OTHER:Flight Dynamics - General - Phase Lag (Δψ), [Gamma (Γ)]
The vertical (Y ~ from 0º to 180º) is thePhase angle. [φ]
The horizontal (X ~ from 0 to 2.2)is the Frequency Ratio. [Ω/ωN undamped] The rotational speed of the rotor over the natural frequency of blade flapping. [i.e. The inverse of the first flapping frequency.]
[c/cc] is the damping ratio, which is damping over the critical damping.
Example: The Sikorsky XH-59A ABC's first flapping frequency ≈ 1.4, therefore its frequency ratio ≈ 0.71
Eventually move the note below on to UniCopter off of this generalized page.
Re UniCopter: As the natural frequency of the blade increases and/or the critical damping increases, the phase angle decreases. If the frequency was infinite then the phase angle will be 0º. The mass required to produce a rigid rotor will be quite large and this large mass will be more effected by gyroscopic precession. Fortunately, the gyroscopic precession of one rotor will be offset by the gyroscopic precession of the other.
The swashplate phasing required is just equal to the delta3 angle.
Δψ = 90° - tan-1(Kp) = 90° - δ3
Change in the azimuth = 90° - tan-1 * [pitch-flap coupling] = 90° - [delta3 angle]
Phase lag angle (Γ) (Δψ) and Control phase angle (γ) may be two names for the same item.
Hinge offset will also reduce the phase angle to some value that is less than 90º. The larger the offset, the smaller the angle.
The phase angle (Γ) (Δψ) is given by tan-1(Sβ)
The stiffness number (Sβ) is given by 8(λβ2 - 1)/ γ
The flap frequency ratio (λβ) is given by Ω / ωN undampened
The Lock number (γ) is given by (ρ * a * c * R^4) / Ib.
Aerodynamic forces do not change the phase lag from 90º[Source ~ RWP1 p.447]
The phase lag "stems from the two components ...... one aerodynamic due to the distribution of airloads from the angular motion, the other from the gyroscopic flapping motion." ~ Padfield, [my bold]
Prouty, (I think it was him) has stated that when the control plane (swashplate) is changed by cyclic input the tip-path-plane will realign itself with the control-plane within 1/2 of a rotor's revolution.
"For blades freely articulated at the center of rotation, or teetering rotors, the response is lagged by exactly 90-degrees in hover; for hingeless rotors, such as the Lynx and Bo105, the phase angle is about 75 to 80-degrees."
It has been stated that the conventional swashplate is an ideal control mechanism for rotors with a 90-phase angle, but it is not ideal for rotors with a phase angle less than 90.
To me it appears that the resultant control (swashplate) phase angle is a compromise between the gyroscopic precession's 90º and the aerodynamic precession's less than 90º.
In other words, if the UniCopter had absolutely rigid blades, the aerodynamic precession's phase lag will be 0-degrees, the gyroscopic precession's phase lag will be 90º and the resultant phase angle will be???
The first point that must be dispelled is that there is no singular lifting force at 90º azimuth, which results in an increased elevation at 180º azimuth. The lifting force exists at all degrees from 1º to 179º. This very point is part of the mathematical proof of gyroscopic precession. The location(s) of this force is also obvious when envisioning aerodynamic precession, since the blade does not snap to some steep pitch at 89º and then snap back down at 91º.
The only thing of interest about 90º is that;
1/ It is the location of the greatest lifting force, between 0º and 180º, and
2/ it is the mid-location of all the degrees that exhibit a lifting force.
The second point is that it has been mathematically and experimentally proven that precession, both aerodynamic and gyroscopic, can have phase angles of less than 90º.
The third point is that flapping hinge offset and delta3 cause the tip path plane to realign itself in less than 180º, after a change in the control plane (swashplate).
The above shows that by the use of a flapping offset hinge or delta3 hinge it is possible to start the lifting at 18º, have maximum lifting force at 99º and alignment of the two planes at 180º. Once the two planes are reoriented, in respect to each other, the remaining 18º become irrelevant.
Even more thoughts:
The following is submitted to provoke interest, boredom, or a scathing rebuttal.
The conditions are;
~ The rotation is CCW, when viewed from above.
~ Azimuth-0º is aft, azimuth-90º is to the right, etc.
~ The flight maneuver is a transition from hover to forward flight.
In the case of A/, the higher blade pitches will be found between azimuth-181º and azimuth-359º, with the highest at azimuth-270º. The 90º phase-lag will result in the disk being high between azimuth-271º and azimuth-89º, with the highest at azimuth-360º. This will cause the rotor disk to 'drag' the helicopter's nose down about its pitch axis.
In the case of B/, the higher blade pitches will be found between azimuth-271º and azimuth-89º, with the highest at azimuth-360º. The 0º phase-lag, in conjunction with the absolutely rigid coupling of the rotor to the fuselage will cause the rotor disk to 'pry' the helicopter's nose down about its pitch axis.
An addendum to the forgoing is that this extremely rigid rotor must have higher than normal strength and thus greater mass. The mass will invoke gyroscopic precession. However, the gyroscopic precession will be relatively small, and in the case of twin counter-rotating main rotors, such as the coaxial and intermeshing configurations, the opposing gyroscopic precessions should provide some static stability.
Γ[Gamma] is Variable Control Input Phase Angle. It is the same as Phase Lag.
More Gamma means less load on the advancing blade and more tip clearance
For reference; [B1' ] Differential (Opposed) Lateral Cyclic is the other method available on the ABC
Γmay have exactly the same meaning as be the same as Δψ and φ
The optimally desired phase angle of a helicopter will vary depending on the forward velocity and other aerodynamic events at that specific time.
DESIGN: Dragonfly ~ Control - Flight - Assembly - Phase Lag
Delta-3 and Phase Angle
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Last Revised: September 1, 2008