Spring 2003 16.61AC1-6 Can linearize about various steady state conditions of fight For steady state fight conditions must have F= Faero+ Gravity+ Thrust=0 and T=0 o So for equilibrium condition, forces balance on the aircraft L=W andt= d Also assume that P=Q=R=U=V=W=0 Impose additional constraints that depend on the fight condition ◇ Steady wings-level flight→==6=业=0 Key Point: While nominal forces and moments balance to zero, motion about the equilibrium condition results in perturbations to the forces/moments Recall from basic fight dynamics that lift Lo=Clao, where o CI= lift coefficient, which is a function of the equilibrium condition ao= nominal angle of attack (angle that the wing meets the air fow) But, as the vehicle moves about the equilibrium condition, would expect that the angle of attack will change Q+△ Thus the lift forces will also be perturbed C(a0+△a)=L+△ Can extend this idea to all dynamic variables and how they influence all aerodynamic forces and moments
Spring 2003 16.61 AC 1–6 • Can linearize about various steady state conditions of flight. – For steady state flight conditions must have F = Faero + Fgravity + Fthrust = 0 and T = 0 ✸ So for equilibrium condition, forces balance on the aircraft L = W and T = D – Also assume that P˙ = Q˙ = R˙ = U˙ = V˙ = W˙ = 0 – Impose additional constraints that depend on the flight condition: ✸ Steady wings-level flight → Φ = Φ =˙ Θ =˙ Ψ=0 ˙ • Key Point: While nominal forces and moments balance to zero, motion about the equilibrium condition results in perturbations to the forces/moments. – Recall from basic flight dynamics that lift Lf 0 = Clα0 , where: ✸ Cl = lift coefficient, which is a function of the equilibrium condition ✸ α0 = nominal angle of attack (angle that the wing meets the air flow). – But, as the vehicle moves about the equilibrium condition, would expect that the angle of attack will change α = α0 + ∆α – Thus the lift forces will also be perturbed Lf = Cl(α0 + ∆α) = Lf 0 + ∆Lf • Can extend this idea to all dynamic variables and how they influence all aerodynamic forces and moments
Spring 2003 16.61AC1-7 Gravity Forces Gravity acts through the CoM in vertical direction(inertial frame +Z) Assume that we have a non-zero pitch angle eo Need to map this force into the body frame Use the Euler angle transformation(2-15) F=71()T()1(0)0= mg sin g cos6 cosΦcos日 For symmetric steady state fight equilibrium, we will typically assume that ≡60,Φ≡亚o=0 sin e Fg=mg cos e Use Euler angles to specify vehicle rotations with respect to the Earth frame 6=Qcos中-Rsin ④=P+Qsin重tan+ R cos p tan 亚=(Qsin更+Rcos)sec Note that if g≈0,then≈Q Recall:≈Roll,e≈ Pitch,and业≈ Heading
Spring 2003 16.61 AC 1–7 Gravity Forces • Gravity acts through the CoM in vertical direction (inertial frame +Z) – Assume that we have a non-zero pitch angle Θ0 – Need to map this force into the body frame – Use the Euler angle transformation (2–15) Fg B = T1(Φ)T2(Θ)T3(Ψ) 0 0 mg = mg − sin Θ sin Φ cos Θ cos Φ cos Θ • For symmetric steady state flight equilibrium, we will typically assume that Θ ≡ Θ0, Φ ≡ Φ0 = 0, so Fg B = mg − sin Θ0 0 cos Θ0 • Use Euler angles to specify vehicle rotations with respect to the Earth frame Θ = ˙ Q cos Φ − R sin Φ Φ = ˙ P + Q sin Φ tan Θ + R cos Φ tan Θ Ψ=( ˙ Q sin Φ + R cos Φ) sec Θ – Note that if Φ ≈ 0, then Θ˙ ≈ Q • Recall: Φ ≈ Roll, Θ ≈ Pitch, and Ψ ≈ Heading