A unified approach for the free vibration analysis of an elastically supported immersed uniform beam carrying an eccentric tip mass with rotary inertia

Although an immersed beam with elastic support and fixed support are usually tackled, separately, to simplify the procedures of analysis, a unified approach will be more convenient for the computer programming in the computer method. The purpose of this paper is to use a unified approach to determine the “exact” lowest several natural frequencies and the associated mode shapes of the (partially or fully) immersed beam in both the elastic- and fixed-support conditions. Furthermore, by modeling the distributed added mass along the immersed part of the beam with a number of concentrated added masses, a point added mass method (PAM) incorporated with the mode-superposition approach is also presented to determine the “approximate” lowest several natural frequencies and the associated mode shapes of the last two types of immersed beam. It is unlike most of the vibrating systems with their boundary conditions independent on the eigenvalues that the boundary conditions of the current immersed beam are frequency dependent due to the existence of frequency-dependent boundary inertial forces and moments. Since the last frequency-dependent boundary conditions significantly affect the orthogonal condition of the mode shapes and so do the applicability of the mode-superposition approach for the vibration analysis, the theory regarding the orthogonal condition for the mode shapes of the current vibrating system will be one of the key points in this paper. The numerical results have been compared with the existing information or the results of finite element method and a good agreement is achieved. Furthermore, to check the last theoretical results, several model tests are also carried out on the scale models of the fixed and elastically supported towers and reasonable agreement is obtained.