Fa2004 16.3334-5 Can linearize about various steady state conditions of flight For steady state flight conditions must have aero gravity 0 and T o So for equilibrium condition, forces balance on the aircraft L=wandT=D Also assume that P=Q=R-U=V=W=0 Impose additional constraints that depend on flight condition ◇ Steady wings-level flight→Φ=∮==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 flight dynamics that lift Lo=Clo ao where O Cl= lift curve slope- function of the equilibrium condition o ao= nominal angle of attack(angle that wing meets air flow But, as the vehicle moves about the equilibrium condition, would expect that the angle of attack will change ao+△a Thus the lift forces will also be perturbed =CLn(a0+△a)=L6+△L Can extend this idea to all dynamic variables and how they influence l aerodynamic forces and moments
Fall 2004 16.333 4–5 • Can linearize about various steady state conditions of flight. – For steady state flight conditions must have F Faero + � Fthrust = 0 and T� = 0 � = � Fgravity + � 3 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 flight condition: 3 Steady wingslevel 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 = CLαα0 where: 0 3 CLα = lift curve slope – function of the equilibrium condition 3 α0 = nominal angle of attack (angle that wing meets air flow) – But, as the vehicle moves about the equilibrium condition, would expect that the angle of attack will change α = α0 + Δα – Thus the lift forces will also be perturbed Lf = CLα(α0 + Δα) = Lf + ΔLf 0 • Can extend this idea to all dynamic variables and how they influence all aerodynamic forces and moments
Fa2004 16.33346 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) sIn B=T()T2On①)0= g sin重cose 9」 COsΦcos日 For symmetric steady state flight equilibrium, we will typically assume that≡60,①≡Φo=0,so SIn b== mg e Use euler angles to specify vehicle rotations with respect to the earth rame =OcosΦ-Rsin Φ=P+Qsin重tan+BcosΦtan (Qin重+Rcos重)sece Note that if更≈0,then≈Q Reca:≈Rol,≈ Pitch,andy≈ Heading
Fall 2004 16.333 4–6 Gravity Forces • Gravity acts through the CoM in vertical direction (inertial frame +Z) – Assume that we have a nonzero pitch angle Θ0 – Need to map this force into the body frame – Use the Euler angle transformation (2–15) ⎡ ⎤ ⎡ ⎤ 0 − sin Θ Fg = T1(Φ)T2(Θ)T3(Ψ) ⎣ 0 ⎦ = mg ⎣ sin Φcos Θ ⎦ B mg cos Φcos Θ • For symmetric steady state flight equilibrium, we will typically assume that Θ ≡ Θ0, Φ ≡ Φ0 = 0, so ⎡ ⎤ − sin Θ0 Fg = mg ⎣ 0 ⎦ B 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 sinΦtan Θ + R cos Φtan Θ Ψ˙ = (Q sinΦ + R cos Φ)sec Θ – Note that if Φ ≈ 0, then Θ˙ ≈ Q • Recall: Φ ≈ Roll, Θ ≈ Pitch, and Ψ ≈ Heading