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Applied Mathematics and Mechanics

, Volume 40, Issue 1, pp 85–96 | Cite as

Free vibration of non-uniform axially functionally graded beams using the asymptotic development method

  • Dongxing CaoEmail author
  • Yanhui Gao
Article

Abstract

The asymptotic development method is applied to analyze the free vibration of non-uniform axially functionally graded (AFG) beams, of which the governing equations are differential equations with variable coefficients. By decomposing the variable flexural stiffness and mass per unit length into reference invariant and variant parts, the perturbation theory is introduced to obtain an approximate analytical formula of the natural frequencies of the non-uniform AFG beams with different boundary conditions. Furthermore, assuming polynomial distributions of Young’s modulus and the mass density, the numerical results of the AFG beams with various taper ratios are obtained and compared with the published literature results. The discussion results illustrate that the proposed method yields an effective estimate of the first three order natural frequencies for the AFG tapered beams. However, the errors increase with the increase in the mode orders especially for the cases with variable heights. In brief, the asymptotic development method is verified to be simple and efficient to analytically study the free vibration of non-uniform AFG beams, and it could be used to analyze any tapered beams with an arbitrary varying cross width.

Key words

axially functionally graded (AFG) beam non-uniform natural frequency asymptotic development method 

Chinese Library Classification

O242 

2010 Mathematics Subject Classification

74S05 

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Copyright information

© Shanghai University and Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.College of Mechanical EngineeringBeijing University of TechnologyBeijingChina
  2. 2.Beijing Key Laboratory of Nonlinear Vibrations and Strength of Mechanical StructuresBeijingChina

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