Category Helicopter Performance, Stability, and Control

Tip Speed

Low tip speeds have the advantage of low noise and good hovering performance. High tip speeds have the advantage of low rotor and drive system weights and high stored energy for autorotative entries and flares.

One of the primary considerations that limits tip speeds on the high side is rotor noise. Tip speeds of more than about 750 ft/sec are considered to be excessively noisy. A lower limit is set by the requirement to store kinetic energy in the rotor in case of a power failure.

Avoiding advancing-tip compressibility and retreating-tip stall also limits the choices of rotor speed. It is generally accepted that advancing tip Mach numbers of more than about 0.92 will produce high blade loads as a result of the Mach tuck

900 800 700

6001— Acceptable Tip Speed Choices 500 400 300 200 100

0 20 40 60 80 100 120 140 160 180 200 220

Forward Speed (knots)

FIGURE 10.4 Constraints on Choice of Tip Speeds

phenomenon described in Chapter 6. It is also generally accepted that for conventional helicopters at maximum speed, the tip speed ratio limit should not exceed 0.5 to avoid retreating blade stall. Figure 10.4 shows how all these constraints limit the tip speed options available and why the maximum speed of "pure” helicopters is about 200 knots (as of this writing).


Once values of disc loading and tip speed are selected, the solidity is the primary main rotor physical parameter to be chosen. There are three possible flight conditions that might establish solidity:

1. Hover at high altitude and temperature: The solidity is selected to achieve the maximum Figure of Merit. This criterion applies primarily to compound helicopters and flying cranes.

2. Maximum speed: The solidity is selected to prevent retreating blade stall at the design maximum speed.

3. High hadfactors: The solidity is selected to prevent retreating blade stall at the design maximum maneuverability requirement.

The basis for all these considerations will be found in Chapters 1, 3, and 5.

Disc Loading

The choice of the main rotor disc loading will be influenced by the following considerations:

Advantages of Low Disc Loading Advantages of High Disc Loadings

Low induced velocities Compact size

Low autorotative rate of descent Low empty weight

Low power required in hover Low hub drag in forward flight

In the early days of helicopter development, designers used low disc loadings because engines were heavy. With the development of the turbine engine, the consideration of engine weight became less important and designers chose higher disc loadings to take advantage of the potentials of smaller overall aircraft size and low empty weight. A possible limit to this trend, however, may be a specific customer requirement not to exceed a given disc loading. The total amount of air displaced by a hovering aircraft is a function of its gross weight, but the ability to ‘placer mine” the landing surface is a function of its disc loading. Thus interference with pilot visibility or the difficulty of concealment in snow or dust, or the ability to tumble equipment at some distance from the hover spot are all related to gross weight, but the ability to entrain gravel, dirt clods, or bushes into the recirculation pattern is related to the disc loading. The limits for operation are not yet well defined. Hovering aircraft with very high disc loadings (20 to 50 lb/ ft2) such as the Vertol Model 76 and the Ling-Temco-Vought XC-142, were proved to be quite satisfactory for operation from grass lawns or asphalt but completely unsuitable for operation above sand or gravel surfaces. It is evident that the intended use of the aircraft will have a bearing on the maximum "environmental” disc loading.

The autorotative rate of descent is a function of the disc loading, as shown in Chapters 2 and 3. The ability to autorotate is recognized as one of the inherent and desirable features of helicopters. Good autorotative capability is extremely important for single-engine helicopters since it is practiced extensively during pilot training, but even multiengine helicopters are required to demonstrate full power- off autorotations and landings by both the military and the FA A. Any rotor will autorotate if the rate of descent is high enough and, in theory at least, a successful landing can be made from any rate of descent if the stored energy in the rotor is sufficient. In practice, however, the pilot’s chances of making a successful landing at high rates of descent are limited by his reaction time and his ability to judge the

precise altitude at which to initiate the landing flare. The same problem applies to airplanes in deadstick landings. Tests made on a B-25 bomber showed that good landings could consistently be made by average pilots if the steady rate of descent in the glide was less than 2,500 ft/min, but that at higher rates of descent, increased pilot skill was required. Although these tests have not been repeated on helicopters, it is felt that the rate of descent of 2,500 ft/min is a valid division line between satisfactory and unsatisfactory for single-engine helicopters in which students will practice autorotations. For multiengined helicopters in which the autorotation capability will be demonstrated by skilled pilots, it is suggested that the boundary be raised to 3,500 ft/min. The energy methods of Chapter 3 can be used to establish the maximum disc loadings that correspond to these limits.


Although every preliminary design team will have a different process for achieving its goals, the following steps are typical and can be used as a guide.

1. Guess at the gross weight and installed power on the basis of existing helicopters with similar performance.

2. Estimate the fuel required using a specific fuel consumption of 0.5 lb/ h. p. hr for piston engines or 0.4 Ib/h. p. hr for turbines applied to the installed power.

Fuel = sfc x h. p.installcd x Mission time

3. Calculate the useful load:

U. L. = crew + payload + fuel

4. Assume a value of the ratio U. L./G. W. based on existing helicopters and trends. Use Figure 10.1 for guidance.

5. Estimate gross weight as:

U. L./G. W.

and compare this value with the original estimate. Modify the estimate of installed power and fuel if the two gross weights are significantly different.

6. Assume a disc loading at the maximum allowable value or at the highest deemed practical, and lay out the configuration based on the rotor radius corresponding to this disc loading and to the estimated gross weight.

7. Make first design decisions for main rotor tip speed, solidity, and twist based on maximum speed or maneuverability requirements.


FIGURE 10.1 Historic Trend of Ratio of Useful Load to Gross Weight

14. Continue with layout and structural design. Modify group weight statement as the design progresses.

15. Make detailed drag and vertical drag estimates based on drawings and model tests if possible.

16. Maintain close coordination between the team members to ensure that design decisions and design compromises are incorporated in the continuing updating of the various related tasks.

If the new design is to be fairly conventional, the first thirteen steps can be programmed on a computer to yield a good starting configuration even before the designer puts a sheet of paper on his board. Figure 10.2 slightly modified from reference 10.1 shows one such scheme.

To find the design gross weight, the input gross weight is varied and the program is used to calculate the weight of fuel required to perform the mission and the amount of weight available for fuel from the equation:

Fuel = G. W. – (E. W. + Payload + Crew + Unusable fuel)

FIGURE 10.2 Block Diagram of typical Computer Program for Initial Steps of Helicopter Preliminary Design

The gross weight that makes the fuel available equal to the fuel required is the design gross weight as shown in Figure 10.3.

As a fallout of this process, the difference in the slopes of the two lines of fuel weight versus gross weight yields the growth factor—the change in gross weight that is forced by a 1-pound increase in the payload or the structural weight. The growth factor, G. F., is:

where the two slopes are taken at the design gross weight. The denominator is always less than unity, so the growth factor is always greater than unity. Determining the magnitude of the growth factor in preliminary design gives the engineers an indication of the feasibility of their design. A growth factor of over 2 is an indication of serious trouble at this stage. After the helicopter is built, of course, a 1-pound increase in payload or structural weight is accepted as a 1-pound increase in gross weight, with a corresponding decrease in performance.


Although a program like the one just outlined is very useful, with or without a computer, it cannot be used to make all the necessary engineering judgments. The selection of some of the configuration parameters is influenced by considerations

not easily quantified. For this reason, it is necessary to discuss the individual parameters and their qualitative effect on performance, size, weight, cost, and operational suitability. As a backup to these discussions, a tabulation of configuration parameters for a number of current helicopters will be found in Appendix B.

Preliminary Design


The preliminary design of a new helicopter is a team effort between the designer, the aerodynamicist, and the weight engineer, with help from other specialists. The effort progresses in cycles of iteration, at the end of which the design converges into its final form, leaving each member of the design team more or less satisfied.

The effort starts with a set of requirements established by the potential customer, by a marketing survey, or by some other means. The requirements that usually have the most influence on the design are:

1. Payload.

2. Range or endurance.

3. Critical hover or vertical climb condition.

4. Maximum speed.

5. Maximum maneuver load factor.

There are always design constraints, either formally stated or understood, that limit the design alternatives in some manner. Some of the most common involve:

1. Compliance with applicable safety standards.

2. Maximum disc loading.

3. Choice of engine from a list of approved engines.

4. Maximum physical size.

5. Maximum noise level.

6. Minimum one-engine-out performance.

7. Minimum autorotative landing capability.

The primary objective of the preliminary design team is to design the smallest, lightest, and least expensive helicopter that simultaneously satisfies all of the requirements and all of the constraints.

Stability Map

The airplane aerodynamicist illustrates the way important derivatives affect the spiral stability and the Dutch roll modes by plotting lines representing zero values for Routh’s discriminant and the constant term, E, in the characteristic equation on a stability map on which the two axes represent the directional stability derivative, dN/dy, and the dihedral derivative, dR/dy. For airplanes, these two derivatives are significant since they can easily be modified during the design by changing the area of the vertical stabilizer and the wing dihedral. For helicopters, the direct relationships between these derivatives and easily changed geometric parameters are not so straightforward, especially in the case of the dihedral effect. As an be seen from the tabulation of the stability derivatives for the example helicopter,

Unstable (Dutch Roll)

contributions to dR/dy come from the main rotor, the tail rotor, the vertical stabilizer, and the fuselage (and from the horizontal stabilizer if it has dihedral.) The contributions are of nearly the same magnitude but of different signs, and none of them approach the power of the dihedral effect that can be obtained relatively easily with a wing. For this reason, the helicopter aerodynamicist is much more likely to use changes in the damping derivatives obtained by stability augmentation systems to improve the lateral-directional flying qualities.

Despite this qualification, the stability map gives a graphic illustration of the effects of the two dominant airplane-type derivatives as shown in Figure 9-23 for the example helicopter at 115 knots. In this case, the critical boundary is that associated with the constant term of the characteristic equation, E, being zero.

As one who flew model airplanes in his youth, I can attest to the fact that an effective way to cure spiral dives on hand-launched gliders is to reduce the directional stability by whittling away some of the area of the vertical stabilizer. A fix to do the same thing can be seen on the Bell 212 of Figure 9.24, which was

Source: Courtesy Rotor & Wing International.

certificated for instrument flight by the FAA. On this aircraft, a vertical destabilizer is installed ahead of the center of gravity to reduce the directional stability. A more common fix is to install a auxiliary yaw damper either as an independent unit or as part of the stability augmentation system (SAS). Increased yaw damping can be used to cure either unstable Dutch roll or spiral dive.


Basic rotor derivatives in hover Main rotor derivatives in hover

Tail rotor derivatives in hover 569

Total derivatives in hover 571

Rotor derivatives in forward flight from charts 574

Basic main rotor derivatives in forward flight 576

Basic tail rotor derivatives in forward flight 578

Main rotor derivatives in forward flight 578

Tail rotor derivatives in forward flight 582

Nondimensional horizontal stabilizer derivatives 586

Horizontal stabilizer derivatives 585

Nondimensional vertical stabilizer derivatives 587

Vertical stabilizer derivatives 587

Nondimensional fuselage derivatives 589

Fuselage derivatives 590

Total derivatives in forward flight 591

Longitudinal characteristic equation in hover 597

Period of longitudinal oscillation from simple equation 598

Yaw damping in hover 605

Longitudinal transfer function in hover 607

Response to longitudinal control step in hover 609

Fully coupled characteristic equation in forward flight 614

Longitudinal characteristic equation in forward flight 615

Longitudinal roots in forward flight 615

Forward flight matrix of equations 617

Longitudinal stability map 619

Root locus of longitudinal modes in forward flight 620

Phugoid period 624

Phugoid period from approximate method 624

Short-period stability map 626

Concave downward compliance 627

Stabilizer sizing 628

Lateral-directional characteristic equation and roots 629

Dutch roll period 630

Lateral-directional stability map 634


The following items can be evaluated by the methods of this chapter.


Characteristic equation in hover 596

Characteristic equation in forward flight 614

Period of oscillation in hover 596

Roots of the characteristic equation in hover 596

Roots of the characteristic equation in forward flight 614

Roots of the longitudinal characteristic equation in forward flight 615

Roots of the phugoid characteristic equation 623

Roots of the short-period characteristic equation 625

Roots of the lateral-directional characteristic equation 629

Roots of the Dutch roll characteristic equation 632

Routh’s discriminant in hover 604

Sizing of horizontal stabilizer 628

Stability derivatives in hover 565

Stability derivatives in forward flight, 574

Time history in hover 607

Transfer function in hover 607

Yaw damping in hover 605

The Short-Period Motion

Some of the roots in the root locus plot of the top portion of Figure 9.16 represent the short-period mode. As its name implies, the time associated with this mode is so short that it can be assumed that no speed change occurs while it is being excited. This allows us to reduce the three-degree-of-freedom analysis to two degrees of freedom for investigation of this mode. The equations in matrix form are:

And the characteristic equation of the short period mode is:

G. W. dZ

G. W. dZ


The bottom portion of Figure 9.16 shows the root locus of the short period as the stabilizer area is increased. It will be seen that these roots are yery similar to the corresponding roots of the three-degree-of-freedom system plotted above them. A stability map for the short-period mode based on the two derivatives representing angle-of-attack stability, дМ/dz, and damping in pitch, dM/dq, is given in Figure 9.19. Damping in pitch is not dramatically increased (negative sign on dM/dq) by increasing stabilizer area, but it can be increased significantly by using a rate gyro that commands changes in main rotor cyclic pitch or in horizontal stabilizer incidence. Both these methods are used on modern military helicopters such as the Sikorsky UH-60 and the Hughes AH-64.

ЭМ, ft lb JT ft/sec

The characteristics of the short-period motion are addressed in paragraph of reference 9.6. This requires that following an aft longitudinal control step, the time histories of normal acceleration and of pitch rate shall become conave downward within 2 seconds. A method for studying whether a given helicopter does or does not satisfy this requirement was first presented in reference 9.13. That analysis resulted in the criterion shown on Figure 9-20, which is similar to the stability map of Figure 919, and for the example helicopter at least would result in choosing about the same size stabilizer.

Yet another stabilizer-sizing study was reported in reference 9.14 as part of the development of the Boeing Vertol YUH-61. Here again the criterion is based on the short-period mode. The study made use of a ground-based simulator in which several pilots evaluated the acceptability of the longitudinal handling qualities as the last, or spring, term in the short-period characteristic equation was varied. This term is approximately:

The airplane aerodynamicist would call this parameter the maneuvering margin. When it is positive, the aircraft will respond to a control input by going to a new steady-state flight condition; but when it is negative, there is no equilibrium condition that will satisfy it.

The simulator pilots were asked to assign a Handling Qualities Rating (sometimes referred to as a Cooper-Harper pilot rating, for its originators) for various values of the parameter. This rating system goes from 1, or "perfect,” to 10, "completely unacceptable.” A rating of 3.5 is considered to be the boundary between "acceptable as is” and "should be fixed.” Figure 9.21 shows the results of the simulation study. Also shown are the three points representing the three horizontal stabilizer areas on the example helicopter. It may be seen that this criterion is compatible with the other two since it indicates that only the two largest stabilizers are acceptable for this helicopter.

When the determinant of the matrix subset representing the lateral-directional equations of motion for the example helicopter at 115 knots is expanded, it yields the characteristic equation:

s4 + 8.460s3 + 17.68s2 + 45.54s + 2.2548 = 0

Again three roots from this equation correspond closely to three of the roots from the fully coupled equation:

Lateral-directional subset (uncoupled): -6.842, -.7841 ± 2.4317/, -.05058 Full system (coupled): -6.602, -.7822 ± 2.4432/, -.03910

Dutch Roll

The complex pair represent an oscillation known as Dutch roll after the motion that two skaters with locked arms make as they travel down the canal. The rear view of a helicopter doing a slightly unstable Dutch roll is given in Figure 9-22 (page 630). In the case of the example helicopter, the calculated Dutch roll motion has a period of 2.6 seconds and is well damped. Many fixed-wing aircraft have unstable Dutch roll characteristics—especially at high altitude. You can observe how the autopilot is controlling this mode on a jet transport by watching the motion of the inboard ailerons, which will be going up and down in a more or less regular manner every 2 to 4 seconds.

Source: Blake AAlansky, "Stability and Control of the YUH-61 A,"JAHS 22-1,1977.

Comparison of the roots from the uncoupled and the coupled systems for the example helicopter at 115 knots show little effect of the coupling on the Dutch roll mode. This is not to be taken as a general rule, however. In many cases, the solution of the fully coupled equations will show significantly different damping than the uncoupled subset. This can be traced primarily to the effect of angle of attack on rotor torque, as represented by the derivative, dN/dz. The typical phasing of the relative motions is shown in Figure 9-22 with the helicopter pitching up as it yaws to the right. At low forward speed, rotor torque decreases with angle of attack, producing a positive damping effect; but at high speed, torque increases, giving negative yaw damping. The sign and magnitude of dN/dz can be obtained from the rotor performance charts in Chapter 3, specifically the charts of CQ/o versus CT/o for different values of collective pitch. At the trim values of |i, CT/o, and 0O, an increase in CT/o caused by an increase in angle of attack (related to X’ in the next chart of the pair) will either have a negative slope, a positive slope, or be almost flat as it is for the example helicopter for |J = .3, Cr/o = .085, and 0^ _50 = 13.5°. At higher speeds the collective pitch would be higher and the coupling would be more powerful, leading to a reduction in Dutch roll damping.

G. W. ..

і *

Approximate equations for the Dutch roll roots can be derived by making some simplifying assumptions as outlined in reference 9-15 when discussing the Dutch roll characteristics of airplanes. The first assumption is that the aircraft is allowed to roll and yaw, but its center of gravity is constrained to follow a straight flight path. This is the same as eliminating all the side force contributions from aerodynamics and roll angle while retaining only the inertial terms in the Y equation.

The lateral-directional determinant can thus be

For the example helicopter, this is:

s2 + 1.565s + 6.29 = 0 and the two Dutch roll roots are:

s = -.7823 ± 2.3826/

which are essentially the same as obtained from the more complete equations. As a matter of fact, the damping and spring terms in this example are primarily due to the damping in yaw derivative, dN/dr, and the directional stability derivative, dN/dy. Since this is probably true for most single-rotor helicopters, an approximation to the characteristic-equation of the Dutch roll mode can be written:

The roots for the example helicopter are:

s = -.7702 ± 2.4662/

Thus again little accuracy has been lost in the simplification. The period of the Dutch roll is approximately 2.5 seconds, which is fairly typical of both helicopters and airplanes of all sizes.

The success of the analysis of the Dutch Roll characteristics using either approximate or more "exact” methods is somewhat compromised by our poor understanding of the flow conditions in which the empennage and tail rotor actually operate. When evaluating the stability derivatives affected by these components, by neccessity we must assume a flow pattern that does not change drastically with small changes in flight conditions. A clue that this is not a good conclusion was illustrated in Figure 8.21. The measured flow distortion behind the rotor and engine installation of the Hughes AH-64 (and presumably, most other helicopters) can only be described as chaotic.

On the AH-64, small changes to the empennage had unexpectedly large effects on the damping of the Dutch roll mode with the stability augmentation system (SAS) turned off. (Turning the SAS on increased the damping in yaw by a factor of 3 and produced good damping even with the worst configuration.) A similar result is reported in reference 9.16, in which the Dutch roll was found to have much more positive damping in a right sideslip than in a left.

Spiral Stability

The roots of the lateral-directional equations for the example helicopter included a small negative real root that represented a time to half in amplitude of about 14

seconds. Reference 9.11 reports that if the time to double amplitude is less than 8 seconds, the helicopter is not satisfactory for flight on instruments.

The time history of the spiral mode is either a non-oscillatory convergence or a divergence. The sign of the constant term in the characteristic equation of the lateral-directional determinant determines which. The equation for E is:

j " dR dN dN dR

1J« Іду dr dy dr _

The spiral mode will be unstable if E is negative. A variety of conditions could cause this to happen. The signs of the various derivatives should be noted:

Approximate Solutions for the Phugoid Mode

The observation that the phugoid involves little change in angle of attack allows its analysis to be made by eliminating the Z equation out of the set of the three longitudinal equations of motion. Retaining only the speed and pitch equations gives:

and the phugoid’s characteristic equation may be written as:

/ /„. dX длЛ 2 1 dX dM

yG. W./g di: + dq J + G. W./g dx dq

For the example helicopter with the three different-sized horizontal stabilizers, the root loci of this mode are given in the second portion of Figure 9.16. It may be seen that although the phugoid period is approximately correct, the simplification has sacrificed reasonableness for the phugoid damping.

An even simpler way of calculating the phugoid frequency an be obtained by applying the same assumption used in hover—that since the pitching motion is occurring about a virtual center far away from the flight path, the aircraft’s moment of inertia about its center of gravity can be ignored. This reduces the

which is the same equation as that derived for hover and gives results almost identical to those. produced either by the full three-degree-of-freedom system or the two-degree-of-freedom system in which the moment of inertia is retained. For illustration, using the 54 square foot horizontal stabilizer on the example helicopter:


Calculated Natural Frequency, rad/tec

Period, sec

Full 3 degrees of freedom



Full 2 degrees of freedom



Approximate 2 degrees of freedom



Since the phugoid motion is primarily an interchange between kinetic and potential energy (speed and altitude), anything that dissipates energy in the process will add damping. A natural energy dissipator is parasite drag. Thus the aerodynamicist is faced with a dilemma because anything he does to clean up the aircraft will result in the phugoid having less damping.

The Yaw Mode in Hover

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 single-degree-of-freedom 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

right-hand 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:




1 dM

I 1 dx l дм


(Note that in the derivation for a single-rotor 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


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



— = —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:


lO) _ dBt

B,(j) дм

я1 dq

The corresponding equation in terms of time is:


This is a response that asymptotically approaches a steady value:


(j_ _ dBl deg/sec

BlJt^go dM deg of cyclic pitch


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 well-documented 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, MIL-H-8501A—reference 9-6—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 of reference 9-6 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, MIL-H-8501A is in the process of revision and will probably lose this size distinction.)



Minimum Response







Maximum Response (deg./sec)









+ 1,000

yfc. W. + 1,000





— 20



+ 1,000

$b. w. +1,000





27/,, 71


$3.W. + 1,000

+ 1,000

2Not a requirement, only a


TABLE 9.18

Summary of MIL-H-8501A 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


‘■*— ‘e T‘ -1

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




The fact that the displacement requirement is similar to a final rate line and that the damping-to-inertia 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, deg/sec

Steady Rate, degree in

A fixed-wing aircraft is symmetrical, and in most flight conditions there is little coupling between its longitudinal degrees of freedom and its lateral-directional 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 single-rotor 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 cross-coupling would be found on a fixed-wing 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 9-19 and 9.20. The matrix has been so arranged that the longitudinal equations form a submatrix in the upper-left-hand corner while the lateral-directional 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 left-hand 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 – of-attack 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 variable-stability 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:


дМ дМ Ж

dz dx dZ


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 nose-down 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 right-hand plane.

Both helicopters and airplanes with enough stabilizer area to give positive angle-of-attack 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 sharp-edged gust. The controls are held fixed so that the aircraft can demonstrate its inherent characteristics. It first shows its short-period 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 of reference 9-6. In brief, they are as given in Table 9.21.

FIGURE 9.17 Longitudinal Motions

TABLE 9.21

Summary of MIL-H-8501A Stability Requirements


Damping Requirement

Visual Flight

Instrument Flight

<5 sec

amplitude in 2 cycles

amplitude in 1 cycle

5-10 sec

At least lightly damped

amplitude in 2 cycles

10-20 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 72-square-foot 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 angle-of-artack 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 angle-of-attack 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 nose-up 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 angle-of-attack stability in level flight at 113 knots; but, as Figure 9.18 illustrates, the margin would be gone in a 1.82-g turn or pull-up. Below this point, the helicopter, upon encountering an up-gust, 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 nose-up unless prevented by an alert pilot.

What Routh’s Discriminant Tells Us

Even without solving for the roots, the characteristic equation can be made to yield useful information by using Routh’s discriminant (R. D.). For a cubic equation:

R. D.(3) = BC — AD

or in this case:

g dM

Iyy dx

Table 9.17 lists the tests of Routh’s discriminant.

TABLE 9.17

Tests of Routh’s Discriminant



Application to Hovering Helicopter

1. All coefficients are positive.

No pure divergence

For most helicopters, both dX/dx and dM/dq are negative, thus В is



is positive, thus D is positive dx

2. R. D. is positive.

No unstable oscillation

Since дМ/дх is generally

3. R. D. = 0

Neutrally stable

positive, this test is failed. Requires that дМ/дх be zero

4. R. D. is negative.


Generally true due to sign of дМ/дх

5. D = 0

6. One coefficient is

Non-oscillatory neutral stability Only true if дМ/дх = 0 Pure divergence or unstable No negative coefficients for



conventional helicopters

Note the strong role that the sign of the speed stability term, дМ/дх, plays in these tests, with a positive value leading to instability. (This is in contrast to the forward flight situation, where a positive sign on the speed derivative is usually necessary for stability, as will be discussed in the next section.) The equation for the derivative shows that it is made up of both hub stiffness (through offset flapping hinges, for instance) and the tilt of the thrust vector with a moment arm proportional to mast height. For a conventional helicopter, both effects have the same sign thus leading to Test 4 predicting an unstable oscillation. If the rotor were mounted under the helicopter, the contribution of the tilt of the thrust vector would change sign. For a teetering rotor with no hub stiffness, the sign of the entire derivative would be reversed as shown in Figure 9.10. Routh’s discriminant would then satisfy Test 2, and no unstable oscillation could occur. Test 6, however, would indicate a pure divergence, represented by the tendency of this helicopter to drift off into translational flight.

Several helicopters have actually been built with the low rotor position. One was the De Lackner "stand on” configuration shown in Figure 9.11. This

FIGURE 9.10 Effect of Rotor Location on Sign of Speed Stability Derivative

helicopter, however, had hingeless blades with high inherent hub stiffness, and consequently its stability did not benefit significantly from its unique rotor location. [13]

thrust as required and that in this case longitudinal control is used to prevent any pitching motion. If the purpose of the analysis is to study cross-coupling problems, these assumptions will have to be dropped and, instead of simply using two or three equations, all six equations should be treated simultaneously. The method of solution is the same as for the more restricted situation. The mathematical manipulation to solve for the roots becomes very tedious if done by hand but very easy if turned over to a modern computer.

Add Simplification



Nearly the same results are obtained if the helicopter is assumed to be constrained vertically so that the Z-Force equation can be eliminated. Now the equations are:

It is often useful to see a more graphical representation of the system. One such is the block diagram of Figure 9.8. An even more graphic illustration is the mechanical analog of Figure 9.9, in which viscous dampers and screw jack actuated beam-rider weights are used to generate appropriate forces and moments.

The characteristic equation is:

, / 1 дХ 1 дм 2 g dM

si ——————– +———- s2 + л——– = 0

G. W./£ dx I„ dq } I„ dx

or, for the example helicopter:

i3 + .724j2 + .115 = 0

whose roots are:

sx = —.87, i2 } = .075 ± .355/

FIGURE 9.8 Block Diagram for Two-Degree-of-Freedom System Representing a Hovering Helicopter

FIGURE 9.9 Mechanical Analog of Hovering Helicopter

Note that these roots are almost the same as those obtained when the Z-Force equation was included. The only root missing represents the damped plunge mode.

The calculated period of the oscillation is 17.7 seconds, and the time to double amplitude is 9.2 seconds. Comparison with the same parameters obtained from the more complete set of equations demonstrates that some simplifying assumptions can often be very useful in reducing the complexity of the analysis of dynamic systems, especially when the modes of motion are only weakly coupled.

The procedure can be carried even a step further if only the period of the oscillation is required. This was first pointed out in reference 9-4 by Hohenemser, who noted that since the helicopter is oscillating about a point far above, the moment of inertia about its own center of gravity can be neglected. (This is analogous to a child on a swing.) If this suggestion is followed, the characteristic equation reduces to:

This equation has the form of that of a single-degree-of-freedom system consisting of a mass on a spring:

ms2 + k = 0

for which the natural frequency is:

or in this case:

For the example helicopter this simple approach gives a period of 15.7 seconds for the longitudinal oscillation in hover. Note that under the assumption of no fuselage moment of inertia, this value applies just as well to the lateral oscillation.

A generalization can be stated about the period of oscillation—it is essentially proportional to the square root of the rotor radius. The demonstration makes use of the following line of reasoning:

In hover

д я. q у

-r—^ = — 0O + 20j — 2

Jp J ‘ 1 ClR

and from Chapter 3, with the tip speed ratio, |i, set to zero:

Thus with only a small white lie concerning the coefficient of the induced velocity ratio:

. 16CT/g

dp a

The damping derivative with zero hinge offset is:

^ai, 16

dq у R

When these are substituted into the equation for the period, we have:

If it is assumed that most modern helicopters have nearly the same values of CT/o and Lock number, y, as the example helicopter, the period can be roughly approximated by:

P = 3-2 V/? sec

(it is interesting to note that the period of a pendulum is approximately V/73 seconds and so, by analogy, the helicopter is swinging from a support 33 radii above it.)

A human can learn to control an unstable vehicle like a hovering helicopter as long as the period is significantly longer than the total time delay in perceiving an error, processing the information in the brain, and moving the appropriate control to correct the error. The period of very small one-man helicopters

approaches the lower limit and of radio-controlled models generally goes below unless some means is used to change their characteristics. For this reason, most models use a version of the Hiller servorotor system that increases the damping by an order of magnitude and hence increases the period enough to make it compatible with the capabilities of the ground-based pilot.