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)

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

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