ENERGY METHODS FOR OPTIMAL TRAJECTORIES

The problem to be considered briefly in this section concerns the al­titude-velocity (or Mach number) schedule, which should be flown to mini­mize the time or fuel required to go from one speed and altitude to another speed and altitude. As pointed out in Reference 7.4, this problem can be solved by the application of variational calculus. However, the result is a formidable computer program. As an alternate method, which is approximate but close to the more exact solution, one can obtain a graphical solution by considering the energy state of the airplane.

As noted previously, Equation 7.14 is an energy relationship for the rate of climb. If we let he denote the total energy, kinetic and potential, per unit weight of the airplane, Equation 7.14 can be written in terms of this specific energy as

dhe V(T-D) dt W

where

(7.62)

dhjdt will be denoted by Ps and is called the excess specific power.

The rate of change of he with respect to fuel weight, Wf, will be denoted by fs and can be written as

r _ dhe

Is ~’dWf

dhjdt dWJdt

The time required to go from one energy level to another will be given by

(7.64)

The path to minimize At at any altitude and airspeed will be the one that gives the maximum rate of change of he for a given Ps value. Therefore, if contours of constant he and constant Ps values are plotted as a function of altitude and Mach number, the path for minimum time will be the locus of points for which the contours are parallel. Similarily, contour plots of constant fs and he values provide an altitude-Mach number schedule for minimum fuel con­sumption.

As an example, consider a hypothetical subsonic turbojet airplane with the thrust and drag given by

T=T0<r

_ pV2f 2(Wlbf 1 D~ 2 + тгре V2

In this case, one can write, for the density ratio <r

В + VB2 + 4AC

a =————– 2———-

where

PofV2 2

The curves of altitude versus airspeed presented in Figure 7.31 were prepared by evaluating a over a range of airspeeds for constant values of Ps. For the standard atmosphere, a and h are related by h =44.3 (1 – o-0235)km. Curves of constant he are also shown in Figure 7.31. The altitude-airspeed schedule for climbing from sea level to Fmax at 11.8 km is indicated by the dashed line in this figure. This line passes through points on the Ps curves where these curves would be tangent to lines of constant he. In this example, where the thrust and drag are well behaved, the result is about as one would expect.

The results are substantially different, however, for an airplane designed to operate through Mach 1, particularly if the thrust is marginal in the transonic region. Such a case is presented in Figure 7.32 (taken from Ref. 7.4). As indicated by the dashed line, in this case the optimum trajectory consists

V, m/s Figure 7.31 Excess specific power and specific energy for a hypothetical sub­sonic airplane.

•i

of a subsonic climb at a nearly constant Mach number to 33,000 ft followed by a descent through the transonic drag rise region to 20,000 ft and a Mach number of 1.25. A climb to 39,000 ft at increasing Mach numbers then ensues up to 39,000 ft and Mach 2.1. The remainder of the climb up to 50,000 ft is accomplished at a nearly constant Mach number, as shown.

Figure 7.32 Excess specific power and specific energy for the F-104 at maxi­mum power and a weight of 18,000 lb(80,064 N). (L. M. Nicolai, Fundamentals of Aircraft Design, L. M. Nicolai, 1975. Reprinted by permission of L. M. Nicolai.)