Hopf Bifurcation of Heated Panels Flutter in Supersonic Flow

Li-Na Cao,Guofeng Yao

doi:10.3390/math7090787

Abstract

A differential equation of panel vibration in supersonic flow is established on the basis of the thin-plate large deflection theory under the assumption of a quasi-steady temperature field. The equation is dimensionless, and the derivation of its second-order Galerkin discretization yields a four-dimensional system. The algebraic criterion of the Hopf bifurcation is applied to study the motion stability of heated panels in supersonic flow. We provide a supplementary explanation for the proof process of a theorem, and analytical expressions of flutter dynamic pressure and panel vibration frequencies are derived. The conclusion is that the algebraic criterion of Hopf bifurcation can be applied in high-dimensional problems with many parameters. Moreover, the computational intensity of the method established in this work is less than that of conventional eigenvalue computation methods using parameter variation.

Highlights

Aircraft siding and skin, which are surrounded and fixed on a skeletal structure by an adhesive or rivet, form the dimensional member of an aerodynamic shape
Chen et al studied the coefficients of the characteristic equation of the first approximation system and its corresponding Hurwitz determinant, which is used to derive the algebraic criterion of Hopf bifurcation
We established a differential equation for the vibration of heated panels in supersonic flow on the basis of the thin-plate large deflection theory under the assumption of a quasi-steady temperature field

Summary

Introduction

Aircraft siding and skin, which are surrounded and fixed on a skeletal structure by an adhesive or rivet, form the dimensional member of an aerodynamic shape. Mathematics 2019, 7, 787 and stability of the panel [11].Yang et al provided a study on the nonlinear thermal flutter of heated curved panels in supersonic air flow using the Newton iterative approach and Runge-Kutta method [12]. Zhang investigated the stability and bifurcation behaviors of a two-dimensional, nonlinear, viscoelastic panel in supersonic flow, using analytical and numerical methods [15]. Chen et al studied the coefficients of the characteristic equation of the first approximation system and its corresponding Hurwitz determinant, which is used to derive the algebraic criterion of Hopf bifurcation This method transforms the problem of searching for the system bifurcation point into the problem of solving the root of the nonlinear equation [17,18]. We provide a supplementary explanation for the proof process of a theorem

Differential Equation of Heated Panel Flutter in Supersonic Flow

Mechanical

Algebraic Criterion of the Hopf Bifurcation

Panel Flutter in Supersonic Flowobtained by Hopf Bifurcation Criterion

Numerical Example

Conclusions