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

Alvaro H S Salas1, Gilder-Cieza Altamirano2, Manuel Sánchez-Chero3

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Researchers generalized He's frequency approach for the damped Duffing equation using a time-varying amplitude. This study also employed homotopy and Lindstedt-Poincaré methods, deriving accurate formulas for Jacobi elliptic functions.

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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.