Consider the free axial motion of a beam with the elastic boundary conditions of Example 7.2, and solve for the first four natural frequencies, modes, and the complete solution, for the following cases:

(a) \(k=E A \mathrm{~N} / \mathrm{m}, L=10 \mathrm{~m}\)

(b) \(k=10 E A \mathrm{~N} / \mathrm{m}, L=1 \mathrm{~m}\)

(c) \(k=E A / 10 \mathrm{~N} / \mathrm{m}, L=1 \mathrm{~m}\)

(d) \(k=E A / 10 \mathrm{~N} / \mathrm{m}, L=10 \mathrm{~m}\).

Transcribed Image Text:

Example 7.2 Axial Vibration of Uniform Beam with Elastic Boundary Consider the axially vibrating beam shown in Fig- ure 7.12, where the right end of the beam is restrained elastically. Derive the eigenvalue problem and solve for the eigenvalues.