Altitude and Glide Path Control

One of the most important problems in the control of flight path is that of following a prescribed line in space, as defined for example by a radio beacon, or when the air­plane flies down the ILS glide slope. We discuss this case by considering first a sim­ple approximate model that reveals the main features, and then examining a more re­alistic, and hence more complicated case.

FLIGHT AT EXACTLY CONSTANT HEIGHT—SPEED STABILITY

The first mathematical model we consider can be regarded as that corresponding to horizontal flight when a “perfect” autopilot controls the angle of attack in such a way as to keep the height error exactly zero. The result will show that the speed variation is stable at high speeds, but unstable at speeds below a critical value near the mini­mum drag speed. Neumark (1950) recounts that this criterion was first discovered in 1910 by Painleve, and that it was at first accepted by aeronautical engineers and sci­entists, but later, on the basis of the theory of the phugoid, which showed no such ef­fect, was rejected as false. In fact, to the extent that pilots can control height error by elevator control alone, that is, to the extent that they approximate the ideal autopilot we have postulated, the instability at low speed will be experienced in manual flight. Since speed variation is the most noticeable feature of this phenomenon, it is com­monly referred to as speed stability.

We could analyze this case by applying (4.9,18) to the stated flight condition. However, it is both simpler and more illuminating to proceed directly from first prin­ciples. The airplane is flying on a horizontal straight line at variable speed V. It is im­plied that a is made to vary, by controlling 8e, in such a way that the lift is kept ex­actly equal to the weight at all times. The equation of motion is clearly

mV=T-D (8.5,1)

where T is the horizontal component of the thrust, and D is the drag. Since the speed cannot change very rapidly, then neither does a, and we can safely ignore any effects of q and a on lift and drag. Consequently, T and D are simply the thrust and drag or-

dinarily used in performance analysis, as displayed in Fig. 8.16. We denote the refer­ence thrust and drag by T0 and D0 and define the stability derivatives

Ту = дт/dv and Dv = dD/dV so that T — D = (T0 + TvAV) – (D0 + DvAV)

Since V = V0 + AV, and T0 = D0, (8.5,1) becomes

mAV = (Tv – Dv)AV (8.5,2)

This first-order differential equation has the solution

AV = aekt

with A = {Tv — Dv)/m (8.5,3)

Tv and Dv are the slopes of the tangents to the thrust and drag curves at their intersec­tion. If they intersect at a point such as P in Fig. 8.16, then Tv < Dv, A is negative, and the motion is stable. If, on the other hand, the flight condition is at a point such as Q, the reverse is the case. A is then >0, and the motion is unstable. If when flying at point Q there is an initial error in the speed, then it will either increase until it reaches the stable point P or it will decrease until the airplane stalls. The stable and unstable regimes are bounded by the speed V*, which is where the thrust curve is tan­gent to the drag curve. V* will be the same as Vmd of Fig. 7.1 if Tv = 0. Hence the appellation “back side of the polar” is used to describe the range V < V*, with refer­ence to the portion of the aircraft polar (the graph of CL vs. CD) for which C, is greater than that for maximum L/D.

Although we have analyzed only the case of horizontal flight, the result is similar for other straight-line flight paths, climbing or descending (see Exercise 8.8). Flight in the unstable regime can indeed occur when an airplane is in a low speed climb or landing approach. This speed instability is therefore not entirely academic, but can present a real operational problem, depending on by what means and how tightly the aircraft is constrained to follow the prescribed flight path. An important point insofar as AFCS design is concerned is that for speeds less than V* it is not possible to lock

exactly onto a straight-line flight path, and at the same time provide stability, using the elevator control alone, no matter how sophisticated the controller! To achieve sta­bility it is mandatory to use a second control. This would most commonly be the throttle, but in principle spoilers that control the drag could also be used.

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