1. Helicopter Flight Dynamics Chapter 3: Rotor Flapping Motion

5. Flapping Motion Equation The summation of these moments should be zero. That is, We have the following equation of rotor flapping motion. Recall the motion equation of mass – spring system By comparing, the natural frequency of flapping motion is: The natural frequency of flapping motion is the same as rotor rotation speed. If the frequency of exciting force is also rotor rotation speed. The rotor flapping motion will be resonance.

6. Coefficients of Flap Motion Assuming: Neglecting the harmonic components larger than two: Assuming the flapping motion of each blade is the same. The geometric meaning of above equation is an upending cone. : Cone angle : Backward tilted angle : Sideward titled angle

7. Aerodynamic Forces on Blade Element Air density Longitudinal cyclic pitch Attack angle Slope of airfoil lift curve Blade twist Chord of airfoil Pitch angle Pitch angle at blade root Longitudinal cyclic pitch Hub plane

8. Aerodynamic Forces of Blade Element Advancing ratio Inflow ratio Induced velocity Equivalent induced velocity Inflow Hub plane Side View Top View

9. Solution of Flap Coefficients

10. Flap Motion Due to Forward Speed Relative speed

11. Flap Motion Due to Forward Speed

12. Flap Motion Due to Pilot Control Blade pitch angle : Cyclic control results in the variation of blade aerodynamic force and flap motion For central articulated rotor, the flap coefficients are Although the variations of pitch angle at advancing and retreating side are same the aerodynamic force is different, the variation of aerodynamic force in the advancing side is greater then retreating side due to the higher relative speed in advancing side. Thus, the flap is slight larger than cyclic pitching angle. In hover, the flap is the same as feathering. We call it as ‘equivalence of flap and feathering” Pilot controls the helicopter by flap motion due to control :

13. Equivalence of Flapping and Feathering Interpretation of flapping and and feathering coefficients

14. Equivalence of Flapping and Feathering longitudinal lateral

15. Flap Motion Due to Fuselage Angular Velocity The angular velocity of fuselage will produce additional flap motion because the angular velocity causes the additional velocity and Coriolis Force at the blade segment. For example, At first, producing additional velocity and attack angle Resulting in the variation of lift and flap Secondly, producing Coriolis force as The Coriolis force applies additional flap moment

17. Exercise Assume the induced velocity distribution of a see-saw rotor helicopter in hover is: The controls applied are: To determine the rotor flapping coefficients:

18. Effect of Flapping Hinge Offset on Rotor Flapping Motion Hinge offset = 0 Hinge offset ≠ 0

23. Effect of Flapping Hinge Offset on Rotor Flapping Damping Flapping damping Damping Average damping ratio If the damping ration is about 84% of central hinge rotor Physical meaning ? Flapping motion produces velocity at blade element , which results in the variation and produces the damping moment

24. Effect of Flapping Hinge Offset on Rotor Flapping Phase Non-resonance, the phase of output to input For linear system For uniform blade Special case

25. Effect of Flapping Hinge Offset on Rotor Flapping Motion Producing hub moments Hub moments comes from centrifugal force and is proportional to flapping hinge offset e

27. Flap Motion of Hinge-less Rotor For hinge-less rotor, the flapping motion is implemented by the elastic deformation of blade root. We can treat the hinge-less rotor as an articulated rotor with equivalent flapping hinge offset. By setting the first order flapping frequency of hinge-less rotor equal to that of the articulated rotor, the equivalent flapping hinge offset is: