ASE 167M - Lecture 9 - ASE 167M Lecture#9 Numerical...

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Equations of Motion for Gliding Flight Maximum Range Conditions Integration of the Glide EOM Numerical Integration Program #2 Equations of Motion for Gliding Flight Maximum Range Conditions Integration of the Glide EOM Numerical Integration What is Numerical Integration? Euler Integration Midpoint Integration Runge-Kutta Integration Program #2 The University of Texas at Austin
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Equations of Motion for Gliding Flight Maximum Range Conditions Integration of the Glide EOM Numerical Integration Program #2 Gliding Flight I From ASE 367K, we have the following equations of motion for unsteady flight in the vertical plane: ˙ x = V cos γ ˙ h = V sin γ ˙ V = g W [ T cos - D - W sin γ ] ˙ γ = g WV [ T sin + L - W cos γ ] ˙ W = - CT The University of Texas at Austin
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Equations of Motion for Gliding Flight Maximum Range Conditions Integration of the Glide EOM Numerical Integration Program #2 Gliding Flight The University of Texas at Austin
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Equations of Motion for Gliding Flight Maximum Range Conditions Integration of the Glide EOM Numerical Integration Program #2 Gliding Flight I We can enforce the glide condition, T = 0 , to get unsteady gliding flight equations of motion ˙ x = V cos γ ˙ h = V sin γ ˙ V = - g W [ D + W sin γ ] ˙ γ = g WV [ L - W cos γ ] ˙ W = 0 The University of Texas at Austin
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Equations of Motion for Gliding Flight Maximum Range Conditions Integration of the Glide EOM Numerical Integration Program #2 Gliding Flight I Enforcing steady conditions, ˙ V = 0 and ˙ γ = 0 , and assuming flat glide path, γ 1 , yields ˙ x = V ˙ h = V γ 0 = D + 0 = L - W The University of Texas at Austin
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Equations of Motion for Gliding Flight Maximum Range Conditions Integration of the Glide EOM Numerical Integration Program #2 Gliding Flight I The goal of gliding flight is to I remain in the air for the maximum possible time, which means that we want the minimum possible rate of descent, ˙ h min I travel as far as possible during the glide, which implies that we want the minimum glide angle, γ min The University of Texas at Austin
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Equations of Motion for Gliding Flight Maximum Range Conditions Integration of the Glide EOM Numerical Integration Program #2 Maximum Range I We wish to integrate the glide equations of motion along the maximum range path since it is typically desirable to maximize ground distance when in gliding situations I How can we achieve maximum range? The University of Texas at Austin
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Equations of Motion for Gliding Flight Maximum Range Conditions Integration of the Glide EOM Numerical Integration Program #2 Maximum Range I Maximum range is achieved by I maximizing E = L/D or minimizing γ I max { L/D } ∼ max { C L /C D } I max { C L /C D } ∼ min { γ } ∼ flattest glide angle I E max = constant I How do we find E max ? I Solve for E in terms of C L I Set the first derivative of that function equal to zero and solve for C ?
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  • Spring '10
  • Staff
  • Midpoint method, Runge–Kutta methods, Numerical ordinary differential equations, Flying and gliding animals, Maximum Range Conditions

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