The vibration modes and frequencies of uniform Euler beams for various boundary end conditions are traditionally expressed in terms of sinβnx, cosβnx, sinhβnx, and coshβnx functions. For modes above the second, however, the numerical evaluation of these modes requires an increasingly large number of significant figures to be kept in order to distinguish small differences between sinhβnx and coshβnx. The present note, gives simple expressions for arbitrarily high-order modes and frequencies of a uniform Euler beam. These expressions not only allow simple numerical calculation but also help to identify the physical nature of the modes.
