Analytical Approximant to a Quadratically Damped Duffing Oscillator.

Area of Science:

• Nonlinear Dynamics
• Mechanical Oscillations
• Applied Mathematics

Background:

• The Duffing oscillator is a fundamental model in nonlinear dynamics.
• Strong quadratic damping presents significant challenges for analytical solutions.

Purpose of the Study:

• To develop an elementary approximate analytical solution for the Duffing oscillator with strong quadratic damping.
• To validate the proposed solution against established numerical and analytical methods.
• To enable estimation of solution axis-crossing points.

Main Methods:

• Derivation of an approximate analytical solution using exponential and trigonometric functions.
• Comparison with the Runge-Kutta numerical solution.
• Application of He's homotopy method and the Krylov-Bogoliubov-Mitropolsky method.

Main Results:

• An elementary approximate analytical solution was successfully developed.
• The analytical approximant shows good agreement with the Runge-Kutta numerical solution.
• The method provides accurate estimation of the solution's horizontal axis crossing points.

Conclusions:

• The proposed approximate analytical solution is effective for the Duffing oscillator with strong quadratic damping.
• This method offers a valuable tool for analyzing nonlinear oscillatory systems.
• The technique facilitates the prediction of critical solution behaviors, such as zero crossings.