Bifurcation analysis of a composite cantilever beam via 1:3 internal resonance

M Sayed,I H Mustafa,S I El-Bendary,D Y Alzaharani,A A Mousa

doi:10.1186/s42787-020-00102-7

Abstract

In this paper, we study a multiple scales perturbation and numerical solution for vibrations analysis and control of a system which simulates the vibrations of a nonlinear composite beam model. System of second order differential equations with nonlinearity due to quadratic and cubic terms, excited by parametric and external excitations, are presented. The controller is implemented to control one frequency at primary and parametric resonance where damage in the mechanical system is probable. Active control is applied to the system. The multiple scales perturbation (MSP) method is implemented to obtain an approximate analytical solution. The stability analysis of the system is obtained by frequency response (FR). Bifurcation analysis is conducted using various control parameters such as natural frequency (ω1), detuning parameter (σ1), feedback signal gain (β), control signal gain (γ), and other parameters. The dynamic behavior of the system is predicted within various ranges of bifurcation parameters. All of the stable steady state (point attractor), stable periodic attractors, unstable steady state, and unstable periodic attractors are determined efficiently using bifurcation analysis. The controller’s influence on system behavior is examined numerically. To validate our results, the approximate analytical solution using the MSP method is compared with the numerical solution using the Runge-Kutta (RK) method of order four.

Highlights

An active control system is defined necessarily in terms of that a mass of external power or energy is required
El-Bassiouny [13], Eissa et al [14,15,16,17], and Jaensch [18] confirmed how active control is functional in vibration attenuation at resonance for different
The main objective of the work is to control the vibration of a nonlinear composite cantilever beam with external and parametric excitation forces

Summary

Introduction

An active control system is defined necessarily in terms of that a mass of external power or energy is required. Vibrations can be abolished via a movement of feedback signals conducted by the mentioned technique. Active constrained layer damping has been effectively implemented as an efficient method to control the vibration of various flexible mechanical structures [2,3,4,5,6,7,8,9]. Eissa and Sayed [10,11,12] examined the active controller’s effect on both spring and simple pendulum at the primary resonance using negative velocity feedback. El-Bassiouny [13], Eissa et al [14,15,16,17], and Jaensch [18] confirmed how active control is functional in vibration attenuation at resonance for different

Objectives

Conclusion