Two theories are presented for the analysis of the optimum burning program for horizontal flight. The first theory is based on Green's theorem which leads to a simple straightforward proof of the necessary and sufficient conditions for the existence of a maximum. A linear relationship between thrust and engine mass flow is assumed. Hibbs' results are generalized by lifting any restriction concerning the shape of the drag polar and by considering a finite maximum burning rate for the engine. Two regions of best range are detected, one subsonic-transonic, the other one supersonic. The influence of the boundary conditions on the optimum technique of flight is discussed. A simple similarity rule for stratospheric solutions is pointed out.The second theory, based on the variational method of Lagrange multipliers, considers the general case of an arbitrary one-to-one correspondence between thrust and propellant mass flow. Particular attention is devoted to the case of a polygonal thrust characteristic. An example of application is worked out.The concept of index value presented in a previous note is here emphasized. Such an index value appears to be a very effective device in controlling the composition of the optimum path for discontinuous Eulerian solutions.Nomenclature a = speed of sound (ft sec -1 )partial derivative of drag with respect to lift, calculated at constant altitude and constant velocity D v = partial derivative of drag with respect to velocity, calculated at constant altitude and constant lift (lb ft" 1 sec) g -acceleration of gravity (ft sec -2 ) h = altitude (ft) K = CDI/CL 2 = ratio of induced drag coefficient to square of lift coefficient L = lift (lb) m = instantaneous mass of the aircraft (lb ft -1 sec 2 ) m* -m exp [F/F«] = modified mass (lb ft -1 sec 2 )Mach number atmospheric pressure (lb ft -2 ) index value (ft sec -1 ) Reynolds number reference surface (ft 2 ) time (sec) thrust (lb) absolute velocity of the solid part of the aircraft, i.e., velocity referred to a reference frame rigidly connected to the earth (ft sec -1 ) equivalent relative exit velocity of the gases, i.e., velocity referred to a reference frame rigidly connected to the solid part of the aircraft (ft sec" 1 ) stalling velocity (ft sec -1 ) instantaneous weight of the aircraft (lb) horizontal distance (ft) -m = instantaneous burning rate of the rocket engine (lb ft -1 sec) ratio of specific heat at constant pressure to specific heat at constant volume = 1.4 for air Lagrange multiplier (sec) V/V e = ratio of flight velocity to equivalent exit velocity of the engine Lagrange multiplier (lb -1 ft 2 sec -2 ) m/m r = nondimensional mass m*/m r = nondimensional modified mass air density (lb ft -4 sec 2 )
A n ACpl -At P 2 = cos L vi + «i J ^ (i _ k) vm J-^v This is in agreement with the solution obtained by Evvard. 5 SUMMARY Three types of edges can border a region: (1) a leading edge, (2) a trailing edge, and (3) an edge bordering another region. For types (2) and (3) the method given in references 2 and 3 applies, without regard to the homogeneous solution to the integral equation. For type (1) an additional analysis similar to that given here must be used. restated the inconsistency of deformation theories and pointed out that a more complex incremental theory than the PrandtlReuss should be used if initial imperfections of the specimens did not prove to be responsible for the mentioned disagreement. Here, it is shown that, if a small initial deflection is taken into account, the incremental theory can afford an explanation of experimental facts in plastic buckling of compressed plates, even in the case where the discrepancy appears most marked.Let s be the thickness and b be the width of the strip. One of its long sides is free; the other is simply supported. Let
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