DUALITY RELATIONS OF BEAMS IN NATURAL VIBRATIONS

The duality relations are studied in the paper for the Euler-Bernoulli beams with homogeneous boundaries in natural vibrations. A pair of beams is first defined as a dual of different cross-sections if they have the same natural frequencies, but different variations of cross-sections. The duals of different cross-sections are analyzed via a dual of displacement description and bending moment description, and the non-uniform beams with homogeneous boundaries can be classified as the following seven duals. They are (1) the dual of a free-free beam and a clamped-clamped beam, (2) the dual of a slipping-free beam and a slipping-clamped beam (and their mirrors), (3) the dual of a hinged-free beam and a hinged-clamped beam (and their mirrors), (4) the dual of two hinged-slipping beams, (5) the dual of two slipping-slipping beams, (6) the dual of two hinged-hinged beams, and (7) the dual of a clamped-free beam and a free-clamped beam. Then, a pair of beams is defined as a dual of identical cross-sections if they have the same natural frequencies and the same variations of cross-sections. It is proved that the first four duals of different cross-sections become the duals of identical cross-sections if and only if the area of cross-section and the inertial moment of cross-section of any beam in those duals take a specific form of exponential function. Afterwards, the first three duals of identical cross-sections are verified to keep the dual relations for uniform beams, whereas the fourth dual is degenerated to a pair of mirrors. Based on the dual of displacement description and slope description, a new dual of uniform beams is found for a slipping-slipping beam and a hinged-hinged beam. Finally presented is an important feature of all the duals of uniform beams. That is, one uniform beam in a dual has statically determinate constraints while the other uniform beam in the same dual has statically indeterminate constraints.