mass, damping and stiffness added for a vibrating structure placed in a potential flow - Bibliography and implementation in the*Code_Aster* - HP-61/95/064
BLEVINS R.D: Formulas for natural frequency and mode shape. Ed. Krieger 1984
SELIGMANN D, MICHEL R: Solving Algorithms for the Quadratic Problem [R5.01.02], Aster Reference Manual.