Relationship between flapping and lagging frequency

Question

In my rotorcraft module at university, we were told that blades leading/lagging is as a result of blade flapping. To conserve angular momentum, an upward flapped blade (CG moves toward rotor hub) will speed up and hence "lead". A downward flapped blade (CG moves outwards) will slow down and hence "lag". I understand this, however if the lead/lag was as a direct consequence of flapping, then shouldn't their frequencies be the same? In the above image (Shown to us in lecture) you can see that the lagging frequency is roughly 1/5 of the flapping frequency. How is this possible? If the blade lag is a direct consequence of flapping surely this is not possible.

Are there more factors that affect lead/lag? Perhaps cyclic pitch control alters the drag of the blades and hence affects the lead/lag? Or could it be to do with the hinge and any associated damping effects.

A comprehensive explanation would be appreciated.

Answer 1

I agree with your assessment. Have you discussed this with your prof/instructor? When performing Rotor Track and Balance, a worn lead/lag damper or dampers will directly affect the 1 per revolution lateral vibration with little to no affect to other rotor rpm orders or suborders. This is most pronounced in helicopter models which utilize hydraulic dampers as opposed to elastomeric or friction type lead/lag dampers. Hope this helps.

Another factor that contributes to Lead/Lag is alternating spanwise distribution of lift as a result of forward flight. As a rotor blade advances, the lift migrates toward the tip. Conversely, as the blade retreats, lift shifts more toward the root of the blade. Still a 1 per revolution effect.

Chordwise blade mass distribution (Chordwise Balance Aka Product Balance) accounts for angle of attack changes with respect to power changes. If one blade climbes or dives a greater amount than the other/s in transition from Ground-Flat-Pitch to Hover, there is a chordwise balance disparity, which, in turn leads to effects in Lead/Lag. Still a 1 per revolution effect.

Answer 2

The smaller lag frequency mentioned is intended to be the natural frequency that the blade will lead/lag when left alone. In other words, if you "plucked" the blade tip (pulled it aft a bit and released), it would vibrate in the lead/lag direction at that frequency.

In actual operation, oscillatory forces acting on the blade will occur at other frequencies. If the rotor is flapping at 1/rev then conservation of momentum will apply a 1/rev oscillatory force in the lead/lag direction.

The total net lead/lag motion of the blade will be a function of both of these and aerodynamic forces, other natural frequencies, ....

As an analogy, consider the A2 string on a guitar. Its natural frequency is 110Hz (if you pluck it and watch it vibrate). However, when someone plays guitar they pluck the string at other frequencies.

Another analogy you could analyze mathematically would be a spring-mass system with natural frequency F1, but with an external force acting on the mass with different frequency F2.

Answer 3

Possibly it has to do with the fact that “flapping” also incorporates not just dyssimetry of lift but also control inputs.

Perhaps the formulation comes from the fact that it is calculated for forwards flight?

Control inputs to tilt the disc forwards cause the blade to flap up relative to a hover state of equal pitch in 360deg. Rotation.

The command input to the blade to pitch up and create forwards flight begins as the blade passes through 90deg. From the nose due to gyroscopic precession requirements. As the blade reaches its maximum flap at 180deg from the nose it begins to descend but is then affected by dyssemetry of lift and tries to flap up again as it passes through 270deg. The point at which it then experiences the maximum dyssemetry.

Thus... the blade experiences more than just a 1 Per/rev for upward flapping events. While its only downward flap event from a neutral hover pitch angle is while it travels from 000 to 90 Deg... as it experiences loss of lift due to its retreating blade airflow.