Consider the stepped shaft consisting of two uniform segments of lengths \(L_{1}\) and \(L_{2}\), torsional rigidities \(G J_{1}\) and \(G J_{2}\), and mass moments of inertia per unit length \(I_{1}\) and \(I_{2}\), shown in Figure 7.52. Set up the eigenvalue problem for the torsional vibration of the system and derive the frequency equation. Show that as \(L_{2} \rightarrow 0\),

\[ \lim _{L_{2} \rightarrow 0} I_{2} L_{2}=I_{D} \]

where \(I_{D}\) is the mass moment of inertia of a disk. Find the frequency equation and the orthogonality condition.

Transcribed Image Text:

GJ, GJ, 12 L Figure 7.52: Stepped shaft undergoing torsion.