Dynamical analysis of a damped harmonic forced duffing oscillator with time delay

Area of Science:

• Nonlinear Dynamics
• Control Systems Engineering
• Applied Mathematics

Background:

• Damped nonlinear excited Duffing oscillators (DO) exhibit complex behaviors.
• Time-delayed control techniques are increasingly important for managing oscillations.
• Precisely modeling and controlling nonlinear systems remains a significant challenge.

Purpose of the Study:

• To develop and analyze a time-delayed controller for a damped nonlinear excited Duffing oscillator.
• To investigate the impact of position and velocity time delays on oscillation reduction.
• To validate a novel control strategy for nonlinear systems.

Main Methods:

• Utilized an advanced Homotopy Perturbation Method (HPM) for approximate solutions.
• Employed the fourth-order Runge-Kutta (RK4) technique for numerical verification.
• Applied multiple scales homotopy technique to derive steady-state amplitude equations.

Main Results:

• The improved HPM accurately approximates solutions for nonlinear oscillators, minimizing error.
• Controller performance is significantly influenced by time delays and gain products.
• Complete vibration reduction is achievable with optimized control and feedback gains.

Conclusions:

• The proposed time-delayed controller is effective in minimizing vibrations in Duffing oscillators.
• Increasing time delay parameters enhances system stability.
• The developed methodology offers a robust approach for diverse nonlinear control problems.