Lie Symmetry Analysis of the Duffing Oscillator for Nonlinear Vibration Control and Stability in Mechanical Systems

Nonlinear vibration effects commonly occur in mechanical engineering systems like flexible structures, rotating machinery and micro-mechanical devices. Precise evaluation and regulation of these vibrations are vital for ensuring systems stability and enhancing mechanical efficiency. Nonetheless, the nonlinear differential equations that regulate vibration dynamics are frequently challenging to solve with standard analytical techniques. Mathematical methods grounded in symmetry offer a strong alternative for simplifying intricate systems while maintaining their fundamental physical characteristics.

This research examines the use of Lie symmetry analysis in nonlinear vibration models, specifically targeting the Duffing oscillator, a commonly utilized nonlinear model in mechanical engineering, representing systems with nonlinear stiffness properties. The study creates a symmetry reduction framework utilizing Lie group transformations on the governing nonlinear ordinary differential equation of the Duffing system. Through the identification of invariant transformations and symmetry generators, the original nonlinear equation can be systematically simplified, facilitating the derivation of invariant solutions and reduced-order dynamical models.

The suggested framework enhances the analytical comprehension of nonlinear vibration characteristics, encompassing resonance phenomena, equilibrium positions and stability criteria. By employing symmetry-based reduction, the intricate dynamics of the Duffing oscillator can be examined more effectively, shedding light on the effects of nonlinear stiffness, damping parameters and external forces on system stability. These results also aid in the development of vibration control strategies for mechanical systems where effects are important.

The findings indicate that Lie symmetry methods provide a methodical mathematical strategy to simplify nonlinear vibration models while maintaining their structural characteristics. This study emphasizes the significance of symmetry techniques in applied mathematics and engineering evaluations and adds to the current research on symmetry-oriented modelling of dynamic systems.