Use the assumed modes method with trial functions [ w_{1}(x)=sin left(pi frac{x}{L} ight) quad w_{2}(x)=sin left(2 pi frac{x}{L} ight) quad w_{3}(x)=sin left(3 pi frac{x}{L} ight)

Use the assumed modes method with trial functions

\[ w_{1}(x)=\sin \left(\pi \frac{x}{L}\right) \quad w_{2}(x)=\sin \left(2 \pi \frac{x}{L}\right) \quad w_{3}(x)=\sin \left(3 \pi \frac{x}{L}\right) \]

to approximate the lowest natural frequency and its corresponding mode shape for a uniform fixed-fixed bar of length \(L\).