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Benchmark UP6

Longitudinal Vibrations of a Mass on a Rubber Spring

Description

Let us consider a cantilever of length $l$ and width and depth $z_0$, fully fixed at $x=0$. The cantilever is made of the Neo-Hookean material of density $\rho$ (considered zero). At position $x=l$ an additional mass $m$ is added, equally distributed to all four nodes located at $x=l$. The cantilever, initally at rest, is subjected to acceleration $g$ in $-x$ direction. The numerical solution is tested on motion of the point at position $x=l$.

Parameters

$\begin{array}{ccc}\hfill m& \hfill =\hfill & 0.1\mathrm{kg}\hfill \\ \hfill g& \hfill =\hfill & 1\mathrm{m}·s^{-2}\hfill \\ \hfill l& \hfill =\hfill & 1\mathrm{m}\hfill \\ \hfill z_0& \hfill =\hfill & 0.01\mathrm{m}\hfill \\ \hfill \kappa & \hfill =\hfill & {10}^8\mathrm{Pa}\hfill \\ \hfill \mu & \hfill =\hfill & 3.\overline{33}·{10}^5\mathrm{Pa}\hfill \\ \hfill \rho & \hfill =\hfill & 0\mathrm{kg}·m^{-3}\hfill \end{array}$

Theory

The analytical solution of the problem, considering small strain limit, yields
$\frac{{\text{d}}^2\lambda }{\text{d}{t}^2}+\frac{3\mu {z_0}^2}{\mathrm{ml}}\lambda +\frac{g}{l}=0$
which gives us frequency $f=\frac{z_0}{\mathrm{2\pi }}\sqrt{\frac{3\mu }{\mathrm{ml}}}=5.033\mathrm{Hz}$.

FE Model

The model consists of twenty-five BRICK24 elements.

Results