Fa2004 16.3334-1 Aircraft Dynamics Note can develop good approximation of key aircraft motion( Phugoid using simple balance between kinetic and potential energies Consider an aircraft in steady level flight with speed Uo and height ho. The motion is perturbed slightly so that U0→U=U70+ ho→h=ho+△h perturbation. It then follows that u N mi t before and after the Assume that e=muz+ mgh is constar e From Newton 's laws we know that. in the vertical direction mh=L-w where weight W= mg and lift L= PSclUZ(S is the wing area) We can then derive the equations of motion of the aircraft mh=l-w OPSCL(U2 PSCL((Uo+u)2-U0x psCi(2uUo)(4 PSCL/94h U ( osce)△h(5) Since h= Ah and for the original equilibrium flight condition L W=3(pSCLU=mg, we get that ScL=2 Combine these result to obtain △+Ω2△h=0,≈V These equations describe an oscillation(called the phugoid oscilla tion of the altitude of the aircraft about it nominal value Only approximate natural frequency (Lanchester ), but value closeFall 2004 16.333 4–1 Aircraft Dynamics • Note can develop good approximation of key aircraft motion (Phugoid) using simple balance between kinetic and potential energies. • Consider an aircraft in steady, level flight with speed U0 and height h0. The motion is perturbed slightly so that U0 → U = U0 + u (1) h0 → h = h0 + Δh (2) • Assume that E = 1mU2 + mgh is constant before and after the 2 perturbation. It then follows that u ≈ −gΔh U0 • From Newton’s laws we know that, in the vertical direction mh¨ = L − W 1 where weight W = mg and lift L = 2ρSCLU2 (S is the wing area). We can then derive the equations of motion of the aircraft: ¨ 1 mh = L − W = ρSCL(U2 − U0 2 ) (3) 2 1 = ρSCL((U0 + u) 2 − U0 2 ) ≈ 1 ρSCL(2uU0)(4) 2 � � 2 gΔh ≈ −ρSCL U0 = −(ρSCLg)Δh (5) U0 ¨ ¨ Since h = Δh and for the original equilibrium flight condition L = W 1 = 2(ρSCL)U2 = mg, we get that 0 � �2 ρSCLg g = 2 m U0 Combine these result to obtain: Δh¨ + Ω2 Δh = 0 , Ω ≈ g √ 2 U0 • These equations describe an oscillation (called the phugoid oscilla￾tion) of the altitude of the aircraft about it nominal value. – Only approximate natural frequency (Lanchester), but value close