The new shift perturbation method with applications to vibrational problems

Abstract

Employing variable horizontal and vertical shifts for the unperturbed solution, a new perturbation method is proposed. The new method is named as the shift perturbation method. The existing perturbation methods are interpreted with respect to the horizontal and vertical shifts for the unperturbed solution. The new method is applied to some well-known vibrational problems. The method is capable of producing admissible approximate solutions for a wide range of problems such as the free and forced vibrations of the Duffing equation, equations with quadratic and cubic nonlinearities, variable amplitude nonlinear problems, and boundary layer problems. A variant of the method also produced valid solutions for large perturbation parameters.

Effect of Angular Speed Variations on the Nonlinear Vibrations of a Rotational Spring-Mass System

“…The primary resonances and the principle parametric resonances are investigated in detail using the Method of Multiple Scales, a perturbation technique [15]. The problem involves amplitude and phase variations in time requiring an advanced perturbation technique such as multiple scales, averaging method or the newly proposed shift-perturbation method [16]. The mentioned advanced techniques are more successful in predicting the dynamical behavior of such systems since they allow for shifts in the horizontal and vertical directions [16].…”

“…The problem involves amplitude and phase variations in time requiring an advanced perturbation technique such as multiple scales, averaging method or the newly proposed shift-perturbation method [16]. The mentioned advanced techniques are more successful in predicting the dynamical behavior of such systems since they allow for shifts in the horizontal and vertical directions [16]. The amplitude and phase variation equations in time are derived for both cases and the steady state solutions are depicted via the frequency response curves.…”