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Free End

The boundary conditions are now

\begin{subequations}\begin{align}\frac{\partial^{2} w(0,t)}{\partial x^{2}} &= 0...
...ightarrow&& \frac{\partial m(0,t)}{\partial x} = 0 \end{align}\end{subequations}

The condition $ m=0$ is the same as for the fixed end which is allowed to pivot. Thus we terminate the series junction in the same way as in that case. For the other condition, $ \frac{\partial m(0,t)}{\partial x} = 0$, there is complete symmetry with the case of the fixed clamped end, where the condition was $ \frac{\partial v(0,t)}{\partial x} = 0$. The junction can be terminated as in Figure 5.4(c), and we choose $ Y_{c,0}=Y_{J,0}-Y_{x^{+},0}$, where $ Y_{J,0}=\frac{\bar{(\rho A)}_{0}}{\mu}$, and $ Y_{x^{+},0}$ follows from the particular type of mesh configuration that we are using. Stability of this boundary condition for the three types of mesh follows as before as well.

Figure 5.4: Boundary terminations for the Euler-Bernoulli system-- (a) fixed end, allowed to pivot; (b) fixed, clamped end; (c) free end.

Stefan Bilbao 2002-01-22