Application to a Panel Flutter Problem

Abstract

In this chapter we apply the results of Chapter 4 to a particular two parameter problem. The equations are

$$ \dot x = Ax\, + \,f(x) $$

(5.1.1)

where

$$ \begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {x = {{\left[ {{x_1},{x_2},{x_3},{x_4}} \right]}^T},\,\,\,\,f(x) = {{\left[ {{f_1}(x),{f_2}(x),{f_3}(x),{f_4}(x)} \right]}^T},} \\ {A = \left[ {\begin{array}{*{20}{c}} 0&1&0&0 \\ {{a_1}}&{{b_1}}&c&0 \\ 0&0&0&1 \\ { - c}&0&{{a_2}}&{{b_2}} \end{array}} \right]} \end{array}} \\ {\,{f_1}(x) = {f_3}(x) = 0,} \end{array}} \\ {{f_2}(x) = {x_1}g(x),} \\ {{f_4}(x) = 4{x_3}g(x),} \end{array}} \\ {2g(x) = - {\pi ^4}(kx_1^2 + \sigma {x_1}{x_2} + 4kx_3^2 + 4\sigma {x_3}{x_4}),} \\ {c = \frac{{8\rho }}{3},\,\,\,\,{a_j} = - {\pi ^2}{j^2}\left[ {{\pi ^2}{j^2} + \Gamma } \right],} \\ {{b_j} = - \left[ {\alpha {\pi ^4}{j^4} + \sqrt \rho \delta } \right];\,\,\,\alpha = 0.005,\,\,\,\delta = 0.1,} \end{array} $$

k < 0, σ < 0 are fixed and ρ,Г are parameters. The above system results from a two mode approximation to a certain partial differential equation which describes the motion of a thin panel.