2.
Introduction <ul><li>The trim, stability and control of helicopter is different from the fixed-wing aircraft. </li></ul><ul><li>Flapping motion of rotor blade is the key in helicopter stability and control. </li></ul><ul><li>Factors of flapping motion: forward speed, controls and angular velocity. </li></ul><ul><li>Pilot controls the helicopter through flapping motion. </li></ul>
3.
Topics <ul><li>Equation of rotor flapping motion </li></ul><ul><li>3 origins of rotor flapping motion </li></ul><ul><li>The effect of flapping hinge offset on the rotor flapping motion. </li></ul><ul><li>Hub moments </li></ul><ul><li>Flapping motion of hinge-less rotor. </li></ul>
4.
Flapping Motion Equation <ul><li>Assume: </li></ul><ul><ul><li>rigid blade; </li></ul></ul><ul><ul><li>flap hinge is at the center of hub </li></ul></ul><ul><li>Thus: </li></ul><ul><li>In the flap plane, the forces applied on the blade segment include aerodynamic, centrifugal, inertia forces and gravity. The moments of these forces about the hinge are: </li></ul>Hub plane Rotor shaft
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.
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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
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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
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Aerodynamic Forces of Blade Element Advancing ratio Inflow ratio Induced velocity Equivalent induced velocity Inflow Hub plane Side View Top View
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
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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
16.
Flap Motion Due to Fuselage Angular Velocity Similarly, the roll rate causes , Lock number: <ul><li>Summary: </li></ul><ul><li>Damping effect: Pitching nose up produces rotor disk tilting forward, rolling right results in rotor disk tilting left. </li></ul><ul><li>Gyro effect: There exists a serious corss-coupling, the magnitude of the cross-coupling is about the half of primary motion, the tilting of rotor disk leads 90 degree with respect to angular velocity. </li></ul><ul><li>The flap motion due to angular velocity varies with forward speed. </li></ul><ul><li>The flap motion due to angular velocity decreases with the increment of rotor RPM </li></ul>
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
19.
Effect of Flapping Hinge Offset on Rotor Flapping Motion <ul><li>We have discussed the flapping motion of rotor without hinge offset </li></ul><ul><ul><li>Flapping frequency is exactly the rotor rotation speed </li></ul></ul><ul><li>How about the flapping motion of rotor with hinge offset ? </li></ul><ul><ul><li>Flapping frequency ? </li></ul></ul><ul><ul><li>Flapping damper ? </li></ul></ul><ul><ul><li>…… </li></ul></ul>
20.
Effect of Flapping Hinge Offset on Flapping Frequency <ul><li>The balance of flapping moments, produced by aerodynamic, centrifugal and inertial forces is about flapping hinge other than rotor rotation center. </li></ul><ul><li>The equation of flapping motion becomes </li></ul>
21.
Effect of Flapping Hinge Offset on Flapping Frequency <ul><li>Natural frequency is </li></ul><ul><li>For blade with uniform mass </li></ul>When <ul><li>Conclusion: </li></ul><ul><li>There exists the effect of hinge offset on the rotor flapping frequency </li></ul><ul><li>The effects is small for the practical helicopter </li></ul>Physical meaning ?
22.
Effect of Flapping Hinge Offset on Rotor Flapping Damping <ul><li>Reviewing mass-damper-spring system </li></ul><ul><li>Damping: c </li></ul><ul><li>Critical damping: </li></ul><ul><li>Damping Coefficient: </li></ul>m k c f(t) x
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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
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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
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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
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Effect of Flapping Hinge Offset on Rotor Flapping Motion <ul><li>Notes about hub moment: </li></ul><ul><li>For flapping motion due to control, the large e produces the great control moment and improves the control capability and maneuverability. </li></ul><ul><li>For flapping motion due to forward speed, the large hub moment provide the velocity stability and dihedral effect. However, increases the instability of attack angle. </li></ul><ul><li>In order to obtain the necessary control power, we need e in a certain extent. While we have to limit the value of e to avoid the instability of attack angle. Generally, the suitable value of e is about 5%R. </li></ul>
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:
28.
Exercise <ul><li>The rotation direction of Black hawk helicopter rotor is anticlockwise from the top view. Please answer the following questions: </li></ul><ul><li>How the rotor flaps when the helicopter comes up gust in hove? How to adjust the controller to keep the hover condition? </li></ul><ul><li>How the blade changes its feathering angle following the pilot’s adjustment? </li></ul><ul><li>During the left and right turn flight with constant speed and altitude, is there any different for pilot control? Why? </li></ul>
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