In Sect. 3.3, the DE for eigenfrequencies is derived and discussed in relation to simply supported BC. Now including rotational end springs to get further BC, the analytical solutions complicate and generalized basic Berry functions with an added tilde notation must be introduced. For each case of no axial force, eigenfrequencies as a function of conservative axial force are obtained. Graphically results are shown for different combinations of rotational spring support. It is noted that the DE may describe the combination of harmonic vibration with continuous support (Winkler support) and with linear magnetic attraction. Relations to Kolousek functions are pointed out.