Renewable Energy, 5th Edition

(5.30)

Inserting (5.25) into (5.30), a second-order differential equation for the determination of τ_r(r) results. The solution depends on materials properties through ρ, Y, and μ and on the state of rotation through ω. Once the radial stress is determined, the tangential one can be evaluated from (5.25).

As an example, consider a plane disc of radius r_max, with a center hole of radius r_min. In this case, the derivatives of b(r) vanish, and the solution to (5.30) and (5.25) is

$\begin{array}{ll}{\tau }_r(r)\hfill & =(3+\mu )\rho {\omega }^2(r_{min}^2+r_{max}^2-r_{min}^2{r}_{max}^2/r^2-r^2)/8\hfill \\ {\tau }_t(r)\hfill & =(3+\mu )\rho {\omega }^2(r_{min}^2+r_{\mathit{max}}^2+r_{min}^2{r}_{max}^2/r^2-(1+3\mu )r^2/(3+\mu ))/8.\hfill \end{array}$ (5.31)

(5.31)

The radial stress rises from zero at the inner ...