Category Helicopter Performance, Stability, and Control

In addition to all of the effects discussed above, the rotor will also respond to changes in sideslip. This is because blade flapping is produced by conditions referenced to the flight path rather than to whatever orientation the fuselage might have at the time.

Imagine the helicopter of Figure 7.7 in forward flight with no sideslip and with the rotor trimmed perpendicular to the shaft. If the flight direction is suddenly changed so that the helicopter is flying directly to the right without changing fuselage heading or control settings, the blade over the tail becomes the advancing blade and the one over the nose the retreating blade. Since the cyclic pitch no longer corresponds to trim conditions, the rotor will flap down on the left side because of the asymmetrical velocity distribution—thus producing a rolling moment to the left.

In practice, of course, sideslip angles are less than the 90° used for illustration, but the trend is the same—the helicopter tends to roll away from the approaching wind. This is the same characteristic found on airplanes with dihedral (both wings slanted up) and is known as the positive dihedral effect on rotors even though the source is different.

It is a desirable characteristic that helps the pilot. With negative dihedral, a sideslip would tend to roll the aircraft into an ever-tightening spiral dive. Too much positive dihedral, on the other hand, also can be undesirable, as will be later pointed out in the discussion of lateral-directional stability in Chapter 9- Positive

FIGURE 7.7 Effect of Sideslip on Rotor Flapping

dihedral manifests itself during flight as a lateral stick displacement required in the direction of sideslip to maintain equilibrium.

The rolling moment due to the dihedral effect is also accompanied by a pitching moment. Again going back to Figure 7.7 and the helicopter when it is flying directly to the right, it may be seen that the blade pointing in the direction of flight was originally the advancing blade and had a low pitch. It still has a low pitch that will cause the rotor to flap down over the nose, producing a nose-down pitching moment.

Similarly, during flight to the left, the blade pointing in the direction of flight has a high pitch and will cause the rotor to flap up over the tail—also producing a nose-down moment. Thus steady sideslip in either direction requires aft stick displacement, an effect that does not exist on an airplane.

Somewhat surprisingly, if the same analysis is made on a rotor turning clockwise when viewed from above, the pitching moment direction is un­changed—nose-down for sideslip in either direction. This pitching effect is not always observable in flight, since other pitching moments may be generated by

changes in airflow conditions on the horizontal stabilizer and tailboom as they move out from behind the fuselage during sideslip.

Gust Response

The blade flapping is also responsible for a helicopter behaving better in gusty air than an airplane does. This is because the rotor blades flap individually in response to the gusts, allowing the rest of the helicopter to have a relatively smooth ride. The wing of an airplane, on the other hand, transmits its unsteady loading directly into the fuselage. This gust alleviation feature has been demonstrated by flying helicopters and airplanes of the same size in formation through gusty air, as reported in reference 7.1. The recording instrumentation showed that the helicopter had a smoother ride. The comparison is similar to that of an automobile with independent wheel suspension compared to one on which the wheels are rigidly mounted to the chassis.

Frequency Ratio

As already discussed, the addition of hinge offset changes the characteristics of the rotor from being a system in resonance to one whose natural frequency is higher than the rotational frequency. An analysis of this system in hover gives equations for the frequency ratio, damping, phase lag, and cross-coupling as a function of the hinge offset. From Figure 7.8, it may be seen that the increment of moment about the hinge due to centrifugal force perpendicular to the shaft is:

AA1c f = mArCl2r(r — ^)(3

2 К

1——–

R

For the example helicopter, the hinge offset is 5% of the radius, and thus the corresponding first flapping frequency ratio is 1.04.

The equation an be rearranged for use in determining the effective hinge offset of a hingeless rotor. For these rotors, the value of (0jCl is generally calculated separately for the purpose of performing rotor dynamic analyses. Once this parameter is known at the operational rotor speed, the effective hinge offset ratio can be found as:

A factor that is important to helicopter flying qualities is the damping moment the rotor produces when the helicopter is subjected to pitch or roll angular velocities

by external means such as gusts. This damping is produced by the tilt of the tip path plane, which lags behind the motion of the shaft by an amount that is proportional to the rate of pitch or roll, as illustrated in Figure 7.6. It has already been shown how the aerodynamics on the blade causes the tip path plane to tend to stabilize itself in an equilibrium position with respect to the shaft. A rotor attached to a shaft that is continuously tilting, therefore, will follow the shaft. The rotor disc may be considered to be a gyro that must be precessed by a moment applied 90° before the direction of tilt. The moment can only be generated by aerodynamic means, which in turn can only be produced by asymmetric flapping velocities. To maintain a steady nose-up pitch rate, for example, the airload must be higher on the blade at |/ = 90° than on the blade at {/ = 270°: this asymmetry is generated by a downward flapping velocity—with respect to the shaft—at |/ = 90° ( and an upward flapping velocity at |/ = 270°. The maximum flapping amplitude with respect to the shaft, therefore, is down at \f = 180° and up at у = 0°, and the

tip path plane follows the motion of the shaft at a lag angle that is proportional to the rate of pitch and the rotor moment of inertia.

During a steady nose-up pitching maneuver, some lateral flapping also is generated as a result of the decreased angle of attack at у = 180° and the increase at у = 0°’ caused by the rate of pitch itself. This difference in angle of attack is compensated for by lateral flapping that produces enough flapping velocity at these two blade positions to cancel out the effect of the pitch rate. The production of both a lateral and a longitudinal flapping by a pure pitch rate is a source of cross­coupling, which applies equally to rotors with and without hinge offset. The cross­coupling ratio, bi/ax is a function of blade inertia, being highest for light blades.

Consider the converse case in which the steady pitch rate is being produced by the pilot in a deliberate maneuver instead of by external means. Since the rate is steady (no acceleration), no hub moment is needed if we ignore the damping moments generated, by the airframe. Under this condition, the longitudinal and lateral cyclic pitch required to maintain the maneuver will be exactly equal to the longitudinal and lateral flapping that would have been produced by a pitch rate caused by external means. This is a consequence of the equivalence of flapping and feathering explained in Chapter 3.

Steady lateral flapping, like steady longitudinal flapping, is caused by asymmetric airloads. In this case, the asymmetry of airloads is on the blades at \f = 0° and |f = 180° and is caused by coning. In forward flight, the coning causes the blade over the nose to be affected by a velocity component of forward speed that is more up with respect to the blade than for the blade over the tail. This asymmetry of vertical velocity at the blades at j/ = 0° and |/ = 180° produces a corresponding asymmetrical angle of attack and airload distribution that causes maximum response 90° later and thus lateral flapping that is upward on the retreating side. The magnitude of this flapping is a function of the coning and tip speed ratio. Because of it, the pilot must move his stick toward the retreating blade as well as forward in order to keep the helicopter in trim as he increases speed.

Cyclic Pitch Change in Forward Flight

Just as in hover, flapping is produced in forward flight by changes in cyclic pitch. For a rotor with no hinge offset, the flapping occurs 90° after. the cyclic pitch input. Because of the distribution of the aerodynamics, a 1° change in lateral cyclic pitch, Av will cause the rotor to flap down to the right by exactly 1э, but a 1° change in the longitudinal pitch, Bu will cause the rotor to flap down in front by slightly more than 1°. Thus the response to longitudinal control is somewhat more sensitive in forward flight than in hover, but the response to lateral control remains the same. If the rotor has hinge offset, the maximum response is less than 90° after the input and the magnitude of the response is a function of the offset, as will be shown by the flapping equations.

Another longitudinal flapping effect is that caused by an increase in collective pitch in forward flight. Both the advancing and retreating blades receive the same change in angle of attack; but the advancing blade, being at a higher dynamic pressure, develops more additional lift than the retreating blade. Both blades flap up a quarter of a revolution later, but the advancing blade flaps up more than the

retreating blade, resulting in both an increase in coning and a net rearward tilt of the tip path plane. This type of flapping is sometimes noticed by the pilot when he decreases collective pitch when entering autorotation. In this case, of course, it is a nose-down tilt of the tip path plane that results. It is also partially responsible for the trend of cyclic stick position with speed. From hover to approximately the speed for minimum power, the collective pitch required for trim decreases, thus causing the tip path plane to want to tilt forward. An aft cyclic stick motion is required to keep the helicopter in trim. This effect may be larger that the aft flapping caused by the increase in forward speed; and, as a consequence, the trimmed stick position may initially move aft or have an unstable gradient. At high speeds, where collective pitch is increasing, the gradient will almost always be stable.

If the shaft is tilted laterally in forward flight, the effect is the same as it is in hover—the tip path plane follows the shaft, and the flapping with respect to the shaft remains unchanged. If, however, the shaft is tilted longitudinally, the nonuniformity of the velocity distribution produces a different situation.

Figure 7.5 illustrates this with a rotor that, for simplicity’s sake, starts from a condition of zero lift on the advancing and retreating blades. Following a sudden nose-up tilt, the immediate result is the same as in hovering: the advancing blade receives a increase in angle of attack and the retreating blade a decrease— producing an unbalanced lift that auses the tip path plane to flap nose up. In forward flight, however, when the blade flaps nose up until it is perpendicular to the shaft, the forces are not yet balanced. The unbalance is due to the forward flight velocity vector. The airflow coming at the rotor as a result of its forward speed modifies both the local angle of attack and the local velocity. Both blades have positive angles of attack, with the angle on the retreating blade actually being the greater, as shown in Figure 7.5c. The lift on the retreating blade, however, is less than on the advancing blade beause the lift is proportional to the product of the angle of attack and the square of the local velocity. This causes the rotor to flap past the perpendicular to the shaft to a more nose-up position where the forces are in balance, as shown in Figure 7.5d. The magnitude of the excessive flapping is approximately proportional to the square of the forward speed. The result is negative angle of attack stability, since the aft flapping generates a nose-up pitching moment about the helicopter’s center of gravity that tends to ause а further increase in the shaft angle of attack. It is an undesirable characteristic, but it an be compensated for rather easily with a horizontal stabilizer of reasonable size.

From this illustration it may be seen that increasing the angle of attack of the shaft also increases the rotor thrust, just as increasing the angle of attack of Л wing increases its lift.

Resultant Velocity Vector

У У

One of the most important of the rotor’s flapping characteristics is that caused by a change in speed. To illustrate, let us examine a lifting rotor in a wind tunnel as the tunnel is started. The tunnel speed adds to the velocity on the advancing blade, thus increasing its lift, but subtracts from the velocity on the retreating blade thus decreasing its lift. The advancing blade accelerates up about its flapping hinge, but at the same time the blade is being rotated toward the nose. As a result, the advancing blade reaches its maximum upward flapping angle over the nose and the retreating biade reaches its maximum downward flapping angle over the tail. The rotor trims itself to an equilibrium position when the flapping velocities on the advancing and retreating blades are just enough to change the angles of attack to compensate for the change in dynamic pressure. Thus the flapping will increase as the tunnel speed increases. The magnitude of the flapping is a function of the lift of the rotor. If the blades initially had no lift, the unbalanced dynamic pressure would cause no flapping. The rearward tilt of the rotor thrust vector produces a nose-up pitching moment with respect to the helicopter’s center of gravity, as shown in Figure 7.4. The resultant change in pitching moment as a function of forward speed is known as speed stability> and is one of the most important differences between a helicopter and an airplane, which has no corresponding change in pitching moment with respect to speed.

In free flight, the change in longitudinal flapping with increasing speed is stabilizing since it produces a nose-up moment that causes the helicopter to pitch up and to slow down to its original speed. In some cases the effect of a horizontal stabilizer carrying positive lift, or the interference effects of the front rotor on the rear rotor of a tandem rotor helicopter, can overpower the natural speed stability

of the rotors and produce negative speed stability. In this case, an increase in speed produces a nose-down pitching moment. This causes the helicopter to go into a dive in which the speed and the nose-down pitching moment increase as a pure divergence. This characteristic is, of course, undesirable from a flying-qualities standpoint; but pilots an learn to fly even such unstable aircraft, or the characteristic can be changed by methods which will be discussed. It will later be shown that in all flight conditions too much positive speed stability is as bad as too much negative speed stability and that in hover the optimum condition is one of neutral speed stability. A manifestation of positive speed stability is the requirement for the pilot to move the cyclic stick forward as he increases speed to keep the helicopter trimmed. If the speed stability is negative, the pilot will push the stick forward to initiate an increase in speed; but when he finally trims at the new speed, the stick will be further aft than when he started.

In Chapter 3 it was shown that a hinged rotor blade without offset is a system in resonance; that is, its natural frequency is identically equal to its rotational

frequency. A characteristic of a system in resonance is that its maximum response follows by exactly a quarter of a cycle its maximum force input. Thus, if cyclic pitch is applied to a rotor of this type, it will have its maximum flapping amplitude 90° later. Although the derivation of Chapter 3 ignored the existence of aerodynamic forces, they do not change the phase lag from 90°. The aerodynamic forces only add damping to the flapping motion and, as shown in Figure 7.3, the phase angle is always 90° for any damping level. For a zero offset rotor in hover, a 10 change in cyclic pitch will result in a 10 change in flapping. The mathematical derivation of this will be given later; from an intuitive point of view, however, it is because the rotor’s stable condition is with no cyclic angle of attack variations with respect to its tip path plane without regard to the relative position between the shaft and the tip path plane. Thus the rotor flaps just enough to cancel out the initial cyclic pitch input and to return it to its initial hover angle of attack configuration with respect to its tip path plane.

If the rotor has hinge offset, the phase angle is somewhat less than 90° and the flapping is not quite numerically equal to the cyclic pitch. This is because as offset is increased, the restoring moment due to centrifugal forces increases faster than the moment of inertia about the flapping hinge; as a result, the flapping natural frequency is higher than the rotational frequency. In short, the system is no longer in resonance. The frequency ratio will be less than unity and the phase angle will be less than 90°, as shown in Figure 7.3. The total magnitude of the flapping

will be slightly less than for a rotor with no hinge offset because of the restraint provided by the nonflapping inner portions of the blades.

If the shaft is tilted while the rotor is hovering in air, aerodynamic forces will be generated that will force the tip path plane to align itself perpendicular to the shaft whether the rotor has hinge offset or not. The sequence of steps leading to this is shown in Figure 7.2.

No Hinge Offset

With Hinge Offset

FIGURE 7.1 Effects of Shaft Tilt in Vacuum

Right Side

First, there is the tilt of the shaft alone as the rotor disc acts as a gyroscope and remains in its original plane. Since the blade feathering is referenced to the shaft, however, the angle of attack of the right-hand blade is increased and that of the left-hand blade decreased by the same amount. This causes the rotor to flap until it is perpendicular to the shaft, where it will again be in equilibrium with a constant angle of attack around the azimuth and the moments will be balanced. This alignment is very rapid, usually taking less than one rotor revolution following a sudden tilt. Because of this, the flapping motion in hover has practically no effect on the stability of the helicopter in terms of holding a given attitude.

QUALITATIVE DISCUSSION OF FLAPPING

Before deriving the helicopter equations of equilibrium and motion, it seems well to develop a general understanding of rotor flapping, a primary factor in helicopter stability and control analysis.

As might be expected, the stability and control characteristics of a helicopter are different from those of an airplane primarily as a rotor is different from a wing. Whereas the wing remains more or less rigidly attached to the airplane airframe, the rotor tip path plane tilts easily with respect to the helicopter airframe in response to changing flight conditions and control inputs. This tilt produces changes in forces and moments at the top of the rotor shaft. A hingeless blade may have no single point at which flapping occurs, but an effective hinge offset can be determined that will give it the same stability and control characteristics as a blade with an actual mechanical hinge at that point. This concept will be used to eliminate any consideration of blade structural stiffness in the following analyses.

Several different flapping characteristics will be discussed in a cause-and – effect manner and then mathematically derived from the laws of physics. These characteristics are summarized in Table 7.1 for rotors with and without hinge

offset. (There are few modern rotors with individual blades hinged with no offset but two-bladed teetering rotors fall into this category for the purposes of this discussion.)

All these flapping characteristics can be explained on the basis that at the flapping hinge (or at an effective flapping hinge in the case of a hingeless rotor) the summation of moments produced by aerodynamic, centrifugal, weight, inertial, and gyroscopic forces must be zero.

Shaft Tilted in a Vacuum

If a rotor with no hinge offset is operating in a vacuum, there are no aerodynamic forces; only centrifugal forces acting in the plane of rotation. These can produce no moments about the flapping hinges. If the shaft is tilted, no changes in moments will be produced and the rotor disc will remain in its original position, as shown in Figure 7.1. If, on the other hand, the rotor has hinge offset, the centrifugal forces acting in the plane of rotation will produce moments about the hinges that will force the blades to align themselves perpendicular to the shaft.

Airfoil lift and drag characteristics as a function of angle of attack and Mach number are often used in a computer as a bivalue table in conjunction with some table look-up scheme. There are occasions, however, when it is desirable to convert the airfoil data to equation form for use in simple hand analysis, in a small computer with limited storage capacity, or even in a large computer for quick comparisons of rotor performance using different airfoils. The simplest analytical expressions based on ignoring the effects of stall and compressibility are:(Note that the drag equations are unsymmetrical with respect to angle of attack and will give unrealistically low values of the drag coefficient for negative angles of attack. It is suggested that, if the negative angle of attack region is of importance in a specific analysis, the drag equations be rewritten in terms of only even powers of a. A cambered airfoil does not have zero lift at zero angle of attack. This can be accounted for by writing the equations in terms of a — aLO or by redefining the angle of attack such that it is zero at zero lift.)

A procedure for writing equations for the lift and drag coefficients for cases in which compressibility and stall cannot be ignored is given next.

For purposes of illustration, the lift and drag data synthesized from whirl tower tests of a rotor with the NACA 0012 airfoil as published in reference 6.66 will be used. These data are shown as solid lines in Figure 6.43. Both the lift and the drag coefficients of the NACA 0012 airfoil change characteristics at a Mach number of about 0.725, where compressibility effects are first evident. Because of this, two sets of equations must be written for the two separate Mach number regimes.

Lift Coefficient below M — .725

Theoretically, the slope of the lift curve should follow the Prandtl-Glauert relationship:

Source: Carpenter, “Lift and Profile-Drag Characteristics of an NACA 0012 Airfoil Section as Derived from Measured Helicopter-Rotor Hovering Performance,” NACA TN 4357, 1958.

a0

a = —,

Figure 6.44 shows, however, that the slope is somewhat lower, being better fit by the equation:

The angle of attack at which the lift coefficient first shows the effects of stall will be defined as dL and is a function of Mach number. For the NACA 0012, dL is essentially linear with Mach number, as shown in Figure 6.44. An approximate equation is:

a£= 15 – 1Ш

Above dL, the lift coefficient can be represented by:

c’*>4 = «a – K,(a – aL)*‘

The exponent, K2, at any Mach number is obtained by plotting the difference between ad and the measured lift coefficient versus (a — a£) on log-log paper. The slope of the straight line faired through the points is the value of the

exponent to use. For the NACA 0012, this procedure, as illustrated in Figure 6.45, gives:

These points lie nearly on a straight line, whose equation—favoring the highest Mach number points—is:

K2 = 2.05 – 0.95 M

The coefficient, Kb at any Mach number is found by evaluating the equation for ct at the angle of attack corresponding to the maximum lift coefficient:

aac, – о

for the NACA 0012 airfoil:

When K{ is plotted against Mach number, as in the top portion of Figure 6.46, it is seen to be a constant plus some power of M. Again using the log-log paper technique, as in the lower portion of Figure 6.4(5, the exponent can be evaluated and an approximate equation can be generated:

fCj = 0.0233 + 0.342 M715

Thus the lift coefficient below 0.725 Mach number and above stall is:

Ci < 0.725, a > aL = І -0Л – 0.0Ш a

– (0.0233 + 0.342ЛТ 15)(a – 15 + 16М)(2 05-°-9Ш)

Lift Coefficient above 0.12b Mach Number

The slope of the lift curve of the NACA 0012 breaks at 0.725 Mach number. The slope above this value is shown in Figure 6.44 to be nearly a. straight line, represented by:

a = 0.677 – 0.744 M

In order to satisfy the experimental lift characteristics by equations, the following constants were evaluated by the same methods used at the lower Mach numbers:

aL= 3.4

K, = 0.575 – 0.144(M – 0.725)044 K2 = 2.05 – 0.95 M

Thus:

c, = (0.677 – 0.744Л1)а

vH>.725,a>3.4 4 I

– [0.0575 – 0.144(M – 0.725)044] [a – 3.4](2 05“095Ar)

Mach Number

FIGURE 6.46 Evaluation of Equation for К^

Figure 6.43 shows the correlation of measured lift coefficient and the generated lift coefficient.

Drag Coefficient below Drag Divergence

At the lowest test Mach number, the incompressible drag coefficient can be represented by a power series of the form:

Cj. = Cj + Cj + Cj Q,2 + . . . +Cj d”

"incomp a0 al d2 an

The coefficients can be evaluated by selecting n control points and solving a set of n simultaneous equations. Before selecting the control points, the angle, aD, at which the individual drag curves break away from the incompressible curve should be established. For the 0012 airfoil, this angle, as shown in Figure 6.44, is approximately:

aD= 17- 23.4M

(Note that the line represented by this equation goes through the drag divergence Mach number of 0.725 at a = 0.) The control points should include a = 0 and aD for the lowest test Mach number and n — 2 other points in between. For the 0012 a satisfactory fit was obtained with the 0.1 Mach number test data using a five-term series evaluated at 0, 2, 6, 10, and 14.7 degrees. This gives (with a in degrees):

cd. = 0.0081 + (-350a + 396a2 – 63.3a} + 3.66a4) x кг6

“mcomp ‘ ‘

(Warning: too many terms in the series may introduce large fluctuations of the curve between the control points).

For Mach numbers above 0.1, the drag coefficient breaks away from the incompressible value as a exceeds aD. The curves have the characteristics represented by the equation:

^=4co„p + K>(a-a’>)’;4

where Kb and KA are evaluated in the same manner in which Kx and K2 were evaluated in the equation for the lift coefficient. For the NACA 0012 airfoil:

Using average values gives:

Kb = 0.00066 K4 = 2.54

Thus:

Cd =cd. + 0.00066 (a – 17 + 23-4АГ)2 54

яЛ{<.725,а>а£) “incomp ‘ /

Drag Coefficient above Drag Divergence

For Mach numbers above drag divergence, another term must be added to account for the drag increment at zero angle of attack. The equation becomes:

= Чкошр + Къ(а " аоҐ4 + K>(M ~ Mdd)K6 For the NACA 0012, the coefficients and exponents that give the best fit are:

aD = o *, = 0.00035 K4 = 2.54 *5 = 21 *6=3.2

Thus:

cdu = cd + 0.00035a234 + 21(M – 0.725)3 2

aM>. 725 “incomp ‘ ‘

The correlation of the drag coefficient generated by this procedure with the test data is shown in Figure 6.43.

Equations Suited for Forward Flight Analysis

The foregoing procedure is adequate for hover performance methods and has been used to prepare the hover charts at the end of Chapter 1. For forward flight, however, the equations should be modified somewhat. Since negative angles of attack are possible on the advancing tip, the equation for the incompressible drag coefficient should be written in terms of even powers of a so that it is symmetrical about a = 0. For the NACA 0012, the four-term series on a in degrees has been found to give satisfactory representation:

cd = 0.0081 + (65.8a2 – 0.226a4 + 0.0046a6) x 10~6

irtcomp v 7

A second modification accounts for the fact that some of the inboard elements on the retreating side will be subjected to angles of attack up to 360°, although at low Mach numbers. Figure 6.47 shows measured NACA 0012 lift and drag coefficients from 0° to 180° taken from reference 6.15. Equations for the lift coefficient an be written by dividing the angle of attack range up into segments:

Lift Coefficient Ci = generated coefficient Ci = 1.15 sin 2a ci = -0.7

o = o. i (a – 180°)

Ci = 0.7

0 = 1.15 sin 2a

Cj = generated coefficient

Similarly, the drag coefficient is represented by:

Moment Characteristics of a Cambered Airfoil

whirl tower and two-dimensional wind tunnel tests. The whirl tower method has the advantage that three-dimensional effects are accounted for, but these tests are relatively expensive. Two-dimensional wind tunnel tests are relatively inexpensive but do not include any three-dimensional effects. The airfoil data that are readily available at this time and are considered suitable for rotor analysis are summarized in Table 6.1. Much of the data is plotted in a consistent manner in reference 6.57.

The existence of several sets of published data for the NACA 0012 airfoil presents an opportunity for a comparison of test variables. Figure 6.51 shows the significant lift and drag characteristics of four sets of data as a function of angle of attack and Mach number. The comparison shows that the maximum lift coefficient and the drag coefficient are both strongly influenced by the chord of the test airfoil. This is a Reynolds number effect, which has been demonstrated in other tests.