0871

Item 0871

DESIGN: UniCopter ~ Rotor - Disk - Lateral Dissymmetry of Lift and Drag? - (3-blades)

Review 1073.html in Obsolete folder to see if these two pages are in agreement
Overview:

Evaluation of the oscillating rolling moments with two 3-blade really rigid rotors, during forward flight, caused by dissymmetry of lift.

Conclusion:

There will be vibration, with 3-blade rotors. The side of the helicopter that has the advancing blade located at 90º azimuth will have greater lift than the other side where the advancing blades are at 30º and 150º azimuth. This is irrespective of whether the rotation is outside-forward or inside-forward.

Potential Solutions:

DESIGN: UniCopter ~ Rotor - Disk - Lateral Dissymmetry of Lift (Drag?) - 4-blade Rotors

UniCopter ~ Control - Flight - Swishring

Evaluation of the oscillating yawing moments during forward flight, caused by the dissymmetry of H-force drag, may be later considered on a separate web page.

Drawing of Lateral Moment Arms of a Single Rotor:

Hover:

Moment on Y-Y: (100 lbs. * 1.0) = (2 * ( 100 lbs. * .5))

If all three blades are providing an equal amount of lift then the center of the lift will be at the centerline of the disk. Brilliant; I didn't need a drawing to prove this.

_____________________________

Perhaps the significance of this is that if the center of lift is behind the centerline of the blade's pitch axis then the blades will be imparting a loading on the pitch arms. During hover, there should be no cyclic oscillation because the sum of the moments of the three blades should equal zero. There will be a force on the collective.

The center of lift on the VR7 blade varies from 41% of chord at Alpha = 0º to 29.3% of chord at Alpha = 12º.

This page was done to assure that the wheels in the 'swash-ring' are always in contact with the same flange of the ring and therefore not having to change their direction of rotation ( at 600 RRPM) 40 times per second. They will contact both flanges if Higher Harmonic Control is implemented. See swishring with revision for one torus with a follower on the top and a follower on the bottom.

Forward Flight: (Lateral Dissymmetry of Lift and Drag)

I think that the single advancing blade at 90º azimuth will have a greater thrust than the other rotor's two advancing blades which will be at 30º and 150º azimuth, Note the moment arm lengths and the air velocity. This will cause a rotational vibration about the longitudinal axis (roll). Higher Harmonic Control will allow the lift at 30º and 150º azimuth to be increase somewhat and the lift at 90º azimuth to be decrease somewhat.

Flight-Control Solution:

Consider adjusting HHC pitch by a spring steel 'warpable' swishring. Adjust by heat (bimetal), electrical (magnet) or mechanically.

The 1P will involve frequent changes, as the cyclic stick is moved. Changes to the HHC will be gradual and infrequent, relative to the primary control, since it is controlled by the forward velocity of the craft. The 3P control may be similar to the SynchroLite's Opposed Lateral Cyclic. For possible means of flight control, see DESIGN: UniCopter ~ Control - Flight - Swishring - Fixed Azimuth Overview

Re Drag: An airfoil that has a higher c_l/c_d than another should have a lower dissymmetry of drag for a given lift.

Calculations: by FORM: Lateral Dissymmetry ~ UniCopter

Default Values:

Slope of lift chord: a = 6.0 / radian

Vertical velocity in hover: v₁ = 25 fps

Density: ρ = 0.002377 slug/ft³@ ISA

Rotor: Ω = 600 RPM

Radius: r = 0.75 * 8.33 = 6.2475 feet

Chord: c = 0.666 feet

Blade segment Δr = (8.33 / 10) = 0.833 feet

Collective = 5.5º

Inputs:

Collective: θ₀

1P Blade Pitch θ_1P, Azimuth of input blade pitch: Ψ_1P.

Higher harmonic 3P blade pitch: θ_3P.

Forward Velocity: V

Cyclic amplitude: θ_1P

Cyclic amplitude azimuth: Ψ_1P

3rd harmonic amplitude: θ_3P

Algorithims:

Tangential (local) Velocity: U = (Ω * r) + V * sin(Ψ * ( π / 180) [ft/sec]

Inflow Angle: φ = v₁ / (U) [radians]

Cyclic amplitude at azimuth: θ_1P@Ψ = θ_1P * sin(Ψ_1P) This will have to be checked.

3rd harmonic amplitude at azimuth: θ_3P@Ψ = θ_3P * sin(Ψ * 3) This will have to be checked.

Blade Pitch: θ = θ_1P@Ψ + θ_3P@Ψ

Blade Pitch: θ = ά + φ [radians]

Angle of Attack: ά = θ - φ [radians]

Lift [L]:

An aerodynamic force caused by air flowing over an airfoil. The vertical component of thrust.

For an airplane wing: L_W = (ρ / 2) * V² * C_L * S . Where S is the area of the wing.

ΔL = (ρ / 2) * (Ω * r)² * a(θ - (v₁ / (Ω * r)) * c * Δr

ΔL = (ρ / 2) * (U)² * a(θ - (v₁ / U)) * c * Δr

ΔL = (ρ / 2) * (U)² * a(θ - φ) * c * Δr

ΔL = (0.002377 / 2) * (U)² * a(θ - φ) * 0.666 * 0.833

ΔL = 0.001188 * (U)² * a(θ - φ) * 0.666 * 0.833

L = (U)² * a(θ - φ) * 0.000659

Thoughts on coding:

Base calculations on r = .75R at all azimuths, even though it will be greater on the retreating side.

FORM: Lateral Dissymmetry ~ UniCopter

The following values assume that the rotor disks have no forward pitch and all the forward thrust is coming from a pusher propeller.

The following thrust values are only for the blade segment at .75R of a helicopter weighing 750 lb GW and are probably not representative of the whole blade.

MPH:	θ₀	θ_3P	90º	Port	30º	Star	150º	Star	270º	Star	210º	Port	330º	Port	PORT	STAR	Total Lift:
			lbs	lb-ft	lbs	lb-ft	lb	lb-ft	lbs	lb-ft	lbs	lb-ft	lbs	lb-ft	Port	Star
0	5.5	0	58	427	58	246	58	246	58	-296	58	-116	58	-116	724	723	348
50	5.4	0.1	79	580	69	294	69	294	38	-195	46	-92	46	-92	772	775	347
100	5.1	0.3	95	704	80	341	80	341	22	-113	33	-67	33	-67	816	817	343
150	4.8	0.5	110	814	92	389	92	389	11	-55	23	-47	23	-47	872	869	351
200	4.3	0.6	119	877	97	414	97	414	3	-16	15	-30	15	-30	888	893	346
250	3.8	0.7	122	900	102	434	102	434	0	-1	9	-18	9	-18	904	901	344

___________________

Port is the port rotor and Star is the starboard rotor.

Violet color represents forces and moments on port side of craft.

Red represents forces and moments on starboard side of craft.

PORT is the sum of the positive rolling moment and STAR is the sum of the negative rolling moment. They must be in balance.

The collective and the 3P-amplitude have been adjusted as the forward velocity increases so as to maintain a constant thrust, plus no rolling moment.

So far it appears that the 3rd harmonic (swashring) only needs a maximum amplitude of 0.7º.

___________________

The following is the same as the previous table excepted that it checks the roll moment at the 150-mph conditions, when the blades are at a different azimuth.

MPH:

θ₀

θ_3P

105º

Port

45º

Star

165º

Star

285º

Star

225º

Port

345º

Port

PORT

STAR

Total Lift:

lbs

lb-ft

lbs

lb-ft

lbs

lb-ft

lbs

lb-ft

lbs

lb-ft

Port

Star

150

4.8

0.5

111

797

105

587

195

-55

-56

-16

854

852

349

Looks good!

None of the forgoing yet considers forward cyclic on the rotors.

For more information that might be related see: DESIGN: UniCopter ~ Rotor - Disk - Cross-coupling

Reference:

Advancing Blade Concept (ABC) Dynamics ~ Vibration, page 5 onward ~ Presentation May 1977

Advancing Blade Concept (ABC) High Speed Development ~ Vibration, page 10-12 ~ Presentation May 1980

	OTHER: Aerodynamics - Vibration - Rotor Induced - Higher Harmonic Control	0793
	DESIGN: UniCopter ~ Control - Flight - Swashring - Fixes Azimuth Overview	1092

Drawing:

Consider 2P and 4P higher harmonic, as alternative.

Direction of Rotation:

Objective:

To get a comparative value of the moment about the craft's roll axis, based upon the direction of rotor rotation (i.e. outside-forward or inside-forward).

Conclusion:

The direction of rotation does not change (improve or worsen) the lateral dissymmetry of lift.

Supporting Calculations:

The FORM: Lateral Dissymmetry ~ UniCopter was used for these calculations.

The current default values (which will not be the final ones) were the same for every blade azimuth.

The forward velocity of 100 mph was selected.

A fixed collective value was used and no cyclic values were used. The assumption is that the prop is providing all forward thrust.

The blade azimuths and approximate moment arms are as shown in the second drawing on this page.

The craft's rolling moment is positive (+) when it is CW when viewed from the rear.

______________________________________

Rotors Turning Outside-Forward.

Rotor:	Azimuth:	Lift: (pounds)	Arm: ⁽¹⁾(feet)	Moment; (ft-lb)	Direction of Roll: (viewed from aft)
Port	90º	109.5067	7.3725' (88.47")	807.3384	CW
Port	210º	38.1645	2.0048' (24.06")	-76.5113	CCW
Port	330º	38.1645	1.9893' (23.87")	-75.9194	CCW
Sum of Port Moments:				654.9077	CW
Starboard	30º	81.6424	4.2496' (50.99")	-346.9485	CCW
Starboard	150º	81.6424	4.2444' (50.93")	-346.5265	CCW
Starboard	270º	22.5509	5.1225' (61.47")	115.5169	CW
Sum of Starboard Moments:				-577.9581	CCW
Resultant Moment:				76.9496	CW

(1) The Arm values are back calculated from the Access generated Lift and Moment.

______________________________________

Rotors Turning Inside-Forward.

This table is the above table with values switched so as to represent the crafts moment about its X-axis if the blades were rotating inside-forward.

Rotor:	Azimuth:	Lift: (pounds)	Arm: ⁽¹⁾(feet)	Moment; (ft-lb)	Direction of Roll: (viewed from aft)
Port	30º	81.6424	1.9893' (23.87")	-162.4112	CCW
Port	150º	81.6424	2.0048' (24.06")	-163.6757	CCW
Port	270º	22.5509	7.3725' (88.47")	166.2565	CW
Sum of Port Moments:				-159.8304	CCW
Starboard	90º	109.5067	5.1225' (61.47")	560.9481	CW
Starboard	210º	38.1645	4.2444' (50.93")	-161.9854	CCW
Starboard	330º	38.1645	4.2496' (50.99")	-162.1838	CCW
Sum of Starboard Moments:				237.7789	CW
Resultant Moment:				77.9485	CW

Comparison with Coaxial

Objective:

To compare the above two intermeshing roll moments with that of an identical coaxial configuration.

Conclusion:

All three rotor configurations have the same lateral dissymmetry of lift (moment).

Rotors Crossing at 90º - 270º:

This is the 60º index, which the Sikorsky first used.
Upper rotor turns CCW
This table is identical to the above tables except that the stagger is 0" .

Rotor:	Azimuth:	Lift: (pounds)	Arm: ⁽¹⁾(feet)	Moment; (ft-lb)	Direction of Roll: (viewed from aft)
Lower	90º	109.5067	6. 25'	684.4169	CW
Lower	210º	38.1645	3.125'	-119.2641	CCW
Lower	330º	38.1645	3.125'	-191.2641	CCW
Sum of Lower Moments:				301.8887	CW
Upper	30º	81.6424	3.125'	-255.1325	CCW
Upper	150º	81.6424	3.125'	-255.1325	CCW
Upper	270º	22.5509	6.25'	140.9432	CW
Sum of Upper Moments:				-369.3219	CCW
Resultant Moment:				67.4332	CW

Vibration:

Lockheed XH-51A article from the Nov 2004 issue of Stu Field's magazine Experimental Helo

The rotor head was of a "hinge-less" design. Prouty further states, the "hinge-less" design suffers from the characterization that "they all shook". Lockheed added a fourth blade and controlled the shaking to a more pilot acceptable level.

Additional Information:

Advancing Blade Concept (ABC) High Speed Development ~ Vibration, page 10-12 ~ Presentation May 1980

See notes on: DESIGN: UniCopter ~ Rotor Disk - Absolutely Rigid Rotor

Last Revised: August 12, 2008