Reference solution
=====================


Calculation method used for the reference solution
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The following coupled problem is solved analytically:

:math:`\{\begin{array}{}m\ddot{y}+ky=F\\ \frac{{\partial }^{2}p}{\partial {y}^{2}}=0\\ (\frac{\partial p}{\partial y})={\rho }_{f}\ddot{y}\end{array}`

with

.. csv-table::

    ":math:`F` hydrodynamic pressure force on the piston"
    ":math:`P` hydrodynamic pressure in the fluid"
    ":math:`m,k`: piston mass and stiffness per radian"


The hydrodynamic pressure field in the fluid is written as:

:math:`p=-{\rho }_{f}\ddot{y}(y-l)`

Hence the pressure force exerted on the piston:

:math:`F\mathrm{=}{\mathrm{\int }}_{0}^{e}pnr\mathit{dr}\mathrm{=}\mathrm{-}{\rho }_{f}\ddot{y}l\frac{{e}^{2}}{2}`

The mass added by radian is: :math:`{m}_{a}={\rho }_{f}l\frac{{e}^{2}}{2}`

the eigenmode of the coupled system is equal to: :math:`f=\frac{1}{2\pi }\sqrt{\frac{k}{m+{m}_{a}}}=\mathrm{27,25}\mathrm{Hz}`


Benchmark results
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Analytics.


Bibliographical references
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1. GIBERT R.J.: Vibrations of structures, Eyrolles (1988).