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L= Qly + PR(Ix - /z ) +(P 2 - R 2 )Jxz E 4,At'= RIz - PIn + PQ( Iy - Ix) + QRJxz (1-32) The equations of linear motion from Eq. 1-15 are E 4Fx=m(U+WQ-VR) E 4Fy=m(V+UR-WP) E 4Fz=m(W+VP-UQ) (1-33) Equations 1-32 and 1-33 are the complete equations of motion for the aircraft. It will next be necessary to linearize the equations and expand the left-hand sides. A" • AIRCRAFT ATTITUDE WITH RESPECT TO THE EARTH 15 These conclusions are based on the assumptions that: 1. OX and OZ are in the plane of symmetry.
The aircraft is a rigid body. ~ 4. The earth is an inertial reference. 5. The perturbations from equilibrium are small. [6. The flow is quasisteady (to be explained in Section 1-7). These equations require that the X axis be aligned with the aircraft velocity vector while the aircraft is in equilibrium flight. The stability derivatives are defined in Table I-land are derived in Section 1-7. d t 26 LONGITUDINAL DYNAMICS It should be remembered that in these equations U = Uo, q = tpuo2 , 'u=ujUo, 'a=wjUo, and 'a=wjUo.
The aircraft is flying in straight and level flight at 40,000 ft SOLUTION OF THE LONGITUDINAL EQUATIONS (STICK FIXED) 37 with a velocity of 600 ft/sec (355 knots), and the compressibility effects will be neglected. 088 iJCD Cx,. 13 iJCL C z ,. 619 m ,. 514 sec From the preceding equations it can be seen that (c /2U)C za and are much smaller than mU / Sq and can be neglected in this case. SubstitJting these values into Eq. 1928)8(8) = 0 (1-104) The only nonzero solution of these simultaneous equations requires that the determinant of the coefficients be zero.