The mode shapes of a uniform fixed-free bar are of the form

\[ \phi_{n}(x)=\sin \left[\frac{(2 n-1) \pi x}{2 L}\right] \quad n=1,2,3, \ldots \]

Use the assumed modes method with \(\phi_{1}(x), \phi_{2}(x), \phi_{3}(x)\) as trial functions to approximate the lowest natural frequency and mode shapes for the tapered bar of Figure P11.6.

Transcribed Image Text:

FIGURE P11.6 L r(x) = ro(1-x) To=5 cm =1 cm/m L=3m E-200x109 N/m p=7500 kg/m