National Academy Science Letters

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

A Heuristic Method to Solve Nonlinear Vibration Problems

  • Satyanarayana BadetiEmail author
  • Somaraju Vempaty
  • Srinivas Suripeddi
Short Communication


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.


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


  1. 1.
    Nayfeh AH (1981) Introduction to perturbation techniques. Wiley, HobokenzbMATHGoogle Scholar
  2. 2.
    Nayfeh AH (1985) Problems in perturbation. Wiley, HobokenzbMATHGoogle Scholar
  3. 3.
    Cole JD (1968) Perturbation methods in applied mathematics. Blaisdell Publishing Company, WalthamzbMATHGoogle Scholar
  4. 4.
    Kevorkian J, Cole JD (1996) Multiple scale and singular perturbation methods. Springer, BerlinCrossRefzbMATHGoogle Scholar
  5. 5.
    Johnson RS (2005) Singular perturbation methods. Springer, BerlinGoogle Scholar
  6. 6.
    Bender CM, Orszag SA (1999) Advanced mathematical methods for scientists and engineers. Springer, BerlinCrossRefzbMATHGoogle Scholar
  7. 7.
    Sheikholeslami M et al (2014) Effects of heat transfer in flow of nanofluids over a permeable stretching wall in a porous medium. J Comput Theor Nanosci 11(2):486–496MathSciNetCrossRefGoogle Scholar
  8. 8.
    Sheikholeslami M, Ganji DD et al (2012) Analytical investigation of Jeffery–Hamel flow with high magnetic field and nanoparticle by Adomian decomposition method. Appl Math Mech 33(1):25–36MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Sheikholeslami M, Ganji DD (2013) Heat transfer of Cu–water nanofluid flow between parallel plates. Powder Technol 235:873–879CrossRefGoogle Scholar
  10. 10.
    Sheikholeslami M, Ganji DD (2015) Nanofluid flow and heat transfer between parallel plates considering Brownian motion using DTM. Comput Methods Appl Mech Eng 283:651–663ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Smith SH (1979) The modification of boundary layers by the imposition of an axial velocity within a rotating fluid. Q J Mech Appl Math 32:135ADSCrossRefzbMATHGoogle Scholar
  12. 12.
    Vempaty S, Rudraiah N (1986) Effect of normal blowing on the hydrodynamic flow between two differentially rotating infinite disks. Indian J Pure Appl Math 17:1412zbMATHGoogle Scholar

Copyright information

© The National Academy of Sciences, India 2018

Authors and Affiliations

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

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