7. Blade motion and rotor control
Transcript of 7. Blade motion and rotor control
Active Aeroelasticity and Rotorcraft Lab.
7. Blade motion and rotor control
2020
Prof. SangJoon Shin
Active Aeroelasticity and Rotorcraft Lab., Seoul National University
I. Equilibrium of hinged blades
II. Control of the hinged rotor in hover
III. Blade flapping motion
IV. Rotor control in forward flight
V. Blade motion in the plane of the disk
Overview
1
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Introduction
Rotor moving edgewise in the air : forward flight
β two standard means available to overcome dissymmetry of lift
1. Hinged at the roots so that no moments can be transmitted
β Control can be achieved by tilting the hub axis until the resultant rotor
vector points in the desired direction
2. Rigidly attached to the shaft but cyclically feathered
β Decrease pitch on advancing side / increasing pitch on retreating side
β Equalize the lift around the disk
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I. Equilibrium of hinged blades
Normal flapping blade⦠effectively mounted to the hub on a universal
joint β free to flap, lead, or lag, but always fixed in pitch
1. Equilibrium about the flapping hinge
Forces acting on the blade in flapping direction
β’ lift, centrifugal forces, weight(negligible)
Elemental centrifugal forces (Fig. 7-3)
m : mass per unit lengthΞ© : rotational speedπ : radius of the elementπ½ : blade flapping angle
π πΆ. πΉ. = (πππ)Ξ©2π cos π½ (1)
Fig. 7-3 Centrifugal force distribution
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I. Equilibrium of hinged blades
β’ Component of centrifugal force
perpendicular to the blade
π πΆ. πΉ. sin π½ = πππΞ©2ππ½
β varies linearly with the radius β
β’ Lift force distribution * π = ππ , the blade mass
(2)
(3)
Untwisted constant-chord blade
Ideally twisted constant-chord
inflow varies linearly with radius
inflow is constant along the radius
πΆ. πΉ.ππππππ‘ =1
3π ππΊ2π π½ =
2
3πΆπΉ π π½
β’ Moment exerted by C.F. about the flapping hinge
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I. Equilibrium of hinged blades
β’ Elemental lift ππΏ
ππ= ππ
π
2πΊ2π2π
β’ For an ideally twisted constant-chord blade
ππ = πΌππΌ = πΌ ππ‘π
πβ
π£
Ξ©π
β πΏπππ‘ = ππππ π‘πππ‘ Γ π
Lift of an ideally twisted blade varies with radius
For an untwisted blade, πΌπ roughly constant, lift varies β π2
2
3π Γ ππππ‘ (for ideal twist)
4
3π Γ ππππ‘ (no twist and no taper)
(4)
(4a)
πΏπππ‘ ππππππ‘
(5)
(6)
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I. Equilibrium of hinged blades
β’ Coning angle Ξ²
Ξ² =πππππ ππππ‘
πΆ.πΉ.(for ideally twisted)
Ξ² =9
8πππππ ππππ‘
πΆ.πΉ.(untwisted, constant-chord)
β Ξ² in hovering ~ Cπ
2. Equilibrium about the drag hinge
component of C.F. perpendicular to the blade toward zero lag (Fig. 7-6)
)π πΆ. πΉ. = ππΊ2πππ(π β π
* π = lag angle* π = angle between no lag position
and line of action of C.F.
(7)
(8)
π = π 1 βπ
π
β’ From Fig. 7-6, ππ = π(π β π)
β΄ π πΆ. πΉ. = ππΊ2ππππ 1 β 1 βπ
π= ππΊ2ππππ β¦ constant along the span (9)
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I. Equilibrium of hinged blades
Moment of the centrifugal force about the lag hinge
πΆ. πΉ.ππππππ‘ = mRππΊ2π π.π.π = MππΊ2π π.π.π
Aerodynamic forces
Denote resultant force as πΉ, point of application as π π.π.
π΄πππππ¦πππππ ππππππ‘ = πΉπ π.π.
Equating with C.F. moment,
Equating the shear forces @ lag hinge (Fig. 7-7)
πππππ’π/π = πΉ πππ π + ππΊ2π π.π. π ππ π = πΉ +ππΊ2π π.π.π
* π π.π.: distance from axis of rotation to the blade c.g.
πΉπ π.π. = ππΊ2π π.π.ππ or πΉ =ππΊ2π π.π.ππ
π π.π.
(10)
(11)
(12)
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I. Equilibrium of hinged blades
(12) β (11)
mean drag angle is a function of π‘ππππ’π/Ξ©2 β πΆπ
Relatively insensitive to change in π π.π.
π =πππππ’π
πΞ©2π π.π.ππ
π π.π.+ 1
Torque/e = πΞ©2π π.π.ππ
π π.π.+πΞ©2π π.π.π = π πΞ©2π π.π.
π
π π.π.+ 1
(13)
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II. Control of the hinged rotor in hover
Sudden rotation of control axis (Fig. 7-11)
Change in pitch angle of the blade
β Lift increase β Blade moves, or
βflapsβ β Continues until the plane
of the blades is again perpendicular
to the control axis @ which position
no cyclic-pitch changes occur
Some delay between a rapid control
angle change and the re-alignment
of the rotor disk
β extremely small
Differences when the rotor is moving
edgewise through the air (Fig. 7-12)
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III. Blade flapping motion
1. Flapping as represented by a Fourier series
Flapping motion
π½ = π0 β π1 πππ π β π1 π ππ π β π2 πππ 2π β π2 π ππ 2πβ¦
π½ : angle between the control axis and the bladeπ : azimuth angle (Fig. 7-13)
(14)
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III. Blade flapping motion
2. Geometrical interpretation of the Fourier coefficient
π0 : flapping angle independent of the blade azimuth angle π in hover
π½ = π0 (Fig. 7-14 β)
π1 : amplitude of a pure cosine motion
π½ = βπ1 πππ π (Fig. 7-15, 7-16 β)
π1 : amplitude of a pure sine motion
π½ = βπ1 π ππ π (Fig. 7-17, 7-18)
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III. Blade flapping motion
(-) sign β result in plus values for the π1 and π1 coefficients
in normal forward flight
π2 : amplitudes of the higher harmonics
π½ = βπ2 πππ 2π (Fig. 7-17, 7-16)
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III. Blade flapping motion
3. Physical explanation of the existence of the component motions
An infinite number of terms in Fourier series exactly describes any arbitrary
motion. However, only a few terms are necessary.
Magnitude of a typical flapping motion in forward flight
π0 = 8.7Β°, π1 = 6.1Β°, π1 = 3.9Β°, π2 = 0.5Β°, π2 = β0.1Β°
β Coning angle, π0 β¦ depend on the magnitudes of 2 primary moments about
the flapping hinge Thrust moment (Fig. 7-21)
C.F. moment
Hover⦠large inflow (induced), loading toward tips,
larger coning angle (9Β°)
Min. power⦠small inflow (small induced), loading
more inboard, smaller coning angle (8Β°)
High speed⦠large inflow (parasite), loading toward
tips, larger coning angle 9Β°
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III. Blade flapping motion
β‘ Backward tilt, π1β¦ π = 90Β° β lift increase β flapping up (Fig. 7-23)
β’ AoA decrease (Fig. 7-24) no unbalanced force for blade with no inertial forces
β’ To consider blade mass and air damping, blade as a dynamic system
β 1 DOF system (Fig. 7-25)
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Force-displacement phase to the frequency of the forced vibration (Fig. 7-26)
III. Blade flapping motion
β’ π· : phase angle between the
max. applied force and max.
displacement
β’π
ππ: ratio of the actual
damping to critical damping
β’ When Ο
Οπ= 1 β phase angle
π = 90Β° and is independent of
the amount of damping
β’ Flapping blade (Fig. 7-2)
(15)ππ =πΎ
πradians/second
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Simple flapping rotor with flapping hinge on the axis of rotation
III. Blade flapping motion
πΆ. πΉ.ππππππ‘ = ΰΆ±0
π
πΊ2π2π½πππ = ππΊ2π½π 2
3
π ππ π‘πππππ ππππππ‘ = πΎπ½ (πΎ = ππΊ2 π 2
3)
πΌ =1
3ππ 2, ππ =
πΎ
πΌ= πΊ2 = πΊ
When hinge offset = h,
Exciting air forcesβ¦ 1/rev βΟ
ππ= 1
β πππππ β πππ πππππππππ‘ πβππ π = 90Β°
β’ Maximum flapping at π = 180Β°
Minimum flapping at π = 0Β°
(17)
(18)
(19)
ππ = πΊ 1 +3
2
β
π (19a)
Fig. 7-2 Flapping blade
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III. Blade flapping motion
β’ Sideward tilt, π1, β¦ may be viewed as arising from coning, π0
Coned rotor (Fig. 7-27a) : Difference in AoA between front and rear of the
blades due to forward speed
No coning (Fig. 7-27b) : effect of forward velocity is identical
Active Aeroelasticity and Rotorcraft Lab., Seoul National University
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III. Blade flapping motion
Fig. 7-28 : force is maximum at π = 180Β°, minimum at π = 0Β°.
β Force-displacement phase of 90Β°
β Max. flapping at π = 270Β°, min. at π = 90Β°
β π + π1 motion because of coning. π1 β¦ same order as π or larger
π1 tilt is very sensitive to variation in inflow
β assumed as uniform for forward flight performance analysis
β However, for low forward speed, π£ ~ quite large at the rear β π1 increase
At higher forward flight speed, inflow decreases, and becomes uniform
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III. Blade flapping motion
β£ Higher harmonics
a2, π2, π3, π3β¦ weaving of the blade in and out of the surface of the core
Presence of the forces which produce higher harmonic motions
Asymmetric flow pattern, reverse flow region πΉπππππππ π£ππππππ‘π¦ β π ππ2 π
Little importance on control and performance, but extremely important for
vibration and stresses
β€ Effect of blade mass on flapping motion
π0β¦ directly affected by blade mass
π1β¦ independent of blade mass since exciting forces act on resonant system
π1β¦ in resonance β independent of blade mass
but exciting forces proportional to π0, which is proportional to blade mass
Blade mass increases to infinity β π1 decreases to zero
Higher harmonics⦠forced vibration well above resonance, goes to zero
when blade mass β
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IV. Rotor control in forward flight
Tip-Path Plane (TPP) tilts backwards and sidewards (by π1 and π1)
w.r.t. control axis / resultant thrust perpendicular to TTP
β govern the control of helicopter
Hover⦠TPP exactly perpendicular to control axis
Forward Flight⦠similar, but not exactly perpendicular (Fig. 7-29)
TPP tilts faster than the control axis tilts (both for forward and rearward
β instability of the rotor w.r.t. AoA -> control is more sensitive as forward
speed increases
Active Aeroelasticity and Rotorcraft Lab., Seoul National University
How to achieve the desired control axis tilt
Physically tilting the rotor shaft (βdirect controlβ) β¦ autogyro
β Mechanically awkward in helicopters β 2 methods to solve
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IV. Rotor control in forward flight
β Rotor hub tilting (Fig. 7-30)
Separation of the shaft axis and
control (hub) axis
The hub axis then becomes the
control axis
β‘ Means for cyclically varying
blade pitch (Fig. 7-31)
Pitch will be always constant
w.r.t. the plane of swash plate
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IV. Rotor control in forward flight
Basic equalities of flapping and feathering
Fig. 7-32β¦ Control axis vertical, TPP tilts rearward by an amount π1
β Low pitch on the advancing side, high on the retreating side
Fig. 7-33β¦ Blade feathering w.r.t. TPP = blade flapping w.r.t. control axis
Fore and aft (π1) flapping w.r.t. control axis
β lateral (π½1) feathering w.r.t. TPP
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IV. Rotor control in forward flight
Geometrical relationships among
β Fig. 7-34
Axis of no feathering (control axis)
Axis of no flapping (TPP)
Intermediate shaft axis
β’ Flapping motion w.r.t. control axis
π½ = π0 β π1 cosπ β π1 sinπ β π2 cos 2π β b2 sin 2π
β’ Feathering motion w.r.t. TPP
π = π΄0 β π΄1 cosπ β π΅1 sinπ β π΄2 cos 2π β B2 sin 2π
⒠Subscripts⦠w.r.t. shaft axis
πΌ = πΌπ β π΅1π π΄0 = π΄0π π0 = π0π
π1 = π1π + π΅1π π1 = π1π β π΄1π
π2 = π2π π2 = π2π
(20)
(21)
* π : Aoa of the perpendicular to the control axis with the relative wind
(22)
(23)
(24)
(25)
(26)
(27)
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IV. Rotor control in forward flight
Fixed resultant force vector in space for
a given weight, parasite drag, speed
(Fig. 7-35) β TPP fixed β flapping motion
completely determined β control axis
determined
β’ Orientation determined : resultant force
vector / TPP / control axis
3 possible shaft angles and feathering
controls for identical flight conditions
(Fig. 7-36)
β’ Fuselage attitude and control position
may vary due to different CG position
β no effect on the rotor control in space,
except secondary influence
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V. Blade motion in the plane of the disk
1. Sources of in-plane blade motionβ¦
Periodic blade motion arises from 2 sources
β periodically varying aerodynamic forcesβ¦ variation in velocity and AoA
①periodically varying mass forces⦠TPP tilt
β’ Fig. 7-37 TPP tilt in hover, control
axis still vertical
β’ Blade flapping by π1π β CG of the
forward blade nearer to the axis of
rotation (shaft axis) β to maintain
the angular momentum, forward
blade must move faster, rearward
blade slower β blades move back
and forth as they rotate
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V. Blade motion in the plane of the disk
Two different axis systems and the resulting forces / motion
Motion w.r.t. shaft axis⦠flapping motion
π½π = π0 β π1π cosπ
β Two periodic torques about the shaft axis
a. Due to lift acting in the plane perpendicular to the shaft
πΏπππ‘ π‘ππππ’π πππππππππ‘ ππππππππππ’πππ π‘π π βπππ‘ = ππππ.π.π1π π ππ π
b. Periodic βmass forceβ torque due to periodically changing MOI
Caused by masses moving radially in a rotating plane (βCoriolis forceβ)
ππ : thrust per bladeππ.π. : radius of the resultant lift on blade
π1π : fore and aft flapping w.r.t. shaft
(29)
(28)
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V. Blade motion in the plane of the disk
2. Coriolis force
Fig. 7-38β¦ Point mass moves radially
outward β tangential velocity increase
β resists tangential acceleration β exert
a force to the right on the rotating plane
Tangential acceleration
π(Ξ©π)
ππ‘= Ξ©
ππ
ππ‘= Ξ©Vradial
Second component β¦ changing direction
of radial velocity vector in space
ππππ‘ππ πππππ‘β Γππ
ππ‘β πΊπππππππ
π ππ π’ππ‘πππ‘ πππππππππ‘πππ = 2πΊπππππππ
πΆπππππππ πππππ πΉππππππππ = 2πππππππππΊ (30)
Active Aeroelasticity and Rotorcraft Lab., Seoul National University
Coriolis torque acting on Hovering rotor (Fig. 7-37)
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V. Blade motion in the plane of the disk
β’ w.r.t. rotor shaft, a blade element moves outward with a velocity
π(π πππ π½π )
ππ‘= βπ π ππ π½π
ππ½π ππ‘
= βππ½π αΆπ½π
π½π = π0 β π1π πππ π, αΆπ½π = π1π πΊ π ππ π
πππππππ = ππ½π αΆπ½π = βππΊ π0π1π π ππ π βπ1π
2
2π ππ 2π
* Negligible order
(31)
(32)
Coriolis torque
πππππ’ππΆπππππππ = βΰΆ±0
π
2πΞ©2π0π1π sinππππ = β2
3ππ 2π0π1π Ξ©
2 sinπ
β’ Uniform blade
π0 =3ππ.π.ππ
π πΊπ 2 β πππππ’ππΆπππππππ = β2ππππ.π. π1π π ππ π
β¦ depends on ππ and π1π , but not on the blade mass
(33)
(34) (35)
Active Aeroelasticity and Rotorcraft Lab., Seoul National University
3. Equation of motion for blade in lag
Spring restoring torque
Equating all torques to the angular acceleration,
Substituting,
Rearranging,
Damping neglected. Possible sources aerodynamic damping
physical dampers at the blade root
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V. Blade motion in the plane of the disk
πΆ. πΉ. π‘ππππ’π =ππ
2ππΊ2π
πππππ β ππΆπππππππ β ππ πππππ = πΌ α·π
ππππ.π.π1π π ππ π β 2ππππ.π.π1π π ππ π βπππΊ2π
2π = πΌ α·π
(+) a1 motion β MOI decrease β lead motion β (-)
Increase lag angle for (+) a1 motion
Always resists the motion
πΌ α·π + πππΊ2π
2π = βππππ.π.π1π π ππ π
(36)
(37)
(38)
(39)
Active Aeroelasticity and Rotorcraft Lab., Seoul National University
For a uniform blade,
For the limiting case of e β 0,
Fig. 7-39β¦ For a rotor with blades hinged
at the center of rotation, Coriolis forces
cause the blades to move always at
constant velocity w.r.t. TPP.
31
Solution of Eqn. (39) β¦ assuming a solution of form ΞΆ = ΞΆ0 sinππ‘
V. Blade motion in the plane of the disk
π0 =βππππ.π.π1π
ππ 2
ππΊ2 β πΌπ2, Ξ© = Ο (41)
(42)πΌ =1
3ππ 2 β π0 =
2
3π0π1π 2
3β
π
π
π = π0π1π π ππ π (43)
Active Aeroelasticity and Rotorcraft Lab., Seoul National University
As the lag hinge is moved outward, the blade motion in TPP is,
In Eqn. (40),
Setting forcing torque to 0, and
solving for frequency,
Variation of blade natural frequency
with lag-hinge distance (e
R) for a
uniform mass blade (Fig. 7-40)
32
V. Blade motion in the plane of the disk
π β 0 β ππππ = π0π1π
3
2
π
π
1β2
3
π
π
β¦ Ο βͺ Ξ©
βπΌπ2π0 +ππ
2πΞ©2π0 = βππππ.π.π1π
βπΌππ2 =
ππ
2ππΊ2, ππ = πΊ
3
2
π
π
(44)
(45)
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4. Lag motion in forward flight
Additional exciting torque⦠periodic variations in blade drag
In all practical cases, periodic in-plane blade motion is quite small 1
2~ 2Β°
Mean lag angle variation w.r.t. flight conditions β¦ much larger 10Β° ~ β 1Β°
5. Higher harmonic in-plane motion
Although usually small compared to the
first harmonics, important source of
vibration (Ch. 12)
In hover, second harmonic component
exists in proportion to π1π
π1π induces in-plane motion twice each
revolution (Fig. 7-41)
V. Blade motion in the plane of the disk
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First harmonic motion β a0π1π (Eqn. (43))
Eqn. (32) β if π0 = 0, only second harmonic in-plane motion exists,
π΄πππππ‘π’ππ =1
2π1π 2
Also, second harmonic motions also arise in forward flight due to the second
harmonic aerodynamic forces
4th harmonic depend on 2nd harmonic flapping
due to Coriolis and aerodynamic
Important for fatigue stresses and rotor vibrations
However, small enough to be safely neglected as far as their effects on the
velocities and air forces encountered by the blade are concerned
V. Blade motion in the plane of the disk