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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 coecients Fall 2004 16.333 41 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 ight with speed U and height h . The motion is perturbed slightly so that U U = U + u (1) h h = h + h (2) Assume that E = 1 mU 2 + mgh is constant before and after the 2 perturbation. It then follows that u g h U From Newtons laws we know that, in the vertical direction mh = L W 1 where weight W = mg and lift L = 2 SC L U 2 ( S is the wing area). We can then derive the equations of motion of the aircraft: 1 mh = L W = SC L ( U 2 U 2 ) (3) 2 1 = SC L (( U + u ) 2 U 2 ) 1 SC L (2 uU )(4) 2 2 g h U = ( SC L g ) h (5) SC L U Since h = h and for the original equilibrium ight condition L = 1 W = 2 ( SC L ) U 2 = mg , we get that 2 SC L g g = 2 m U Combine these result to obtain: h + 2 h = , 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 42 The basic dynamics are: I I F = mv c and T = H 1 B v c Transport Thm. F = v c + BI m B BI T = H + 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 into the body frame: BI = Pi + Qj + Rk v c = Ui + V j + Wk P U BI 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 43 The overall equations of motion are then: 1 B F = v c + BI 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 + 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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