Category Dynamics of Flight

The characteristic lightly damped, low-frequency oscillation in speed, pitch attitude, and altitude that was identified in Chap. 6, was seen in Chap. 7 to lead to large peaks in the frequency-response curves (Figs. 7.14 to 7.18) and long transients (Fig. 7.20). Similarly, in the control-fixed case, there are large undamped responses in this mode to disturbances such as atmospheric turbulence. These variations in speed, height, and attitude are in fact not in evidence in actual flight; the pilot (human or automatic) effectively suppresses them, maintaining flight at more or less constant speed and height. The logic by which this process of suppression takes place is not unique. In principle it can be achieved by using feedback signals derived from any one or a combination of pitch attitude в, altitude h, speed v, and their derivatives. In practice, the availability and accuracy of the state information determines what feedback is used.

Since the phugoid oscillation cannot occur if the pitch angle в is not allowed to change (except when commanded to), a pitch-attitude-hold feature in the autopilot would be expected to suppress the phugoid. This feature is commonly present in air­plane autopilots. We shall therefore look at the design of an attitude hold system for the jet transport of our previous examples. Pitch attitude is readily available from ei­ther the real horizon (human pilot) or the vertical gyro (autopilot). Consider the con­troller illustrated in Fig. 8.5. From (8.2,1) we see that the overall transfer function is

в Ges(s)J(s)

Я" = , ( І(Л (8-3,1)

0C 1 + GHSe(s)J(s)

If we write GeSe(s) = N(s)/D(s), and J(s) = N'(s)/D’(s), then the characteristic equa­tion is

D(s)D'(s) + N(s)N'(s) = 0 (8.3,2)

To proceed further we need explicit expressions for the above transfer functions. Since в is an important variable in both the short period and phugoid modes, it might be expected that neither of the two approximate transfer functions for GeSe derived in Sec. 7.7 would serve by itself. We therefore use the exact transfer function derived

from the full system of linearized longitudinal equations of motion. Then N(s) is given by (7.7,2) and D(s) by (6.2,2). The result is

^ -(1.158s[21] + 0.3545s + 0.003873)

GV – 54 + 0_750468s3 + 0.935494s2 + 9.463025 X 10~[22]s + 4.195875 X 10~3

(8.3,3)

As to J(s), a reasonable general form for this application is

к і

J(s) = — + k2 + k3s (8.3,4)

s ‘

For obvious reasons, the three terms on the right hand side are called, respectively, integral control, proportional control and rate control, because of the way they oper­ate on the error e. The particular form of the controlled system, here GeSe(s), deter­mines which of jk„ k2, k3 need to be nonzero, and what their magnitudes should be for good performance. Integral control has the characteristic of a memory, and steady – state errors cannot persist when it is present. Rate control has the characteristic of an­ticipating the future values of the error and thus generates lead in the control actua­tion. In using (8.3,4), we have neglected the dynamics of the elevator servo actuator and control surface, which would typically be approximated by the first-order trans­fer function 1/(1 + rs). Since the characteristic time of the servo actuator system, r, is usually a small fraction of a second, and we are interested here in much longer times, this is a reasonable approximation.

For the example airplane at the chosen flight condition it turns out that we need all three terms of (8.3,4) to get a good control design. This might not always be the case. Let us first look at the use of proportional control only, in which case J is a con­stant gain, k2. To select its magnitude, we use a root locus plot2 of the system, Fig. 8.6, in which the locus of the roots of the characteristic equation of the closed loop system are plotted for variable gain k2. We see that at a gain of about —0.5 the phugoid mode is nearly critically damped, that is, it is about to split into two real roots. At this gain, the phugoid oscillation is effectively eliminated. We note that at

the same gain the short period roots have moved in the direction of lower damping. The response of the aircraft to a unit step command in pitch angle with only propor­tional control is shown in Fig. 8.7a. It is clear that this is not an acceptable response. There is a large steady-state error (steady-state error is a feature of proportional con­trol) and the short-period oscillation leads to excessive hunting. The steady-state er­ror could be reduced by increasing k2 (see Exercise 8.2), but this would further de­crease the short period damping.

We digress briefly to explore the reason for the damping behavior. It was noted previously (Sec. 6.8 and Exercise 6.4) that the term of next-to-highest degree in the characteristic equation gives “the sum of the dampings.” That is, the coefficient of s3 in (8.3,3) is the sum of the real parts of the short-period and phugoid roots. Now when J = k2 the closed loop characteristic equation (8.3,2) becomes D(s) + k2N(s). That is we add a second degree numerator to a fourth degree denominator, leaving the coefficient of s3 unchanged. Thus any increase in the phugoid damping can only come at the expense of that of the short-period mode. This is exactly what is seen in Fig. 8.6. The shifts of the two roots in the real direction are equal and opposite.

To eliminate the steady-state error, we use integral control and choose

1 + (8.3,5)

The result is shown in Fig. 8.7b. The steady-state error has been eliminated, but the short-period oscillation is now even less damped. Now the damping of the short – period mode is governed principally by Mq [see (6.3,14)], so in order to improve it we should provide a synthetic increase to Mq. A signal proportional to q is readily ob­tained from a pitch-rate gyro. Since q = в in the system model we are using, we ac­complish this by adding a third term to J:

s + 1 + у j (8.3,6)

The result, shown in Fig. 8.7c is an acceptable controller, with little overshoot and no steady-state error. A commanded pitch attitude change is accomplished in about 10 sec. Note that in this illustration, all the constants in (8.3,4) are the same, that is, -0.5. Fine-tuning of these could be used to modify the behavior to reduce the over­shoot or speed up the response. Throughout this maneuver, the elevator angle remains less than its steady-state value (which it approaches asymptotically), so that the gains used are indeed much smaller than the elevator control is physically capable of pro­viding (see Exercise 8.2).

The preceding analysis does not reveal the underlying physics of why k2 damps the phugoid. This can be understood as follows. An angle в in the low frequency phugoid implies vertical velocity (i. e., h = VO). Now a positive elevator angle pro­portional to a slowly changing в implies a negative increment in angle of attack and hence in the lift as well. Thus k2 leads to a vertical force (downward) 180° out of phase with the vertical velocity (upward), exactly what is required for damping.

Although the phugoid oscillation has been suppressed quite successfully by the strategy employed above, it should be remarked that in the example case neither the speed nor the altitude has been controlled. As a consequence, the speed drifts rather slowly back to its original value, and the altitude to a new steady state.

Finally it should be noted that a controller design that is correct for one flight condition, in this case high speed at high altitude, may not be acceptable at all speeds and altitudes, for example, landing approach. In the real world of AFCS design, this problem leads, as in most engineering design, to compromises between conflicting requirements. If the economics of the airplane justifies it, gain scheduling can be adopted; that is, the control gains are made to be functions of speed, altitude, and configuration.

For linear invariant systems such as we have discussed above, the methods available for assessing stability include those used with open loop systems. One such method is to formulate the governing differential equations, find the characteristic equation of the system, and solve for its roots. Another is to find the transfer function from input to output and determine its poles. With the powerful computing methods available, it is feasible to plot loci of the roots (or poles) as one or more of the significant design parameters are varied, as we shall see in examples to follow. For the multivariable high-order systems that commonly occur in aerospace practise this is a very useful technique.

Let us now consider the stability of the loop associated with one particular in – put/output pair in the light of (8.1,6a). Since p = 1 and F and H are scalars, the trans­fer function is

This should be contrasted with the characteristic equation for the airframe alone, which is D(s) = 0, where D is the denominator of G(s). The block diagram corre­sponding to (8.2,1) is shown in Fig. 8.2. FH is the open loop transfer function, that is, the ratio of feedback to error, z/ё. Its absolute value FH is the open loop gain.

The stability of the system can be assessed from the frequency response F(ico)H(i(o). It is clear that if there is a frequency and open loop gain for which FH = -1 then un-

der those conditions the denominator of (8.2,1) is zero and Gyr(ico) is infinite. When these conditions hold, the feedback signal that is returned to the junction point is pre­cisely the negative of the error signal that generated it. This means that the system can oscillate at this frequency without any input. This is exactly the situation with the whistling public address system. For then the acoustic signal that returns to the mi­crophone from the loudspeakers, in response to an input pulse, is equal in strength to the originating pulse. Clearly the point (—1,0) of the complex plane has special sig­nificance. Nyquist (1932) has shown how the relationship of the frequency response curve (the Nyquist diagram) of FH to this special point indicates stability (see Fig. 8.3). In brief, if the loop gain is <1 when the phase angle is 180°, or if the phase is < 180° when the gain is unity, then the system is stable. The amounts by which the curve misses the critical point define two measures of stability, the gain margin and phase margin, illustrated on the Nichols diagram (see McLean, 1990) of Fig. 8.4. The

examples of Figs. 8.3 and 8.4 are for the open loop transfer function

K(s + 0.125)
.v2(0.15.t2 + 0.8.У + 1)

We have already alluded to the general nature of feedback control, and the need to provide sensors that ascertain the state of the vehicle. When human pilots are in con­trol, their eyes and kinesthetic senses, aided by the standard flight information dis­played by their instruments, provide this information. (In addition, of course, their brains supply the logical and computational operations needed, and their neuro-mus­cular systems all or part of the actuation.) In the absence of human control, when the vehicle is under the command of an autopilot, the sensors must, of course, be physi­cal devices. As already mentioned, some of the state information needed is measured by the standard flight instruments—air speed, altitude, rate of climb, heading, etc. This information may or may not be of a quality and in a form suitable for incorpora­tion into an automatic control system. In any event it is not generally enough. When both guidance and attitude-stabilization needs are considered, the state information needed may include:

Position and velocity vectors relative to a suitable reference frame.

Vehicle attitude (в, ф, ф).

Rotation rates (p, q, r).

Aerodynamic angles (a, /3).

Acceleration components of a reference point in the vehicle.

The above is not an exhaustive list. A wide variety of devices are in use to measure these variables, from Pitot-static tubes to sophisticated inertial-guidance platforms. Gyroscopes, accelerometers, magnetic and gyro compasses, angle of attack and sideslip vanes, and other devices all find applications as sensors. The most common form of sensor output is an electrical signal, but fluidic devices have also been used. Although in the following examples we tend to assume that the desired variable can be measured independently, linearly, and without time lag, this is of course an ideal­ization that is only approached but never reached in practice. Every sensing device, together with its associated transducer and amplifier, is itself a dynamic system with characteristic frequency response, noise, nonlinearity, and cross-coupling. These at­tributes cannot finally be ignored in the design of real systems, although one can use­fully do so in preliminary work. As an example of cross-coupling effects, consider the sideslip sensor assumed to be available in the gust alleviation system of Sec. 8.9. Assume, as might well be the case, that it consists of a sideslip vane mounted on a boom projecting forward from the nose. Such a device would in general respond not only to (3 but also to atmospheric turbulence (side gusts), to roll and yaw rates, and to lateral acceleration av at the vane hinge. Thus the output signal would in fact be a complicated mathematical function of several state variables, representing several

feedback loops. The objective in sensor design is, of course, to minimize all the un­wanted extraneous effects, and to provide sufficiently high frequency response and low noise in the sensing system.

This brief discussion serves only to draw attention to the important design and analytical problems related to sensors, and to point out that their real characteristics, as opposed to their idealizations, need finally to be taken into account in design.

It is frequently helpful to view a feedback loop as simply a method of altering one of the airplane’s inherent stability derivatives. When one of the main damping deriva­tives, Lp, M4, or Nr, is too small, or when one of the two main stiffnesses Ma or Nfj is not of the magnitude desired, they can be synthetically altered by feedback of the ap­propriate control. Specifically let x be any nondimensional state variable, and let a control surface be displaced in response to this variable according to the law

Д 8 = к Ax; к = const

(Here к is a simplified representation of all the sensor and control system dynamics!) Then a typical aerodynamic force or moment coefficient Ca will be incremented by

ДСа = CasA8 = CaJcAx

This is the same as adding a synthetic increment

A Cax = kCas

to the aerodynamic derivative CUr. Thus if x be yaw rate and 8 be rudder angle, then the synthetic increment in the yaw-damping derivative is

(8.1,9)

which might be the kind of change required to correct a lateral dynamics problem. This example is in fact the basis of the often-applied “yaw damper,” a stability-aug­
mentation feature. Again, if x be the roll angle and 8 the aileron, we get the entirely new derivative

С1ф = кСІВи (8.1,10)

the presence of which can profoundly change the lateral characteristics (see Exercise

8.1).

8.1 General Remarks

The development of closed loop control has been one of the major technological achievements of the twentieth century. This technology is a vital ingredient in count­less industrial, commercial, and even domestic products. It is a central feature of air­craft, spacecraft, and all robotics. Perhaps the earliest known example of this kind of control is the fly-ball governor that James Watt used in his steam engine in 1784 to regulate the speed of the engine. This was followed by automatic control of torpedoes in the nineteenth century (Bollay, 1951), and later by the dramatic demonstration of the gyroscopic autopilot by Sperry in 1910, highly relevant in the present context. Still later, and the precursor to the development of a general theoretical approach, was the application of negative feedback to improve radio amplifiers in the 1930s. The art of automatic control was quite advanced by the time of the landmark four­teenth Wright Brothers lecture (Bollay, 1951)’. Most of what is now known as “clas­sical” control theory—the work of Routh, Nyquist, Bode, Evans, and others was de­scribed in that lecture. From that time on the marriage of control concepts with analogue and digital computation led to explosive growth in the sophistication of the technology and the ubiquity of its applications.

Although open-loop responses of aircraft, of the kind studied in some depth in Chap. 7, are very revealing in bringing out inherent vehicle dynamics, they do not in themselves usually represent real operating conditions. Every phase of the flight of an airplane can be regarded as the accomplishment of a set task—that is, flight on a specified trajectory. That trajectory may simply be a straight horizontal line traversed at constant speed, or it may be a turn, a transition from one symmetric flight path to another, a landing flare, following an ILS or navigation radio beacon, homing on a moving target, etc. All of these situations are characterized by a common feature, namely, the presence of a desired state, steady or transient, and of departures from it that are designated as errors. These errors are of course a consequence of the un­steady nature of the real environment and of the imperfect nature of the physical sys­tem comprising the vehicle, its instruments, its controls, and its guidance system (whether human or automatic). The correction of errors implies a knowledge of them, that is, of error-measuring (or state-measuring) devices, and the consequent actuation of the controls in such a manner as to reduce them. This is the case whether control is by human or by automatic pilot. In the former case—the human pilot—the state in­formation sensed is a complicated blend of visual and motion cues, and instrument readings. The logic by which this information is converted into control action is only

‘In 1951 most aeronautical engineers were using slide rules and had not heard of a transfer function!

imperfectly understood, but our knowledge of the physiological “mechanism” that in­tervenes between logical output and control actuation is somewhat better. In the latter case—the automatic control—the sensed information, the control logic, and the dy­namics of the control components are usually well known, so that system perfor­mance is in principle quite predictable. The process of using state information to gov­ern the control inputs is known as closing the loop, and the resulting system as a closed-loop control or feedback control. The terms regulator and servomechanism describe particular applications of the feedback principle. Figure 8.1 shows a general block diagram describing the feedback situation in a flight control system. This dia­gram models a linear invariant system, which is of course an approximation to real nonlinear time-varying systems. The approximation is a very useful one, however, and is used extensively in the design and analysis of flight control systems. In the di­agram the arrows show the direction of information flow; the lowercase symbols are vectors (i. e. column matrices), all functions of time; and the uppercase symbols are matrices (in general rectangular). The vectors have the following meanings:

r: reference, input or command signal, dimensions (pX 1) z: feedback signal, dimensions (pXl) e: error, or actuating, signal, dimensions (pX 1) c: control signal, dimensions (mXl)

g: gust vector (describing atmospheric disturbances), dimensions (/X1) x: airplane state vector, dimensions (nX 1) y: output vector, dimensions (qX) n: sensor noise vector, dimensions (qX 1)

Of the above, x and c are the same state and control vectors used in previous chapters, r is the system input, which might come from the pilot’s controller, from an external navigation or fire control system, or from some other source. It is the com­mand that the airplane is required to follow. The signal e drives the system to make z

follow r. It will be zero when z = r. The makeup of the output vector у is arbitrary, constructed to suit the requirements of the particular control objective. It could be as simple as just one element of x, for example. The feedback signal z is also at the dis­cretion of the designer via the choice of feedback transfer function H(.v). The choices made for D(s), E(.v) and Hi. v) collectively determine how much the feedback signal differs from the state. With certain choices z can be made to be simply a subset of x, and it is then the state that is commanded to follow r.

The vector g describes the local state of motion of the atmosphere. This state may consist of either or both discrete gusts and random turbulence. It is three-dimen­sional and varies both in space and time. Its description is inevitably complex, and to go into it in depth here would take us beyond the scope of this text. For a more com­plete discussion of g and its closely coupled companion G’ the student should con­sult Etkin (1972) and Etkin (1981).

In real physical systems the state has to be measured by devices (sensors) such as, for example, gyroscopes and Pitot tubes, which are inevitably imperfect. This im­perfection is commonly modeled by the noise vector n, usually treated as a random function of time.

The equations that correspond to the diagram are (recall that overbars represent Laplace transforms):

The feedback matrix H(.v) represents any analytical operations performed on the out­put signal. The transfer function matrix J(s) represents any operations performed on the error signal e, as well as the servo actuators that drive the aerodynamic control surfaces, including the inertial and aerodynamic forces (hinge moments) that act on them. The servo actuators might be hydraulic jacks, electric motors, or other devices. This matrix will be a significant element of the system whenever there are power- assisted controls or when the aircraft has a fly-by-wire or fly-by-light AFCS.

From (8.1,1) we can derive expressions for the three main transfer function ma­trices. By eliminating x, e, c, and z we get

[I + (DG + E)JH]y = (DG + E)Jr – (DG + E)JHn + DG’g (8.1,4) from which the desired transfer functions are

Gvr = [I + (DG + E)JH] (DG + E)J (a)

Gyn = – [I + (DG + E)JH] -1 (DG + E)JH (b) (8.1,5)

Gvg = [I + (DG + E) JHI DG (c)

The matrices that appear in (8.1,5) have the following dimensions:

D(q X n); G(n X m); E(g X m); J(m X p); H(p X q); G'(n X l) The forward-path transfer function, from e to y, is

F(.v) = (DG + E)J; dimensions (q X p) so the preceding transfer functions can be rewritten as

Gyr = (I + FH)-‘F

Gyn = (I + FH) FH

Gyr = (I + FH) ‘DG’

Note that F and H are both scalars for a single-input, single-output system.

When the linear system model is being formulated in state space, instead of in Laplace transforms, then one procedure that can be used (see Sec. 8.8) is to generate an augmented form of (8.1,2). In general this is done by writing time domain equa­tions for J and H, adding new variables to x, and augmenting the matrices A and В accordingly. An alternative technique for using differential equations is illustrated in Sec. 8.5. There is a major advantage to formulating the system model as a set of dif­ferential equations. Not only can they be used to determine transfer functions, but when they are integrated numerically it is possible, indeed frequently easy, to add a wide variety of nonlinearities. These include second degree inertia terms, dead bands and control limits (see Sec. 8.5), Coulomb friction, and nonlinear aerodynamics given as analytic functions or as lookup tables.

We saw in the last section how to include nonlinear gravity effects in control re­sponse and how such effects manifest themselves in the response of a relatively se­date vehicle. Of the other two categories of nonlinearity—aerodynamic and inertial— little in a general way can be said about the first. Aerodynamic characteristics, especially for flexible vehicles at high subsonic Mach numbers, are too varied and complex to admit of useful generalizations. A very elaborate (and very costly!) aero­dynamic model is required for full and accurate simulation or computation. Not so, however, for the second category of nonlinearity. There is a class of problems, all generically connected, known by names such as roll resonance, spin-yaw coupling, inertia coupling, and so on (Heppe and Celinker, 1957; Phillips, 1948; Pinsker, 1958) that pertain to large-angle motions, or even violent instabilities, that can occur on missiles, launch vehicles, and slender aircraft performing rapid rolling maneuvers. These have their source in the pq and pr terms that occur in the pitching and yawing moment equations. A detailed analysis of these motions would take us beyond the scope of this text. Some is given in Etkin (1972), and much more is given in the cited references. One very important conclusion, due to Phillips (1948), is that there is a band of roll rates for airplanes within which the airplane is unstable. At lower roll rates, the usual stability criteria apply. At rates above the band the airplane is gyrosta – bilized in the way a spinning shell or top is. The lower of the critical roll rates for a normally stable airplane is given approximately by the lesser of

If the roll rate in a maneuver approaches or exceeds this value the possibility of a dangerous instability exists.

The preceding analysis shows how a lateral response starts, but not how it continues. For that we need solutions to the governing differential equations. As remarked ear­
lier, at the beginning of this chapter, control responses can rapidly build up large val­ues of some variables, invalidating the linear equations that we have used so far. There is a compromise available that includes only some nonlinear effects that is use­ful for transport and general aviation airplanes, which are not subjected to violent maneuvers. The compromise is to retain a linear representation of the inertia and aerodynamic effects, but to put in an exact representation of the gravity forces. This allows the angles ф, 9, and ф to take on any values. As we shall see in the following example, the solution obtained is then limited by the airplane speed growing beyond the range of linear validity, that is, it is an aerodynamic nonlinearity that then con­trols the useful range of the solution. When the procedure that led to (4.9,18 and 19) is repeated without the small angle approximations we get the following for 90 = 0 (see Exercise 7.8):

The data for the B747 jet transport previously used was incorporated into the preced­ing equations. A step aileron input of —15° was applied at time zero, the other con­trols being kept fixed, and the solution was calculated using a fourth-order Runge – Kutta algorithm. The results are shown in Fig. 7.30. The main feature is the rapid acquisition of roll rate, shown in Fig. 130b, and its integration into a steadily grow­ing angle of bank (Fig. 7.30c) that reaches almost 90° in half a minute. Sideslip, yaw rate, and yaw angle all remain small throughout the time span shown. As the airplane rolls, with its lift remaining approximately equal to its weight, the vertical component of aerodynamic force rapidly diminishes, and a downward net force leads to negative 9 and an increase in speed. After 30 seconds, the speed has increased by about 10% of m0, and the linear aerodynamics becomes increasingly inaccurate. The maxi­mum rotation rate is p = .05 rad/s, which corresponds to p = 0.01. This is small enough that the neglect of the nonlinear inertia terms in the equations of motion is justified.

(e) Attitude angles

Figure 7.30 Response of jet transport to aileron angle; Sa = —15°. (a) Velocity components, (b) Angular velocity components, (c) Attitude angles.

We have seen that useful lateral steady states are produced only by certain definite combinations of the control deflections. It is evident then that our interest in the re­sponse to a single lateral control should be focused primarily on the initial behavior. The equations of motion provide some insight on this question directly. Following a step input of one of the two controls the state variables at t = 0+ are all still zero, and from (4.9,19) we can deduce that their initial rates of change are related to the control angles by

v = ^sA

р = ад + ад (7.11,1)

r = NSa8a + M SA

The initial sideslip rate v is thus seen to be governed solely by the rudder and, since °}JSr > 0, is seen to be positive (slip to the right) when Sr is positive (left rud­der). Of somewhat more interest is the rotation generated. The initial angular acceler­ation is the vector

d) = ip + kr (7.11,2)

The direction of this vector is the initial axis of rotation, and this is of interest. It lies in the xz plane, the plane of symmetry of the airplane, as illustrated in Fig. 7.28a. The angle £ it makes with the x axis is, of course,

£ = tan-1 — (7.11,3)

P

Let us consider the case of “pure” controls, that is, those with no aerodynamic cross­coupling, so that LSp = NSa = 0. The ailerons then produce pure rolling moment and

the rudder produces pure yawing moment. In that case we get for 8n — 0 the angle gR for response to rudder from

and similarly for response to aileron:

tan i, = IJI. (7.11,5)

The angles are seen to depend very much on the product of inertia Lx. When it is zero, the result is as intuitively expected, the rotation that develops is about either the x axis (aileron deflected) or the z axis (rudder deflected). For a vehicle such as the jet transport of previous examples, with IXp = 0.4IZp, the values of Ix, lz, /., given by (4.5,11) yield the results shown in Fig. 7.29. The relations are also shown to scale in Fig. 7.28b for є = 20° (high angle of attack). It can be seen that there is a tendency for the vehicle to rotate about the principal x axis, rather than about the axis of the aerodynamic moment. This is simply because IJ1Z is appreciably less than unity. Now the jet transport of our example is by no means “slender,” in that it is of large span and has wing-mounted engines. For an SST or a slender missile, the trend shown is much accentuated, until in the limit as aspect ratio —* 0, both tan and tan tend to tan e, and the vehicle rotates initially about the xp axis no matter what control is used!

Approximate transfer functions that can be written out explicitly, and that reveal the main aerodynamic influences in a particular frequency range, can be very useful in designing control systems. In Sec. 6.8 we presented two approximate second-order systems that simulate the complete fourth-order system insofar as the characteristic modes are concerned. These same approximations can be used to get approximate transfer functions for control response.

SPIRAL/ROLL APPROXIMATION

When aerodynamic control terms are added to (6.8,9) and the Laplace transform is taken, the result is

In (7.10,1) the <31 and Я derivatives are as defined in Sec. 6.8, and the 5£s derivatives are

(7.10,2)

where 8 is either <5„ or 8r. From (7.10,1) we get the desired transfer functions. The de­nominators are all the same, obtained from (6.8,11) as

Cs2 + Ds + E

and the numerators are (again using 8 for either 8a or Sr):

NvS = a3s3 + a2s2 + axs + a0 Мф8 = bys + b0 NrS = d2s2 + dts + d0 Nps =

7.10 Approximate Lateral Transfer Functions 251

The coefficients in these relations are:

a3 = a2 = —QlgliEp + Яг) — u0Ns

а, = – ад) – u0(%sXp – + Xsg

fl() = g&rX8 – £sNr) (7.10,5)

б, = <tJ820; b0 = u0(XsNv – $VXS) + ~ ад)

d2 = <*)SXV – dx = WuJfP – %РЮ d0 = g(iesXv – $VNS)

DUTCH ROLL APPROXIMATION

Following the analysis of Sec. 6.8 and adding control terms to the aerodynamics the reduced system equations are

v = qjvv – u0r + Д%

r = Xvu + Nrr + ANC

From (7.10,6) we derive the canonical equation

With the system matrices given by (7.10,7) the approximate transfer functions are found in the form of (7.2,8) (see Exercise 7.7) with

The accuracy of the preceding approximations is illustrated for the example jet trans­port on Figs. 7.26 and 7.27. Two general observations can be made: (1) The Dutch Roll approximation is exact in the limit of high frequency, and (2) the spiral/roll ap­proximation is exact as ю -» 0. In this respect the situation is entirely analogous to that of the longitudinal case, with the spiral/roll corresponding to the phugoid, and the Dutch Roll to the short-period mode. There are ranges of frequency in the middle where neither approximation is good. We repeat that lateral approximations must be used with caution, and that only the exact equations can be relied on to give accurate results.

t

The procedure for calculating the response of the airplane to sinusoidal movement of the rudder or aileron is similar to that used for longitudinal response in Sec. 7.6. The

aerodynamics associated with the two lateral controls are given by a set of control de­rivatives:

(7.9,1)

The Laplace transform of the system equation (4.9,19) is then

where В is

(7.9,3)

О о

NUMERICAL EXAMPLE

For our numerical example we use the same jet transport and flight condition as in Sec. 7.6. A is given by (6.7,1) and the control derivatives are given in Table 7.3, from which, with the definitions of Table 7.1, the elements of В are calculated to be

The eight transfer functions are then as in (7.2,8), where f(s) is the characteristic polynomial of (6.7,2) (with s instead of A) and with the numerators as follows:

NvSa = 2.896s2 + 6.542s + 0.6220 (a)

NvSr = 5.642s3 + 379.4s2 + 167.9s – 5.934 (b)

NpSa = 0.1431s3 + 0.02730s2 + 0.1102s (c)

NpSr = 0.1144s3 – 0.1997s2 – 1.368s (d)

NrSa = -0.003741s3 – 0.002708s2 – 0.0001394s + 0.004539 (e) (7.9,5)

NrSr = -0.4859s3 – 0.2327s2 – 0.009018s – 0.05647 (f)

Ыф8а = ,1431s2 + 0.02730s + 0.1102 (g)

АГф8г = 0.1144s2 – 0.1997s – 1.368 (h)

From the transfer functions G^s) = Nij(s)/f(s), the frequency response functions Gj/iw) were calculated for both aileron and rudder inputs. The results for v. ф and r are shown on Figs. 7.26 and 7.27. The most significant feature in all of these re­sponses is the peak in the amplitude at the Dutch Roll frequency, and the associated sharp drop in phase angle.

At zero frequency we see from (7.9,5c and d) that the roll rate amplitude is zero for both inputs. All the other variables have finite values at ш — 0. Even for moderate

control angles, however, the steady-state values of /3 = v/u0 and ф are very large (see Exercise 7.10). Hence the linearity assumption severely constrains these zero fre­quency solutions. If, however, we postulate that the control angles are so small that the linearity conditions are met, then there is a steady state with constant values of ф, A and r. This can only be a horizontal turn in which the angular velocity vector is

= [0 qss rJT qss = fl sin ф rss = П cos ф