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National Academy Science Letters

, Volume 41, Issue 4, pp 225–231 | Cite as

A Heuristic Method to Solve Nonlinear Vibration Problems

  • Satyanarayana Badeti
  • Somaraju Vempaty
  • Srinivas Suripeddi
Short Communication
  • 19 Downloads

Abstract

We propose a heuristic method to obtain the solutions, at least to the lowest order, for linear and nonlinear vibration problems governed by a small parameter ɛ. For a linear or nonlinear oscillator, we assume a perturbation expansion for the dependent variable u(tɛ) as in regular perturbation method, but choose a solution of the form a(t) cos (ω0t + β(t)) for the lowest order term u0 to take care of the frequency–amplitude interaction. It is then in general true that the frequency correction to lowest order is O(ɛ2) or O(ɛ) depending on whether a(t) = O(ɛ) or zero respectively. This physical feature is made use of to obtain directly the secular terms in O(ɛ) and O(ɛ2) governing equations and hence obtain the amplitude a(t) and frequency drift β(t) to at least lowest order. The efficacy of the method is tested and illustrated with several examples. Also numerical values obtained using this method are compared with the numerical solution obtained with Differential transform method and Homotopy analysis method for one typical problem.

Keywords

Linear and nonlinear vibrations Perturbation methods Secular terms Dispersive Diffusive and dispersive–diffusive derivatives 

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Copyright information

© The National Academy of Sciences, India 2018

Authors and Affiliations

  • Satyanarayana Badeti
    • 1
  • Somaraju Vempaty
    • 2
  • Srinivas Suripeddi
    • 1
  1. 1.Department of MathematicsVIT-AP UniversityAmaravatiIndia
  2. 2.GVP-LIAS College of EngineeringVishakhapatnamIndia

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