This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: 16.333 Lecture 4 Aircraft Dynamics Aircraft nonlinear EOM • • Linearization – dynamics Linearization – forces & moments • • Stability derivatives and coeﬃcients Fall 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 U and height h . The motion is perturbed slightly so that U U = U + u (1) → h h = h + Δ h (2) → • Assume that E = 1 mU 2 + mgh is constant before and after the 2 perturbation. It then follows that u ≈ − g Δ h U • From Newton’s laws we know that, in the vertical direction ¨ mh = L − W 1 where weight W = mg and lift L = 2 ρSC L U 2 ( S is the wing area). We can then derive the equations of motion of the aircraft: ¨ 1 mh = L − W = ρSC L ( U 2 − U 2 ) (3) 2 1 = ρSC L (( U + u ) 2 − U 2 ) ≈ 1 ρSC L (2 uU )(4) 2 2 g Δ h U = − ( ρSC L g )Δ h (5) ≈ − ρSC L U ¨ ¨ Since h = Δ h and for the original equilibrium ﬂight condition L = 1 W = 2 ( ρSC L ) U 2 = mg , we get that 2 ρSC L g g = 2 m U Combine these result to obtain: ¨ Δ h + Ω 2 Δ h = , Ω ≈ g √ 2 U • 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. k ) Fall 2004 16.333 4–2 • The basic dynamics are: ˙ I ˙ I F = mv c and T = H 1 ˙ B ω × v c Transport Thm. F = v c + BI ⇒ m ˙ B BI ⇒ T = H + ω × H • Basic assumptions are: 1. Earth is an inertial reference frame 2. A/C is a rigid body 3. Body frame B fixed to the aircraft ( i,j, BI • Instantaneous mapping of v c and ω into the body frame: BI ω = Pi + Qj + Rk v c = Ui + V j + Wk ⎡ ⎤ ⎡ ⎤ P U ⎦ ⇒ BI ω B = ⎣ Q ⎦ ⇒ ( v c ) B = ⎣ V R W • By symmetry, we can show that I xy = I yz = 0 , but value of I xz depends on specific frame selected. Instantaneous mapping of the angular momentum H = H x i + H y j + H z k into the Body Frame given by ⎡ ⎤ ⎡ ⎤⎡ ⎤ H x I xx I xz P H B = ⎣ H y ⎦ = ⎣ I yy ⎦⎣ Q ⎦ H z I xz I zz R 1 Fall 2004 16.333 4–3 • The overall equations of motion are then: 1 ˙ B F = v c + BI ω × v c m ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎡ ⎤ X U ˙ − R Q U ⇒ m ⎣ Y ⎦ = ⎣ V ˙ ⎦ + ⎣ − P ⎦⎣ V ⎦ R ˙ Z W − Q P W ⎡ ⎤ U ˙ + QW − RV ˙ ⎦ = ⎣ V + RU − P W ˙ W + P V − QU ˙ B BI T = H + ω × H ⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎡ ⎤⎡ ⎤ ˙ L I xx P ˙ + I xz R − R Q I xx I xz P ⇒ ⎣ M ⎦ = ⎣ ˙ I yy Q ˙ ⎦ + ⎣ − P ⎦⎣ I yy ⎦⎣ Q ⎦ R P I xz I zz R N I zz R + I xz P ˙ − Q ⎡ ⎤ ˙ I xx P ˙ + I xz R + QR ( I zz − I yy ) + P QI xz ⎦ = ⎣ I yy Q ˙ + P R ( I xx − I zz ) + ( R 2 − P 2 ) I xz ˙ I zz R + I xz P ˙ + P Q ( I yy − I xx ) − QRI xz • Clearly these equations are very nonlinear and complicated, and we have not even said...
View
Full Document
 Fall '04
 AlexandreMegretski
 Dynamics, Aerodynamics, equilibrium condition, Ixz, Ixz 0 Izz, stability axes

Click to edit the document details