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Structural Engineering and Mechanics
  Volume 28, Number 6, April20 2008, pages 659-676

Free vibration analysis of a uniform beam carrying multiple spring-mass systems with masses of the springs considered
Jia-Jang Wu

    The reports regarding the free vibration analysis of uniform beams carrying single or multiple spring-mass systems are plenty, however, among which, those with inertia effect of the helical spring(s) considered are limited. In this paper, by taking the mass of the helical spring into consideration, the stiffness and mass matrices of a spring-mass system and an equivalent mass that may be used to replace the effect of a spring-mass system are derived. By means of the last element stiffness and mass matrices, the natural frequencies and mode shapes for a uniform cantilever beam carrying any number of springmass
systems (or loaded beam) are determined using the conventional finite element method (FEM). Similarly, by means of the last equivalent mass, the natural frequencies and mode shapes of the same loaded beam are also determined using the presented equivalent mass method (EMM), where the cantilever beam elastically mounted by a number of lumped masses is replaced by the same beam rigidly attached by the same number of equivalent masses. Good agreement between the numerical results of FEM and those of EMM and/or those of the existing literature confirms the reliability of the presented approaches.
Key Words
    mass of spring; cantilever beam; spring-mass system; equivalent mass; natural frequency; mode shape.
Jia-Jang Wu: Dept. of Marine Engineering, National Kaohsiung Marine University, No. 142, Hai-Chuan Road,
Nan-Tzu, Kaohsiung 811, Taiwan, Republic of China

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