Solve for the first three modes and frequencies of the nonuniform longitudinally vibrating beam of Figure 7.47 beginning with the known equation of motion. The following properties hold:

\[ \begin{aligned} E A(x) & =2 E A(1-x / L) \\ \text { and } m(x) & =2 m(1-x / L) \end{aligned} \]

(a) Solve this problem for arbitrary \(k\), and

(b) then reduce the solution for the case where \(k=0\). How far can this problem be solved analytically?

Transcribed Image Text:

u(x,t) m(x), EA(x) mm L Figure 7.47: Elastic constraint of a longitudinally vibrat- ing beam. At position x, the beam elongates an amount u(x,t). At the right end there is a spring of stiffness k.