rotary wing aircraft handbooks and history volume 10 stability and control of rotary wing aircraft
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PB-111521TRANSCRIPT
J 1521
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Stability and Control
of Rotary Wing Aircraft
Rotary Wing Aircraft Handbooks and History
Vol. 10
Distributed by OTS in the Interest of Industry
This report is a reprint of an original document resulting from Government-sponsored
research. It is made available by OTS through the cooperation of the originating agency.
Quotations should credit the authors and the originating agency. No responsibility is
assumed for completeness or accuracy of this report. Where patent questions appear to
be involved, the usual preliminary search is suggested. If Copyrighted material appears,
permission for use should be requested of the copyright owners. Any security restrictions
that may have applied to this report have been removed.
UNITED STATES DEPARTMENT OF COMMERCE
OFFICE OF TECHNICAL SERVICES
UNIVERSITY OF MICHIGAN LIBRARIES
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ROTARY WING AIRCRAFT
HANDBOOKS AND HISTORY
STABILITY AND CONTROL
• OF
ROTARY WING AIRCRAFT
BY
WILLIAM E. COBEY
VOLUME 10
ONE OF A SERIES OF 18 VOLUMES EDITED BY
EUGENE K. LIBERATORE
PREWITT AIRCRAFT COMPANY
CLIFTON HEIGHTS, PENNSYLVANIA
AND PREPARED FOR
WRIGHT AIR DEVELOPMENT CENTER
AIR RESEARCH AND DEVELOPMENT COMMAND
UNITED STATES AIR FORCE
WRIGHT-PATTERSON AIR FORCE BASE, OHIO
UNDER CONTRACT NO. W33-038 ac-21804 (20695)
DISTRIBUTED BY
U.S. DEPARTMENT OF COMMERCE
BUSINESS AND DEFENSE SERVICES ADMINISTRATION
OFFICE OF TECHNICAL SERVICES
WASHINGTON 25, D. C.
1954
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Ingln. Library
111521
TL
116
CONTENTS
Pages
I. Introduction l
n. Symbols 2-3
m. Stability of Aircraft 4-6
IV. Character of Stability 7-ll
V. Helicopter Stability l2 - 22
VI. Comparison of Helicopter and Airplane Stability 23 - 28
VH. Rotor Stability 29 - 5l
VIE. Abstracts of Papers on Helicopter Stability and 52 - 63
Control
IX. Index 64
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INTRODUCTION
A. Scope
The subject of helicopter stability is here treated in a manner that
should prove interesting to design engineers and to those technical people
who desire a thorough understanding of the principles of stability without
undertaking the subject on a higher mathematical level. The works of
Hohenemser, Miller, Sissingh, Kelley and others, representing years of
research, have advanced the theory of helicopter stability and control to
its present state of high development. Abstracts of the more important
papers are given in the aopendix to this volume.
B. Preparation
This volume was orepared by William E. Cobey, Prewitt Aircraft Company.
The project of which this volume is a part was initiated by the Air
Technical Intelligence Center. It was continued to completion by the
Wright Air Development Center, under suDervision of Messrs. B. Lindenbaum
and W. Oleksak.
Distributed in the Interest of Industry
by the
U. S. DEPARTMENT OK COMMERCE
OFFICE OF TECHNICAL SERVICES
WASHINGTON 25, D. C.
With the Cooperation of the
Originating Agency
AU secrecy restrictions on the contents of this document have been lifted. Quo-
tations from «his report should credit the authors and originating agency. No
responsibility is assumed for the completeness or accuracy of this report. Where
patent questions are involved, the usual preliminary search is suggested. If a copy-
right notice appears on the document, the customary request for quotation
or use should be made directly to the copyright bolder.
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II. LIST OF SYMBOLS
8
e
x
44
K> Hi
Ks
Ct
p
R
W
b
c
R
C
Z
Angle of attack of blade (measured from zero lift)
Mean angle of attack (no flapping)
Amplitude of periodic portion of *J" (due to flapping)
Tip speed ratio w
Blade azimuth angle, radians
Factor of backward tilt of tip path plane introduced because of
flap reducing devices. Backward tilt a 4**1"
Total thrust
Coefficient of thrust component in plane of rotor disc
Thrust coefficient
Tip speed
Disc area
Air density
Mean blade drag coefficient
Solidity = Blade Area
Disc Area
Blade radius
Gross weight
Centrifugal force on each blade
Number of blades
Blade chord
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3
n.
LIST OF SYMBOLS
>>
<
o
<•<
o
as
e
a.
Distance from center line of rotation to horizontal pin
Inertia moment of blade (about the hinge)
X
J
Inertia moment of aircraft in pitch (without blades)
«u
s
Distance from rotor to center of gravity
Slope of lift curve
On
Blade section moment coefficient about the aerodynamic center
V
V
Speed of rotor center
Angle of rotation of aircraft in pitch about center of gravity
%
»
*
Acceleration of gravity
M
Longitudinal moment
Sn
Force normal to rotor shaft
uT
Rotational velocity (angular)
X
Radius of blade element
*s
o<
Angle of shaft with vertical
Axis of no feathering
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4
m. STABILITY OF AIRCRAFT
The study of stability problems of a general nature takes into considera-
tion the following sciences:
A. Aerodynamics
B. Dynamics of Periodic Motion
AERODYNAMICS
A study of aerodynamics is necessary to determine the forces and
moments acting on an aircraft in flight. It is necessary to know such things
as what forces and moments arise due to the deviation of the body from its
trim position in flight. The fundamentals are given in the discussion to follow.
THE DYNAMICS OF PERIODIC MOTION
Under the influence of periodic motion, an aircraft will be acted upon
by dynamic forces and moments. These forces and moments are brought
into play by virtue of the motion, and have unique properties depending on
whether the forces are sensitive to a displacement, a velocity or an accelera-
tion.
DIFFERENTIAL EQUATIONS
Some knowledge of differential equations is required in the study of
stability. For instance,in writing the equations for equilibrium of forces or
moments on a body, the forces or moments will fall into three general groups.
These are classified according to their origin such as those due to displace-
ment of the body from trim, those due to velocity of the body in moving from
trim, and those due to acceleration of the body. For example, a dart under-
going a pitching motion is acted upon by all three types of moments as it is
pitching about its center of gravity.
Displacement Forces on a Dart in Pitching Motion:
e l. Moments Created by Displacement
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Figure l shows a dart displaced from trim by the angle°C . An aerodynamic
force F is created due to this displacement which, acting on lever arm S,
becomes a restoring moment FS. F varies with the amount of displacement
(oc),from F - O at trim condition,to F - Max. when ot is maximum.
This moment FS depends for its existence on©< and is directly proportional to
o< . This moment could be written:* , v
where M©< is the .constant of proportionality between the moment M and the
displacement«c . This constant (Me* ) is called the partial derivative of M
with respect to«»c and is sometimes written M. This explains the physical
hex.
meaning of partial derivatives used in stability studies.
Velocity Forces on a Dart in Pitching Motion
Figure 2. Moments Created by Velocity
The dart, pitching with an angular velocity ci , experiences a side velocity
equal to S x o( on its tail surface. Combined with the forward velocity of the
dart (V),the tail surface experiences a resultant velocity Vjj. By geometry,
the angle of attack of the relative windas^and it follows that the force F is
equal to a constant times the angle of attack for a given velocity (V)
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and the moment about the e.g. becomes
M = K ■ .
V
If K, S and V are constants^they can be combined and the above equation will
read:
M = (m 6c)°^ »
where M of = JKS?, and is the constant of proportionality between and M,
and is also the partial derivative of moment with respect to o( . Another
way of expressing Mot is .
Acceleration Forces on a Dart in Pitching Motion
If the angular velocity of the dart is being changed by the fore-going
moments, inertia forces are developed within the body tending to resist
these changes in angular velocity.
Suppose the dart is undergoing periodic motion in a pitching sense.
An acceleration is necessary to reverse the motion at both ends of its swing.
The dart will then be acted upon by an inertia moment equal to the moment of
inertia of the body in a pitch sense times the angular acceleration, which can
be written:
M = I ci
This is analogous to the moment required to bring a flywheel to rest and
reverse its motion. The inertia of the flywheel resists the change in velocity
and a force is required to accomplish the change.
The fore-going treatment of moments arising from displacement,
velocity and acceleration in a rotary sense is sufficient to describe the
stabilizing forces in a dart type of body. More complicated aircraft are sen-
sitive also to changes in translational motion. Similar expressions can be
written for the translational motions and are included in the general equations
of motion given later.
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IV. CHARACTER OF AIRCRAFT STABILITY
In order to determine whether an aircraft is stable, it must first be
trimmed. Stability is related to the behavior of an aircraft after it is disturbed
slightly from the trimmed condition. Stability is referred to as stick-fixed
or stick-free stability, depending on whether the control is held fixed in its
trim position after the disturbance,or is left free. The behavior of an air-
craft after such a disturbance may consist of a divergence, a convergence or an
increasing or decreasing oscillation, thus:
Fi'q. 4a
STATICALLY UNSTABLE
DIVERGENT MOTION
FIG. 4b
CONVERGENT OR
APERIODIC MOTION
(DYNAMICALLY STABLE)
FIG. 4c
DYNAMICALLY UNSTABLE
INCREASING OSCILLATION
FI6-4d
DYNAMICALLY STABLE
DECREASING OSCILLATION
STATIC STABILITY
Static stability is expressed in terms of this behavior as follows:
An aircraft is statically stable if,when disturbed slightly from its trimmed
condition^ will initially tend to return to its trimmed condition. Figures 4 (b),
4 (c) and 4 (d) represent statically stable motion in that the initial tendency is
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to return to neutral. An aircraft is statically unstable if,when it is disturbed
slightly from its trimmed condition, it performs a divergence. Figure 4(a)
illustrates a statically unstable condition. The vertical coordinate represents
displacement and the horizontal coordinate represents time. The curve in-
dicates that the displacement increases with time and shows no tendency to
return to the neutral position. Such a position is statically unstable and cannot
be dynamically stable.
DYNAMIC STABILITY
The dynamic stability includes the consideration of static stability
plus what happens to the body in its motion following a disturbance. It is
inevitable that once the disturbance has been created and motion ensues, dynam-
ic forces will be brought to bear on the body and will influence its subsequent
motion. The study of this motion is called the dynamic stability. The dynamic
stability of an aircraft may be defined as follows: An aircraft is dynamically
stable if,after a disturbance, it performs a decreasing oscillation; and an air-
craft is dynamically unstable if^after a disturbance, it performs an oscillation of
increasing amplitude. Figure 4 (b) and 4 (d) illustrate examples of dynamical-
ly stable motion. Figure 4 (c) illustrates an example of a dynamically un-
stable motion. All of these motions, however, have the property of being
statically stable because they do show an initial tendency to return to the
trimmed condition. The only difference is that the subsequent motion is built
up in the case of dynamically unstable condition and is decayed,or reduced,in
the case of the dynamically stable condition. The dynamically stable motions
represented by Figure 4 (b) and 4 (d) differ further in that one is oscillatory
in character while the other is non-oscillatory (or aperiodic ; Figure 4b).
The aperiodic motion is said to have critical damping.
FUNDAMENTAL DEFINITIONS
a. Periodic Motion
Generally speaking, periodic motion is a vibratory or oscillating
motion which repeats itself in every respect at fixed intervals of time. The
simplest kind of periodic motion is harmonic motion. Aperiodic motion is
non-oscillating.
b. Displacement
The distance a body moves from some fixed reference is called the
displacement. Displacement may be either angular or linear.
c. Amplitude
The maximum value of the displacement is called the
amplitude (see Xe , Figure 3).
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d. Cycle
A cycle of motion is that part of the motion that takes place in one
period of time. ^
£_\ / \ TIME,!
PERIOD, T( TIME
REQ'D TO COM-
5 nG 3 PLETE ONE CYCLEt
e. Period
The length of time between repeating parts of the motion is called
the period.
f. Time to Half Amplitude
In the consideration of stable motion - where the amplitude is
reducing with time - the amount of stability can be measured by the time
elapsed during which the amplitude of vibration is reduced by one-half. This
is denoted Tjj and called the Time of Half Amplitude.
g. Time to Double Amplitude
In unstable motion - where the amplitude is increasing with time - a
measure of the instability is the time elapsed during which the motion increases
to twice the amplitude. This is denoted Tp and is called Time to Double
Amplitude
h. Stick Fixed Stability
Stick fixed stability is the character of the aircraft motion when the
controls are held in the neutral trim condition while the motion is in progress.
i. Stick Free Stability
Stick free, stability is the character of the aircraft motion when the
controls are permitted to move freely (hands off) during the ensuing motion.
It is understood that the helicopter is properly trimmed before the controls are
freed.
j. Degrees of Freedom (No relation to stick free stability)
One Degree of Freedom
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TO
A mechanical system is said to have one degree of freedom if its
geometrical position can be expressed at any instant by one number only. Take,
for example,a piston moving in a cylinder: Its position can be specified at any
time by giving the distance from the cylinder end.
Several Degrees of Freedom
Generally, if it takes n numbers to specify the position of a
mechanical system, that system is said to have n degrees of freedom.
Rigid Body
A rigid body moving freely through space has six degrees of freedom
(three translations and three rotations). Consequently, it takes six
£3 coordinates to express its position. These coordinates are usually denoted
as x, y, z,&t$f\> . The origin is taken as the center of gravity and the axes
are fixed rigidly to the body so that x is forward, y to the right and z upward.
Hinged Parts
If the body is not rigid, but made up of two rigid parts connected
together with universal hinges , the number of degrees of freedom is in-
creased by two, namely two rotations about hinges. Take,for example,a
hinged rotor blade. Two additional degrees of freedom are added for each
blade which can move about a flapping hinge and a vertical hinge.
Elasticity
A completely elastic body has an infinite number of degrees of
freedom.
Helicopter, Simplifying assumptions
As Hohenemser points out in NACA TM-907, to set up exact motion
equations for the helicopter with hinged blades would result in a very com-
plicated system, as each universal hinged joint, even when excluding bending
flexibility, involves two additional degrees of freedom. His simplifying
assumptions in this instance include:
a. The air forces and mass forces on the blades are determined
on the assumption that the rotor tip plane preserves its
position relative to the fuselage during the motions of the
aircraft '(e.g., no freedom between rotor and fuselage).
b. The inclination of the rotor tip path plane relative to
the fuselage is defined on the assumption of a uniform
distribution of air forces and mass forces on the blades.
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rr
This presumes that the aircraft motions are so slow that the blade
undergoes a succession of steady changes. Inasmuch as the forces on the ro-
tors are defined by the motion of the aircraft, blade hinging does not increase
the number of degrees of freedom. The example cited in the reference, namely
side-by-side rotor helicopter, is therefore reduced to two degrees of freedom:
l. Longitudinal Translation
2. Pitching Motion about the Transverse Axis
EQUATIONS OF MOTION
The treatment of a rigorous stability investigation involves the following
plan of study.
First, an equation of the forces arising out of a disturbance is written.
These include the forces generated by a displacement, forces generated by
the impressed velocity of motion and the forces generated by the impressed
acceleration of motion. They will be in equilibrium and have a sum equal to
zero. A similar equation is written for the moments. These equations have
the following general form:
Forces
Moments
The first equation written above is the sum of the forces acting on the aircraft.
The first term is an inertia term which is the product of the mass (if) and
the acceleration of the motion ( x ). The second term is the so-called velocity
or damping force which is made up of the product of the velocity ( x ) and the
damping coefficient (Fx). The third term is the so-called displacement force,
or spring-type force, which is made up of the displacement-( x ) and the dis-
placement coefficient (Fx). The second equation given above is for the sum
of the moments. As in the force equation, the first term is the inertia term or
the rotary acceleration () times the moment of inertia (I).
An excellent sample of the procedure of calculating a stability case
is given in NACA TM-907 "Dynamic Stability of a Helicopter with Hinged Rotor
Blades", by K. Hohenemser. This is an investigation of the dynamic longi-
tudinal stability of a hovering helicopter. A lateral rotor type such as the
FW-6l is considered in particular, but the general procedure applies to a
single rotor helicopter.
The investigation is for two degrees of freedom, namely rotation about
e.g. in a pitching sense.and forward translation.
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V. HELICOPTER STABILITY
There are two fundamental stability phenomena associated with heli-
copters which must be understood in order to form a good picture of heli-
copter stability:
A. Stability of Motion. When a rotor is moved horizontally, it tilts
in a direction opposite to the motion, bringing a stabilizing moment to bear
which tends to resist the motion.
B. Stability of Position. When a helicopter is angularly displaced
while hovering, the resultant forces move the helicopter horizontally (in
the direction of rotor tilt). The resulting restoring moment (from stabil-
ity of motion) gives the helicopter a measure of static stability. The subse
quent oscillation is, however,- dynamically unstable unless sufficient damp-
ing is present.
Stability of Motion
A. Blades flapping at axis of rotation.
Figure l
Hovering
Thrust "T" passes through
the center of gravity (CG)
Figure 2
In Forward Flight
Thrust "T" moves forward of
the center of gravity (CG)
producing a stabilizing nose
up moment on the helicopter.
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When a helicopter passes from hovering into forward flight, forces
come about which tilt the tip path plane of the rotor aft, causing the thrust
vector "T" to move ahead of the CG. This movement of "T" creates a nose
up moment which tends to retard the forward motion and pitch the helicopter
aft. Figures l and 2 illustrate this for rotors whose flapping hinges are at
the center of rotation.
Figures 3 and 4 illustrate that the placement of the flapping hinges
outboard cause an added increment of thrust vector movement. The source
of this added movement is explained later.
B. Blades flapping with offset horizontal hinges.
It
Figure 3
Hovering
Thrust "T" passes through
the center of gravity (CG).
Note that blade now flaps
about a hinge offset from
the axis of rotation by the
distance "e".
Figure 4
In Forward Flight
Thrust "T" moves forward a
greater distance* than il-
lustrated in Figure 2 due
to the added thrust offset
"a" at the hub. The moment
produced by the change in
thrust location produces a
stabilizing moment on the
helicopter.
* See detail explanation in
Figures 5 and 6.
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j4
C. The control effects of the offset hinge arrangements of Figures
3 and 4 are quite different from the control effects of the blade mounting
of Figures l and 2.
In the case of the arrangements illustrated in Figures l and 2,
control can only be had when the rotor is operating with a thrust load,
as otherwise there is no load to produce a moment about the center of
gravity.
The offset hinges illustrated in Figures 3 and 4 provide control
moments depending upon the centrifugal tension in the blades in combina-
tion with flapping. This moment is in addition to the control effects il-
lustrated in Figures l and 2, and is illustrated below:
Taking the offset horizontal blade hinge illustrated for the hovering
helicopter of Figure 3 as a beginning, then consider the following:
CT A
B
CT
CT A
b B QT
Figure 5
Figure 6
Offset Hinges
Blades caused to flap with
introduction of control.
The centrifugal tension of
the opposite blades "A" and
"B" oppose each other in
balanced relationship.
Centrifugal tension of blade
"A" is offset from the cen-
trifugal tension of blade "B"
by the distance "b" creating
a control moment CT x b.
Aerodynamic Aspects of Stability of Motion
The following discussion is a treatment of the aerodynamics of rotors
in translational flight and explains the phenomena which influence stability
of motion.
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Flow Over Blade in Plane of Rotor
The flow across and through the blade varies as it travels from one
phase position to another. Figure 7 shows the rotational velocity distribu-
tion on a rotor blade in two opposite positions in hovering.
Distribution of Velocity in Hovering is as
shown in Figure 7. Distribution is triangu-
lar from root to tip and velocity at any
section is proportional to the radius of that
section. When the helicopter is operating
in forward flight an additional velocity must
be considered. Figure 8 illustrates the flow
over the blades in the left and right positions
when in forward flight.
Distribution of Velocity (laterally) in Forward Flight is as shown in Figure
ffi In addition to rotational velocity, forward speed, V, is added giving(»)R
+ Von advancing side of rotor and6)R - V on retreating side of rotor.
On the high velocity side of the rotor, the
rotational velocity is added to the forward
velocity, and on the low velocity side of the
rotor, the forward velocity is subtracted from
the rotational velocity.
It can also be seen that the blade on
the high velocity side of the rotor will pro-
duce lift right into the hub where the velocity
(uft)is the forward velocity. This distribution of
air flow causes the lift on the blade to move
in closer to the hub with a shorter moment
arm from the center of the blade lift to the
center of the hub. In combination with the centrifugal forces,this distribution
tends to bow the blade down. Considering only the velocity factor, a greater
lift will be produced on the advancing side of the rotor than on the retreating
side of the rotor.
On the low velocity side df the rotor, air flows into the trailing edge of
the inboard part of the blades. This velocity diminishes from the forward
velocity at the hub to zero velocity at a point along the blades where the ro-
tational velocity equals the forward velocity. From this point outboard, the
velocity over the blade increases at the same rate as on the side of the rotor
advancing into the wind. Thus the air lift load,being proportional to velocity
squared, shifts outboard on the blade when traversing the downwind side of
the rotor. This outward shift of the lift forces, in combination with the
triangularly shaped distribution of the centrifugal forces on a straight flapping
GOR+V,
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16
blade, tends to bow the blade up. Considering only the velocity factor, a
lower lift will be produced on the downwind side of the rotor. Figure 9
illustrates the relative velocity and direction of air flow over the blades of
a rotor in eight different phase positions when the forward velocity is 0.3 of
the rotational tip speed (a ratio representative of a forward speed of about
87 knots or 100 mph).
Figure 9
Lift Distribution
Figure l0 is an elevation illustration
of the lift distribution on the blades of a
helicopter rotor in forward flight. The
blade moving upwind is on the right of the
figure and the blade moving downwind is on the left side of the figure. It
may be noted that the lift moments on opposite sides of the rotor are substan-
tially equal.
Figure l0
Rolling moment caused by unequal distribution of airload.
This is brought about through blade flapping for hinged blades (including
teetering rotors) and through mechanical feathering for rigidly mounted blades.
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rr
Flapping Due to Velocity Differentials
Thus the upward flapping on the upwind (high velocity) side of the
rotor automatically reduces the angle of attack of the blade elements when
operating in this region. On the downwind (low velocity) side of the rotor
the blades flap down, thereby increasing the angles of attack of the blade
elements when operating in this region. In the case of the rigidly mounted
system,the blade angles of attack must be mechanically lowered on the up-
wind side of the rotor and increased on the downwind side of the rotor so as
to maintain lateral equilibrium. The quantitative amount of such feathering
would be substantially the same as the change in blade angle of attack pro-
duced by flapping.
It is common knowledge that in a system where a disturbing force
is applied at the natural frequency of the system, maximum displacement
occurs at 1/4 the cycle past the application of the maximum force. In the
case of the highlv aerodynamically damped flapping rotor blade, the maximum
displacing force is applied to the blade on the upwind (high velocity) side of
the rotor and the blade reaches its maximum upward displacement in a forward
position. This condition is illustrated in Figure l3* (1), page 20.
Treatment of Side Gusts, etc.
The treatment of side gusts, side slips or yawed flight is resolved into
simple vector addition of velocities. When a rotor encounters a side gust,
the effect is to change the distribution of velocity over the disc and hence the
mode of flapping of the blades. The character of these elfects can be studied
by considering the changes in relative velocity approaching the rotor. This
is accomplished by vector addition of the forward velocity and the side gust
velocity. Figure ll ( a through d ) shows a step by step treatment of these
velocities. Figure ll (a) shows the rotor in horizontal flight where the hori-
zontal velocity is V and the rotational velocity is COR. Also pictured is the
unequal distribution of velocities on the advancing and retreating sides of the
rotor (as previously described). Figure ll (b) shows the rotor of ll (a) being
acted upon by a side gust Vs.. Figure ll (c) shows vector addition of Trans-
lational Velocitv V and Gust Velocity Vg.\ Figure ll (d) shows Resultant
Velocity Vr in magnitude aiid direction (as derived in Figure ll (c) and also
the new distribution of velocity across the rotor disc.
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FIG. II la) Ft G.I 11(b)
FIG. II(c) FIG. II(d)
We have just completed a discussion of the effects of variations in
blade velocity in the plane of the rotor and how thev indirectly alter the
angles of attack of the blade elements through automatic blade flapping or
through mechanical feathering in the case of the rigid rotor.
Flow Through Rotor Due to Coning
It can be seen that any differential in flow through the rotor (normal
to rotor disc) for different regions of the rotor will also create unsymmetric-
al changes in the angles of attack of the blade elements.
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19
Fig I2
Figure l2 shows a side elevation of a rotor advancing at velocity V. Blade B
is shown in the most advanced position and blade A is shown on the trailing
side of the rotor. Each of the blades has an average coning angle B0 from
a disc normal to axis a - a. The entire disc is rotated in the
direction of advance by the angle
The velocity V will have a component of flow parallel to the blade
(which will not affect the angle of attack of the blade) and it will have a com-
ponent of flow normal to the blade which will appreciably affect the angle of
attack of the blade. The present presentation will show how the component of
the forward velocitv affects the angle of attack of the blade elements. In the
forward,position the flow up through the rotor normal to the blade is:
Vb, =(B0-4)V
In the aft position,the flow down through the rotor normal to the blade is:
Vb2:(Bo+* V
The differential between the upward flow through the blade at the front of the
rotor and downward flow through the blade at the aft side of the rotor is:
'b,
+ Vv
2B0V
The effect of this differential flow through the forward and aft portions of the
rotor is to increase the angles of attack and lift of the blade elements as the
blade passes the forward side of the rotor and to decrease the angles of
attack and lift of the blade elements as they pass the aft side of the rotor.
To summarize, the coning angle,in combination with forward velocitv,
creates higher blade lift in the forward region of the rotor. Like the effects
on flapping of differential velocity on opposite sides of the rotor, coning angle
,with forward velocitv,creates side flapping. With other less influential
factors neglected and considering onlv a flapping hinge which is 90° to the
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20
longitudinal blade axis, the high point of flapping will occur in a quadrant
lying between straight forward and the downwind side of the rotor,as illustrat-
ed in Figure l3.
Ot)R
FIGURE I3.
Stability of Position
When a helicopter is angularly displaced while hovering, the resultant
forces move the helicopter horizontally in the direction of the rotor tilt.
The resulting restoring moment (from stability of motion) gives the helicopter
a measure of static stability. However, the helicopter is wholly lacking in
damping, and the motion which ensues becomes catastrophic after about two
cycles unless corrective control is made by the pilot. The following series
of figures (l through 8 incl.) show the aerodynamic forces and moments which ;
are brought to bear on the helicopter throughout a complete cycle of disturbance:
from hovering. This represents the characteristic statically stable, dynamic- i
ally unstable motion of the helicopter.
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1
2
3
4
Hovering helicopter
becomes angularly
displaced to right.
Angle "a" = Angle "b"
Vs = O
Force "s" is result-
ant of Thrust "T"
and Weight "WM, and
causes ship to move
to right.
Helicopter moving
to right at vel-
ocity (V ) creat-
ing righ&ng mo-
ment (M) from un-
equal coning angle
(Angle "a'Vangle "b").
Moment (M) creates
angular velocity (tu).
Helicopter moving to
right at velocity (Vs)
creating righting mo-
ment M from unequal
coning (angle "a"> angle "b").
Moment (M) continues to
create angular acceleration
( oe ) and increasing angular
Velocity (U»).
READ LEFT TO RIGHT
Helicopter hovering
with angular velocity (**J),
angle "a" = angle "b",
Vs = O , M = O •
Angular velocity
is a maximum. Note
that the condition here
is the same as the
original condition of
Figure 1, except that
in this case the helicopter
is initially rotating to the
left.
INSTABILITY OF POSITION
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8
7
Vs = O M = O
The helicopter has now re-
turned 10 its oi iginal start-
ing poin:. But now it is
angularly iisplaced to the
right a greater amount
than originally. Originally,
the helicopter had no ro-
tational velocity,but now it
has a rotational velocity in
the direction of motion
which will cause the heli-
copter to traverse a great-
er distance with accompany-
ing greater angle of roll
until the motion becomes
catastrophic.
Helicopter moving to
the left at velocity
(-Vs) With ship on
even keel and rotating
to the right with in-
creasing rotational
velocity (CO ),being
accelerated by rota-
tional acceleration
(OC) resulting fr.om
moment (M)jangle
"a" < angle "b".
READ
6
5
4 p\\ •
-Vs
-Vs
Helicopter angularly
displaced to left and
moving at velocity
(-Vs) creating moment
(M) and acceleration
(OC). Angle "a" <
angle "b". Angular
velocity "CO " - O.
The ship will be dis-
placed with maximum
angle to the left in
this condition,since
rotation to the left
has just stopped.
Helicopter moving to
the left at velocity
(-Vs) creating moment
(M) and reduction in
angular velocity ((JJ ).
But angular velocity
(CU),having been created
throughout the entire
right swing of the ship,
(Figures 1 to 4 inc.)
persists in continuing
rolling the ship to the
leftjangle "a" < angle "b".
RIGHTTO LEFT
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23
VI. COMPARISON OF HELICOPTER AND AIRPLANE STABILITY
Helicopters of today possess some measure of static stability in
all phases of flight but are almost wholly lacking in the necessary elements
of damping required to produce dynamically stable flight. This section
points out the essential differences between airplanes and helicopters with
regard, to dynamic stability.
LONGITUDINAL STABILITY
l. Airplane
In an airplane the stabilizer acts both to maintain the plane in
longitudinal balance and to dampen longitudinal oscillations.
Figure l.
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When the airplane moves from position A to position B,the small stabilizer
force Fj is increased due to the increase in its angle of attack. A part of
the angle of attack change comes from the change in angle as illustrated at
0 (the static stabilizing force) and another increase in angle of attack comes
from the vertical downward velocity (Vy) of the stabilizer (the damping force).
The damping force may be evaluated as illustrated in Figure 2.
Figure 2.
The stabilizer moving down with velocity (Vv) creates a relative
upwardly directed air velocity (V ) which,in combination with the forward
velocity V, establishes an increment of increased angle of attack of the
stabilizer <^)), where ^) = Arc tan Vv/V. The equivalent change in lift
coefficient of the surface (A CjJ (assuming arc tan Vv/V is equivalent to
Vv/V for the small angles being considered) is: (57.4VV/V) "a",where "a"
is the slope of the lift curve for the stabilizers. The change in lift (damping
force) is: AF = (l/2 ^A 57.4VV/V) aV2 = 57.4 f AaVvV/2 (l)
It is interesting to note that this longitudinal damping force is proportional
to forward velocity (V). In hovering helicopter,where forward speed is zero,
the equation becomes: F = l/2 ACjVv21where Cl is flat plate drag of the
surface and may be taken as l.35. The ratio of stabilizer damping in forward
flight to stabilizer damping hovering is:
2 x 57.4 AaVvV 57.4aV
2 x ACLV/ = CLVV
For an aspect ratio of (3),a ■ . 068
Substituting values: Ratio (R) of forward speed to hovering damping for a
stabilizer is: R = 57.4 x. 068 V = 2.9V/V --(2)
1.35Vy
where V is forward velocity and Vv represents a vertical velocity on the
stabilizer resulting from maneuvered vertical motion of the stabilizer
relative to the earth.
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25
It may be seen that vertical gusts will create a disturbing force on the
stabilizer in accordance with the above equations.
II. Helicopter
As shown above, the horizontal stabilizer of a helicopter acts as an
effective damper on longitudinal oscillations in forward flight but becomes
inadequate in low speed flight and in hovering. For example,if Vy =30 ft/s<
and V =200 f t/aecjhen R becomes 2.9 x 200/30 = l9.3. The horizontal
stabilizer may also be used to create stabilizing longitudinal moments on the
helicopter.
In a fore and aft rotor helicopter,the rotors act to dampen pitching
moments and?neglecting special downwash considerations, longitudinal
stability can be attained.
The inertia in a pitching sense of fore and aft rotor helicopters tends
to be greater than that of single or side by side rotor machines and therefore
the damping requirements for this configuration tend to become greater for
equivalent ease of piloting. The damping, however, is very high due to the
large area of the rotor and the high velocity of the blades. If a small horizon-
tal rotor is placed at the tail of the helicopter, it should show good damping
for pitching motions of the machine both in hovering and forward flight. In
this case, the rotational velocity of the blades is effectively equivalent to V
in equation 2.
Lateral Stability
l. A. Airplane (at the instant of initiating aileron deflection).
Figure 3.
MOMENT DUF TO
All fron nrn fhtiom
FIG. 4
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26
The initial moment (M^) is Pja + ^2^-
The acceleration is Pja-f* P2b,where I is the moment of inertia
I
about the center of gravity.
B. Airplane (with ailerons deflected after roll has been initiated).
The moment is P,a + P2b - F^c - F«d,where F & F£ are aerodynamic
damping forces which are generated as a result of the angular velocity )
of the machine. When the moments created by the damping forces F^ and F2
are equivalent to the moments generated by the control forces Pj and P£7a
steady (unaccelerated) rate of roll will be established.
II. A. Helicopter (at the instant of initiating lateral rotor control).
FIG, 6
The initial moment (M) is Le.
Like the airplane,the angular acceleration is Le/I,where I is the lateral
moment of inertia of the helicopter about its center of gravity.
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27
B. Helicopter (with rotor control after roll has been initiated). Re-
ferring to Figure 6, the moment (M) is substantially the same as it was
initially since the blades line themselves with their deflected axis of ro-
tation twice each revolution. Furthermore, the increment of angular rota-
tion of the helicopter relative to the blades,which occurs during the interval
of time that the blades are in the lateral positions,is accounted for by a
rotation of the blade about the horizontal hinge. This effect is very small
in all hinged rotors,although some slight damping can be obtained with out*
board horizontal hinges or with rigid blades.
To Summarize: The single rotor helicopter is almost completely lacking
in damping from roll as compared to an airplane. This lack of damping
establishes the requirement for a control force to.be followed by a reverse
control to stop the initiated roll.
in. If the rotors of a helicopter are displaced laterally, then the damping
in roll is altered appreciably.
FIG. 7
A. The initial control moment (M) is P^ a 4» P2b.
B. After the roll has been initiated,the resulting moment (M) is P\a. +
P2b - Fja — F2b. When the velocity through the rotor due to roll is
equivalent to the rate of advance due to the differential pitch change at each
rotor,then roll reaches a steady state.
If the controls are neutralized after initiating the roll^then the damping
forces Fj and F2 will decelerate the roll as in an airplane.
Helicopters having rotors displaced laterally usually have greater
lateral inertia which requires greater control forces and damping for equiva-
lent control characteristics with smaller inertia machines.
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Directional Stability 28
In an airplane the principal directional stability effects are attained by
locating the fin area so that the center of pressure of the fin lies aft of the
center of gravity of the airplane.
In a helicopter the same concepts of directional stability hold with
minor exceptions as follows:
A. In the case of the tail rotor helicopter, changes in power to the
main rotor must be accompanied by changes in the thrust of the tail rotor.
On the other hand, the tail rotor acts to provide excellent damping forces
in a yaw sense.
B. In the closely spaced intermeshing rotors of the synchropter con-
figuration, any side slip velocity through the cocked rotors creates yaw
moments. A side velocity as from a gust, yawed flight, or side slip creates
a flow down through one rotor and up through the other rotor. When the
blades are set at high angles of attack,an upward velocity creates an increase
in rotor torque and a downward velocity through the rotor creates a decrease
in torque. Because the oppositely turning rotors are geared together,this
torque is additive and manifests itself as a yawing moment on the ship. In
a synchropter where the outboard tips are moving aft (like a breast stroke
swimmer),this yaw moment acts in a stabilizing manner and the helicopter
has good directional stability.
When the blades are set at autorotative angles of attack, downward flow
through the rotor tends to retard the rotor and^conversely^pward flow tends
to accelerate the rotor.' Therefore, in autorotation of a breast stroke type
synchropter,the yawing moments created by side flow through the rotors
create an unstable yaw moment which must be corrected by the addition of
more fin area.
C. The fore and aft type helicopter would be expected to have yaw
characteristics similar to an airplane,except that its moment of inertia in
yaw may be relatively high tending to retard the rate of deviation. Unless
lateral rotor control is introduced concurrently with changes in longitudinal
trim for CG shifts or for maneuvers,yaw forces would be expected to develop.
D. In the side by side rotor configuration,where lateral control is
affected through differential collective pitch, the yaw moment resulting from
roll control will depend upon the direction of rotation of the rotors. With
aft moving outboard blade tips, a differential change in collective pitch on
the two rotors creates change in the torque on the rotors so as to produce a
yaw in the direction of the bank. On the other hand, the rotors having the
increased pitch will also have greater drag and this creates an unstable
yaw irrespective of the direction of rotation of the rotors. With the rotors
spaced for non-intermeshing,the adverse yaw with roll is likely to be greater
than the correcting differential rotor torque.
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29 I
VH. ROTOR STABILITY
l. A. Rigid Rotors
V
111111
Figure l
Figure l is a plan view of a rotor moving in a direction from C to D and
turning as indicated by the arrows.
A blade moving upwind on the advancing side of the rotor is acted
upon by the air speed "V",plus the rotational speed " *•> R". A blade moving
downwind on the retreating side of the rotor as at A is acted upon by the
rotational speed " 0> R", minus the air speed "V".
The differential velocities on opposite sides of the rotor create an ex-
cessive lift on the advancing side of the rotor relative to the retreating side
of the rotor unless the angles of attack of the blade elements are differentially
altered by suitable means such as by flapping or by feathering. Bodily
decreasing the pitch on the upwind side of the rotor and simultaneously
increasing the blade pitch on the downwind side of the rotor may be accom-
plished by feathering the blades with the aid of a swash plate or equivalent.
In the case of flapping,the blades move into higher coning angles on the
upwind or advancing side of the rotor,thereby creating an air flow down thru
the rotor with consequent reduction in the angles of attack of the blade
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30
elements. Conversely,the blades flap down on the downwind or retreating
side of the rotor creating an upwind thru the blade elements with an accom-
panying increase in angles of attack.
The angle of attack of the blade in the advance position "D" of Figure 1
is increased relative to the blade in the trailing position "C" causing nose up
moments,when flapping, inertia, and (CT)N forces are neglected.
Change in blade lift(fore and aft)due to angle of attack change resulting
from combination of forward speed and coning angle is illustrated in Figure 2.
It may be noted that the forward velocity V creates an upward velocity
component and increasing angle of attack on the blade in the forward position
"D" and a downward velocity component and decrease in angle of attack on
the blade in the aft position "C". The air load distribution is similar to that
illustrated in Figure 3,unless relief is had thru feathering or flapping as
described above.
l. B. Flapping Rotors
Air Load Distribution
When the blades are operating on the advancing side of the rotor, the
combination of flapping or feathering and forward speed tends to bring the
longitudinal center of pressure of the air forces inboard. Conversely, when
the blades are operating on the retreating side of the rotor, the longitudinal
center of pressure of the air forces acting on the blades tends to move out-
board.
Figures 4, 5 and 6 show:
(l) The average distribution of the air forces acting on a
rectangular rotor blade,
(2) The distribution of air forces acting on the advancing side
of the rotor,
V
Figure 2
Figure 3
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31
RESULTANT
FIG. 4
AVERAGE AIR LOAD
DISTRIBUTION.
RESULTANT C.F.
b cos BJ
FIG. 8 NORMAL COMP.
OF CF OPPOSES AIR
LOAD.
MALL
FIG. 5
ADVANCING SIDE
(HIGH V, L0W6 )
FIG. 9
ADVANCING SIDE.
LARGE
FIG. 6
RETREATING SIDE
(LOW V, HIGH 9)
FIG. I0
RETREATING SIDE
15
RESULTANT C.F.
FIG. 7
FIGURES 9*IO SHOW
THE RELATIVE EFFECT
OF BENDING ON NORMAL
COMP. OF CF. THE
BENDING IS CAUSED BY
AIR FORCES.
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(3) And the distribution of air forces acting on the retreating side
of the rotor, respectively. The center of pressure of the air
forces is indicated by a heavy vector that shifts inboard on the
advancing side and outboard on the retreating side.
Distribution of Centrifugal Forces
The normal components of the centrifugal forces acting on the blade
elements at a given rotor speed are a function of the blade coning angle and the
mass distribution along the span of the blade,if the deflection or curvature of
the blade is neglected. Figure 7 shows the distribution of the centrifugal
loads along the span for a straight blade having uniform mass distribution
from root to tip. With uniform mass distribution, the centrifugal force on a
mass element is proportional to the radius, and hence the centrifugal forces
are triangularly distributed along the blade, and their center of percussion falls
at two-thirds of the distance from the axis of rotation to the blade tip. Figure
8 shows the two components of the resultant centrifugal force which are acting
at the center of percussion of the blade. It will be noted that one component is
perpendicular. The perpendicular (or normal) component'opposes the air load,
which tends to lift the blade into a coned position.
Inertia Forces
As the blade flaps up and down,or as it swings about its vertical hinge,
inertia forces act in opposition to the accelerating forces; i. e., when the
upward flapping velocity of the blade is being increased,as in the aft segment
of the rotor, the inertia forces act down,and in the forward segment of the
rotor,where the blade is being accelerated down,the inertia forces act up.
Like wise,when the blade swings about a vertical hinge in a lagged posit ion, the
blade is being accelerated forward,opposed by the inertia forces. Conversely,
when the blade is in a forward position and being accelerated aft,the inertia
forces act forward in opposition.
The inertia loads described above act as a function of the acceleration
of the blade elements. For a flapping,straight blade,the inertia loads increase
directly with blade radius,creating a triangular loading similar to the centrifu-
gal load distribution. When the blade is being deflected longitudinally,the
blade elements are no longer being accelerated proportional to the radius
from the oscillating hinge,and the distribution of the accelerating forces will
not be triangular.
Displacement of the Center of Percussion due to Blade Bending
If the blade, while coning, remained straight, there would be no change
in the position of the center of percussion. If the center of pressure of the
air load always coincided with the center of percussion of the centrifugal and
inertia forces, there would be little or no bending moments in the blade,and
no bending deflections in the plane of flapping. In actuality,the shifting of the
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center of pressure of the air forces, as illustrated in Figures 5 and 6, introduces
bending deflections in the blades in the flapping plane. Bending deflections in
the flapping plane of the blades create a nonuniform variation in the distribu-
tion of the normal components of the elemental centrifugal forces acting on the
blade.
Figure 9 shows that the normal component of the centrifugal tension is
reduced toward the blade tip when the blade is bowed down. When the blade
takes this shape, the center of percussion of the normal components of the
centrifugal tension moves inboard. It may be noted that an inboard shift of the
air load,which would produce the type of blade bending illustrated (Figure 9),
would tend to offset the inboard shifting of the normal components of the centrifu-
gal force. For the above reasons,a blade that is relatively flexible in the flap-
ping plane will be subjected to smaller bending moments than a more rigid blade.
Figure l0 shows that the normal component of centrifugal tension is
increased toward the blade tip when the blade is bowed up. In this case, the
center of percussion of the normal components of the centrifugal force moves
outboard. When the blade takes the shape shown in Figure l0, the air lift
loads have moved outboard. Since the normal components of the centrifugal
force increase toward the blade tip with increase in bow-up of the blade, the
disproportionate increase in air lift toward the tip is automatically compensat-
ed by the increase in normal component of the centrifugal forces at the blade
tip. An ideally designed blade might be one in which the maximum bending
forces in flight would be equal - positive and negative.
l. C. (l) TORSIONAL LOADS DUE TO LOCAL RELATIONSHIP
BETWEEN CHORDWISE CENTER OF GRAVITY (CG)
AND CHORDWISE AERODYNAMIC CENTER (AC)
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Consider Blade"b" hinged at "H" and rotating about the axis a-a in coned
position B0 from a plane perpendicular to axis a-a. An element at "A" o/
will be acted upon by centrifugal tension CT. This centrifugal force may be
considered in its two components (CTL acting parallel to the blade and
(CT)N which acts normal to the Made to oppose the air lift "L".
Consider now the forces acting on the elemental section d-d of Figure 11,
as shown in Figure 12.
Figure 12.
+ CmKV
The lift force "L" acts upward through the Aerodynamic Center (AC),and the
normal component of the centrifugal forces (CT)jj,as well as all inertia forces,
act through the chordwise center of gravity (CG),to oppose the lift forces "L".
In a flapping blade,the inertia forces I (which may be positive or negative),plus
the normal component of the centrifugal force (CT)N,exactly equal the lift
force "L" (neglecting longitudinal blade bending and blade weight ). The gravi-
tational forces act on the blade at Wb.
Whenever the blade airfoil section has other than a zero moment coeffi-
cient (CjgO), a twisting moment will be introduced which is proportional to
the moment coefficient Cm, the blade chord, and the speed of the blade element
squared (V^). If the moment coefficient is negative, the moment will tend to
lower the nose and raise the trailing edge and,conversely,a positive moment
coefficient will tend to raise the nose and lower the trailing edge. It may be
noted that this velocity sensitive force varies between the advancing and re-
treating side of the rotor. An additional moment is created on the blade ele-
ment resulting from moment arm "X" between the lift "L" at the aerodynamic
center and the (CT)^ and inertia forces "I" acting at the chordwise center of
gravity (CG). When the AC is forward of the chordwise CG, the moment is
positive and when the aerodynamic center is aft of the chordwise center of
gravity, the moment is negative (nosedown). In addition, a torsional force
F^ acts to maintain the blade flat in the plane of rotation. This force is asso-
ciated with the sectional mass distribution, the blade weight, the radius and
rotor speed squared.
2
Total moment of blade element: M = xL + CmKV + Ft.
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35
When a blade is loaded, as illustrated in Figure 6yshear loads are
transmitted from one section to adjacent sections (longitudinally) of the blade
and manifest themselves in bowing the blade. In a rigid rotor,such shear
forces are appreciable and may amount to half the lift forces,but on flapping
rotors,the shear forces are usually much smaller. These shear forces
are represented by vector Fs in Figure l3. It may be noted that the only
difference between Figure l2 and Figure l3 is that inertia vector force (I) Is
equivalently reduced. In addition, structure mayor may not coincide
with the chordwise aerodynamic center (AC). When there is a mismatch with
the above, an additional moment is created which is equivalent to Fg x d.
CmKV
+ CmKV
Figure l3
At the inboard end of a hinged blade, any remaining value of Fg must neces-
sarily pass thru the blade pitch change axis,or else it will create a torsional
moment equivalent to Fs x d', where d' is the distance between the pitch axis
and the shear force Fs. This offset is illustrated in Figure 23.
l. C. (2) EFFECT OF LONGITUDINAL TRANSFER OF TORSIONAL
FORCES DUE TO LONGITUDINAL BLADE BENDING
So long as the blade remains straight,unbalanced torsional forces
distributed along the blade may be considered summated from the blade tip
inboard. That Is to say,the resulting torsion on an inboard blade attachment
will be the summation of all the elemental torsional moments distributed along
the blade. Thus a blade having the chordwise center of gravity at the aero-
dynamic center and a moment coefficient of zero should have no aerodynamic
torsional moments transferred to the hub or controls. This same effect may
be attained provided the summation of moments is zero. For example, a
forward located blade tip weight might create balance for an otherwise out of
|l torsional balance blade. If the blade is bent as illustrated in Figure l4, the
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torsional moments acting on the outboard region of the blade have components
in a plane containing the longitudinal axis of the blade. This minor moment
creates a very small moment tending to bend the blade in the plane of rotation
Likewise, the reduction of the elemental blade torque due to consideration of
the cosine of the angle of blade deflection would obviously alter the torsional
moments to a very minor degree.
Figure l4 illustrates the effect described above. Consider the torque forces
acting in plane d - d of element "A" on blade "B" whose pitch change axis
is at b - b. The torque M acting in plane d-d may be resolved into a com-
ponent Mc parallel to the pitch axis b - b and another component M^ which is
perpendicular to pitch axis b - b. Since the unbalanced torsional moments
on the blade elements are likely to be small, the errors involved in neglect-
ing this effect may be considered small except in cases of unusually large
blade bending in combination with appreciable blade torsional unbalance.
l. D. 3. TORSIONAL MOMENTS ABOUT THE PITCH CHANGE AXIS
If the condition stated above, i. e., L = (CT)n + I is not a true
equation then the inequality of forces will be transferred along the blade to
adjacent sections until equilibrium exists. In this case, longitudinal bending
will be present causing the blade to bow up or down.
Bending will also occur in the plane of rotation. Although the loads in
the plane of rotation are smaller, their unsymmetrical distribution will cause
blade bending that will put compression in the trailing edge of the blade (See
Figures l5 and l6).
b
d
Figure l4
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37
Figure l5
The unsymmetrical distribution of these forces arises from the fact
that the air drag loads vary radially a§ the square of the speed and are
opposed by the in-plane component of centrifugal force which is substantially
constant. This substantially equal distribution of the in-plane driving com-
ponents of the centrifugal forces results from the product of the elemental
centrifugal forces which increase directly with blade radius and the reduction
with radius in component angle illustrated at 0>and ^of Figure l5. The
elemental centrifugal force originating at the axis of rotation "A" creates
centrifugal tension oh elements B and C acting along the line A-B or A-C.
These elemental forces are reduced to a component parallel to the blade and
a driving component normal to the blade. The resulting distribution of the
driving components is illustrated for a blade of constant mass distribution.
In one case, the maximum bending moment in the plane of rotation from this
source was approximately l/4 of the blade driving torque.
This type loading creates the type bending illustrated below: The
system is in equilibrium by unsymmetrical forces "0" and "F'^and reaction
at the vertical hinge "R".
B
Figure 16
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The magnitude of the reaction at B is approximately ll/3 x torque
moment per blade t R. The force F is approximately 3/2 x B and the force
R is approximately B/2.
l. D. 4. TORSIONAL MOMENTS ABOUT THE PITCH CHANGE
AXIS DUE TO BENDING IN THE FLAPPING AND
ROTATION PLANES
It may readily be seen that a hinged blade bent in the plane of flapping
from combined air, centrifugal and inertia forces will create a shear load at
the inboard flapping hinge denoted by "R" on Figure l7. The magnitude of the
reaction is the load required to maintain a static blade in its deflected position.
In the case of both teetering and rigid rotors, the root bending moments are
transmitted across the hub to the blade or blades on the opposite side of the
rotor. In these cases,the root bending which varies with lift may be appreciable
Figure l7
The combination of the in-plane loads illustrated in Figure l6 with blade
deflecting in the plane of flapping illustrated in Figure l7 produces torque
moments about the blade pitch axis. Figure l8 (b) represents a view looking
parallel to the blade pitch axis. Taking moments of the forces P and B,
the resulting moment about the pitch axis a-a is M = F x g - Bxh. The
direction of the torque is dependent upon whether the blade is bowed up or
down.
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FIG. I8 a
PITCH AXIS
(*-•)
FIG. I8b
(view A-A)
ADDITIONAL LOADING CONDITIONS CAUSING TORSIONAL MOMENTS AT
THE BLADE ROOT
l. Coriolis Effect
It is well known that when the inertia of a rotating body is reduced,its
rotational speed will increase in accordance with the equation;
Kinetic Energy = l/2 TW2 = constant.
When a rotor blade bows up or flaps up,its moment of inertia about the
axis of rotation is decreased by virtue of its movement inboard toward the
axis of rotation.
So long as the blade remains straight,the effect is to cause cyclic
oscillations about the vertical blade hinge. In a bowed blade! some of the blade
elements are moved relatively closer to the axis of rotation than other elements
of the blade. This disproportionate shift of the blade elements inboard causes
disproportionate angular acceleration of the blade elements with consequent
twisting of the blade. For example, a_rotating blade changing its shape from
straight,as illustrated in Figure 4, to bowed up at the tip,as illustrated in
Figure 6,will have its tip accelerated forward relative to the remainder of
the blade. The effect of this loading in practice is to create a force in the
control system acting toward the retreating side of the rotor, lowering the
blade pitch angles accordingly.
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Conversely ,when the blade tip is in the process of being bowed down,
as illustrated in Figure 5, the blade tip will be decelerated relative to the
remainder of the blade, thereby also creating a decrease in blade angle. This
phenomena may account for the twice per revolution frequencies which are
sometimes found in three bladed rotors. It may also account for some forms
of stick shake.
DAMPER EFFECT
It is customary to provide friction or viscous damping for oscillations
of the blade about the vertical blade hinge. Whenever the forces along the
blade which create these moments do not act directly on the centerline of the
blade pitch change axis, they create torsional loads in the blade root which
are transferred into the control system. Generally speaking, if the blade
pitch control axis lies in the inboard end of the blade (Figure l9),or in the
extension link (Figure 20),the moments in the control system will be small,
but when the blade pitch axis is in the hub (Figure 2l), the component of damper
moment which is transferred into the control system may be appreciable.
For example, a blade (of Figures l9 or 20) bowed so that its mass
center lies 6 inches away from the blade pitch pivot axis and l5 feet from the
vertical hinge will cause oscillating control moments "C" from the damper
moment "M" of (. 5/l5) M or . 033M = C. With the blade (illustrated in
Figure 2l) coned 5° out of alignment with the blade pitch axis,neglecting blade
bow,the oscillating control moments "C" from the damper moment "M" is
M sin 5° = . 087M = C
2. DETERMINATION OF BLADE MOTION ABOUT HINGES AND TORSION
IN BLADE PITCH AXIS
When a blade is pivoted at the root through universal joint type pivots
, i.e., the two pivot axes at 90° to the assembly and to each other,with the
blade extending straight out from the hub, the action about the hinge is simple
to determine. But when two or more hinges permit angular motion of the
blade in the same plane^the problem is complicated by the effects of centrifugal
tension and torsion acting on the pivots. A chain held at each end will swing
as a catenary with angular deflection at each link, but in a child's chain-
swing substantially all angular deflection occurs at the link adjacent the mem-
ber fixed to the limb. The same sort of thing happens when a blade acted
upon by centrifugal tension and attached to the hub by a multiplicity of pivots
is permitted blade motion in a given plane. Barring restraining forces, the
innermost pivot will account for all of the angular requirements of the blade.
EFFECT OF BLADE HINGE ARRANGEMENTS
For blades having customary horizontal, vertical and pitch change pivots,
the analysis is relatively simple,but when cocked or multiple hinges are
employed,the analysis becomes more complicated.
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7T
The standard hinge arrangements are illustrated on Figures l9, 20 and
2l. Figure l9 shows an arrangement where the blade pitch change axis is
located in the root of the blade. In this case, unsymmetrical forces acting
along the blade fall directly on the blade pitch axis,except for blade deflection
as explained on Page l0. Centrifugal tension tends to cause all forces except
pure torsion to create oscillations about pivots A - A and B - B.
Figure 20 illustrates an arrangement where the pitch change axis lies
between the horizontal hinge A - A and the vertical hinge B - B. So long as the
pitch change axis C - C extends along the longitudinal blade axis, the arrange-
ment illustrated in Figure 20 has the same effect as the arrangement illustrat-
ed in Figure l9. When the blade of Figure 20 is moved to a lagging or leading
position about pivot B - B (see inset Figure 22), the blade pitch axis C - C no
longer extends along the longitudinal blade axis and the moment of inertia of the
blade about the pitch change axis is increased. The blade being free to move
in a flapping direction about the horizontal hinge permits the center of flapping
inertia to remain at substantially the same elevation. This means that in
addition to the inertia due to motion about the longitudinal blade axis, "see-
saw" inertia about the longitudinal center of percussion of the blade is also
introduced. For example,increasing the pitch of a lagged blade will cause the
inboard end of the blade to rise and the tip of the blade will be depressed while
the entire blade is rising due to the influence of the increased blade pitch. In
this case, the longitudinal inertia forces introduced tend to initially oppose
the control forces about the control pivots.
Figure 2l illustrates an arrangement where the control pivot C - C forms
a part of the hub and the horizontal and vertical hinges A - A and B - B act as
a universal joint outboard. This case is similar to the arrangement illustrated
in Figure 20 for motions of the blade in the plane of the rotor. However, with
the horizontal hinge A - A also outboard of the blade pitch pivot C - (^addition-
al forces are introduced whenever the coned position of the blade does not line
up with the blade pitch axis C - C. When the blade is coned above the pitch
axis C - C and moving forward,a blade damper (not shown) at the vertical
hinge B - B introduces a pitch reducing moment or,conversely,an aft moving
overconed blade will create a pitch increasing moment. When the blade lies
below the pitch axis C - C|the above damper components acting about the
pitch axis are reversed. The magnitude of the moments is equivalent to the
product of the damper moment and the sine of the angle formed between the
longitudinal blade axis and the blade pitch axis C - C. The torsional loads are
transferred through the universal joint pivots A - A and B - B,as in shafting.
A longitudinal bending moment is introduced into the adjacent shafts which is
proportional to the deflected angle and the applied moment.
Figure 2l shows an arrangement where the blade pitch axis C - C forms
a part of the hub and an elongated universal joint (Pivots A - A and B - B) lies
outboard of the blade pitch axis C - C. This arrangement is similar to the
arrangement shown in Figure 20 except that an additional load is involved. It
may readily be seen that control motion about the blade pitch axis C - C will
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53
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ROTARY WING AIRCRAFT
HANDBOOKS AND HISTORY
STABILITY AND CONTROL
• OF
ROTARY WING AIRCRAFT
BY
WILLIAM E. COBEY
VOLUME 10
ONE OF A SERIES OF 18 VOLUMES EDITED BY
EUGENE K. LIBERATORE
PREWITT AIRCRAFT COMPANY
CLIFTON HEIGHTS, PENNSYLVANIA
AND PREPARED FOR
WRIGHT AIR DEVELOPMENT CENTER
AIR RESEARCH AND DEVELOPMENT COMMAND
UNITED STATES AIR FORCE
WRIGHT-PATTERSON AIR FORCE BASE, OHIO
UNDER CONTRACT NO. W33-038 ac-21804 (20695)
DISTRIBUTED BY
U.S. DEPARTMENT OF COMMERCE
BUSINESS AND DEFENSE SERVICES ADMINISTRATION
OFFICE OF TECHNICAL SERVICES
WASHINGTON 25, D. C.
1954
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44
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45
cause conical motion of the blade at the vertical hinge B - B. With this
motion, the blade will turn about its center of percussion and the inertia of
blade pitch change motion will be increased. It may also be noted that the
inplane blade bending described on Page 36, having a reaction at the vertical
hinge B - B,will also create a pitch decreasing moment for the arrangement
shown in a coned position in Figure 2l.
Figure 23 illustrates the effect of moving the blade vertical hinge
B - B attachment point out of alignment with the loci of the blade aerodynamic
forces and chordwise center of gravity locations for the blade elements. As
explained more fully on Page 34, the external forces on a rotor blade include:
(1) the air forces acting at the aerodynamic center and include
the moment coefficient when it is other than zero.
(2) The normal component of the centrifugal forces along with
the inertia forces which act at the chordwise center of gravity.
Thus blade bending moments are created from the above forces and when
the vertical hinge B-B is located along the extension of these originating forces
distributed along the blade, illustrated at d-d, no torsional moments will be
introduced from this source. However, when the inboard blade connection at
the vertical hinge is offset from the loci d-d a distance d' as illustrated in
Figure 23, an additional moment is introduced into the control system which
is the product of the offset d'measured between the loci d-d and the blade
pitch axis c-c,and the shear force acting at the inboard end of the blade at
d-d. The value of the shear force is based on the force required to create
the blade root bending moment. In a hinged rotor, these moments are rel-
atively small.
In the case of a rigid or teetering rotor, the root bending moments are
appreciable and the loads which may be fed back into the control system due
to offsetting the pitch change axis from the loci of the lift and inertia forces
may be excessive.
TYPES OF CONTROL
Direct Control
Direct control is the name for any hub tilting arrangement whereby the
rotor hub is tilted to obtain the desired thrust vector tilt for control.
Collective Pitch Control
This signifies changing the pitch angles of all blades of a rotor a like
amount. To increase collective pitch by five degrees means to increase pitch
angle of all blades by five degrees. An increase in collective pitch increases
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*5
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47
thrust and torque. Both effects are used to produce control moments.
Cyclic Pitch Control
This means increasing the pitch of the blades on one side of the rotor
and decreasing it on the other side. This tilts the thrust vector in the
direction of the lowered angle.
Differential Pitch Control
Refers to pitch changes made on two different rotors, for example,
Differential Collective Pitch means that one rotor is increased in collective
pitch when the other is decreased. This is used in side by side rotor systems
to produce rolling moments,and in tandem helicopters to produce pitching
moments. In a similar manner, Differential Cyclic Pitch Control means
putting cyclic pitch into one rotor in one direction and into the other rotor in
the opposite direction. This results in tilting the thrust vectors of the two
rotors in opposite directions and is generally used to produce a yawing moment.
O NEUTRAL
Figure' 24 Cyclic Pitch Control
Figure 25 Differential Collective Pitch Control
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48
Figure 26 Differential Cyclic Pitch Control
TRANSMISSION OF BLADE FORCES TO CONTROL SYSTEM AND TO BODY
l. Fixed Hub (Autogiro Surface Controls or Equivalent Helicopter)
In this arrangement,all blade forces,including blade hinge reactions and
blade torque,are transferred to the body. Depending upon the nature of these
forces,the rotor will tend to stabilize or destabilize the machine. In the usual
case, vibratory forces also act on the machine. The magnitude of the vibratory
force is a function of the magnitude of the disturbing force and the relation-
ship between the frequency of the force and the natural frequency of the
structure being vibrated.
2. Direct Control (Entire hub is moved about trunnions for control purposes)
In this arrangement,all blade forces,including hinge reactions and
blade torque,are transferred into the hub trunnions and any unbalanced moments
about the hub trunnions are transferred into the control system. When the
blades are at rest against the droop stops, the inertia of the entire rotor is in
the control system.
It may be noted that when torque power is being transmitted to blades
through a hub (which is permitted angular displacement relative to the body )
moments resulting from the product of the torque moment and angular dis-
placement of the hub are transmitted into the hub. This greatly complicates
the use of a direct control system for transmission driven helicopters.
It may be noted that a direct control hub acted upon by small stabilizing
forces may cause the control stick to be moved in a desired direction. In this
manner, small stabilizing forces may be automatically amplified to produce
large corrective control forces.
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3. Feathering Control (Where hub is fixed, through bearings, to the body
and the blades are permitted controlled angular freedom.)
In this arrangement, hinge reaction loads are transferred thru the hub
and its bearings to the body,and all moments about the blade pitch change
axis are transmitted into the control system. If the forces acting on the
hinges are stable, thentlike the fixed hub, these forces will tend to stabilize
the machine. In a non-reversible control system,the blade torque forces would
also act on the body to stabilize or destabilize the machine according to the
blade torque forces,as in a fixed hub.
In the usual directly connected control system,any blade torque forces
are transferred to the control system as either steady forces or as vibratory
forces.
Varying Forces at Vertical Hinge
When a rotor blade oscillates about its vertical hinge, the loads applied
to the hinge vary.
Figure 27 illustrates the swinging of a blade about its vertical hinge
when a rotorcraft is in flight.
Figure 28 illustrates the increased centrifugal tension resulting from
increased rotational velocity when the blade is swinging forward. In this case,
the rotational.velocity of the blade is a function of rotor speed 6),, and rotation-
al velocity l**2 about the vertical hinge B.
The value of is 0Ri/R2»where 0 is in radiansjR^ is radius from
vertical hinge to center of percussion,and R2 is radius from ^ of
rotation to center of percussion. For a blade oscillating± l° about the
vertical hinge ( assuming that R\ - R2),the centrifugal tension is increased
(. 0l74+G)j)^ /fri p. For a rotor operating at a rotational speed of 10 rad/sec,
the increase in centrifugal force is (l0.0l74)2/l02 = 1.0035,or approximately
l/3 of 1%.
When the blade is swinging aft as illustrated in Figure 29, the centrifugal
force acting on the vertical hinge is reduced in accordance with the relation-
ship: Q !2/C 0+
When the blade is at either extremity of the oscillating motion- about the
vertical hinge,as illustrated in its lagged position in Figure 30,an additional
reaction is created at the vertical hinge B. The centrifugal tension "CT" may
be divided into inertia component "I" and (CTL,which lies parallel to the blade.
Again,the component of centrifugal tension which is parallel to the blade (CT)
may be divided into a component which extends toward the mean position of
the blade (CT)^ and a force "F" acting at the vertical hinge and normal to the
mean position of the blade.
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ROTATIONAL
VELOCITYOF TIP«tO, +U)2
CENTRIFUGAL FORCE
COMPONENTS-
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When the blade is in a forward position,the force "F" is reversed and
its magnitude is proportional to the angular displacement "-0-". Figure 3l
illustrates an elevation view of rotor blade "D" coned about horizontal hinge
"A". Blade "D" is bowed up at its center creating reaction "P" at horizontal
hinge "A". The combined centrifugal and lift force acting at the horizontal
hinge is directed along the * of the blade shank shown as (CT)'. When the
centrifugal force (CT)' is combined with the bending reaction "P" the resultant
force is shown at "R". When the resulting force is divided into a force (CT)r
acting normal to the axis of rotation and a lift force "L" acting parallel to the
axis of rotation, it may be readily seen that the lift produced on the machine
is substantially proportional to the blade coning angle and is not directly
associated with the air lift on the blade.
FIG 31 CENTRIFUGAL AND
BENDING FORCES COMBINED
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"" 52
Vm. ABSTRACTS OF PAPERS ON HELICOPTER
STABILITY AND CONTROL
(Note: The same symbols are used throughout the first eight ab-
stracts and are explained when occurring for the first time.)
- l -
Title: Theory of Helicopter Control in Hovering Flight
By: K. H. Hohenemser
Where Published: AMC T-2 Translation
Date of Publication: September l946
The attitude changes of a hovering helicopter after a sudden control
deflection are determined. A more elaborate analysis taking account of the
individual blade motion is in close agreement with the simplifying assumption
that the rotor cone as a whole follows the control plane with a certain time
lag. If *>| is the sudden deflection of the control plane (plane of zero cyclic
pitch), the tip path plane deflection ft follows according to:
&3 hinges and blade torsional elasticity may be considered by using a modified
blade inertia coefficient . The lag between control deflection and the follow
up of the tip path plane is very small in most cases.
Symbols a. 1-
0~ Blade inertia coefficient
R* Rotor Radius
Slope of blade lift coefficient versus angle of attack
f38 Air Density
Xg*Blade moment of inertia about hinge axis
^ «time expressed in number of revolutions
c s chord
Title: Longitudinal Stability of the Helicopter in Forward
Flight
By: K. H. Hohenemser
Where Published: AMC T-2 Translation
Date of Publication:
August l946
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5T
The dynamic longitudinal stability of the helicopter is determined under
the assumption of small disturbances and neglecting the rotor moment of
inertia. The frequency equation of the longitudinal motion is a quartic:
A\4+BX3+C>f+l>>4E=0
where the coefficients A"*0E are approximately determined by the dimension-
less moment of inertia of the aircraft
and by the four derivatives: •
Velocity Stability (
Static Stability (- M*fc)
Damping in Pitch (— Ruay^
Vertical Damping (-XV^)
A*Xy _ _
C - Muiy Z^i - M>/z>*
Approximate stability criterion:
O
Approximate dimensionless frequency of phugoid oscillation:
-Flu»y+m Mvx/Zvr
Definition of derivatives in terms of moment coefficientC^ and lift coefficient
Cl for constant rotor torque: . * -
Cf\ includes contributions by the rotor and by the tail surface. The rotor
part has been derived from rotor model tests in a wind tunnel, except forCnusy
which has been determined theoretically by assuming an inclination derivative
of the lift vector of ^ , -
As the above given stability criterion shows, a certain minimum static stability
(negative MVz ) is necessary to obtain a stable helicopter. The rotor con-
tributes^ forward flight) negative static stability, therefore a helicopter is
longitudinally unstable without a horizontal tail or without other stabilizing
devices.
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Ingln. Library
111521
TL
116
CONTENTS
Pages
I. Introduction l
n. Symbols 2-3
m. Stability of Aircraft 4-6
IV. Character of Stability 7-ll
V. Helicopter Stability l2 - 22
VI. Comparison of Helicopter and Airplane Stability 23 - 28
VH. Rotor Stability 29 - 5l
VIE. Abstracts of Papers on Helicopter Stability and 52 - 63
Control
IX. Index 64
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54
Symbols
g - M
Disturbance velocities in X (forward) and Z (downward)
direction
Disturbance (angular velocity) about Y (transverse)
axis, positive if nose up
Pitching moment, positive if tail heavy
Downward force
Derivatives
Symbols with a bar are expressed in a system of units
where the rotor radius R is the unit of length, the
aircraft massVf1 the unit of mass and*" % the unit
of time with A = disc area^ im = ^yf>AR
Rotor torque
Advance ratio
Angle of attack of rotor plane of zero cyclic pitch
Derivations
Angular rotor speed
Aircraft moment of inertia about Y (transverse) axis
- J -
q
Title:
Stability in Hovering of the Helicopter with Central
Rotor Location
By:
K. H. Hohenemser
Where Published:
AMC T-2 Translation
Date of Publication:
August l946
An emperical term derived from tests with an oscillating rotor model is
included in the stability equations of the hovering helicopter. Oscillation time
and damping coefficient of the long period oscillation of the helicopter are:
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f = 2 7r|/C^T 55
Where
Stability is improved by a large^Pfahich means large , large hinge offset
e and small aircraft moment of inertia Xy .
Symbols
^ Backward inclination of thrust vector with respect to rotor axis
^■1^ Derivatives
"B^jt Symbols with a bar are expressed in a system of units where the
ystem oi
rotor radiusR is the unit of length and"!^.1 is the unit of time.
Distance between aircraft e.g. and rotor center
w Aircraft weight
C Centrifugal force at blade root
G Off-set of horizontal hinge
b Number of blades
C|» Thrust coefficient
O Blade pitch angle
-4-
Title: Lateral Stability of the Helicopter in Steady Forward
Flight
By: K. H. Hohenemser
Where Published: AMC T-2 Translation
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56
Date of Publication:
August l946
The equations for the lateral helicopter stability are the same as for the
airplane lateral stability. The derivatives are partly taken from wind tunnel
model tests, partly estimated by approximate analytical processes.
Spiral stability is always present. Dynamic lateral stability is indicated
if a left rolling moment is produced by a side slip to the right and if the damp-
ing in roll, the damping in yaw and the directional stability are positive. For
centrally located rotor axes these conditions are fulfilled. In a synchropter
rotor, however, the static directional stability is negative if the right rotor
turns clockwise seen from above. The investigation indicates that for all types
of helicopters, except for the synchropter type just mentioned, lateral stability
may be expected, contrary to the longitudinal stability which is a serious
problem for most helicopter types.
Title:
By:
Where Published:
- 5 -
Dynamic Stability of Helicopter with Hinged Rotor
Blades
K. H. Hohenemser
NACA Technical Memo No. 907
Date of Publication: September l939
Theory of hovering stability is almost identical to that presented in the
more recent paper No. 3, except for the empirical term included in paper
No. 3. According to the earlier paper,hovering stability of the helicopter is
not possible. According to the later paper,hovering stability is obtainable by
a suitable choice of parameters.
Title:
By:
Where Published:
Date of Publication:
- 6 -
Contribution to the Problem of Helicopter Stability
K. H. Hohenemser
Fourth Annual Forum - American Helicopter Society
October l948
The main rotor control member (swash plate) is assumed to move under the
influence of gyroscopic and air forces produced by stabilizing devices like
Young's rotating bar, Hiller's servo rotor or a plain gyroscope connected to
the rotor control. The control equation is
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ST
where'Wx indicates the aerodynamic effect,C% the gyroscopic effect and C>s
the effect of viscous damping on the control system. Frequency and damping
coefficient of the long period oscillation of the helicopter are, if the gyro-
scopic effect of the rotor proper is neglected:
Stability is improved by a large gyroscopic and viscous damping of the control
system (Cf andCv ), by a large Mp (large hinge off-set) and small aircraft
moment of inertia ly. Without gyroscopic or viscous damping of the control
system,the helicopter is unstable (negative S ). By letting the rotor control
system move freely under the influence of gyroscopic or viscous forces, any
desired degree of stability is obtainable.
Symbols
Deflection of the central plane (plane of zero cyclic pitch)
Wit sr*7 Derivative
4> Attitude angle of helicopter, positive if nose up
Gyroscopic constant
Cv Viscous constant
Title of Paper
By:
Where Published:
Date of Publication:
Automatic Stabilization of Helicopters
G. J. Sissingh
Journal of Helicopter Association of Great Britain,
Volume 2, No. 3
October l948
The stability characteristics of the hovering helicopter are discussed
and the results of a stability analysis are presented. The Sikorsky R-4B
helicopter is taken as an example. Without stabilizing device, the helicopter
is unstable in hovering and the amplitude of the helicopter oscillation is
doubled in five seconds. With a certain type of autopilot providing both pro-
portional and rate control, the helicopter is stable and the amplitude is halved
in three seconds. With the Bell stabilizer, the amplitude is halved in 37
seconds, and with the double bar stabilizer, proposed by the author, the ampli-
tude is halved in eight seconds if the relative damping of the two stabilizing
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58
bars is adjusted for optimum efficiency.
- 8 -
Title of Paper:
By:
Where Published:
Contribution to the Dynamic Stability of a Rotary Wing
Aircraft with Articulated Blades, Part I.
G. J. Sissingh
AMC T-2 Translation
Date of Publication: August l946
Analytical expressions for the aerodynamic and mass forces acting on
the blades are developed for the following condition:
The helicopter moves in the forward, sideward and downward direction
with given translational speeds and given translational accelerations. At the
same time, the helicopter rotates about the three coordinate axes with given
rotational speeds and given rotational accelerations. The blades are assumed
to flap with one per rev. only and the induced flow is assumed to be constant
over the rotor disc. The analytical expressions for the forces, hinge moment
and flapping oscillations are very elaborate and no attempt is made to simplify
these expressions or to compare the exact values with simple approximations
used in previous papers about dynamic stability of the helicopter.
Title of Paper:
By:
Where Published:
- 9 -
Contribution to the Problem of the Dynamic Stability
of a Rotary Wing Aircraft with Articulated Blades.
Part II,
G. J. Sissingh
AMC T-2 Translation
Date of Publication: December l946
The analysis of the hovering stability of the helicopter with hinged rotor
blades, as presented in paper No. 5 is refined by taking into account several
terms which have been neglected in the earlier analysis. While the frequency
of the long period helicopter oscillation remains the same, the refined analysis
results in a reduced amplification of this oscillation.
- l0 -
Title:
A Method for Improving the Inherent Stability and
Control Characteristics of Helicopters
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59
By. R. H. Miller
Summary
Hie problem of helicopter control and stability is examined with a view
to establishing whether satisfactory inherent stability and control characteris-
tics may be achieved without major redesign modifications and without re-
course to automatic control devices. It is shown.that the possibility does
exist of improving both the damping and the static stability of a hovering
helicopter by a relatively minor modification of the blade mass and aerodynamic
chafacteristicSjtogether with the use of springs and dampers in the control
system. This should result in considerably improved blind flying characteris-
tics and a reduction in excessive control sensitivity without sacrificing man-
euverability. The control characteristic of such an inherently stabilized
helicopter is evaluated by means of the transient response characteristics
to abrupt control manipulation.
Contents
Section I
Discussion
I
Section II -
Theoretical Development
ll
Equations of Motion
ll
Transient Response
l7
Design Considerations
2l
Bell stabilizer Bar
24
Hiller Control Rotor
26
Kaman Servo Control
27
References
29
-ll -
Title: Some Aspects of the Helicopter Stability and Control
Problem
By: R. H. Miller
Summary
This paper contains a general discussion of some of the major factors
influencing the handling characteristics of helicopters and in particular
demonstrates the importance of increasing the damping in pitch of the heli-
copter.
It is shown that this may be done fairly simply by offsetting blade chord-
wise center of gravity from feathering axis and by providing suitable restraint
about the feathering axis. The dynamics of the blade when such a modification
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60
is incorporated are considered in some detail, and it is shown that the
spanwise distribution of such an unbalance is of considerable importance.
In particular, the chordwise C. G. should be offset in such a way that the
static moment about the feathering axis remains zero. The product of inertia
about the flapping and feathering axis determines the degree of stabilization.
- l2 -
Title: Helicopter Control and Stability in Hovering Flight
By: R. H. Miller
Summary
l. The complete equations of motion of a helicopter in hovering flight have
been developed first with no simplifying assumptions, except that the effects
of blade flexibility have been neglected. It has been shown that these equations
may be considerably simplified in certain cases by:-
a) Neglecting the effects of accelerations of the tip path plane ,since
the response of the blades to cyclical pitch changes is rapid .
b) Neglecting some of the effects of the offset between flapping hinge
and center of rotation when this offset is small.
c) Neglecting, in the case of the single rotor helicopter, the coupling
between pitch and roll.
n. The simplified equations have then been used to analyse the stability
and control characteristics of :-
l) A single rotor helicopter
2) A helicopter with coaxial rotors or side by side in pitch, or
tandem in roll, all of which represent similar cases.
3) A dual rotor helicopter with a large offset of the flapping hinge
4) A dual rotor helicopter with rigid non-flapping-blades
III. The handling characteristics of the various types of helicopters have
been evaluated by obtaining the response in pitch of the helicopters to abrupt-
control displacements and these responses compared with that of a typical
airplane. The factors influencing the handling characteristics, such as "static"
stability, damping, etc., have been discussed with a view to establishing the
underlying causes for certain undesirable features found in the handling
characteristics of helicopters.
IV. A conventional automatic pilot has then been included in the analysis
and it is shown that, with a correct choice of auto pilot parameters, the control
characteristics may be greatly improved. The analysis has been made both
by obtaining directly the response to abrupt control manipulation, as in the case
of the unstabilized helicopter, and also by the frequency response method.
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V. Finally, it is shown that the possibility exists of stabilizing the helrcopg^J
-ter without using an autopilot by modifying the blade chordwise mass distribu-
tion and providing suitable restraint about the feathering hinge.
- l3 -
Title of Paper:
By:
Where Published:
Helicopter Stability with Young's Lifting Rotor
Bartram Kelley
S.A.E. Journal
Date of Publication: December l945
The paper starts with a description and history of instability, and
refers to N. A. C. A. Technical Memo. 907 by K. Hohenemser, where helicop-
ter stability derivatives were first published. The experimental work of
Arthur M. Young is then described, leading up to a description and discussion
of the stabilizer bar. Appendix I contains the general analysis showing two
natural frequencies and the necessary and sufficient condition that both should
be non-divergent. Appendix II shows that the condition is satisfied in an
actual case.
Formulae without the context would not be useful.
- l4 -
Title of Paper
By:
Where Published:
Stability and Control Characteristics of a Simplified
Helicopter
Charles M. Seibel
Preprint, SAE National Personal Aircraft Meeting,
Wichita, Kansas, May, l947.
Date of Publication: May l947
The paper presents stability equations and a discussion of the stabi.::,
of a helicopter which obtains its "cyclic" control by means of changing the
center of gravity in flight. The Seibel S-3 Helicopter was built and flown with
this system. The design was considered satisfactory for a small helicoptpr
800 pound gross weight or less.
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UNIVERSITY OF MICHIGAN
3 9015 01391 1832
* 62
- l5 -
Title of Paper The Response of Helicopters with Articulated
Rotors to Cyclic Blade Pitch Control
By: A. F. Donovan and M. Goland
Where Published: Journal of Aeronautical Science
Date of Publication: October, l944
An analytical study of helicopter controllability is made. Results
are obtained which permit calculation of the response of hovering rotary
wing aircraft to arbitrary control displacements. The type of helicopter
specifically dealt with is equipped with a single lifting rotor, whose blades
are hinged so as to possess freedom in flapping. Control is effected by the
conventional means of cyclically varying the pitch of the rotor blades.
Seven degrees of freedom are considered as follows:
(a), (b) Longitudinal and transverse motion of the center of gravity
of the fuselage
(c), (d) Rolling and pitching about the center of gravity of the fuselage
(e) Tilt of the rotor cone in a pitching sense relative to the
fuselage
(f) Tilt of the cone in a rolling sense relative to the fuselage
(g) Coning of the rotor blades
The response of a typical helicopter to an imposed abrupt cyclic
pitch is traced through the first several seconds of motion, starting from
the hovering state. The results are shown in graphic form. Because of
the procedure used in linearizing the equations, the results given are valid
only for the first two seconds approximately after application of the controls.
In the linearization it was assumed that all angles, angular rates and trans-
lational velocities involved in the equations of motion were small quantities
and that consequently products of these quantities could be neglected. This
assumption is analogous to that customarily made in studying airplane
controllability where it is usually assumed that the airplane's speed does
not change during the maneuver. In the airplane case,this assumption leads
un to what is termed the short period stability and the short period response,
chz Accordingly, this approximation presents what might be termed the short
period response of the helicopter. The helicopter has neutral short period
stability, a change in velocity being necessary to produce any stabilizing
moments.
In the paper,the stability characteristics were presented using the
system of equation obtained. In view of the linearizing assumptions em-
ployed,these stability characteristics have no meaning physically. Specific-
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