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.

EXAMPLE HELICOPTER CALCULATIONS

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

HOW TO’S

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

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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