Global Dynamics of a Compressor Blade with Resonances

Xiaoxia Bian,Fengxian An,Fangqi Chen

doi:10.1155/2016/3275750

Abstract

The global bifurcations and chaotic dynamics of a thin-walled compressor blade for the resonant case of 2 : 1 internal resonance and primary resonance are investigated. With the aid of the normal theory, the desired form associated with a double zero and a pair of pure imaginary eigenvalues for the global perturbation method is obtained. Based on the simpler form, the method developed by Kovacic and Wiggins is used to find the existence of a Shilnikov-type homoclinic orbit. The results obtained here indicate that the orbit homoclinic to certain invariant sets for the resonance case which may lead to chaos in the sense of Smale horseshoes for the system. The chaotic motions of the rotating compressor blade are also found by using numerical simulation.

Highlights

Compressor blades are widely used in many fields of aerospace, aeronautic engineering, and mechanical industry due to their excellent mechanical properties
Yao et al [10] performed a nonlinear dynamic analysis of the rotating blade with varying rotating speed under high-temperature supersonic gas flow; they [11] explored the contributions of nonlinearity, damping, and rotating speed to the steady-state nonlinear responses of the rotating blade, and they investigated the effects of the rotating speed on nonlinear oscillations of the blade
The study is focused on coexistence of 2 : 1 internal resonance and primary resonance

Summary

Introduction

Compressor blades are widely used in many fields of aerospace, aeronautic engineering, and mechanical industry due to their excellent mechanical properties. The energy-phase method proposed by Haller and Wiggins [25, 26] detected the existence of single-pulse and multipulse homoclinic orbits in a class of near Hamilton systems. Applying the latter two methods, there were many applications to investigate the global behaviors (see, e.g., [16,17,18,19,20,21,22]). We obtain a sufficient condition for the existence of Shilnikov-type homoclinic orbit of a compressor blade with 2 : 1 internal resonance and primary resonance using normal form theory and global perturbation method.

Formulation of the Problem

Dynamics of the Unperturbed System

Dynamics of the Perturbed System

Numerical Simulation of Chaotic Motions

Conclusions