Spring 2003 16.61AC1-2 Aircraft Dynamics First note that it is possible to develop a very good approximation of a key motion of an aircraft(called the Phugoid mode) using a very simple balance between the kinetic and potential energies Consider an aircraft in steady, level fight with speed Uo and height ho The motion is perturbed slightly so that U=U0+ h=ho+△h (2) Assume that E=5mu+ mgh is constant before and after the pertur bation. It then follows that g△h 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= PSCL(U4-U6 2SC(C+02-6)≈20C2(2u)(4) g△h ( PSCL9)△h Since h= Ah and for the original equilibrium fight condition L=w (pSCL)UZ=mg, we get that ptcL 2 Combine these result to obtain △+92△h=0,g≈√2 These equations describe an oscillation(called the phugoid oscillation of the altitude of the aircraft about it nominal value o Only approximate natural frequency, but value very closeSpring 2003 16.61 AC 1–2 Aircraft Dynamics • First note that it is possible to develop a very good approximation of a key motion of an aircraft (called the Phugoid mode) using a very simple balance between the 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 + ∆h (2) – Assume that E = 1 2mU2 + mgh is constant before and after the pertur￾bation. It then follows that u ≈ −g∆h U0 – From Newton’s laws we know that, in the vertical direction mh¨ = L − W where weight W = mg and lift L = 1 2ρSCLU2 (S is the wing area). We can then derive the equations of motion of the aircraft: mh¨ = L − W = 1 2 ρSCL(U2 − U2 0 ) (3) = 1 2 ρSCL((U0 + u) 2 − U2 0 ) ≈ 1 2 ρSCL(2uU0) (4) ≈ −ρSCL
g∆h U0 U0 = −(ρSCLg)∆h (5) Since h¨ = ∆h¨ and for the original equilibrium flight condition L = W = 1 2 (ρSCL)U2 0 = mg, we get that ρSCLg m = 2
g U0 2 Combine these result to obtain: ∆h¨ + Ω2 ∆h = 0 , Ω ≈ g U0 √ 2 – These equations describe an oscillation (called the phugoid oscillation) of the altitude of the aircraft about it nominal value. ✸ Only approximate natural frequency, but value very close.