2. Benchmark solution#
2.1. Calculation method used for the reference solution#
Analytical calculation:
When the structure vibrates in the fluid, it changes the pressure field which obeys a Laplace equation with Von Neuman boundary conditions [R4.07.03].
In our case, taking into account the symmetries of the problem, the pressure field only depends on the variable \(x\) and verifies:
\(\mathrm{\{}\begin{array}{cc}\frac{{\mathrm{\partial }}^{2}p}{\mathrm{\partial }{x}^{2}}\mathrm{=}0& {(\frac{\mathrm{\partial }p}{\mathrm{\partial }x})}_{x\mathrm{=}0}\mathrm{=}\mathrm{-}{\rho }_{f}{\ddot{x}}_{S}\mathrm{.}n\\ p\mathrm{=}0\text{en}x\mathrm{=}L& \end{array}\)
It can thus be seen that the pressure field is an affine function of the abscissa \(x\). The two pressure boundary conditions involve: \(p\mathrm{=}\mathrm{-}{\rho }_{f}{\ddot{x}}_{S}\mathrm{.}n(x\mathrm{-}L)\)
The force of pressure that is exerted on the structure is written as:
\(F\mathrm{=}\underset{\Gamma }{\mathrm{\int }}p(0)nd\Gamma \mathrm{=}\underset{\Gamma }{\mathrm{\int }}{\rho }_{f}L({\ddot{x}}_{S}\mathrm{.}n)nd\Gamma\)
As the problem is one-dimensional, this force can be expressed algebraically according to the following acceleration component \(\mathrm{Ox}\) of the structure:
\(F=-\ddot{x}\underset{\Gamma }{\int }{\rho }_{f}Ld\Gamma =-{\rho }_{f}Ld\ddot{x}=-{m}_{a}\ddot{x}\) with \({m}_{a}={\rho }_{f}Ld\)
It is the linear added mass of the fluid on the structure: we note that it corresponds to the mass of fluid in the column, that is to say to the mass of fluid moved by the piston.
The equation of movement of the piston projected on \(\mathrm{Ox}\) is written (free vibration not damped considering the presence of the fluid):
\(m\ddot{x}+Kx\mathrm{=}F\mathrm{=}\mathrm{-}{m}_{a}\ddot{x}\mathrm{\iff }(m+{m}_{a})\ddot{x}+Kx\mathrm{=}0\)
The natural frequency of this submerged system is therefore written as:
\(f\mathrm{=}\frac{1}{2\pi }\sqrt{\frac{K}{m+{m}_{a}}}\)
The effect of the fluid is therefore to lower the natural frequency of the air system.
Practically, in Aster, the added mass matrix is determined on the modal basis of the structure in a vacuum: To calculate the added mass given above, we limit ourselves to calculating the natural mode of the piston-spring system, which corresponds to a translational movement standardized to the unit: we therefore truncate the modal base of the structure to a single mode in air (operator CALC_MODES option “PLUS_PETITE”). Thanks to this mode, the mass added to the piston is determined.
\(K={10}^{5}N/m\) \({m}_{a}=200\mathrm{kg}/m\) \(m=78\mathrm{kg}/m\)
The natural frequency of the submerged piston-spring system is therefore \(f\mathrm{=}3.018\mathit{Hz}\)
2.2. Benchmark results#
Analytics
2.3. Bibliographical references#
J GIBERT - Vibrations of Structures - Interactions with fluids. Eyrolles (1988).