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(t, ɛ) 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
This is a preview of subscription content, log in to check access.
Nayfeh AH (1981) Introduction to perturbation techniques. Wiley, HobokenMATHGoogle Scholar
Johnson RS (2005) Singular perturbation methods. Springer, BerlinGoogle Scholar
Bender CM, Orszag SA (1999) Advanced mathematical methods for scientists and engineers. Springer, BerlinCrossRefMATHGoogle Scholar
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
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):25–36MathSciNetCrossRefMATHGoogle Scholar
Sheikholeslami M, Ganji DD (2013) Heat transfer of Cu–water nanofluid flow between parallel plates. Powder Technol 235:873–879CrossRefGoogle Scholar
Sheikholeslami M, Ganji DD (2015) Nanofluid flow and heat transfer between parallel plates considering Brownian motion using DTM. Comput Methods Appl Mech Eng 283:651–663ADSMathSciNetCrossRefGoogle Scholar
Smith SH (1979) The modification of boundary layers by the imposition of an axial velocity within a rotating fluid. Q J Mech Appl Math 32:135ADSCrossRefMATHGoogle Scholar
Vempaty S, Rudraiah N (1986) Effect of normal blowing on the hydrodynamic flow between two differentially rotating infinite disks. Indian J Pure Appl Math 17:1412MATHGoogle Scholar