There is another simple case, however, for which the motion can be assumed to be essentially decoupled, and that is hover motion about the yaw axis. In this case the helicopter can be considered to be a singledegreeoffreedom system representing a mass and damper combination. The yawing moment equation without control input is:
. dN
Т/ + ~fffr~ 0
This has only one root representing a heavy damping:
dN 1
or for the example helicopter:
s = —.38
which indicates a pure convergence that damps to half amplitude in 1.82 seconds.
The previous section developed the basis for the study of the stability characteristics of the hovering helicopter. We will now address its control characteristics. One useful piece of information in this regard is the transfer function, which relates the response of the helicopter to an individual control input.
The transfer function is obtained as the ratio between two determinants. The denominator is the determinant already used to generate the characteristic equation, and the numerator is identical except that the control column on the
righthand side of the equations of motion is substituted for the column representing the degree of freedom of interest. As an example, let us return to the analysis of the longitudinal degrees of freedom without plunge motion in hover and obtain the transfer function for pitch attitude, 0, due to longitudinal cyclic pitch, Bx. The two pertinent equations of motion with only the nonzero derivatives retained are:
The transfer funaion of pitch attitude due to longitudinal cyclic pitch in determinant form is:
dM
dx
or
1 dM
I 1 dx l дм
ж+і;у^
(Note that in the derivation for a singlerotor helicopter, some terms in the numerator cancel themselves out just as they did in the derivation of the characteristic equation, which becomes the denominator.)
The operation represented by s is differentiation with respect to time, so the transfer function of the pitch rate, q(s) [or 0 (s)], can be written by multiplying the transfer function of pitch attitude by s:
1 dM 2
1(*) __ _________________________________
Bt(s) /1 ax 1 дм g дм
у+х,~дї
This equation can be made to produce a time history of pitch rate as a response to cyclic pitch. Most modern engineering computers now have canned programs for doing this, but it is of some interest to know that noncomputer methods exist, both for the historical perspective on how it was done in the "old days,” and to make simple checks of computer results. The method that will be illustrated is the Heaviside Expansion, which for this application is:
1 Mo) , в, o(o)
where N(s) and D(s) are, respectively, the numerator and the denominator of the transfer function, and s, is a root of the characteristic equation. In this case for the example helicopter:
= 3 J3 + 1.448j2
and
or
— = —6.78 ^—————
B, 3j + 1.448
Using the three roots:
3(.075 – .355/) + 1.448_
When the algebra is done and trignometric terms substituted for the complex variables, the result is:
— = 5.78e_ 874′6.85e 075’sin(20.34r + 57.54) В і
where the angles are in degrees. Figure 9.12 shows the time history obtained from this equation for the helicopter free both to pitch and to have horizontal translation.
If the helicopter had been mounted on trunions so that only pitching motion were permitted, the transfer function would reduce to:
dM
lO) _ dBt
B,(j) дм
я1 dq
The corresponding equation in terms of time is:
X
This is a response that asymptotically approaches a steady value:
dM
(j_ _ dBl deg/sec
BlJt^go dM deg of cyclic pitch
dq
and has a time constant of:
For the example helicopter, this time history is plotted on Figure 9.12 along with the more unconstrained system. It may be seen that the two time histories are essentially identical during the first quarter cycle of the oscillation. After that point, the effects of horizontal translation become dominant.
FIGURE 9.12 Response of Example Helicopter to Longitudinal Control Step in Hover

Guidelines for Response
Pilots have found that there are both maximum and minimum limits on the response to control motion for desirable flying qualities. If the response per inch of control motion is too small, the pilot will find the helicopter too sluggish; if the response is too large, he will complain of oversensitivity because even very small inadvertent control motions will produce large responses. (It is a welldocumented observation, however, that pilot opinion changes with experience in a given helicopter design. What might be judged to be oversensitivity initially often later becomes sluggishness as the pilot becomes more experienced in the machine.) Many flight and simulator studies have been made to determine the limits. One of the first was done in the late 1950s with a small, variable stability helicopter making instrument landing system (ILS) approaches. This program is reported in reference 9.5 and the results are summarized in Figure 9.13 as regions on the plot
FIGURE 9.13 Control Power and Damping for Acceptable Handling Qualities
of damping versus control power, which produced varying degrees of satisfactory flying qualities.
At the time of these tests, it was felt that small helicopters should be more responsive than large helicopters, and this reasoning was used when the Military Specification for helicopter flying qualities, MILH8501A—reference 96—was being written. These requirements specified the damping and the response to one – inch control steps for both visual and instrument flight conditions in all three axes. Paragraphs 3.2.13, 3.2.14, 3.3.5, 3.3.15, 3.3.18, 3.3.19, and 3.6.1.1 of reference 96 can be summarized as in Table 9.18.
A study of the results of later flight test programs such as those reported in references 9.7, 9.8, and 9.9 indicate that size is not really a factor and that all helicopters should have about the same control characteristics. (At the time of this writing, MILH8501A is in the process of revision and will probably lose this size distinction.)

Time,
sec

Minimum Response
(fy)


Damping
(ftlb/rad/sec)

Axis

Visual

Instrument

Maximum Response (deg./sec)

Visual

Instrument

Longitudinal

1

45

lb


«V

‘V

+ 1,000

yfc. W. + 1,000


Lateral

.5

27

32

— 20

18/„7

25V

+ 1,000

$b. w. +1,000

Directional

1

110

110


27/,, 71

3 ZIIJ7

$3.W. + 1,000

+ 1,000


2Not a requirement, only a

preference.





TABLE 9.18
Summary of MILH8501A Response Requirements

The displacement requirements can be converted into combinations of the two parameters of Figure 9.13 by treating each of the moment equations of motion as single degrees of freedom. For example, the longitudinal equation reduces to;
дМ – дМ *
г— В, = 1„Ъ ~ "Г" 0
дВх уу dq
For illustration, the response and damping requirements of reference 9.6 for the example helicopter are superimposed on the envelopes of Figure 9.13.
Calculated points for the example helicopter are also plotted. These indicate that this aircraft would satisfy the instrument flight requirements, while perhaps not being optimum for the ILS approach task on which Figure 9.13 was based unless equipped with some stability augmentation equipment.
Note that the response curves at zero damping go through a point obtained from the simple equation from high school physics:
s = at2
or in this case:
Cont. Pow./Inch 0/Inch
Inertia ){В/Ы) Iі
For very high damping, the terms inside the bracket approach unity, and the line becomes asymptotic to the ray defined by:
/ Cont. Pow./Inch ^ Damping
Inertia j (D/1^ Inertia
Rays also take on another meaning related to steady velocity, since the equation for angular rate is:
As time increases, the rate takes on its steady, constant value, which plots as a ray from the origin.
Cont. Pow./Inch
Inertia
Damping
Inertia
The fact that the displacement requirement is similar to a final rate line and that the dampingtoinertia ratio is the inverse of the time constant allows the maps of Figure 9.13 to be approximated in another format as combinations of the
final rate and the time constant. This alternative format is shown in Figure 9.14 as simplified approximations of the boundaries that are generally accepted today as the result of several flight test and simulator studies such as those reported in references 9.10 and 9.11. This format is useful in that flight test data in the form of time histories following step control inputs can yield the information required to judge the flying qualities directly.
Takes Too Long to Respond

Steady Rate, degree in
A fixedwing aircraft is symmetrical, and in most flight conditions there is little coupling between its longitudinal degrees of freedom and its lateraldirectional degrees of freedom. In the preceding discussion of a helicopter in hover, the same concept was used where it was assumed that there was no significant coupling. In forward flight, on the other hand, there are several obvious sources of crosscoupling, and it is not clear that they can be ignored. For a singlerotor shaft – driven helicopter, they include the yawing moment produced by main rotor torque as a function of both forward speed and rotor angle of attack; the pitching moment due to blade flapping during roll maneuvers and the rolling moment during pitch maneuvers; and the yawing moment caused by changes in tail rotor thrust during changes in forward speed. No similar sources of crosscoupling would be found on a fixedwing aircraft. In the interest of rigorousness—if not of simplicity—the analysis will first be done on the combined equations of motion and then on the two uncoupled subsets.
The six equations of motion can be written in matrix form, as in Tables 919 and 9.20. The matrix has been so arranged that the longitudinal equations form a submatrix in the upperlefthand corner while the lateraldirectional equations are in the lower right. The other two corners represent the coupling between the primary submatrices. Table 9.20 gives the numerical matrices representing the example helicopter at 115 knots.
Expanding the lefthand determinant produces the coupled system’s characteristic equation:
s8 + 10.02s7 + 28.88s6 + 48.98s5 + 26.28s4 – 137.88s3 4.627s2 + 4.315s2 + .1675 = 0
In order of decreasing damping, the roots are:
6.602, 2.907, .7822 ± 2.4432/, .1710, 0391, .1828, 1.085
The positive roots, of course, denote that the example helicopter is quite unstable—a discovery that should come as no surprise after the discussion in Chapter 8 of the inadequacy of its horizontal stabilizer area to give positive angle – ofattack stability. The roots in this form give no clue to which types of motion are stable and which are unstable. That information could be obtained from the equations with some available mathematical techniques, but for our purposes the same thing can be done by separately studying the longitudinal and the lateral – directional submatrices.
If only the longitudinal subset determinant is expanded for the example helicopter at 115 knots, the resultant characteristic equation is:
s4 + 1.545s3 – 2.618s2 + .0228 + .0949 = 0 The Routh’s discriminant is:
R. D. = .32
According to the discriminant tests, the example helicopter is longitudinally unstable in this flight condition. The characteristic equation has four real roots which match up well with those from the fully coupled equations.
Longitudinal subset (uncoupled): 2.564, .1782, .2106, .9867
Full system (coupled): 2.907, .1710, .1828, 1.085
The positive roots produce a pure divergence, with the largest one governing and making the amplitude double in less than one second. As discussed earlier, the horizontal stabilizer of only 18 square feet on the example helicopter is not large enough. Many early helicopters had no stabilizers at all; and, although they could be flown by alert pilots in conditions giving them good cues, they were difficult to fly when the pilots were distracted or did not have a good view of the horizon. A flight test program using a variablestability helicopter reported in reference 911 indicates that for flight on instruments, a time to doubb amplitude of less than about 8 seconds is unacceptable.
If this were an actual design program, the example helicopter would undoubtedly be given a bigger tail to put it on a more competitive footing with other modern designs. An alternative approach would involve an electronic auxiliary control system with various degrees of complexity to make up for the lack of inherent stability.
A guide to the resizing of the horizontal stabilizer can be generated as a stability map using two of the most important derivatives: one defining angle of attack stability, dM/dz, and one defining speed stability, дМ/дх, as variables. The effect of combinations of these two derivatives on Routh’s discriminant will define stable and unstable regions. The first step in preparing the stability map is to express the characteristic determinant as before, but leaving the two derivatives as variables:
The resulting characteristic equation is:
The loci of the two derivatives that make the discriminant vanish is the boundary between positive and negative stability. This is shown in Figure 9.15 along with the combinations that make the coefficient of the constant term, E, equal to zero. This defines the boundary between stable oscillations and unstable divergences. It is where:
dZ
дМ дМ Ж
dz dx dZ
дх
Also shown in Figure 9.15 is a boundary in the right unstable region between oscillations and divergences. This was determined by finding combinations of the two derivatives that made the roots of the characteristic equation switch from complex to real.
Note that the type of stability map of Figure 9.15 is unique to helicopters because for airplanes in trimmed level flight, the speed stability derivative is essentially zero. (Envision an airplane model in a wind tunnel with the elevator angle adjusted to make the model have no pitching moment. Then, unless compressibility is a factor, the moment will remain zero as the tunnel speed is changed. This is not true for a helicopter model whose rotor tip path plane will tilt as the tunnel speed is changed.)
One of the uses of the stability map is to predict the effect of increasing the area of the horizontal stabilizer. In this case, it may be seen that doubling the area would improve the longitudinal flying qualities by moving the example helicopter from a region of pure divergences to one of unstable oscillations, and that tripling the area would stabilize the aircraft. Stabilizer incidence can also be used to move the point on the map since it changes the speed stability parameter, дМ/дх, increasing it as incidence is decreased. It may be seen that the minimum increase in stabilizer area to achieve stability would involve increasing the incidence to take advantage of the corner of stability near the origin. In practice, of course, approaching the lower divergence boundary would introduce a risk of going unstable. Another consideration would be the possible problems of high oscillatory
FIGURE 9.15 Longitudinal Stability Map for Example Helicopter at 115 Knots

blade loads if the big download on the stabilizer required excessive nosedown flapping to balance the helicopter.
Changing the size of the horizontal stabilizer will change many other derivatives in addition to the two on the stability map. The full effect is shown in a different format in the top portion of Figure 9.16 as the locus of roots of the characteristic equation as the stabilizer area is increased. Roots that have no imaginary components represent either pure divergences or convergences, and roots with imaginary components represent oscillations—unstable if they are in the righthand plane.
Both helicopters and airplanes with enough stabilizer area to give positive angleofattack stability will exhibit oscillations in forward flight. The oscillation
FIGURE 9.16 Root Locus Plots as Horizontal Stabilizer Area Is Increased

typically has a period of 10 to 30 seconds and primarily involves an interchange between forward speed and altitude—that is, between kinetic and potential energy at a nearly constant angle of attack.
This oscillation was first observed by W. F. Lanchester, a pioneer British aerodynamicist working with model gliders at about the same time that the Wrights were doing their first testing. Lanchester, in naming the motion, chose phugoid, based on a Greek verb that he thought meant "to fly.” Actually the verb means "to flee,” but we have happily used the word ever since.
Figure 9.17 shows the flight path of a helicopter following a brief encounter with a sharpedged gust. The controls are held fixed so that the aircraft can demonstrate its inherent characteristics. It first shows its shortperiod response, which disappears rapidly because it is well damped in this example. The helicopter then goes into its phugoid motion, shown as slightly unstable in this illustration.
The assumption that the analysis can be based on the uncoupled equations has been shown in reference 9.12 to overestimate slightly the damping of the phugoid mode. This can be traced primarily to the omission of the coupling that gives pitching moments as a function of roll rate represented by the derivative, дМ/dp. During an uncontrolled phugoid in flight, the helicopter will have a rolling motion phased with the pitching motion in such a way that the damping of the system is slightly reduced.
The imaginary component of the root is the frequency of the oscillation in radians per second. With a stabilizer area of 72 square feet, the example helicopter has a frequency of 0.37 radians per second or a period of just over 17 seconds. Even though this point is unstable, doubling in amplitude in about 10 seconds, it would still be considered satisfactory for visual flight but not for instrument flight. The acceptability of oscillations in the two flight regimes is indicated by the specifications found in paragraphs 3.2.11 and 3.6.1.2 of reference 96. In brief, they are as given in Table 9.21.
FIGURE 9.17 Longitudinal Motions
TABLE 9.21
Summary of MILH8501A Stability Requirements
Period

Damping Requirement

Visual Flight

Instrument Flight

<5 sec

amplitude in 2 cycles

amplitude in 1 cycle

510 sec

At least lightly damped

amplitude in 2 cycles

1020 sec

Not double in 10 sec

At least lightly damped

>20 sec

No requirement

Not double in 20 sec


The justification for not imposing a requirement in visual flight for periods above 20 seconds is that the time is so long that the pilot instinctively corrects for any instability with his normal control motions.
The 72squarefoot stabilizer on the example helicopter would allow the aircraft to satisfy the visual flight requirement, but a somewhat larger tail (or an auxiliary stability system) would be needed for instrument flight where the cues are not as good and the pilot has other duties that require his attention.
In an actual design project, of course, the empennage parameters should be selected to give good flying qualities not only at one flight condition but throughout the entire flight envelope as well. A full analysis of the example helicopter then would involve more than just the 115 knots at sea level that has been chosen for illustration. It is true, however, that a stabilizer area chosen to satisfy the requirements at one forward speed would not be dramatically different from the one that satisfies them at any other speed. This is because even though the upsetting effect on angleofartack stability of the rotor flapping is proportional to tip speed ratio squared—as can be seen from the equations for longitudinal flapping in Chapter 3—the correcting effect of the horizontal stabilizer is proportional to velocity squared, thus maintaining an overall balance for the entire helicopter.
It should be recognized, however, that a helicopter that is stable in level flight will probably be unstable at some higher load factor at the same speed. This is in contrast to an airplane, whose angleofattack stability is nearly invarient with wing angle of attack. The difference is due to the contribution of the rearward tilt of the thrust vector; the higher the thrust, the stronger the destabilizing moment due to noseup flapping. The airframe, on the other hand, maintains a constant stabilizing influence. •
The use of a large horizontal stabilizer of 90 square feet on the example helicopter would be enough to provide a margin of positive angleofattack stability in level flight at 113 knots; but, as Figure 9.18 illustrates, the margin would be gone in a 1.82g turn or pullup. Below this point, the helicopter, upon encountering an upgust, would pitch down by itself because of the stabilizing effect of the large horizontal stabilizer. Above this point, however, the rotor would overpower the stabilizer and would pitch the helicopter noseup unless prevented by an alert pilot.