Dynamics of vibrating beams using first-order theory based on Legendre polynomial expansion

  • A. CzekanskiEmail author
  • V. V. Zozulya


First-order models are used in the analysis of the tension–compression and transverse bending modes of beam vibration. The equation of motion for each mode and the expressions for boundary conditions are obtained using the generalized variational principle. Systems of partial differential equations for the longitudinal and bending modes of vibrating beams are reduced to a single fourth-order equation, and frequency equations are obtained. The problem of free and forced vibrations of beams that are simply supported at both ends is presented. An analysis and comparison with well-known theories is performed using computer algebra system Mathematica.


First-order beam theory Vibration Natural frequency Legendre polynomial 



We gratefully acknowledge the financial support provided by the Committee of Science and Technology of Mexico (CONACYT) through a Research Grant (Ciencia Básica, Reference No. 256458). The authors thank Professor Isaac Elishakoff, from Florida Atlantic University, for sending papers [13, 32] and for providing motivation for the research performed.


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Authors and Affiliations

  1. 1.Department of Mechanical Engineering, Lassonde School of EngineeringYork UniversityTorontoCanada
  2. 2.Centro de Investigación Científica de Yucatán, A.C.MéridaMexico

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