Revised dynamic characteristics of uniform beams with free - hinged and free - free end conditions

The uniform beams with free-hinged ends and with free-free ends have very slack bonds and, therefore, in accordance with Rayleigh’s theorem, their lowermost eigenvalue must be lesser compared to the lower most eigenvalue of beams with clamped-free and clamped-hinged ends. In spite of the physical evidence, the magnitudes of lowermost eigenvalue of beams with slack bonds, available in all publications, are larger. This contradiction signifies that there are the missing modes of vibration with the lowermost eigenvalue for beams with free-free and free-hinged ends. The rigorous analysis of uniform beams vibration with free-free and free-hinged ends conditions defines these missing lowermost natural frequencies and normal modes and ascertains the frequencies and modes for all uniform beams with various end conditions into the ordered system. The lowermost mode of vibration of a beam with free-free ends, caused by ocean choppiness and determined in this investigation, is paramount for estimation of the ships structural strength, particularly important for the tankers.