2. Reference solution#

2.1. Calculation method used for the reference solution#

The following coupled problem is solved analytically:

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

with

\(F\) hydrodynamic pressure force on the piston

\(P\) hydrodynamic pressure in the fluid

\(m,k\): piston mass and stiffness per radian

The hydrodynamic pressure field in the fluid is written as:

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

Hence the pressure force exerted on the piston:

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

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

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

Analytics.