# An innovative inspection of a cantilever beam exposed to principal parametric excitation

## TL;DR

This paper presents a new method to reduce vibrations in cantilever beams under parametric excitation using a nonlinear feedback law and non-perturbative approach.

## Contribution

A novel non-perturbative method is introduced to approximate solutions for nonlinear parametric systems with high precision and reduced complexity.

## Key findings

- The proposed method achieves high numerical accuracy and simplifies solving nonlinear parametric problems.
- System stability increases with higher linear and nonlinear damping coefficients.
- The Poincaré map, phase portrait, and bifurcation analysis reveal detailed system behavior.

## Abstract

The inspection of a cantilever beam subjected to parametric stimulation is essential in engineering structures such as bridges, aircraft wings, and micro electromechanical systems (MEMS). It is demonstrated that nonlinearities restrict the rise in accessibility. The existing issue mitigates vibrations in a structure exposed to primary parametric stimulation. It utilizes knowledge to develop a simple nonlinear feedback law designed to reduce vibrations of the first mode of a cantilever beam. The fundamental methodology relies on the non-perturbative approach (NPA), which is grounded in He’s frequency formula (HFF). This approach simply converts a weakly nonlinear oscillator of a second nonlinear ordinary differential equation (ODE) into a linear one. Consequently, the goal is to depart from traditional perturbation methods and get unrestricted approximation solutions of small amplitude parametric components. Furthermore, the method is extended to find the best solutions for large amplitude nonlinear fluctuations of coupled system. The generated parametric equation shows good agreement with the original equation when validated using the Mathematica Software (MS). The stability behavior is investigated across several contexts. The current methodology is founded on explicit principles, is suitable, user-friendly, and yields remarkably high numerical precision. The present approach reduces the mathematical difficulty, making it beneficial of the mathematical execution of nonlinear parametric issues. It is found that the system is stable with the increase of both linear and nonlinear damping coefficients, while it becomes less stable as excitation force’s parameters increase. Furthermore, the Poincaré map, phase portrait, and bifurcation are analyzed, which collectively provide a comprehensive depiction of the system’s behavior at different phases.