Cheillini integrability and quadratically damped oscillators

TL;DR

This paper introduces a novel method using Jacobi's last multiplier and Cheillini's integrability condition to analyze quadratically damped Lienard equations, providing explicit solutions via Lambert W-function.

Contribution

It presents a new analytical approach for solving Lienard equations with quadratic damping, including explicit solutions using Lambert W-function.

Findings

Derived a closed-form solution for the transcendental characteristic equation.

Established a connection between Cheillini integrability and quadratic damping.

Provided a new framework for analyzing strongly damped oscillators.

Abstract

In this paper a new approach to study an equation of the Lienard type with a strong quadratic damping is proposed based on Jacobi's last multiplier and Cheillini's integrability condition. We obtain a closed form solution of the transcendental characteristic equation of the Lienard type equation using the Lambert W-function.