=+in which case the characteristic values ????s are located at their intersection. Figure 4.56 shows the determination of the characteristic values ????s for ???? = M m = 0.25. The intersections of the curve f2(????) with the horizontal axis are the roots of the clamped– free beam while the asymptotes cross it at the roots of the clamped–pinned beam. The roots of the beam with tip mass are bracketed by them as physically expected.

4.7 Proposed exercises