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

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

3.3.11.1 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: