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By Raymond L. Bisplinghoff, Holt Ashley, Robert L.Halfman

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H+/3=0 7. AIRCRAFT AND ENGINE CHARACTERISTICS The translational motion of an aircraft having variable mass and operating in a three-dimensional space over a flat Earth is described by the seven equations (20), (27), and (28). Contained in these relations are the five functions fi = D, f2 = Q, f3 = L, f4 = T, f5 = P (29) which depend on the characteristics of the aircraft and the engine. Hence, in order to understand the nature of the differential system, a discussion of these functions is necessary.

1 Approximate derivation of the equation of motion of a rocket. A simple, approximate derivation of the equation governing the translational motion of a rocket is possible, if the linear momentum theorem is employed in combination with the following idealized scheme (Fig. 1): (a) At time instant t, all the particles located within the geometric boundary ABCA of the rocket have the same absolute velocity V,, where 0 is any point of the solid walls of the rocket. 24 ELEMENTS OF DYNAMICS -... , [CHAP.

16)) one obtains the following expression for the acceleration of the aircraft relative to the Earth: dv - dt = Vi, + v di, dt where the dot sign denotes a derivative with respect to time. I n consideration of Poisson’s formulas, the time rate of change of the unit vector tangent to the flight path and the evolutory velocity are related by di, _ - owx i, dt = r j , - q,k, (25) Consequently, if the equation of forces is combined with Eqs. (21) through (25) , the following scalar equations are derived: T COSECOS v - D - m g s i n ~- mV = 0 TcosEsinv - Q + m g s i n p c o s Y - mVr, = 0 (26) T s i n e + L - mgcospcosy - mVqw = 0 * N o special significance is implied in the signs appearing on the right-hand side of Eq.

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