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Structural Engineering and Mechanics
  Volume 29, Number 5, July30 2008 , pages 531-550
DOI: https://doi.org/10.12989/sem.2008.29.5.531
 


On the natural frequencies and mode shapes of a multi-span and multi-step beam carrying a number of concentrated elements
Hsien-Yuan Lin

 
Abstract
    This paper adopts the numerical assembly method (NAM) to determine the exact solutions of natural frequencies and mode shapes of a multi-span and multi-step beam carrying a number of various concentrated elements including point masses, rotary inertias, linear springs, rotational springs and springmass systems. First, the coefficient matrix for an intermediate station with various concentrated elements, cross-section change and/or pinned support and the ones for the left-end and right-end supports of a beam are derived. Next, the overall coefficient matrix for the entire beam is obtained using the numerical assembly technique of the conventional finite element method (FEM). Finally, the exact solutions for the natural frequencies of the vibrating system are determined by equating the determinant of the last overall coefficient matrix to zero and the associated mode shapes are obtained by substituting the corresponding values of integration constants into the associated eigenfunctions.
 
Key Words
    multi-step beam; exact solution; natural frequency; mode shape.
 
Address
Hsien-Yuan Lin: Dept. of Mechanical Engineering, Cheng Shiu University, Kaohsiung 833, Taiwan, Republic of China
 

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