Example $16\mathrm{.}5$

We now consider the determination of the natural frequencies of vibration for a beam fixed at both ends as shown in Figure $16-16.$ The beam has mass density $\backslash (\backslash$rho $\backslash ),$ modulus of elasticity $\backslash ($ E $\backslash ),$ cross$-$sectional area $\backslash ($ A $\backslash ),$ area moment of inertia $\backslash ($ I $\backslash ),$ and length $\backslash (2$ L $\backslash ).$ For simplicity of the longhand calculations, the beam is discretized into $($a$)$ two beam elements of length $\backslash ($ L $\backslash )($Figure $16-16($a$))$ and then $($b$)$ three beam elements of length $\backslash ($ L $\backslash )$ each $($Figure $16-16($b$)).$