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s12_ps09_fall03

# s12_ps09_fall03 - Unied Engineering I Fall 2003 Problem...

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Unified Engineering I Fall 2003 Problem S12 (Signals and Systems) The longitudinal dynamics of an aircraft ﬂying are given in state-space form as dh dt = V sin γ (1) dV T D dt = g sin γ + m m (2) dt = L mg mV (3) where h = altitude V = velocity of aircraft γ = ﬂight path angle g = acceleration due to gravity L = lift T = thrust D = drag m = mass of aircraft We assume that (1) the thrust is constant, (2) the aircraft ﬂies at constant coeeficient of lift, and (3) the aircraft ﬂies at constant drag coeeficient. These assumptions are not bad for the case whent he pilot releases all the controls, and makes no adjustments, say, for altitude variations. In this case, we can rewrite the equations as dh = V sin γ (4) dt dV T D 0 V 2 = g sin γ + (5) dt m m V 0 2 L 0 ( V/V 0 ) 2 mg = (6) dt mV where L 0 and D 0 are the lift and drag at the nominal velocity, V 0 . If the aircraft is intially in trim, it must also be true that T = D 0 (7) L 0 = mg (8) Equations (4)–(6) are the equations of motion in state-space form, although the equations are nonlinear.

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