Analytical Approximant to a Quadratically Damped Forced Cubic-Quintic Duffing Oscillator

Area of Science:

• Nonlinear Dynamics
• Mechanical Vibrations
• Applied Mathematics

Background:

• The cubic-quintic Duffing oscillator is a fundamental model in nonlinear dynamics.
• Understanding its behavior under strong damping and forcing is crucial for many physical systems.
• Existing analytical solutions are often limited in scope or accuracy.

Purpose of the Study:

• To develop an elementary approximate analytical solution for the cubic-quintic Duffing oscillator.
• To validate the accuracy of the proposed analytical approximant.
• To utilize the solution for estimating critical points of the system's response.

Main Methods:

• Formulation of an approximate analytical solution using exponential and trigonometric functions.
• Numerical simulation using the Runge-Kutta method for comparison.
• Analysis of solution behavior, specifically focusing on axis-crossing points.

Main Results:

• An elementary approximate analytical solution was successfully derived for the cubic-quintic Duffing oscillator.
• The analytical approximant showed good agreement with the Runge-Kutta numerical solution.
• The approximant effectively estimates the points where the oscillator's solution crosses the horizontal axis.

Conclusions:

• The proposed analytical solution offers a valuable and accessible method for studying the cubic-quintic Duffing oscillator.
• This approach provides reliable estimations for key dynamic behaviors, including equilibrium crossings.
• The findings contribute to a better understanding of damped and forced nonlinear oscillatory systems.