Category Dynamics of. Atmospheric Flight


The Dutch-roll oscillation may from a piloting standpoint be termed a nuisance factor. Its oscillatory nature is not purposely induced to perform any maneuver, and its presence may hinder the maintenance of precise flight-path control. Originally attempts were made to correlate pilot opinion with the ratio фЦЗ of the eigenvector and the damping of the oscillation. However, when it was found that pilots desired more damping for a given фІ@ at lower flight speeds, the parameter ф/(и0р) or ф/v was introduced to replace фЦЗ. Additional studies indicated that the altitude was also important with more damping being desired at higher altitude. This lead to the us

of plp0 or <f>/vE. Further refinement then replaced cycles to half amplitude

by the inverse of the time-to-half-amplitude, 1/Tj. Figure 12.20 illustrates the pilot rating boundaries plotted on a l/T^ vs. <f>/vE diagram. This is typical for fighter-type aircraft.

As is often the case in the field of handling qualities, this is not the final answer. In fact some results can be shown to correlate better with bank angle response to rudder input and root-mean-square bank angle response to random gust inputs.


When investigating open-loop roll control it is appropriate to consider the ratio of the roll time constant TR to some typical maneuver time tm, and/or the maximum roll acceleration following a unit step aileron input. The Laplace transform of the roll acceleration following a unit step aileron input (Д5Я = 1 js) can be found from (12.9,3)


TgS + 1

The maximum roll acceleration occurs at t = 0, and from the initial value theorem is

Figure 12.19 gives the pilot rating boundaries obtained from roll response studies of fighter-type aircraft (ref. 12.15). The lower boundary on these iso-opinion curves is blamed on oversensitivity of the controls and probably

Fig. 12.19 Pilot rating of open-loop roll re­sponse (from ref. 12.15).

has the same basis as the poor ratings achieved with overly large values of the CAP discussed in the section on longitudinal handling qualities.


The pilot model of Sec. 12.2 has been used by Ashkenas (ref. 12.21) to study the handling qualities associated with the closed-loop control of bank angle. This application demonstrates the use of pilot models in analyzing the pilot/aircraft system. Figure 12.16 presents the closed-loop situation. It is assumed that the pilot is functioning in a compensatory fashion to

Fig. 12.16 Compensatory closed-loop roll control.

control external disturbances [represented by %($)] to the vehicle’s bank angle. [Note that this control situation is similar to that of Fig. 12.3b with »i(<) = —*(*)•] In an attempt to achieve the equalization outlined in Sec. 12.2 the pilot adopts a form of describing function that reduces the combined transfer function of the pilot and aircraft as nearly as possible to Kjs. This results in an attempt by the pilot to generate a lead equalization term to cancel the 1 j(TBs + 1) lag present in the aircraft. In addition, if the analysis is restricted to frequencies near the system crossover frequency, it is found that to a reasonable approximation all the dynamics associated with the pilot’s neuromuscular system can be lumped in with the effective time delay as te. This is found to be sufficient for the present application. The forward-loop transfer function is thus of the form

Г(-~) j K»e~T£S(Tvs + }) • A*T*

Ada S(^7KS + 1)

which reduces to

Kve• АфТп

if the pilot can generate TL = TR. It is found that human pilots are generally limited to TL < 5 sec because of physiological factors. In addition, as TR is reduced to zero it is found that pilots do not attempt to keep TL equal to TR – It appears that as soon as the phase lag contributed by TR becomes acceptably small the pilot no longer feels the need to compensate for it. Figure 12.17 shows the TL adopted by pilots for a range of TR’s.

In this isolated control situation, it would appear that the pilot rating could depend upon closed-loop system performance, the gain generated by the pilot, Kp, and TL. Since the forward-loop transfer function always appears

to be approximately Kjs, all systems studied will tend to have similar response characteristics. If an experiment is performed wherein TR is varied and the pilot is allowed to select the system gain Аф at each step so as to be optimum in his opinion, then the rating assigned to each configuration should be mainly influenced by the TL required of the pilot. The results of such an experiment (ref. 12.21) are given in Fig. 12.18. Here ДR is the increase in pilot rating associated with TL above the basic rating for the complete vehicle. The rating becomes less favorable as the pilot is required to generate

Fig. 12.18 Effect of TR on pilot rating (from ref. 12.21).

lead (the generation of lead can be thought of as an attempt to anticipate the future input signal).

The optimum gain Аф selected by the pilot for a particular value of TR is assumed to be uniquely related to the pilot gain generated at the crossover frequency, coc. At crossover Y(icoc) • fjA5a(icoc) = 1, and for a particular value of TR, the optimum value of pilot gain, | F(«oc)|opt, is assumed to be unique. Based on these assumptions the gain Аф selected by the pilot can be found from (12.9,4) to be

A =


If соф — od and £ф — so that the two quadratic terms in (12.9,1) cancel, or if only the initial vehicle roll response is considered, then for cases where ljTs is negligible the roll-to-aileron transfer function reduces to

which corresponds to the single-degree-of-freedom approximation (9.7,7). It has been found that this transfer function affects pilot ratings significantly. When considering this response it is convenient to look at closed-loop and open-loop control situations separately. Closed-loop control tasks involve the continuous monitoring of system error by the pilot and his responses to this stimulus. Examples of this type of control include formation flying, instru­ment flight, and landing. Open-loop control differs in that a previously – learned pattern is utilized to respond to a particular flight situation. No continuous monitoring of system error as such is involved and often the maneuver is of very short duration. Examples of this form of control are obstacle avoidance, rapid turn entry, and recovery from sudden upsets.


Generally speaking, lateral-directional control is more complex than longitudinal control. This, of course, is due to the fact that two axes of rotation are involved, leading to cross-coupling effects and the use of two primary control surfaces. As a result many groups of parameters are presently being studied to determine their correlation with pilot ratings. The following is intended to introduce the reader to some of these handling qualities parameters and to indicate the trends of research.

The primary lateral-directional control task facing the pilot is the control of bank angle through the aileron control system. The transfer function relating bank angle response to aileron input can be derived from (5.11,10) by putting ДLc = Ls ASa, ANc = Ns ASa and solving for the ratio <f>/A5a. The
result is

Here the factors in the denominator represent the spiral mode (time constant Ts), the roll mode (time constant TH), and the lateral oscillation of radian frequency (o)a) and damping (ld). The values of these four constants come from the solution of the eigenvalue problem, discussed at some length in Chapter 9, where approximate solutions for them are also given. The user of the approxi­mations should note their restricted range of validity. The numerator constants are given below with the aerodynamic transfer functions replaced by the corresponding stability derivatives, and with Yv = Yr = yc = 0.




A partial list of parameter groups used in handling qualities studies includes <*>ФІша, Ійсяй, TR, Ts, фІ0, ф/пЕ, andp where (vE = Wp/p0)-


The spiral mode time constant, Ts, determines the aircraft’s tendency to maintain a given course when cruising. It is generally found that in the case of a divergent spiral mode, pilots will rate the aircraft as satisfactory provided that |TS| > 20 sec.


The pilot commands longitudinal vehicle response mainly through control column inputs. Hence it is found that the characteristics of the control system

Fig. 12.15 Effect of stick force and stick movement per g on pilot opinion (from ref. 12.15).

influence the handling qualities of the vehicle. An otherwise satisfactory – vehicle can he rated as poor due to a control system that does not “feel” right to the pilot. Figure 12.15 shows the manner in which pilot rating varied with stick movement per g and stick force per g in an aircraft with an irre­versible control system. It is seen that there is only a relatively small region where a satisfactory rating is achieved, indicating the importance of the proper selection of control system characteristics. The studies which produced these results also determined that pilots do not object to break-out or frictional forces if they are not large when compared to the stick force per g.


Substantial disagreement among results based on correlating pilot ratings with short-period damping and natural frequency (ref. 12.16) has resulted

Fig. 12.12 Longitudinal short-period oscillation—pilot opinion contours (from ref. 12.15).

in a search for more meaningful parameters. One such was derived by noting that the pilot’s opinion of an aircraft’s longitudinal dynamics is very much influenced by the response of the vehicle to control inputs. This in turn depends on terms in both the numerator and denominator of the longitudinal transfer functions, whereas the short-period characteristics appear in the denominator only [see (10.2,11)]. An important transfer function is the approximate one relating pitch rate response to elevator angle input, given by (10.2,11c and b). (See also Figs. 10.6, 10.3.) If we neglect CLi, 0L(l, and Cm ■ in (10.2,126) and convert to dimensional form, we get the approximation

g = MsL* 1 + mVsjLx g

A5e mIyV (e2 + 2toons + con2) ‘ ’

where q is in rad/sec, and s corresponds to djdt, not djdt. mn (in rad/sec) and £ are, of course, the approximate short-period frequency and damping, respectively. The quantity (LJrnV) in the numerator is the lead time constant in this response and has been identified as an important parameter for longitudinal handling qualities (ref. 12.31). In ref. 12.16 it is argued that the appropriate correlation of pilot ratings is with the parameters shown in Fig. 12.13. It is stated that when the aircraft load :factor response to angle of

514 Dynamics of atmospheric flight na < 15.0

Fig. 12.13 Pilot ratings based on LalmVojn and nalcon (from ref. 12.16).

attack (nx = (dLjW)/da = LJW) is less than 15 g/rad, pilot opinion correlates well with

—^2— and


The importance of LjmV can easily he inferred. Figure 10.6 shows that the early part of the response to elevator separates clearly into two phases— an initial pitch-up to a nearly steady Да, and a subsequent flight-path curvature associated with the lift increment AL = La Да. The magnitude of the curvature is approximately ДLjmV = (LjmV) Да. The changeover in correlating parameter at about nx of 15 appears to be due to the pilot’s concern to control load factor at large na, whereas he concentrates on flight path at low na. Figure 12.13 shows iso-opinion curves based од the use of these parameters.

An additional parameter has been developed based on the consideration of pilot comments and the physiology of the pilot (ref. 12.17). It is called the

“Control Anticipation Parameter” or GAP. The GAP is defined to be the ratio of the instantaneous angular acceleration in pitch to the steady-state change in load factor when the pilot applies a step input to the longitudinal control. Thus

CAP = -^


This theory is based on the fact that in order to make precise adjustments to the flight path, the pilot must infer from the initial attitude response of the vehicle, the ultimate response of the flight path. It is found that the best cue for sensing attitude response is the initial angular acceleration in pitch (q0) which the pilot senses through his inner ear. For precision control tasks the pertinent steady-state parameter is taken to be the change in steady-state load factor (Дnss), which is related to flight-path curvature (see Sec. 6.10).

It is found that if an aircraft has a CAP which is too small, the pilot tends to overcontrol and rates the pitch response as sluggish. This comes about as follows.

When the flight path requires adjustment the pilot moves the controls and monitors the effect of this action by noting the size of the q0 generated. If the CAP is too small no q0 will be detected because it is below the threshold of the pilot’s inner ear. Consequently he will apply more control input until a q is finally sensed. The result is an extremely large Anss and the desired response is exceeded.

On the other hand, if the CAP is too large, the pilot tends to undershoot his desired flight-path corrections, and rates the response as fast, abrupt, and too sensitive. This occurs because any slight pitch control inputs from the pilot generate a large q0 which is interpreted as the prelude to a gross change in vehicle state and not the small desired change. As a result the pilot tends to reduce or reverse his pitch control input to avoid this, resulting in a steady – state response that is too small.

The CAP can easily be derived from relations previously given. q0 is simply the initial pitching moment divided by Iy, i. e.

. MsAd,

1f> = —~—


The steady-state load factor is obtained from (10.2,9), in conjunction with the short-period approximation (10.2,11) (note that Gvs = 0 in this approxi­mation). The aerodynamic transfer functions are replaced by stability derivatives, we let s = 0, and neglect CL and CL& to get the approximate

result for the static gains:

After conversion to dimensional form we get

The acceptable range in CAP extends upward from about 15 deg/sec2/g. The upper limit has not been determined, with good pilot ratings obtained from 25-50 deg/sec2/gr. Figure 12.14 compares the pitch response of two different jet fighters. Under the conditions which prevailed for this test, the F-105A with a CAP of 16 deg/sec2/g received an adverse rating while the F-84F was rated as “good” with respect to formation flying.


In addition to the vehicle’s attitude response, the pilot also considers the speed stability of the aircraft when rating its handling qualities. This is especially true when performing such rectilinear maneuvers as the landing approach. In Sec. 11.5 it was shown that the aircraft response to a disturbance Д F0 in forward speed could be written as AV0etlT. The response is convergent for T negative. Although no clear criterion for speed stability exists it appears that if in all other respects the aircraft is rated as satisfactory, then the pilot will rate the speed response as satisfactory if it is convergent with a time to half amplitude less than 35 sec. However, it is found that under certain conditions a vehicle can be rated as acceptable even if the speed response is divergent, provided that the time to double amplitude is greater than 17 sec.


In investigating the handling qualities related to longitudinal dynamics, many workers in the field separate the problem into two parts, associated

Fig. 12.11 Effect of phugoid damping, on pilot rating (from ref. 12.15).

with the short-period response and phugoid response. Attempts are then made to correlate pilot opinion with the various parameters or with the characteristics of these two modes.

First consider the phugoid response. This mode was discussed at length in Chapter 9, and approximations to the period and damping were given in Sec. 9.2. For conventional fixed-wing airplanes the period is very long and not a significant factor in pilot rating. The damping is important however, and some experimental results (ref. 12.15) are shown on Fig. 12.11. These were obtained in flight under instrument conditions. As the damping of the phugoid mode decreases more attention must be devoted to controlling the associated low-frequency motion, which can be excited by movement of the aircraft controls or by gusts. It is seen that, generally speaking, a divergent phugoid mode (a negative £p) must be avoided. The same study that produced these results found that under visual flight conditions, a reduction in the damping from.32 to —.12 had little influence on pilot ratings.

Studies of the effect of the short-period response on pilot ratings have been made using variable stability aircraft (ref. 12.15). Although a range of results have been noted for various tasks and aircraft, the general pattern is as illustrated on Fig. 12.12. It shows a typical plot of pilot “iso-opinion” curves from such an experiment. The solid lines represent curves of constant pilot rating as the values of o)n and £ are altered. The regions of satisfactory, acceptable, poor, and unacceptable handling qualities are indicated along with the pilot comments for the various areas in the unacceptable region.


Research into aircraft handling qualities is aimed in part at ascertaining which vehicle parameters influence pilot acceptance. It is obvious that the number of possible combinations of parameters is staggering, and conse­quently attempts are made to study one particular aspect of the vehicle while maintaining all others in a “satisfactory” configuration. Thus the task is formulated in a fashion which is amenable to study. The risk involved in this technique is that important interaction effects can he overlooked. For example, it is found that the degree of difficulty a pilot finds in controlling an aircraft’s lateral-directional mode influences his rating of the longitudinal dynamics. Such facts must be taken into account when interpreting test results. Another possible bias exists in handling qualities results obtained in the past because most of the work has been done in conjunction with fighter aircraft.


Research in the field of aircraft handling qualities is undertaken for two primary reasons. These are (i) to formulate a set of design criteria which if met will ensure that a new flight vehicle will have adequate handhng qualities and (ii) to better understand how the various vehicle and mission parameters affect the human pilot. These problems are tackled by means of experi­mental programs involving trained pilots and actual aircraft or flight simulators, or through theoretical analyses involving human-pilot describing functions. Most of the recent research has been experimental work carried out with flight simulators.

The flight simulator is a device that creates the illusion of flight to a certain extent for a pilot seated in its cockpit. This is achieved partly by con­structing the cockpit to appear like that of the real aircraft. The simulator is then programmed to respond to the actuation of the controls in a fashion which resembles the response of the actual vehicle. This is accomplished by programming the vehicle’s equations of motion on an analog or digital computer, using the pilot’s control movements as the inputs to the computer system and driving the response system of the simulator with the computer output. The realism achieved with a given simulator depends to a great extent upon the visual and motion cues provided by the response system. The motion response of the simulator can range from none at all for fixed-base simulators, through limited motion in some degrees of freedom, to complete six-degree – of-freedom motion with a variable stability aircraft, which is in fact a flying simulator. The visual cues provided can include instrument displays, closed – circuit television representations of the outside world, or the full visual and instrument display provided by a variable stability aircraft. Figure 12.10 depicts a typical simulator system.

The advantages offered by the flight simulator to researchers in the field of handling qualities are many. With the simulator it is possible to isolate a single system parameter for study, allowing it to vary while holding all other parameters fixed. Situations that would involve an element of danger if a real aircraft were utilized can be simulated with no risk to life or equipment. The lower cost of operating the simulator and the control over environ­mental factors such as turbulence also favor the simulator. However, care must be exercised in interpreting the results of simulator studies. Since the simulator is usually only an engineering approximation to the actual flight system, the pilot must extrapolate his experience in the simulator in order to relate it to an actual flight situation. The ability of a pilot to do this and hence achieve meaningful handling qualities ratings depends upon his previous flight and simulator experience. In addition, care must be taken to

provide the pilot with the pertinent stimuli. For example, it would not make sense to use a fixed-base simulator to rate a vehicle in the performance of a mission which normally requires the pilot to sense vehicle motions.