Solution to a Damped Duffing Equation Using He's Frequency Approach

Area of Science:

• Nonlinear Dynamics
• Applied Mathematics
• Theoretical Physics

Background:

• The damped Duffing equation is a fundamental model in nonlinear dynamics, crucial for understanding phenomena in mechanical vibrations and electrical circuits.
• Existing analytical methods for solving the damped Duffing equation face challenges with strong nonlinearities and damping effects.
• He's frequency approach offers a powerful tool for analyzing nonlinear oscillators, but requires generalization for complex scenarios.

Purpose of the Study:

• To generalize He's frequency approach by incorporating a time-varying amplitude for solving the damped Duffing equation.
• To compare the generalized approach with established methods like the homotopy analysis method and the Lindstedt-Poincaré method.
• To derive highly accurate approximation formulas for the Jacobi elliptic function 'cn'.

Main Methods:

• Generalization of He's frequency approach with a time-varying amplitude.
• Application of the homotopy analysis method.
• Application of the Lindstedt-Poincaré method.
• Derivation of approximation formulas using Chebyshev and Pade approximation techniques.

Main Results:

• Successful generalization of He's frequency approach for the damped Duffing equation.
• High-accuracy formulas for approximating the Jacobi elliptic function 'cn' were derived.
• The proposed methods provide effective analytical solutions for the damped Duffing equation.

Conclusions:

• The generalized He's frequency approach offers a robust and accurate method for solving the damped Duffing equation.
• The derived formulas for Jacobi elliptic functions enhance the analytical toolkit for nonlinear system analysis.
• This work contributes to a deeper understanding of nonlinear oscillatory systems and their solutions.