Approximate Equations for the Longitudinal Modes
The numerical solutions for the modes, although they certainly show their properties, do not give much physical insight into their genesis. Now each oscillatory mode is equivalent to some second-order mass-spring-damper system, and each nonoscillatory mode is equivalent to some mass-damper system. To understand the modes, and the influence on them of the main flight and vehicle parameters, it is helpful to know what contributes to the equivalent masses, springs, and dampers. To achieve this requires analytical solutions, which are simply not available for the full system of equations. Hence we are interested in getting approximate analytical solutions, if they can reasonably represent the modes. Additionally, approximate models of the individual modes are frequently useful in the design of automatic flight control systems (McRuer et al., 1973). In the following we present some such approximations and the methods of arriving at them.
There are two approaches generally used to arrive at these approximations. One is to write out a literal expression for the characteristic equation and, by studying the order of magnitude of the terms in it, to arrive at approximate linear or quadratic factors. For example, if the characteristic equation (6.1,13) is known to have a “small” real root, an approximation to it may be obtained by neglecting all the higher powers of A, that is,
DA + E = 0
Or if there is a “large” complex root, it may be approximated by keeping only the first three terms, that is,
AA2 + BA + C = 0
This method is frequently useful, and is sometimes the only reasonable way to get an approximation.
The second method, which has the advantage of providing more physical insight, proceeds from a foreknowledge of the modal characteristics to arrive at approximate system equations of lower order than the exact ones. For the longitudinal modes we use the second method (see below), and for the lateral modes (see Sec. 6.8) both methods are needed.
It should be noted that no simple analytical approximations can be relied on to give accurate results under all circumstances. Machine solutions of the exact matrix is the only certain way. The value of the approximations is indicated by examples in the following.
To proceed now to the phugoid and short-period modes, we saw in Fig. 6.3 that some state variables are negligibly small in each of the two modes. This fact suggests
certain approximations to them based on reduced sets of equations of motion arrived at by physical reasoning. These approximations, which are quite useful, are developed below.
Lanchester’s original solution (Lanchester, 1908) for the phugoid used the assumptions that aT = 0, Дa = 0 and T — D = 0 (see Fig. 2.1). It follows that there is no net aerodynamic force tangent to the flight path, and hence no work done on the vehicle except by gravity. The motion is then one of constant total energy, as suggested previously. This simplification makes it possible to treat the most general case with large disturbances in speed and flight-path angle (see Miele, 1962, p. 271 et seq.) Here we content ourselves with a treatment of only the corresponding small-disturbance case, for comparison with the exact numerical result given earlier. The energy condition is
E = mV2 — mgzE = const
or V2 = ul + 2 gzE (6.3,1)
where the origin of FE is so chosen that V = u0 when zE = 0. With a constant, and in addition neglecting the effect of q on CL, then C, is constant at the value for steady horizontal flight, that is, CL = CLo = CWo, and L = CWohpV2S or, in view of (6.3,1),
L — C w(t2pu ()S + (CWopgS)zE = W + kzE (6.3,2)
Thus the lift is seen to vary linearly with the height in such a manner as always to drive the vehicle back to its reference height, the “spring constant” being
к = CWopgS (6.3,3)
The equation of motion in the vertical direction is clearly, when T — D = 0, and у = angle of climb (see Fig. 2.1)
W — L cos у = miE
or for small y,
W — L = m zE
On combining (6.3,2) and (6.3,4) we get
mzE + kzE = 0
Since CWo = mg/lpu^S, this becomes
T= тЛ/2 — = 0.138uo (6.3,5)
when u0 is in fps, a beautifully simple result, suggesting that the phugoid period depends only on the speed of flight, and not at all on the airplane or the altitude! This