Green functions for boundary value problems about divergence of elongated plate

Abstract

Bucking of a thin flexible elongated plate in supersonic gas flow along Ox-axis, compressed or extended by boundary stresses at the edges x = 0 and x = 1 is governed by a boundary value problem for nonlinear ordinary integro-differential equation with two bifurcation parameters: external stress and Mach number. The linearized equation for the described boundary problem of aeroelasticity [2] χ 2 w (4)= 0 with boundary conditions B or B representing spectral two-point boundary value problem. Here χ 2 rigidity of the plate, T < 0 (T > 0), proportional to compressing stress (extending stress), σ > 0, proportional to Mach number are dimensionless bifurcation parameters. At the investigation of these boundary value problems the following possibilities arise:In the first case T is greater then 0. The characteristic equation (Ch.Eq.) f 0 (λ) = χ 2 λ 4 − T λ 2 + σλ = 0 has one negative root −α and two positive roots β 2 > β 1 > 0 (α = β 1 + β 2 ). For d = 0 T is greater than 0 again. Here there are two equal roots β 1 = β 2 = β > 0 and one negative root −α. Third case which is possible at both extension (T > 0) and compression (T < 0) of the plate, the roots are a pair of complex conjugate γ ± δi (γ, δ > 0) and −α < 0 (α=2γ). Here for the buckling investigation it is convenient to introduce the following. It is not difficult to see that the values3 χ 2 respond to the plate extension, and the values u > √ 3 ⇒ σ > 8γ 3 χ 2 respond to the plate compression. The value u= √ 3 implies T =0, i.e. the extension/compression absence. The value u = 0 corresponds to d = 0. At the investigation of Ch.Eq. with two parameters T and σ Sturm method for roots separation was used.