Eur. Phys. J. Plus (2021) 136: 107
https://doi.org/10.1140/epjp/s13360-020-01018-y

Regular Article

Exact analysis of the nonlinear vibration of an archetypal non-natural oscillator

Applied Mechanics and Design (AMD) Research Group, Department of Mechanical Engineering, Faculty of Engineering, University of Port Harcourt, Port Harcourt, Nigeria

^a akuro.big-alabo@uniport.edu.ng

Received: 16 July 2020
Accepted: 11 December 2020
Published online: 19 January 2021

Abstract

The exact periodic solution of an archetypal non-natural oscillator with quadratic inertia and cubic nonlinear static stiffness has been derived in terms of the elliptic integral of the third kind. The periodic solution was derived based on a new form of elliptic integral that arose naturally from the model of the archetypal non-natural oscillator. It was shown that this new form of elliptic integral is an alternate form of the elliptic integral of the third kind and it has several advantages over the classical Legendre form. Hence, some new properties of elliptic integrals were derived based on the new elliptic integral. A bifurcation analysis of the present oscillator revealed the possible combinations of static stiffness parameters that can produce periodic solutions and the corresponding initial amplitude range where these periodic solutions exist. The bifurcation analysis informed investigations on the frequency–amplitude response which included the effect of the different static stiffness parameters. It was observed that the frequency increased for an increase in the positive cubic nonlinear stiffness parameter and vice versa. Additionally, the large-amplitude displacement history showed the same profile notwithstanding the combination and magnitudes of the stiffness parameters.