# Effects of strong parametric excitation on cantilever beam: non-perturbative approach

## TL;DR

This study explores how parametric excitation affects the stability and chaotic behavior of a cantilever beam using a non-perturbative method.

## Contribution

The paper introduces a non-perturbative approach using He’s frequency formula to analyze nonlinear oscillations with high numerical precision.

## Key findings

- The non-perturbative method accurately converts nonlinear equations into linear ones with excellent agreement.
- Bifurcation diagrams and Lyapunov exponents reveal chaotic and periodic oscillations under parametric excitation.
- The method simplifies complexity and provides insights into stability thresholds and resonance conditions.

## Abstract

The impact of primary parametric excitations on the bifurcation behavior and chaotic oscillations of a cantilever beam construction is examined in the existing study. The results provide valuable insights into dynamic transitions, resonance conditions, and stability thresholds. This innovation is crucial in technical applications, including aerospace and civil engineering, as slight parametric variations to stimulate complex nonlinear behavior endanger structural safety. The fundamental methodology relies on the non-perturbative approach, primarily developed by the confidential He’s frequency formula. This methodology is adopted to convert a weak oscillator of a nonlinear ordinary differential equation into a linear one. An excellent agreement is obtained between the two equations. The current approach is appropriate, based on basic ideas, and produces peculiarly high numerical precision. The stability performance is assessed in various scenarios. The current method reduces assessed complexity, and the explanation is significant in the mathematical execution of nonlinear parametric issues. The dynamics of nonlinear simulation are examined via bifurcation illustrations, analytical essential elements that affect system behavior. The largest Lyapunov exponent elucidates chaotic and periodic oscillations, providing insight into long-term stability and the genesis of chaos.