Extended framework of Hamilton's principle applied to Duffing oscillation

TL;DR

This paper introduces an extended Hamilton's principle framework for Duffing oscillation, enabling a variational formulation that accurately incorporates initial conditions and facilitates the development of finite element methods.

Contribution

It presents a novel variational formulation of the Duffing equation using an extended Hamilton's principle that accounts for initial conditions and supports finite element method development.

Findings

The formulation recovers all governing differential equations as Euler-Lagrange equations.

A simple linear temporal finite element method is developed.

Numerical examples verify the method's performance and efficiency.

Abstract

The paper begins with a novel variational formulation of Duffing equation using the extended framework of Hamilton's principle (EHP). This formulation properly accounts for initial conditions, and it recovers all the governing differential equations as its Euler-Lagrange equation. Thus, it provides elegant structure for the development of versatile temporal finite element methods. Herein, the simplest temporal finite element method is presented by adopting linear temporal shape functions. Numerical examples are included to verify and investigate performance of non-iterative algorithm in the developed method.