Prouty Helicopter Aerodynamics 2 - Chapter 75 - Some Coupling Considerations
In explaining how a rotor works, we often talk about how the flapping response follows the maximum aerodynamic input a quarter of a revolution later. For our scientific-minded friends, we relate this to a system in resonance or to a gyroscope. It turns out that this is a very good rule-of-thumb that adequately covers most situations, but there are several “Yes, buts”.
Enter Hinge Offset
One of the these is that the resonance theory strictly applies only to rotors with no hinge offset. Thus, the explanation was valid up to about 1950 since all autogyro and helicopter rotors up to that time had been designed that way. The Sikorsky S-52 was the first production helicopter I know of that had hinge offset. The only motivation at the time was that the design of the hub could be made simpler than on the zero-offset S-51. When the test pilots first got in the S-52, they were astounded by the improved control power compared to what they had been used to. We now think of this as the prime motivation for using flapping hinge offset.
With zero offset, as on teetering rotors, if you are on the ground and apply minimum cyclic pitch to the blade directly over the tail boom, the resulting flapping response will be seen to be down on the right side. This also initially happens in the air when the rolling motion is first accelerating. If the helicopter has offset flapping hinges, or a hingeless rotor where flexibility takes the place of an actual flapping hinge, the response angle is not 90 gegrees but something less since the dynamic and centrifugal force effects are not quite balanced and this causes the rotor to operate somewhat off its natural frequency.
Thus, when the minimum cyclic pitch is applied over the tail boom, the rotor produces a large left rolling moment but a small nose-up moment as well (we are talking about American helicopters, of course). The response angle, however, is seldom less than 85 degrees and the resulting cross-coupling is practically unnoticeable to the pilot.
There are exceptions to this. The Eurocopter BO-105, designed in the 1960s, has a very stiff hingeless rotor that acts as if it had a flapping hinge at 13.6% of the rotor radius. Its calculated response angle is 73 degrees and since the moment of inertia in roll is much less than in pitch, the cross-coupling is such that by pulling straight back on the stick, the roll response is more than the pitch response!
As awkward as this might seem, pilots quickly adapt to this characteristic. There are, however, some other adverse effects to so much stiffness and so the next two projects in this line, the BK-117 and the EC-135, have softer hingeless rotors with effective hinge offsets of 12% and 11.8% respectively.
When a Steady Rate is Achieved
What we have been talking about so far is acceleration cross-coupling. A different physical law governs rate cross-coupling. Imagine a helicopter in which the pilot is making the helicopter roll to the right at a constant rate. For this case, there is no rolling moment and therefore no flapping with respect to the shaft; otherwise the roll motion would not be at a constant rate but would be accelerating. (We are ignoring the small aerodynamic roll damping associated with the rest of the airframe.)
To get this steady right roll state, the pilot is holding his stick to the right to obtain an airload distribution that will precess the rotor down to the right as a gyroscope. The airframe will respond by rolling to the right along with the rotor. Since there is no flapping in this steady condition, it does not matter how the blades are attached to the hub.
Where does the rate cross-coupling come from? Compare the environment of the blades on the right and left side. On the right, the blade is going down through the air and thus has a higher angle of attack than the blade on the left, which is going up. This should produce some up-flapping toward the nose, but if the maneuver is such that the helicopter has only rolling motion as we specified, the pilot must be putting in some forward stick to suppress this flapping and its resulting pitching moment.
This results in the minimum cyclic pitch being imposed, not when the blade is directly to over the tail boom, but somewhat later.
The resulting value of this cross-coupling is a function of the ratio of aerodynamic forces to dynamic forces. This ratio is known as the Lock number after a British engineer who developed equations for autogyros in the 1930s. Since the blade flapping inertia is in the denominator of the equation, heavy blades have low Lock numbers and light blades, high. For current helicopters, the range for main rotor blades is from about four to twelve. For a conventional Lock number of eight, the longitudinal cyclic pitch required to keep from pitching is just half the lateral pitch that is being used to maintain the roll rate. The lighter the blade, the more longitudinal cyclic pitch must be used.
The Other Side Of The Coin
The above discussion applies to maneuvering where the pilot is making the rotor do his bidding and the airframe goes along. There is yet another consideration. It is when the airframe is moving and dragging the rotor along with it. This happens, for instance, when the pilot neutralizes his control after a maneuver but the helicopter still has some rolling or pitching rate or when the helicopter has been upset by a gust.
In these cases, the rotor flaps with respect to the shaft to produce a damping moment. For the case of no hinge offset, the cross-coupling effect is the mirror-image of the one that applies in a steady rate case. If the aircraft is rolling to the right without a pilot command, the magnitude of lateral flapping producing a damping moment to the left is equal to the cyclic pitch that the pilot would have used to maintain that right roll rate. This is a result of the fact that flapping and cyclic pitch are related in a one-to-one ratio. For the same reason, there is a corresponding longitudinal flapping producing a noseup pitching moment which is about half the lateral flapping. In other words, there is cross-coupling even in this case.
The situation is a little different if the rotor has offset flapping hinges. Not only is the damping moment larger, but it can be shown that the cross-coupling diminishes with offset and that for conventional blades, a 15% offset would eliminate this coupling entirely so that a rolling velocity would produce no longitudinal flapping. (My thanks to Tom Hanson for pointing this out to me.)
Because all of these cross-coupling effects are different, there is no single adjustment to the control system that will simultaneously eliminate all of them. In most helicopters that I know of, moving the stick directly to the right produces minimum cyclic pitch over the tail boom and the decoupling is left to the skilled pilot. I have been told, however, of two where the designers have attempted to minimize cross-coupling by changing the geometry of the control system within the fuselage. One is Frank Robinson who has designed the control systems on his helicopters such that when the stick is pushed straight forward, the maximum pitch displacement occurs at an azimuth angle of 73 degrees instead of at 90 degrees.