R. & M. No. 2813
(12,466)
A.R.C. Technical Report
MINISTRY
AERONAUTICAL
' REPORTS
OF SUPPLY
RESEARCH
AND
COUNCIL
MEMORANDA
The Simple Harmonic Motion ofa
Helicopter
Rotor with Hinged Blades
By
J. K. ZBROZEK,Dipl.Eng.
Crown C@yright Reserved
LONDON"
HER
MAJESTY'S
STATIONERY
1955
PRICE
3S 6d
NET
OFFICE
The Simple Harmonic Motion of a Helicopter
Rotor with Hinged Blades
By
J. K. ZBxozE~:, I)ipl.Eng.
COMMUNICATED BY THE PRINCIPAL DIRECTOR OF SCIENTIFIC RESEARCH (AIR),
~/~INISTRY OF SUPPLY
Reports and Memoranda No. 2 813 *
April,
1949
Summary.--In simple harmonic oscillation of the helicopter with hinged blades, the tip-path plane is tilted with
respect to the shaft in the plane of oscillation and in the plane perpendicular to it. The angles of tilt can be expressed as functions of angular velocity and acceleration. The influence' of the acceleration term on the dynamic
stability of the helicopter is small.
The expressions for angles of tilt due to angular velocity can be simplified to the expressions obtained in previous
work under assumptions of quasi-static conditions.
1. I~troductio~.--It has been shown in Ref. 1 that when the rotor shaft of a helicopter tilts
with constant pitching velocity, the tip-path plane lags behind the shaft and also tilts Sideways.
These angles of lag and sidetilt were found to be proportional to the pitching velocity of the
shaft and the following expressions were derived:
Oal
_
_
Of
1 .
~q -- 9 ,
~b I
16
.
yD .
.
..
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
(1)
.
.
(2)
where al is the angle of lag between the rotor shaft and the tip-path plane axis measured in the
pitching plane and q is the pitching velocity ; bl is the angle of sidetilt, y is Lock's inertia number
and ~ is the rotational velocity of the rotor.
In I~. & M. 25091 the value of aa~/aq as given by equation (1) was used in the analysis of
dynamic stability. Strictly speaking equation (1) applies only to a motion with constant rate
of pitch. Where the motion of the helicopter approximates to a simple harmonic motion the
use of the derivative given by equation (1) may not be justified in the study of the dynamic
stability.
The present note gives a more detailed analysis of rotor derivatives and shows t h a t the
expressions (1) and (2) for rotary derivatives are accurate enough for any practical application.
However, the analysis indicates the existence of acceleration derivatives, but their influence
on the dynamic stability of the helicopter is small.
* R.A.E. Report Aero. 2319, received 14th July, 1949.
1
2. The Approach to the Pfoblem.--It is assumed t h a t the helicopter is hovering with its shaft
vertical and tip-path plane horizontal. Suddenly the helicopter and its shaft begin to oscillate
in a pitching plane about the rotor centre according to the equation:
0 =
A sin ~t;
.
.
.
.
.
.
.
.
.
.
.
.
(3)
where A is some arbitrary amplitude and v the circular frequency of rotor shaft oscillation.
The main difficulty in the present problem is estimating the magnitude and direction of the
flow through the disc. Lacking any data it is assumed t h a t the flow through the disc is in the
direction opposite to the instantaneous direction of lift. In other words no allowance for any
downwash lag is made.
It is further assumed t h a t the blade hing e has zero offset, and the hinge bearing has no friction.
The diagram of axes and angles is shown in Fig. 1. The reference axes are chosen as horizontal
and vertical, i.e., the axis of the rotor in its initial conditions. All the values are measured from
these axes, positive in an anti-clockwise direction for Fig. 1. The following quantities are
defined:
0 the angle of the shaft tilt measured from the vertical and given by equation (3)
/3 the flapping angle measured between blade axis and horizon
/la the angle of lag, between the shaft and the tip-path plane axis, positive in positive direction
of 0.
The equation of motion is written, assuming t h a t all the quantities are measured from
OX-plane.
Due to the blade hinge being fixed relative to the shaft, the tilt of the shaft in the pitching
plane b y an angle 0 produces, in the horizontal plane, feathering in the form:
-- 0 sin W
. . . . . . . . . . . .
(4)
where w = fat is the blade azimuth angle measured in the 0X-plane from the downwind blade
position. Tile angle of shaft tilt 0 is a function of time given by equation (3) and hence the
feathering is given by:
A sin vt sin fat
. . . . . . . . . .
(5)
@' = @0 -- A sin vt sin fat
. . . . . . . . . .
(6)
--
and the blade pitch angle by:
where ~'0 is the collective pitch angle.
The moments acting on the blade about the flapping hinge are as follows:
Moment
due to centrifugM force:
-- 1#2/~
.
.
.
.
.
.
.
.
.
.
.
.
.
(7)
Moment due to angular acceleration of blade in flapping motion:
-
z,p
.
.
.
.
2
.
.
.
.
.
.
.
.
.
.
.
(s)
The a e r o d y n a m i c m o m e n t :
I~½pa@o-- A s i n v t s i n ~ t
.Qr
where U is velocity of flow t h r o u g h and perpendicular to t h e disc. It is assumed t h a t t h e
coning angle a0 is small a n d t h a t cos a0 can be t a k e n as unity. The flow velocity U is t a k e n
to be constant. I n t e g r a t i n g t h e aerodynamic m o m e n t along the blade and introducing Lock's
inertia number, the aerodynamic m o m e n t can be expressed as:
D
A sin ~,t cos t?t)
. . . . . . . . . .
(10)
. . . . . .
(11)
where
Y --
PagR4
Ii
. . . . . .
is Lock's inertia number• g is a m e a n chord t h e exact value of which can be defined by comparison of equations (9) and (10), but with sufficient accuracy t h e m e a n chord can be defined as '
(12)
f~r:~dr
is the coefficient of the flow through ~the disc defined as:
=
u/~R
.
.
.
.
.
.
.
.
.
.
(13)
.
I n the evaluation of integral (9) no allowance has been m a d e for tip and root losses.
N e g l e c t i n g the gravity term, the equation of m o t i o n of t h e blade is obtained from t h e expressions (7), (8) and (10)
fi + ~),t?~ + ~92~
1
2
~),9
(~6 -- ~Z) -- ~,~9eA sin vt sin t?t . . . . . . .
(14)
R e m e m b e r i n g t h a t t h e coning angle a0 is given b y R. & M. 1127 ~ as
ao =
~7(Oo -~z);
.
.
.
.
.
.
.
.
.
(15)
.
and introducing trigonometrical substitutions equation (14) can be put into t h e following
alternative form for ease of integration
fi + }~X?t) + Y2~ = a0~2 + ~~ A-f2
r 2 cos (~? + v)t -- ¢~A7£2 ~ cos (f2 - - v)t.
..
(16)
The complete solution of equation (16) takes the form:
/3 = ao + e~{C~ cos vFt + C2 sin ~ t }
+ C8 cos (t9 + ~)t + C4 sin (sO + ~,)t
+ "C5 cos ( n - - ~)t + C6 s i n ( n -
~)t ;
. . . . . . . .
where C1, C.,, etc., are constants of integration and are functions of A, ~?, ~ and ~, only.
3
(17>
The first part of equation (17) shows that due to an initial disturbance the blade will osciliate
with frequency given by
..
-(is)
This oscillation is very heavily damped and the damping factor is given by:
2~ - -
yY2
16 .
.
.
.
.
.
.
.
.
.
.
.
(19)
After the blade motion due to the initial disturbance is damped out, which in practice takes
about ~ sec, the mean value of blade flapping angle approaches ao, the coning angle. The
second part of equation (17) shows t h a t the blade will oscillate steadily about its flapping hinge
with two frequencies, ~? + v and ~ -- v.
After some re-arrangement of equation (17) and omitting the rapidly damped part of the
motion, the expression for steady blade flapping measured from horizontal will be in the form:
fi' = a0 + (C~ cos vt + Ca sin vt) cos ~ + (C9 cos ,t + C10 sin vt) sin ~ ;
....
(20)
where ~0 = ~?t and C~, Ca, etc., are constants and functions of A, ~9, , and 7.
The coefficient of cos ? gives the longitudinal tilt of the tip-path plane and the coefficient of
sin ~0 the lateral tilt of the tip-path plane, both measured from horizontal.
The angle between the shaft and 0Z-axis as given b y equation (3) is
0=
Asinvt
. . . . . . . . . . . .
(3)
and the angle of the longitudinal laG i.e., the angle between the tip-path plane axis and shaft
axis (positive in positive direction of 0) i s : - A a = (C7 cos vt + Ca sin vt) -- A sin yr.
(21)
Remembering that:
0=
A sin vt
O=
A v cos vt
0=
- - A v 2sinvt
]
(22)
and putting
v/t) = ~
. . . . . . . . . . . .
(23)
the angle between the shaft and tip-path plane axis, measured in the plane of the shaft oscillation
is given b y the expression:
Aa-11
4
(24)
The corresponding lateral tilt of t i p - p a t h plane is given by:
1 (yy{(,) ~
_,)1
1 (~,'~((r'@_~.,.) + ~(4 _~,)11 o.
o ~ \g/i \g/
;~
..
".
'
.
.
.
and the tilt of the rotor tip-path plane is positive towards the advancing blade.
(24) and (25), D is an abbreviation for t h e following expression:
D =
(1
-
V')" + 2V'
(4 - - a r e + 5~) + ~(4 --
(2s)
.
.
In equations
..
..
(26)
3. Discussion.--3.1. Longitudinal Disc Tilt. The equation (24) can be rewritten:
~a-
16AO + 1 f(16"~2
go
~
i\7/
-
1}to;
. . . . . . . .
(27)
where A a n d / " are functions of ~, a n d ~ only, and for small values of 5 = v/fa can be t a k e n as
unity. The full expressions for A a n d / 1 are a s follows:-,j-~,
•. . . . . . .
@ ' ( ! (~)~(1 -- V') -- (~)"(4 -- 3V' + 25 ~) -- 52(4 -- 5~) 2}
8/ (
f/rV
;
....
(29)
-- 4~J X D
where D is given b y the equation (26).
The numerical values of these functions are given in Fig. 2, and were calculated for a range
of values of ~ a n d y. The values of ~ as found in practice are usually very small ; for example
in t h e case of the Sikorsky R4-B helicopter, which has ~ = 13, t h e values of ~ are about 0 . 0 2
for longitudinal a n d about 0.05 for lateral oscillations. The value of ~ is seldom expected to
exceed 0.05 for full-scale rotors. Also, for most of t h e small helicopter models the value of
= ~/fa remains of t h e same order due to t h e large values of ga necessary if t h e corresponding
tip speed is, to be m a i n t a i n e d on t h e model: I t can be seen from Fig. 2 t h a t within t h e practical
range of values t h e ratio of oscillations, ~ has only a small effect u p o n t h e values of rotor derivatives. Only for .very h e a v y blades, y < 6, can t h e Values of A a n d / ~ differ appreciably from
unity, and in those cases for t h e purpose of stability calculations t h e corrections of Fig. 2 can b e
used.
For normal values of blade inertia n u m b e r t h e following a p p r o x i m a t i o n is justified:
A = r = 1 .
.
.
.
.
.
.
.
.
.
.
.
.
.. .
.
(3o)
a n d thus simplify the equation (27) to t h e form:
Aa-,,-_160
y~
I
+
£22 ( \ y .]
5
-
1}o
•
.
.
.
.
(a)
The derivative of the tilt of the tip-path plane relative to the shaft with respect to angular
velocity of pitch is:
aal _
~q
16
yf2 '
. . . . . . . . . .
(32)
the tip-path plane lagging behind the shaft. The equation (32) is in agreement with equation (1),
which was obtained from the simplified analysis of Ref. 1.
I
The derivative of the tilt of the tip-path plane relative to the shaft with respect to angular
acceleration of pitch is:
a<
l I(lS '
- >L\7/-t)=
!
1
v T/
(33)
The approximate expression for the acceleration derivative (33) and the expression for the
correction factor F (29) are valid only for 7 =# 16 ; for the values of ~ equal or near 16 the full
expression (24) for the acceleration derivative has to be used.
I t is interesting to note t h a t the angular velocity derivative, equation (32), is always negative
and always produces a damping of the angular motion of the helicopter. The sign of the angular
acceleration derivative however depends on the value of blade inertia number. For practical
values of blade inertia number (~ < 16) the acceleration derivative is positive, as shown b y
equation (33), i . e . , the apparent displacement derivative is negative, and it can be shown t h a t
the positive acceleration term has a stabilising effect on the helicopter stability. However the
numerical calculations based on the Sikorsky R4-B helicopter show a very small effect due to
this acceleration derivative. It has to be borne in mind t h a t the present analysis does not take
into account any effect of the time lag of the aerodynamic forces acting on the blade. This
effect could be of considerable importance in determining the blade motion.
For very light blades, y > 16, the acceleration derivative changes sign and becomes negative*.
It is worth noting t h a t according to equation (18) for values of inertia number y > 16 the disturbed
flapping motion of blade ceases to be oscillatory and becomes a pure subsidence.
To give a better illustration of the disc and shaft behaviour in simple harmonic motion, the
diagrams of Fig. 3 were prepared. The thick line represents motion of the rotor shaft according
to the equation"
0=Asin~,t;
.:
• .
.
.
.
.
.
.
.
.
.
(3)
.
The dotted line shows motion of the tip-path plane with respect to the horizon. The angle c~
is the angle between tip-path plane axis and the vertical. The motion of the tip-path plane can
be described by all equation"
"
= CA sin (vt + 6) ;'
•.
.
.
.
.
.
.
.
.
(34)
The constants # and ¢ can be found from equation (20) and are functions of ; and y only.
For the practical range of values of y and 7, the motion of the rotor disc is shown in Fig. 3,
top diagram. The ratio of amplitudes, # is slightly less t h a n unity, but for practical purposes
can be taken as unity. The phase angle ¢ is small and negative. For infinitely heavy blades,
* T h e e x a c t v a l u e of y a t which t h e a c c e l e r a t i o n d e r i v a t i v e becomes n e g a t i v e is s l i g h t l y less t h a n 16 a n d d e p e n d s
on t h e v a l u e of ~.
6
--> 0, Fig. 3, bottom diagram, the amplitude ratio # approaches zero value, and at the same time
the phase angle ¢ --->90 deg. For the limiting case when ~ = 0 and # = 0, the tip-path plane
remains horizontal, irrespective of the shaft motion, which is quite obvious for physical reasons.
For very light blades, when the inertia number ~ is large, the amplitude ratio ~ is increasing and
becomes larger than u n i t y (Fig. 3, bottom diagram). Increasing the value of ~ to infinity brings
the value of # back to unity. The phase angle ¢ is decreasing steadily with increasing ~,, and
for ), --> ~ the phase angle approaches zero. For infinitely light blades, # = 1 and ¢ = 0, the
rotor disc follows exactly the motion of the shaft.
I t might be of interest to point out that the vertical shift between the thick and dotted lines
of Fig. 3 is proportional to the tilt of the disc with respect to the shaft, and at the point vt = 0
this s h i f t i s proportional to the velocity derivative and at the point vt = ~/2 to the acceleration
derivative. Bearing this in mind, it can be deduced immediately from Fig. 3, that the angular
velocity derivative always remains negative, but the acceleration derivative m a y change s i g n
depending on the value of the inertia number.
To give some illustrations of the values involved, Fig. 4 was prepared, where the value of the
amplitudes ratio, ~, is plotted against inertia number, ~, for different values of frequency ratio, 5.
The scale to the right is too small ±o show the shape of the ~-curve for verY large values of 9,, and
the dotted line shows only the tendency of the ~-curve.
Remembering that the values of ~ and r likely to meet in practice are 5 < 0.05 and ~ > 5,
it can be seen t h a t tile value of ¢ differs from u n i t y b y less than 1 per cent.
3.2. Lateral T i l t . - - F o r the practical range of values of ~ = ~/£2 the equation (25) for the lateral
tilt of the tip-path plane can be simplified to:
= Lo
g?
1 240 = 1_6 + _3_1
0 . . . . . . . . .
X?:~ y
D
2 Q ~q
(35)
During the longitudinal oscillations of the helicopter with frequency v, the tip-path plane
tilts not only in the plane of oscillations, but in the plane perpendicular to it as well. This angle
of tilt, A b, can be split up into two parts as in equation (35), first that due to angular velocity
and second that due to the angular acceleration of the helicopter.
The corresponding derivatives of lateral tilt can be written:
Ob~
~q
1
~9
(36)
The value of 3bl/aq for constant rate of helicopter pitch was deduced in Ref. 1, Appendix II,
from kinematic considerations, and is in agreement with present result.
Similarly the acceleration derivative of the sidetilt is:
~b~=_
~q
1 24_3
~2 9,
l Oal
2 t? Oq
....
(37)
....
4. Co~clusio~s.--The mathematical analysis of the dynamics of the flapping blade during
-simple harmonic motion of the helicopter shows the existence of angular velocity and angular
acceleration derivatives. The effect of the accleration derivative on the dynamic stability
of the helicopter is quite small, and for the usual values of blade inertia number it can be neglected.
The effect of the frequency ratio 7 = v/X? on the values of the rotor derivatives is also small,
mainly due to the fact t h a t for all helicopters and for helicopter models b~tilt to dynamic a~d
aerody~¢amic scale, t h e value of ~ is likely to be much less than 0.1.
It seems that only for the case of helicopters with very heavy blades, say ~ < 8, need the
effects of the acceleration term and of !requency ratio be taken into account.
It has to be borne in mind that the analysis given here does not take into account any changes
in slipstream structure due to the oscillation of the helicopter ; and further, the analysis covers
only simple harmonic motion in hovering.
It is proposed to analyse other modes of helicopter motion in hovering, say an exponential
mode of motion, corresponding to motion due to stick displacement.
A further step will be the extension of the present analysis to forward flight.
REFERENCES
No.
Author
Title, etc.
1
J . K . Zbrozek
..
..
2
C . N . H . Lock
..
• . F u r t h e r Development of Autogiro Theory.
3
J. K. Zbrozek
4
Miehl
..
•
°
Investigation of Lateral and Directional Behaviour of Single Rotor
Helicopter (Hoverfly Mk. I). R. & M. 2509. June, 1948.
R. & M. 1127.
March, 1927.
..
Control Response of Single Rotor Helicopter.
published). March, 1949.
..
The Aerodynamics of the Lifting Airscrew with Hinged Blades in Curvilinear Motion. Trans. Inst. Aero-Hydro@namics, No. 465, pp. 1-60.
R.T.P. Translation No. 1533/M.A.P. 1940.
8
A.R.C. 12,465 (To be
List of Symbols
a
ao
Lift slope of blade section
Coning angle
a~
Coefficient of Fourier's series for flapping angle, and corresponding to backward
tilt of the rotor disc
b~
Coefficient of Fourier's series for flapping angle, and corresponding to lateral
tilt of the rotor disc, positive towards tile advancing blade.
B
Tip-loss coefficient ; usually defined: B = 1 -- c/2R
Blade chord
c
Mean blade chord
I1
Blade moment of inertia about flapping hinge
Pitching velocity of helicopter
q=O
t
Time
R
Rotor radius
U
Velocity of flow through the disc
Angle between rotor disc and horizon
Flapping angle, measured between longitudinal axis of the blade and horizon
pagR ~
0
Blade inertia number
Angle of shaft tilt in pitching plane and measured from vertical
Pitch angle of rotor blade ; measured from blade airfoil chord
~9 0 .
U
X2R
~F
VF
9~
Collective pitch
Coefficient of flow through the disc
Damping coefficient of disturbed flapping motion
Frequency of disturbed flapping motion
Circular frequency of shaft oscillation
Frequency ratio
P
= X2t
X2
A
T
qb
$
Air density
Blade azimuth angle, measured from rearmost position
Rotational velocity of rotor
Correction to angular velocity derivative
Correction to angular acceleration derivative
Ratio of disc to shaft amplitudes
Phase angle between shaft and disc motions
9
APPENDIX
The Compariso~ with Calculations for Co~stant A~gular
Velocity a~d Co~sta~t A~gular A cceleratio~,
Since the completion of the present report, the attention of the author was drawn to the work
of Miehl, Ref. 4. Miehl extended Lock's analysis (R. & M. 11272) to steady curvilinear flight,
and his investigations lead to the following expressions for the rotor tilt derivatives:
~al _
16
1
?
B (B 2 - - 1/S
(43)
~bl
1
1
/
~q
fa 1 -- ff~/2B ~
J
where B is the usual tip-loss coefficient. The formulae (43) are in agreement with the present
report if we assume no tip loss, B = 1, and very slow oscillation, ~ =- 0. " It can be seen from
equation (43) that the forward speed has a very small effect on the values of the angular velocity
derivatives.
Miehl (Ref. 4) in his work is considering one particular case of blade motion during uniformly
accelerated rotation of helicopter in hovering. The formulae for the rotor tilt derivatives due
to constant acceleration are in agreement with the present report, assuming frequency ratio,
~=-0.
The comparison of the theoretical calculations with experimental values obtained from model
tests, Ref. 4, have shown that in order to obtain satisfactory agreement it was necessary to modify
the distribution of the induced velocities. It was assumed in Ref. 4 that the increments of
induced velocity are proportional to the new mass forces present in curvilinear accelerated flight.
The calculations have shown that the re-distribution of induced velocities has a small effect
on the values of angular derivatives, and is increasing slightly the value of ~al/~q, the value of
ObjOq not being affected. However, re-distribution of induced velocities has a considerable
effect on the acceleration derivative, and increases the value of ~al/O(t; the value of Obj~O is
not affected. The assumed distribution of induced velocities gave good agreement between
theory and experiment.
10
I,I
A
j
l
l'O
~-
¥=I0
0-9
z
O-B
0.7
o.I
0"2 :~
I.I
l'ZoN
/-,
- ~ = I0
I'0
-~--8
0.9
FIG. I.
Diagram of rotor in simple harmonic motion.
"-~'=6
0,1
]
0."
....
o
FIG. 2.
11
i
o.t
0.2 ;~
Corrections to velocity and acceleration derivatives.
g
0,:. ,SHAFT Dt,,SPLAEEh-IENTt0
/
/
PRACTICAL_VALuEs
"~= 0, I
7
~"\
9=0 A~D 0.01
H¸
¢~
,SHAFT DISPLAcE~E.NT. e
t" ~ ~
/ ' T I P PATH PLANE
5HAFT
0.9
e =
A
Sin
"~
"rip PATH PLANE o( = ~A Sin ('0L +
/
.A"-~TIP PATH PLANE.
///"
DISPLACEMENTpo~
"~
\\
~'~
Y
o*8
IO
15
¢)
20
~"
.
FIG. 4.
Ratio of tip-path plane to shaft amplitudes in
simple harmonic oscillation.
Attitude v e r s u s time c u r v e s of rotor
shaft and tip-path plane axis in steady
harmonic motion.
FIG. 3.
(5115)
Wt. 17/680
K.7
10/Ss
12
H.P.Co.
S4-261
PRINTED
IN
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