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lecture_4 - 16.333 Lecture 4 Aircraft Dynamics Aircraft...

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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...
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lecture_4 - 16.333 Lecture 4 Aircraft Dynamics Aircraft...

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