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Application to a Panel Flutter Problem

  • Jack Carr
Part of the Applied Mathematical Sciences book series (AMS, volume 35)

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.

Keywords

Nonlinear Term Linear Term Local Behavior Parameter Problem Centre Manifold 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag New York Inc. 1982

Authors and Affiliations

  • Jack Carr
    • 1
  1. 1.Department of MathematicsHeriot-Watt UniversityEdinburghScotland

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