Type III: Mixed Network

We set

$$Y_{x^{-},i}=Y_{x^{+},i} = Y_{const}\hspace{1.0in} Y_{t,i} = 2Y_{const}$$

(5.10) and (5.9) then require that we set

$$Y_{c,i} = \frac{2(\overline{\rho A})_{i}}{\mu}-4Y_{const}\hspace{1.0in}\tilde{Z}_{c,i} = \frac{2}{\mu(\overline{EI})_{i}}-\frac{4}{Y_{const}}$$

The optimal choice of $Y_{const} = \sqrt{\min_{i}(\overline{\rho A})_{i})\max_{i}(\overline{EI})_{i}}$ yields the stability bound

$$\mu\leq\frac{1}{2}\sqrt{\frac{\min_{i}(\overline{\rho A})_{i})}{\... ...overline{EI})_{i})}}\leq\frac{1}{2}\min_{i}\sqrt{\frac{(\rho A)_{i}}{(EI)_{i}}}$$

and we are thus forced to use a smaller time step than is required by either of the above arrangements.