Category Helicopter Performance, Stability, and Control

Fortunately, the airframe derivatives are somewhat more straightforward to evaluate than the rotor derivatives. Sometime in the latter stages of the design process, wind tunnel results should be available for determining static derivatives directly. Up to that time, the equations of Chapter 8 can be used with the following methods.

Horizontal Stabilizer

Equations for the steady X and Z forces of the horizontal stabilizer were presented in Chapter 8. They can be used almost as is for obtaining derivatives (see tables

9.10 and 9.11). The exception is the addition of a term to the equation for stabilizer angle of attack to account for the time required for the main rotor downwash to reach the stabilizer during nonsteady conditions. The equation now becomes:

TABLE 9.10

Nondimensional Horizontal Stabilizer Derivatives

Value for

Example Helicopter

-.00515 -.00076 .00077 .00108 .00076 .00331 -.00014

and the time to double amplitude is:

_ .693

It is not necessary to solve for the roots of the characteristic equation to determine whether the system is stable or unstable. Instead, we have only to examine the combination of coefficients known as Routh’s discriminant (R. D.). For a cubic equation of the form:

As* + Bs2 + Cs + D = 0 Routh’s discriminant, R. D. (3), is:

R. D.(3) = BC — AD For a quartic equation of the form:

As4 + Bs* + Cs2 + Ds + E = 0 Routh’s discriminant, R. D. (4), is:

R. D.(4)=BCD – AD2-B2E
(D must be positive)

For a quintic equation of the form:

As5 + Bs4 + Cs* + Ds2 + Es + F = 0

Routh’s discriminant, R. D. (5), is:

R. D. (5) = D(BC – AD)(BE – AF) – B(BF – AF)2 – F(BC – AD)2

Several tests of the discriminant are used to produce clues about the types of time history that are characteristic of the system:

The equilibrium equations of Chapter 8 can be converted into equations of motion by accounting for forces and moments corresponding to inertia effects associated with accelerations, either linear or angular, and combinations of velocities.

G W

XM + XT + XH + XV + XF= G. W. sin0 + —^(x – yr + zq)

£

^ G. W.

YM + YT + Yv + YP = – G. W. sin Ф +——————– (y + xr-zp)

і

G. W.

ZM + ZT + ZH + Zv + Zp – —G. W. cos 0 +——————- (z — x q + уp)

S

Rm + TMhM + ZMyM + YThT f Yvhv + YFhF + RF = lxxp — qr{lyy — Izz)

MM — XMhM + ZJM + MT — XThT + ZTlT — XHhH + ZHlH — Xvhv + MF + Z pip — Xphp = lyyq — pr (I zz — Ixx)

NM — YMlM — YTlT — Yyly + Nf — Ypip = Izzr — pq(Ixx — Iyy)

Figure 9.5 shows the sign convention used.

Thee centrifugal forces and the gyroscopic moments in the foregoing equations fall directly out of the analysis of the dynamics of a rigid body that can

simultaneously translate and rotate along and about all three axes. Many of us, however, need some graphical help to accept these terms. Figure 9.6 is an attempt to provide this help.

From a rigorous standpoint, the set of six equations of motion should be augmented with three more equations representing the coning, longitudinal flapping, and lateral flapping of the rotor, which is not attached very rigidly to the airframe. As was shown in Chapter 7, however, the time constant for the flapping of conventional rotor blades corresponds to one-quarter to one-half of a rotor

revolution. This rapid response justifies the use of the quasi-static assumption, which eliminates blade motions as separate degrees of freedom and simulates replacing the rotor with a black box at the top of the mast, which essentially produces forces and moments instantaneously in response to changes in flight condition or control inputs.

There are some studies, however, in which the coning and cyclic flapping of the rotor on a short time basis cannot be ignored. These include the prediction of the immediate response to a gust or to a control input or the design of a high-gain

stability and control augmentation system (SCAS). For these situations, methods for writing the expanded set of equations are outlined in reference 9.1.

Accepting the quasi-static assumption, the set of six equations is a perfectly general representation of the helicopter in flight and can be used in sophisticated computer programs and flight simulators in which the instantaneous values of the aerodynamic terms are continually calculated as a function of the flight conditions. Even without a computer, the equations of motion can be used in a meaningful analysis by converting them into linear partial-differential small-perturbation equations and treating them as the equations of a multi-degree-of-freedom spring-damper – weight system.

The use of the linear differential equation method of analysis relies on the assumption that the change in the conditions of a dynamic system can be represented by superposition of the individual linear effects of changes in the independent variables. In such a complicated system as an aircraft in flight, the linearity is only rigorously true for very small changes or "small perturbations” from the trim values of the independent variables; but experience has shown that the assumption is valid enough to make the method very useful for the stability analysis of all types of aircraft.

Using the longitudinal force equilibrium equation as an example, the conversion goes like this:

1. Gather all terms on the left-hand side into a single aerodynamic term, X, and use small-angle assumptions on the right-hand side.

G. W.

—— (x-yr + zq)

Note that in this analysis the linear displacements along the body axes, X, Y, and Z are used as prime degrees of freedom. It is more common to see the corresponding velocities x, у, and z (or u, v, and w) used, since the forces and moments are functions of velocities and accelerations but not of the displacements of the aircraft in flight. Two special problems, however, have influenced the choice of displacements instead of velocities: flight simulation of manual or automatic hover over a spot, and the dynamic analysis of a helicopter mounted in a wind tunnel on moderately flexible supports. Although neither of these problems will be addressed here, the present choice of variables allows them to be studied when they arise while putting practically no unnecessary complication into more routine

studies.

2. Rewrite the left-hand side in terms of stability derivatives multiplied by pertinent changes in the six degrees of freedom and the four control inputs. Since the aerodynamics are affected only by velocities (with an exception to be covered later), only the first derivatives with respect to time appear in the equations, and all these variables as well as control motions are understood to now be changes from the initial trim conditions.

Note that the terms associated with the control inputs—main rotor collective pitch, tail rotor collective pitch, and rotor cyclic pitch—have all been gathered on the right-hand side in order to separate free from forced motion. The aerodynamic terms may be seen to be functions only of velocities with the exceptions of dZ/dz and dM./dz, which will later be shown to be associated with the time required for a change in rotor downwash to reach the horizontal stabilizer.

There are three methods for studying the flying qualities of the helicopter using these equations. The first method, discussed in the previous section, involves substituting the general solution in the form:

x = x(j)e"

From this, the characteristic equation can be formed; evaluated by Routh’s discriminant; and/or solved for its roots, which can be related. to periods and damping. This classic method will be illustrated in the remainder of this chapter. A second method involves programming the equations on a so-called analog computer. This uses electronic components consisting of capacitors, resistors, and

inductances to represent spring, damper, and mass terms, respectively, in the equations as if they were written for mechanical systems. Voltages within the circuits represent various degrees of freedom and can be used to produce time histories in response to simulated gusts or control motions. This method was in wide use from about 1955 to 1975, but has now largely been supplanted by the third method, using digital computers, which can both do the classical analysis and produce time histories more accurately than can the analog computers.

In any case, it is necessary first to write the equations of motion in numerical form by evaluating all the derivatives.

The stability derivatives can be evaluated by several methods. The most accurate is, naturally, also the costliest unless it is to be used over and over. That method consists of programming a computer to do the type of numerical calculations illustrated in Chapter 8 to solve first for the trim conditions and then for the changes in aerodynamic forces and moments due to sequential unit variations in each degree of freedom and control input. The heart of this type of program is the subroutine that analyzes the main and tail rotors. Programs with various levels of sophistication are operational in all the major helicopter companies and research organizations. A description of one will be found in reference 9-2.

A noncomputer means of obtaining values for the stability derivatives of the main and tail rotors consists of differentiating equations for the aerodynamic coefficients and flapping angles such as those found in Chapter 3. This procedure, widely used before computer technology matured, still has value for quick analysis and for verifying the more complex computer programs as they are being developed. A description of this method, along with results in the form of charts, is given in reference 9-3.

In the following pages, a third method will be used. It makes use of the rotor performance charts in Chapters 1 and 3 to determine changes to the non – dimensional rotor coefficients as basic parameters are varied.

In many analyses, the derivatives are presented as the ratio of the change in forces divided by the mass of the aircraft and the change in moments about each axis divided by the appropriate moment of inertia. That approach has two advantages: it makes the equations of motion look more compact, and it provides derivatives that can be compared with respect to magnitude from one aircraft to another even if they vary widely in size. This latter reason seems to be of more significance for fixed-wing aircraft than for helicopters, for which many different fuselages an be supported by rotors that are essentially alike. One disadvantage with this system is that these derivatives change magnitude with gross weight and with loadings that change the moments of inertia. In this book, we will leave the derivatives in their dimensional form in order to avoid making them any more abstract than they already are.

Rotor Derivatives near Hover

In the case of hover, only the aerodynamics of the two rotors are important, and the required derivatives can be simply evaluated using the charts and equations already presented in Chapters 1, 2, 3, and 7. The significant rotor derivatives are listed in Tables 9.1 through 9.4 with numerical values corresponding to the example helicopter. Table 9.1 gives the basic rotor derivatives. Tables 9.2 and 9.3 are separate tables for the dimensional derivatives of the main and tail rotors. Finally Table 9.4 is a summary table for the entire helicopter. These tables can serve as checklists for the analysis of any other helicopter.

The basic partial derivatives for the example helicopter have been obtained from the performance charts of Chapter 3 in level flight at 115 knots (jl = .3) and are tabulated in Table 9.5 in the form:

d Dependent Variable d Independent Variable

The derivatives with respect to |i have been obtained by taking the difference between charts for p = .25 and p = .35. The others are from the charts for jl = .30. Since the rotor parameters are plotted against CT/<5, it is used as an intermediate variable, as illustrated by Figure 9.7.

Other simple derivatives needed in the analysis are listed in tables 9.6 and 9.7 along with values for the example helicopter at 115 knots.

The system we have been using for illustration is known as a single-degree-of – freedom system since only the value of a single variable, x, is needed to describe its instantaneous position. Aircraft—and, indeed, most other dynamic systems—have

more than a single degree of freedom. For example, an airplane has six degrees of freedom, representing three linear displacements—forward, upward, and side­ward—and three angular displacements in pitch, roll, and yaw. A helicopter has at least three more, representing rotor coning, longitudinal flapping, and lateral flapping. For illustration of the principles, we will deal with a two-degree-of – freedom system obtained by adding one more spring, weight, and damper to our original system. Figure 9-4 shows the new system.

To define the instantaneous position of this system, we must know both x and y, so this is a two-degree-of-freedom system, and there are two simultaneous equations of equilibrium representing all the forces acting on both weights.

kxx — kyy + cxx — Cyj + mxx = 0
k у + cyy + my{x + у) = 0

Using the substitution,

* = *(j)e"
y=y(i)e"

the two equations of equilibrium become:

(kx + scx + s2mx)x(s) + {~ky – scy)y(s) = 0
s2myx(s) + (ky + scy + s2my)y(s) = 0

or in matrix form:

The roots can be found by expanding the determinant and setting it to zero:

sAmxmy + sb[cymx + my(cx + с,)] + s2[kymx + cycx + my(kx + ^)]

+ skycx + k+ kxky = 0

This is known as the characteristic equation of the system, and its roots are sometimes called eigen values. There is no simple analytical method for solving the quartic equation as there is for the quadratic that represents the single-degree-of-freedom system; but after the values of the spring, damper, and mass constants are inserted, it can be solved by numerical methods. Depending on the values of the constants, the four roots can be either four real numbers, two real numbers and one pair of complex numbers, or two pairs of complex numbers. The existence of two pairs of complex roots indicates that the system has two separate natural frequencies. The time histories for the displacement of either of the masses can independently have any of the characteristics of the six time histories shown in Figure 9.1.

Much of the analysis of helicopter flying qualities is involved with determining the response to a gust or to a control input. To illustrate the basic technique used for this type of analysis, let us suddenly impose a constant force on the system of Figure 9.3. Now the equation of equilibrium is:

kx + cx + mx = F

After using the substitution for x, this equation becomes

k + cs + ms2 = F(s)

The solution to this equation contains the roots of the homogeneous equation—that is, the equation with F(s) = 0, plus two new terms involving constants of integration:

x = x^jje’1′ + х2(г)е’2′ + Cx + C2t

where Jj and s2 are the same roots obtained earlier. The constants x^j), x2(j), C„ and C2 can be evaluated from conditions when time is equal to zero and when time is equal to infinity. At / = 0,

x = 0

x = 0

for a system with positive damping. At / = 00,

x = F/k

Using the last condition first, it is obvious that C2 must be zero and that since sx and s2 have negative real parts for a system with positive damping: at zero time:

Note that these equations are identical to those obtained for the free system except that xj2 has been replaced by —F/2k. The final solution is:

A typical time history based on this equation for a system with a damped oscillation is as follows:

It is often desirable to describe the response in terms of the time required to achieve 63% of its final value. For this purpose, it is only necessary to work with real portions of the solution:

for

Almost all dynamic systems an be represented by combinations of springs, weights, and dampers. It is natural, therefore, to use this system to illustrate basic dynamic principles, since it can easily be visualized and even set up experimentally.

Figure 9.2 shows a simple spring-weight-damper system consisting of one of each of the elements. The variable of interest, x, is the distance between the instantaneous position of the center of the weight, xh and its static position, x„ when there is no load in the spring. At any one time, the forces acting on the weight in the x direction are of three types: the spring force, Fs, the damper force, Fd, and the inertia force, F[. In the absence of any external forces, the summation of these three forces must be zero:

Fs + FD + Fi= 0

Referring to Figure 9.2, this equation can be written:

kx + cx + mx = 0

where k is the spring rate in lb/ft; c is the damper rate in lb/(ft/sec); m is the mass in lb/^t/sec2); x is the displacement from the static position in ft; x is the velocity in ft/sec; and x is the acceleration in ft/sec2.

This is a classical differential equation, which can be solved by using the substitution:

This complicated-looking solution to an apparently simple system represents the six types of time histories shown in Figure 9.1 through various combinations of relative magnitudes of the constants, k, c, and да.

Case 1

k

c = 0, — > 0

The exponential terms can be written in terms of trignometric terms using Euler’s equations:

e, z = cos z + / sin z

and

e, z = cos z. — і sin г

Thus

This is a steady, or neutrally stable, oscillation with a half amplitude of x0 and a frequency of yjk/m radians per second, or a period of 2n/J(k/m) seconds as shown in the sketch.

This is a damped, or stable, oscillation with a frequency of у/(k/m) — [{c/l)/m]2 radians per second whose envelope decays asymtotically. The rate of decay to half amplitude is given by the equation:

or the time to half amplitude is:

In.5 .693

‘m ~~ с/г ~ с/г

m m

Case 3

. k I c/2 c < 0, – > — m m

(Note: Physically, there are no simple negative dampers, but in some systems a source of external energy gives the same effect.)

This is a negatively damped, or unstable, oscillation, with a frequency of yj(k/m) — [(c/2)/mf radians per second, whose envelope expands to double its amplitude every 1.386 mjc seconds.

In this case, the exponents of both e’s are real and negative, so that the motion is a pure convergence with time as in the sketch.

Note that the special subcase of [(с/2)/т^ — k/m represents the dividing line between convergence with oscillation and convergence without oscillation. The value of damping that satisfies this condition is called critical damping. This concept allows a useful alternative form of the equilibrium equation to be written by defining a damping ratio,

The concept of critical damping is sometimes used as a criterion for dynamic systems that might go unstable under certain circumstances by specifying a damping in terms of the critical damping. For example, a value of £ of 0.06,’or 6% of critical damping, is often used as a goal.

Case 5a

k < о

(Note: A simple system with a negative spring is a toggle switch poised at dead center.) Schematically, this system looks like this:

Since at least one e has a positive real exponent, the resultant time history is a pure divergence:

Case 5b

This solution has two es with positive exponents, so it is also a pure divergence:

Case 6

k = o

or

1

*0

Л___________

Time

Note that this solution, based on zero initial velocity, is good whether the damping is positive or negative. If damping were negative, however, any initial velocity would result in a pure divergence.

INTRODUCTION

When designing a helicopter for good performance, the engineer deals almost entirely with laws of physics that are reasonably constant and more or less well understood. When designing for good stability and control, on the other hand, the engineer must also deal with the capabilities of pilots, which are variable and are only partially understood. The problem becomes one of machine-man matching. If the characteristics of the machine are not matched to the natural capabilities of the man, either the machine cannot be operated or the capabilities of the man must be upgraded by training. One example of the problem is the difference between the training and talent required to successfully ride a tricycle, a bicycle, and a unicycle. In this case, the increasing ability required is due primarily to the progressive deterioration in stability. Another example is the difference between attempting a docking maneuver with rowboat, a cabin cruiser, and an ocean liner. In this example, instability is no problem, but the ability to generate accelerations in the required directions becomes more and more degraded until, in the case of the ocean liner, the ability to dock without doing damage is completely inadequate and the job must be relinquished to tugboats.

Thus there are two important elements in machine-man matching: stability and response to control inputs. The best machine-man matching in the aircraft field involves an aircraft that simultaneously has high stability and a rapid and positive response to the pilot’s control inputs. Many early helicopters were not good examples of machine-man matching. In general, a helicopter without special stability augmentation provisions not only is unstable, but its response to control inputs is slow, with maximum results appearing some time after the pilot starts the control input. These characteristics give the pilot a combination of fear of the instability and impatience with the slow response. In many cases the student pilot overcontrols and actually finds himself contributing to the instability rather than damping it. A classic remark made by a student following his first attempt to hover was, "It’s like riding a pogo stick over a floor covered with greasy ball bearings.” Fortunately, pilots an be trained to fly even unstable helicopters and at the same time, the helicopter itself can be tamed by various means.

DEFINITIONS

Stability is the tendency of an object to return to its original conditions following a disturbance. A marble in a bowl is stable, but a marble balanced on top of an inverted bowl is unstable since once disturbed it will go away from its original position with ever increasing speed. The in-between case is a marble on a flat plate, which has no tendency either to return or to leave: it is thus neutrally stable. Static stability is measured by the force or moment per unit of displacement that acts to restore the object to its original position. Dynamic stability is measured by the time required to return to its original position following a unit displacement. The stability characteristics of a system can be categorized by the type of time history it has following a displacement. Figure 9.1 shows six types of time histories characteristic of aircraft.

Control is the ability to apply forces and moments to the aircraft to maintain it in a steady flight condition in gusty air or to perform a desired maneuver. Two terms are used to define control further: control power is the measure of the total moment or force available to the pilot for maneuvering from a steady trimmed flight condition or for compensating for large gust disturbances; control sensitivity is the measure of either aircraft acceleration or steady velocity produced by a unit of control motion. It is of importance in defining the precision of control.

A situation that exists in almost all types of aircraft is that increased stability of the basic airframe results in decreased controllability. As a general rule, pilots prefer controllability over stability, since it permits them to get out of tight spots that even very stable aircraft might get into. In this, they are following the lead of the Wright Brothers, whose "Flyer” was very unstable but also very controll­able.

In forward flight there are two natural trim conditions: zero bank angle and zero sideslip. Since very few helicopters have sideslip indicators, pilots tend to fly at zero bank angle, where they are most comfortable. In these: cases, the pilots find the sideslip angle that makes the sideforces on the fuselage equal and opposite to the tail rotor thrust.

For test purposes, instrumented helicopters generally have sideslip indi­cators, and the pilot may be asked to fly at zero sideslip to minimize drag and maximize performance. In actual flight, of course, the helicopter is usually in a condition of some bank angle and some sideslip.

The relationships of bank angle versus sideslip angle are imbedded in the three equations of lateral-directional equilibrium of Table 8.11.

The solution of these equations results in relationships of Ф, and Tr as a function of the sideslip angle, p. The pilot can sense the roll angle, Ф, but not the other two variables. These show up indirectly as lateral and directional control positions. To calculate these positions, the sideslip is accounted for in the following manner:

Lateral:

Ay (Ay + bp_o by^ + By sin P

Directional:

(Use the method of Chapter 3—"Tail Rotor”—to find 0^.)

Using the rigging plots of Appendix A, the results for the example helicopter at 115 knots have been obtained and are plotted on Figure 8.31. These indicate that this helicopter has three types of positive stability: positive directional stability, positive dihedral effect, and a positive sideforce character­istic.

The amount of pedal position required to hold a sideslip angle indicates the magnitude of the directional stability and tail rotor control power. Although a

TABLE 8.11

Lateral-Directional Equilibrium Equations in Forward Flight

Y-Equilibrium Equation

Component Source

Main Rotor Y-Force

Main Rotor H-Force Tail Rotor Thrust

Tilt of Vert. Stab. Drag

Fuselage Side Force

Yf

Tilt of Fuse. Drag

Tilt of Gross Weight Equilibrium

helicopter might resist flying with sideslip because of high directional stability, only a little pedal displacement may be needed to hold sideslip if large changes in tail rotor thrust can be obtained with small pedal movements.

The change in lateral control position to hold a sideslip angle is an indication of the dihedral effect. Positive dihedral results in a roll to the left when the helicopter inadvertantly slips to the right—just as on an airplane with both wings tilted up. Positive dihedral is desirable to insure dynamic lateral-directional stability.

The change in roll attitude to hold a given sideslip is an indication of the sideforce characteristics of the helicopter. Strong sideforce characteristics help the pilot make coordinated turns by giving him a "seat of the pants” feel whether he is skidding to the outside of the turn or sliding to the inside.

EXAMPLE HELICOPTER CALCULATIONS

page

Horizontal stabilizer characteristics 501

Vertical stabilizer characteristics 511

Longitudinal trim in hover 517

Longitudinal trim in forward flight 522

Speed stability 527

Angle of attack stability 529

Power effects on trim 530

Lateral trim in hover. 532

Directional trim in hover 534

Lateral-directional trim in forward flight 538

HOW TO’S

page

Aerodynamics of the fuselage 512

Aerodynamics of the horizontal stabilizer 488

Aerodynamics of the vertical stabilizer 502

Angle of attack stability 529

Directional trim in hover 534

Lateral-directional trim in forward flight 535

Lateral trim in hover 532

Longitudinal trim in forward flight 517

Longitudinal trim in hover 516

Power effects on trim 530

Speed stability 525

The lateral-directional equilibrium equations are those involving Y-force, rolling moment, and yawing moment—the Y, R, and N equations. The aircraft

components that are significant in these equations are the main rotor, tail rotor, vertical stabilizer, and the fuselage.

Hover

In hover, the airframe aerodynamic effects may be neglected so that only the two rotors enter into the equations. For example, Figure 8.28 shows the relationships that result in the equilibrium equations:

TJls + TT=-G. W. Ф

(dRM ^ dbx

N QM — 1тТт — 0

These equations yield some interesting results for special cases as presented in Table 8.9 and illustrated by Figure 8.29.

The example helicopter has the pertinent parameters given in Table 8.10 These values, used in the equations for bx and Ф, give:

bls = —.027 rad = —1.5 deg ф =—0.050 rad =—2.9 deg

Besides the roll angle and the lateral flapping, the tail rotor pitch and its corresponding pedal position required to maintain heading in hover can be calculated from the conditions that satisfy the yawing moment equilibrium equation:

Qm ~ Ir^r ~ 0

Before one does a numerical calculation, an interesting relationship can be obtained by rewriting the equation in terms of nondimensionalized coefficients, which gives:

Ст/&T _ (ПК)м Rm CQ/oM~ AiT (ClR)2T (lT-lM)

Thus for a given helicopter, this ratio is constant and independent of altitude. If the small effects of compressibility are ignored, the constancy of this ratio also defines another ratio that is independent of altitude: Q0t/Ct/cm, since with this assumption and for given rotor geometries, CT/cT is only a function of 0Ot and CQ/oM is only a function of CT/cM. The practical application of this is that the tail rotor pitch as a function of main rotor thrust/sohdity coefficient need only be calculated for sea-level, standard conditions and then used for other altitudes simply by accounting for the effect of density ratio on CT/cM. For the example helicopter, the work was done as an intermediate step while preparing Figure 4.30 of Chapter 4. Figure 8.30 shows the result, which can be used to guide the

CjkTM

FIGURE 8.30 Tail Rotor Pitch Required in Hover

designers in choosing the maximum tail rotor pitch travel. For the example helicopter at its design gross weight of 20,000 lb, Figure 4.35 gives standard day hover ceilings of 12,400 ft out of ground effect and 15,000 ft in ground effect. Using Figure 8.30, it may be determined that, in each case, a tail rotor pitch of 22.2 degrees is required to hover. A prudent designer would, of course, allow some margin for maneuvering.

If the shift in longitudinal control position while going from a full-power climb to steady autorotation at the same forward speed is excessive, pilots will complain. A typical minimum value written into the flying qualities specifications is 3 inches. The shift is primarily due to the change in aerodynamic pitching moments on the airframe—especially those contributed by the fuselage and the horizontal stabilizer as the angle of attack changes from one flight condition to the other. The calculations for the example helicopter in autorotation, level flight, and climb at

2,0 feet per minute—all at 115 knots—are given in Table 8.8.

For this helicopter, the change in Вx + ax required to trim the rotor aerodynamically is almost equal to the change in ax required to trim the airframe pitching moments so that practically no stick motion is needed when going from one trim condition to another. This is one benefit of having a small horizontal stabilizer. If it were large enough to produce positive angle of attack stability, the shift in control position would have been much greater.

The stability with respect to angle of attack is evaluated in flight by trimming for level flight and then entering a steady, descending turn at the same speed without changing collective pitch. The extra rotor thrust required to support the weight and centrifugal force will come from the increased angle of attack produced by the rate of descent. If during the turn, the longitudinal control is aft of the level flight position, the helicopter is said to have positive maneuver stability, which is another way of saying that it has positive angle of attack stability. The procedure for calculating trim conditions can be used to investigate the sign and magnitude of this stability. The first step is to calculate the trim conditions in a dive at constant collective pitch (using the method outlined in Chapter 3) of a helicopter whose effective gross weight is equal to its actual gross weight multiplied by the load factor, corresponding to the turn. The second step is to correct the calculated longitudinal cyclic pitch to account for the effect of pitch rate (using the procedure in Chapter 7). This process has been done for the example helicopter in a 1.3-g descending turn at 115 knots with collective held at the level flight value. The results are presented in Table 8.7.

The more forward position of the stick in the turn than in level flight is an indication that this particular helicopter does not have a large enough horizontal stabilizer to give it positive angle-of-attack stability. This might be considered to be a design deficiency that would be corrected in an actual program by increasing the stabilizer area or by installing a "black box” stability augmentation system that could use a signal from an angle of attack sensor to move the swashplate nose down or to change the stabilizer incidence nose up so that the pilot would have to use aft stick to compensate. For a helicopter with either inherent or artifically achieved

positive angle of attack stability, an inadvertant pitching motion which produced a nose-up angle of attack would result in a stabilizing nose-down control response without pilot action. Design decisions in this matter involve tradeoffs between the weight of an adequately sized stabilizer and the weight of an auxiliary stabilizing system. Also entering into the study are considerations of safety involving possible shutdowns or hardovers of the black box system.