Question

Consider a thick-walled cylinder of inner radius $r_i$ and outer radius $r_o$ and thickness t . This cylinder is subject to a centrifugal force () from its longitudinal axis.

If we know the radial stress $\sigma_r$,the hoop stress $\sigma_{\theta }$ ,the hoop strain $\epsilon_{\theta }$ , the radial strain $\epsilon_{r}$ :

$\sigma_{r} = \frac{3 + \nu}{8}\rho\omega^2\left(r_o^2+ r_i^2 - \frac {r_i^2r_o^2}{r^2}-r^2 \right )$

$\sigma_{\theta} = \frac{3 + \nu}{8}\rho\omega^2\left(r_o^2+ r_i^2 + \frac {r_i^2r_o^2}{r^2}-\frac{1+3\nu}{3+\nu}r^2 \right )$

$\epsilon_{\theta} = \frac{1}{E}\left(\sigma_{\theta} - \nu \sigma_r \right )$$= \frac{u_r}{ r}$

$\epsilon_{r} = \frac{1}{E}\left(\sigma_{r} - \nu \sigma_{\theta} \right )$

where $\nu$ is the Poisson ration and $u_r$ is the radial displacement.

How to compute the TOTAL POTENTIAL ENERGY |TPE| = $\| \mathcal{U} - \mathcal{P}\|$ ?

N.B. : $\mathcal{P}$ is the work done by external forces

         $\mathcal{U}$ is the elastic internal potential energy

E = mw2T = constant rotation sped дит

Answer 1

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