..
'
NONLINEAR
DYNAMICS
OF
A
HELICOPTER
MODEL
IN
GROUNTJ
RESONANCE
D.M.Tang *
and
E.H.Dow
ell
**
Depa
rt
ment
of
Mechanical
Engineering
and
Material
Sci
ence
Duke
University,
Du
rham,
NC
27706
Abstract
An
approximate
theoretical
method
is
presented
which
determ
in
es
the
limit
cycle
behavior
of
a
hel
icop
ter
model
which
has
one
or
two
nonlinear
dampers.
The
relatio
ns
hip
duri
ng
unstable
ground
resonance
oscillatio
ns
be
tween
lagging
motion
of
the
blades
an
d
fus
elage
motion
is
discussed.
An
experiment
has
been
carried
out
on
using
a
helicopter
scale
model.
The
experimenta
l
results
agree
with
those
of
the
theoret
ical
analysis.
H
I
J s
x,J
sz
t
(A-mo
del)
nO
Po
fa
(ll-model)
n
2
--
n
a
P
2
--P
a
fb
Notation
geomet
rical
length
of
the
mode,
see
Fig.l
mass moment
of
inertia
of
blade
relative
to
drag
hinge
mass moment
of
inertia
of
fuse-
lage
about
X
and
Z
axis
through
A
ma
ss
moment
of
inertia
of
shaft
about
X
and
Z
axis
through
A
coupling
spring
coefficient
be-
tween
fuselage
and
shaft
distance
from
the
axis
of
rota-
tion
center
to
drag
hinge
center
hydrauli
c damping moment
coef-
ficient
nondimen
sional
hydraulic
damping
coefficient
s
dry
frictio
n moment
coefficient
nondimension
al
dr
y
friction
damping
coefficients
mass
of
the
blade
number
of
blade
s
damping
coefficient
of
blade
radius
of
the
rotor
blade
static
mass
moment
relative
to
drag
hinge
time
fuselage
damping
coeffi
cient
natura
l
frequency
of
fuselage
hinged
mass
ratio
2
n
S2H2
/2I\Jsz+Jfz+
n
mbH
)
damping
coefficients
na
tural
frequencies
hinged
mass
ratio
nS
2
H2/2I
(Jsz+nmbH
2)
*
Vi
~
itin<J
Schola
r,
I
anjing
Aeronautical
Institute
, Ch
in
a .
**
Dean ,
School
of
En
g
in
ee
rin
g,
D
u
~e
Uni v e r
si
ty
, !Jur .
Ll
T.
1 ,
ll
C.
(e-model)
nx,nz
Px,P
z
fuselage
damping
coefficients
natural
frequencies
of
fuselage
hi~ge
d
mass
ratio
t c
nS H2/2I(Jo+nmbH2)
8
fx
,8
fz
angular
motion
coordin
at
e
of
fuselage
about
X
and
Z\
axis
an
gular
motion
coor
d '
r.
ate
of
sh
aft
about
X
and
Z a x
ls
an
gu
lar
amplitu
de
of
fuselage
ab
out
X
and
Z
axis
8
fxO
,SfzO
S
sxO'S
s
zO
an
gular
amplitude
of
shaft
ab
out
X
and
Z
axis
45
an
gular
orientation
of
different
ro
tor
blades
an
gular
deflect
ion
of
k-th
blade
na
tural
frequency
of
rotating
blade
relative
to
drag
hinge
di
stance
of
center
of
gravity
of
the
blade
from
drag
hinge
coord
inate
of
the
c
enter
of
gra-
vity
of
the
bla
de
system
in
fixed
reference
frame
nondime
nsional
bla
de
parameter
JLv . hS
/I
r
otor
speed
Introduction
H
elicopter
ground
resonance
is
a
se
lf-excited
vibration
phenomenon
rather
t '
lan
a
forced
vibration.
The
safety
standards
for
helicopter
strength
require
that
a
helicopter
with
an
articulated
roto
r
must
have
enough
stable
margin
whe n
it
is
in
contact
with
the
ground,
espe
cially,
in
case
of
rough
landing.
Such
stan
dards
are
based
on
linear
th
eory,
but
i n
fact
the
most
common
design
of
the
l
anding
gear
has
nonlinear
damping
c
haracteristics.
How
much
of
an
effect
on
g
round
resonance
will
be
induced?
Does
the
n
onlinear
damping
charact
eristi~
tend
to
i
ncrease
stability
or
not?
These
questions
are
waiting
for
investigators
to
solve.
The
present
au
t
hors
[1,2)
and
Tongue (3)
have
en
gaged
in
such
research.
The
purpose
of
this
paper
is
to
further
study
such
p
roblems.
The
nonlinear
dampi ng
in
the
landing
gear
of
a
helicopter
brings
into
action
both
roll
and
pitch
motion
of
the
fuselag~.
In
a
series
of
pUblications
[3-l3J
about
he
licopter
ground
resonance
instability,
this
aspe
ct
was
not
taken
into
account
by
many
authors.
Also
in
the
past,
many
investi9Ators
ha
ve
used
Coleman'
classical
multi-blade
coor
oi
nate
theory
to
simplify
. I
..
the
st
udy
of
the
rotor
blade
dynamics.
Most
~revious
authors
have
considered
linear
dynamic
models.
In
the
present
paper,
an
approximate
method
has
been
used
to
calaulate
the
limit
cycle
behavior
of
a
given
helicopter
model
which
has
a
nonlinearity
in
the
landing
gear
such
as
a
hydraulic
resistance
which
is
approximately
proport
ional
to
the
square
of
velocity
or
dry
friction
resistance.
An
experimental
investigation
has
been
carried
out
as
well.
The
test
apparatus
is
an
improved
version
of
Bielawa's
rotor
model
[4].
The
experimental
results
are
in
good
agreement
with
those
of
the
theoretical
analysis.
In
the
present
paper,
we
study
the
effect
of
combined
roll
and
pitch
fuselage
motions,
and
consider
the
relationship
between
the
lagging
motion
of
the
blades
and
the
fuselage
motion.
Also
discussed
are
the
differences
in
the
unstable
boundary
between
the
linear
and
the
nonlinear
model.
We
believe
that
these
results
will
be
helpful
in
further
understanding
the
physical
essence
of
helicopte
r
ground
resonance
instability,
and
in
protecting
against
such
an
i
nsta
bil
ity
•
Approximate
method
of
DOnline!!
.!.
analysis
The
analysis
is
based
on a
helicopter
model
with
an
articulated
rotor.
A
sketch
of
this
modd
is
shown
in
Fig.l.
We
will
assume
that
tlle
blades
in
the
plane
of
rotation
are
rigid
but
articulated;
the
spring
and
damping
coefficients
of
the
fuselage
and
the
shaft
are
isotropic,
but
their
mass
moments
of
inertia
are
anisotr
opic.
Nonlinear
damping
includes
hydrau
lic
resistance
which
is
proportional
to
ve
locity
squared
and
also
dry
friction
resi~t~nce;
these
can,be
represented
by
Mh 1919
and
Md
signe,
respectively.
The
equat10ns
of
motion
for
the
model
are
six
nonlinear
differential
equations
with
unknown
functions
1 ,
s,
efx'
Sfz,
Sax
and
8s
~'
These
are
as
follows
(see
Ref.[l]):
. • 2
~
••
"'\+2
n
b'l+c.>s~-2Q()
r.
bt.
)=nSH9
sz
/2I
~
+2nb~
+W:~+20(~+nb'l)
=-nSH9
sx/
2I
., , 2 2 '
esz+2
n2
9S?
+(P2+P4)9sz-2
n
49fz
_p
2
e =
sHiVJ
4
fz
sz
••
• 2 2 '
8s~+2
n
19sx+(PI+P3)8sx-2n38fX
(1)
-P-9
=-SH!/ J
3
fx
sx
.. , 2 2 ' 2
8fx
+2ns9fx+(PS+P7)9fx-2n79sx-P78sx
+mhxef~efxl+mdxsi
g
n9fx=0
..
)..
2 2
)..
2
efz
+2n6~fz+(P
6+P
8)efz-2n8~sz-P88sz
+mhZefZ~fzl
+mdZsi9n~fz=0
In
order
to
simplify
the
problem,
we
will
discuss
two
special
cases:
1.
A model
with
a
single
nonlinear
damper.
We
suppose
that
the
motion
in
the
X
direction
is
constrained,
i.e,
es~
-Sfx
-0,
and
the
subscript
z
in
S
SZ
and
Ofz
is
thus
eliminated,
and
define
that:
Ss=m
hz
9
s
' 9
f
=m
hz
9
f
~
=
~mhZSH
/ J
,
~
=
~mh
SH
/ J
sz
z
xz
'i'ill:::;'
(1)
be
co
r.1
Gs
:
;.:
'2'
..
1'1.
+2n
b
1[
+Wsif-22(f+nb~)
=~9s
46
~
+2n
b
t
+uJ~~+2
n(~+
nb1f)
=0
ij
+2n2~
+ (P
2
2
+P
4
2
HJ
-2n
~
_p2
~
=
~
5 5 5 4 f 4 f '(
(2)
.. , 2
2~'"
).
9f+2n6~f+(P6+P8)9f+~f9(+.si9n8f
:..
2
-2n
8
9
-P8~s=0
where
For
the
nonl
inear
terms
Of I
8t
I
and
sign9t
,
we
apply
an
approximate
procedure
which
is
called
quasi-linearization.
Let:
it
'"
,a
f
I
cosAt
Representing
the
above
nonlinear
terms
by
a
Fourier
series
and
retaining
the
first
harmonic
produces(see
Ref.
[3],[14])
~fl~fl=-
8/3n*A
2
19
f
d
2
sin~t
sign
Q
f=-4
/
]I'
*sin"t
Let:
7i
=7)
e
iAt
o
to'
- i
>.
t
;:)
=SOe
~
....
""9
i>l.t
"5
sOe
Once
again
let:
,. ,.
i"t
9
f
-9
fO
e
Subst
i t
ute
these
and
require
coeffi
cients
to
solutions.
The
equations
are:
real
part:
expressions
into
Eqs.
(2),
the
determinant
of
be
zero
for
nontrivial
final
characteristic
aQ
fO'"
8
-b~;O
>.7
-c~fO"
6
+
(d9
f
2
0
+,b)"
5+
e
\;
A 4
2 3 2
fO
-(f~fO+,d)A
-9~fOA
+'fA+h~fO=O
(3)
i
h1
ugi lla r y
?a
r t :
9
2
,, 8
~
7
~2
6...
5
a
l
fo
+b
l
fO"
-(cI~fO+bOa)A
-d
1
9
fO
A
(
-2
) 4
...
3 2 2
e
1
9
fO
+,c
l
"
+f
1
9
fO
"
-(919fO·,e1)"
-h1~fO"-
91=0 (4)
...
.
..
"
The
coefficients
a,b
, c
•••
etc.
of
the
above
Egs.
are
given
in
hppendix
B
of
Ref.
[1).
-
It'
-
For
the
case
of
Ksf
~
(X),
9~
~
\jf
'"
9,
i.e
th:?
shaft
is
connected
together
wit
h
fuselage
,
the
differential
equations
become a
sim
ple
r
nonlinear
mathematica
l model
for
an
alysing
helicopter
gr
ou,l d
resonance
•
(5)
wher
e:
nb=~/PO'
Ws
=U;/P
o
,
nO
=
nO/P
O
'
~=Q/PO
For
this
case,
the
final
chura
ct
e
risti
c
equation
is
the
sa
me
as
in
Ref[3).
Eg.
(32.
or
(4)
is
e
ighth
order
in
".
Also
,
the
9fO
a
ngul
a r
am
plit(;oe
novI
appe
ars
explici
tly
.
This
means
that
the
l i m
it
cycle
amplitude
of
the
nonlinear
syst
em
is
directl
y
relate
d
to
",
the
fre~uency
of
oscill
ation
,
and
both
must
be
solved
for
simul
t a n
eo
u
sly
.
The
Broyden
method
[15J was
used
to
solve
the
above
si
ro
ul
taneous
equi\tions.
Let
functions
fl(gfo,A)
and
f2(9"fO,A)
be
equal
to
the
left
han
d
sides
of
Eg.(3)
and
Eq.
(4),
resp
ecti
v
ely
.
fl
(lifO')
' ) .. 0
f2
(9
fC
,A)
= 0
2.
A model
with
damper s .
He
sup
pose:
two
Ksf
=
OO
,
9sx·9fx
=
9x
'
9sz
=
9fz
=
9z
an
d
define:
nonlinear
J
O
=(J
x
+J
z
)/2,
i
x
=9
x
'
~z =9
z'
~
=
~SH/JO
l=rSH
/J
o
'
Jx=Jfx+Jsx
'
Jz=Jfz+Jsz
'
~=Jo
/Jx
E:3
=
J
o
/J
z
The
general
equations
of
motion
for
th
e
mo
del
are
given
by
Eqa
.
(6).
-
--
_________
"T
,
Eqs.
(6)
is
a
set
of
nonline
a r
differ
tial
equations
for
deter
m
ining
four
un
y.no\'l
n
func
tions
't ,
c,
ex
and
9
z
.
Usin
g
the
retaini
ng
we
take
method
only
the
of
har
m
onic
balance
and
funda
me
nt
al
ha rm
onic,
(7)
Her e
'/Z,
'I" ,
'I,
are
res
pec
tiv
el
y
th
e
phase
diff
ere
nc
e
of
q ,
1i
and
$'
with
r
espect
to
e
z x
tlo
\
~
He
consider
the
t e r m
sign~
anu
~
I
~
I •
As . was men
tione
d p
reviou
sly
[2),
\'Ie
can
obt
a
in
the
fol
lO\-
l
in
g
app
ro
ximate
r
esu
lt:
For",
~zO-::'
O
,:..
- 4 - -
singe
=-r-
(e
cos.\t-e
sin"t
)
z
JI~ZO
ZS
zc
,.:.
8
A2
,e
19
.-
~
e
(9
sinAt-9
cosAt)
Z z
3JI
zO
ZC
zs
In
s
erting
Eq
. (
7)
into
Eq
.
(6)
~I
e
obtain
a
s
ys
te
lll
of
ei
gh t
siu
-,
ul
t a n
eous
no
nl
in
car
al
g
ebraic
eq
uatio
ns
for
det
e r m
ini
ng t '
,e
c
uantitie
s ,
'i
e ,
~
s
'?
c
'
~
s
'
9
x
O' 9
zc
,
6
zs
and
A •
By
eli
m
in
a
tion
of
the
first
four
unknowns ,
the
ec;uat
ion
s
can
be
rewritten
morE:
co
m
pactly
as
a m
atrix
eCj
u
at
ion
wit
l
fou
r
unknow ns ,
QXO;i3
zc
,e
zs
'
".
Let:
(8)
where
CV1·
[-0.-:
=;
-:
c
-b
(p
2
/ A
2
_
1)9
/c
3
+2n
9 /
AE
3+
z
zc
Z
zs
- - -
2-
8mhz9z09
zs/3
1rE
3+4mdz
9zs/nA
9
z 0
t
3
lu)=
(p
2
/
...
2
_1)6
/
£3-
2n 9 /
At
3
-
z
zs
z
ze
- - -
2-
8m
h
9 09
/3~E3-4md
9 /
KA
8
OE3
z z
ze
z
ze
z
(1_P~/A2)
6
xo
/ C
2
-
;.
2-2
2n
x
8
xo
"E2+4m
dx,1r
~"
+8mhx8xo
/3
1rE2
a=(w!1,,2_
l
)/f
l
b=2l),ltl"
2
c=2Q
l),/
f
l'~
d=20/f
l
"
In
order
IF) must
Broyden's
equation.
to
solve
Sq.(8),
the
elements
of
be
zero.
Here
we
also
apply
method
to
solve
the
above
Experimental
Inyestigation
1.
Experimental
equipment
An
overall
view
of
the
test
model
and
recording
equipment
used
in
this
study
is
shown
in
Figs.2
and
3.
The
details
of
the
various
components
of
the
model
appear
in
Ref.
[1)
and
[2).
The A-model
has
only
one
degree
of
freedom
for
fuse
l
age
motion.
The
B-model
has
two
dearees
of freedom
for
shaft
and
fuselage
motion.
The C-model
has
two
degrees
of
freedom
for
the
[011
and
the
pitch
motion
of
the
fuselage.
The
characteristic
curves
of
the
hydraulic
and
dry
friction
damping
are
shown
in
Fig.4(a)
and
(b).
The method
of
determining
these
parameters
is
described
in
Ref.[l).
2.
Measurement
system
The X
and
Z\
angular
displacements
and
velocities
of
the
fuselage
and
the
shaft
are
obtained
by
RVDT,
R30D,
velocity
transducers
located
near
the
dry
friction
dampers,
and
a
LVDT
located
near
the
gimbal
support
assembly.
The
output
voltage
of
the
transformers
and
velocity
tra
nsducers
is
proportional
to
angular
disp
lacement
and
velocity
of
the
roll
and
the
pitch
of
the
fuselage
and
the
shaft.
The
output
from
the
transformers
are
amplified
and
recorded
on a
multiple
channel
tape
recorder,
HP3968A.
In
order
to
obtain
a one
per
revolution
signal
and
the
lagging
motion
of
the
individual
blades
during
rotation
of
the
blades,
we
mount a
l3-ch
brush
and
slip
rin
g
assembly
on
the
shaft
between
the
electric
motor
and hub,
see
Fig.5.
One
of
these
slip
rings
is
not
a
close
d
ring.
It
gives
an
impulsive
signal
once
per
revolution,
so
a
very
accurate
measurement
of
rotor
speed
can
be
provided.
Three
angula
r
transducers,
R30A,
are
mounted on
the
drag
hinges.
The
signals
for
lagging
moti
on
of
each
blade
were
amplified
~nd
reco
rded
through
brush
and
slip
rlng
assemblies.
The 8
channel
signals
were
recorded
simultaneously
on a
tape
recorder,
and
further
analysed
by
a
Frequency
Spectral
Analyzer,
HP3582A.
Finally,
the
phase
plane
plots
and
frequency
spectral
plots
were
plotted
by
a
X-Y
recorder.
48
The
stability
te
st
needs
an
initial
disturbance,
so
two
electro-magnets
were
mounted on
the
test
equipment.
Those
are
used
to
generate
a
disturbance
in
the
direction
of
roll
or
pitch
of
the
fuselage.
The
device
for
angular
amplitude
calibration
i s shown
in
Fig.6.
An
electric
motor
with
stepless
variable
speed
provides
a
vibrat1~n
source,
and
drives
a
cam
which
can
generate
a
sine
wave,
such
that
t
he
relationship
betweer
output
voltage
from
the
transducers
and
the
actual
angle
can
be
determined.
3.
Frequency
response
test
The
purpose
of
the
frequency
response
test
is
to
determ
i ne
the
natural
frequencies
of
the
system,
and
to
provide
an
independent
cheCk on
the
system
parameters.
Before
the
test,
the
drag
hinges
are
fixed,
and
a
known
block
matJs
(3
oz)
is
put
on a
de
ag
hinge.
A
centrifugal
force
caused
by
the
bias
mass
will
excite
the
system,
and
the
frequency
response
versus
rotor
eryeed
can
be
found.
4.
Stability
test
Testing
the
model f or
aelf-excited
instability
regions
was
accomplished
by
slowly
var
ying
the
rotor
speed
until
instability
was
observed
in
response
to
an
initial
disturba
nce.
Results
and
Discussion
Fig.7
shows
the
limit
cycle
behavior
when
only
the
hydraulic
nonlinearity
is
present
in
the
A-model . The
figure
shows
that
the
maximum
limit
cycle
amplitude
occurs
near
the
critical
rotor
speed
of
the
linear
system,
and
that
the
dominant
response
occurs
in
the
region,
(1.7-2.0)
*Q.
Furthermore,
there
is
an
abrupt
change
in
response
from
(1.9-1.95)*2
in
Fig.7.
This
means
that
the
amplitude
response
is
very
sensitive
to
small
changes
in
r
otor
speed
in
this
range
of
rotor
speed.
Fi
g.8
shows
th
e
limit
cycle
behavior
with
both
hydraulic
and
dry
friction
nonlinearities.
The c
urve
has
a
shape
similar
to
that
of
Fig.7,
but
the
response
amplitude
is
smaller
than
that
in
Fig.7.
Physically
the
action
of
dry
friction
is
to
increase
the
equivalent
viscous
damping
in
the
landing
gear.
The
instability
region
beco
m ~s
narrower,
and
the
response
amplitude
reduces.
Fig.9
shows
the
limit
cycle
behavior
of
the
B-model.
There
are
two
instability
regions
in
the
figure.
The
first
one
is
dominated
by
the
shaft
motion.
Because
the
effect
of
non.linear
damping
of
the
fuselage
on
the
shaft
is
weak,
the
shaft
has
a
large
limit
cycle
amplitude
(for
> •
\
•
aome
range
of
Q beyond
the
measurable
angle).
In
contrast
with
the
first
region,
in
the
second
instability
region
there
is
not
a
large
limit
cycle
amplitude
because
of
the
direct
effects
of
nonlinear
damping
of
the
fuselage
on
the
fuselage
motion.
As
far
as
the
C-model
is
concerned,
Fig.10
shows
the
results
of
frequency
spec~ral
analysis
of
the
fuselage
motion
at
a r
otor
speed
of
l.2(HZ).
The
mode
shape
0f
the
unstable
motion
is
dominated
by
rJll
of
the
fuselage,
and
the
os
::
illation
frequency
is
equal
to
the
r~~l
natural
frequenc1
of
the
~u~elage.
S
L
nul
t~neousl~,
there
is
a
small
llm1t
cyclr.
ampl1tude
1n
the
pitch
motion~
its
oscillation
frequen
cy
is
not
equal
to
the
pitch
natural
frequency,
but
is
equal
to
the
roll
natural
frequency.
The
phase
angle
between
roll
and
pitch
is
equal
to
Sl~
corresponding
to
a
frequ
ency
of
O.84(HZ).
The
phase
curve
in
the
vicinity
of
O.84(HZ)
is
almost
parallel
to
the
frequenc
y
axis,
and
the
phase
angle
is
quite
ineensitive
to
frequency.
F
ig.11-12
show
theoretical
and
experimental
results
of
the
limit
cycle
amplitu
de
and
phase
angle
behavior.
A
reasona
ble
experimental
verification
of
the
theory
is
achieved.
Because
of
the
exper
imental
complexity
and
calculation
approximatio
n,
the
quantitative
accuracy
of
the
verification
is
not
very
high.
However,
the
general
features
of
the
experime
ntal
results
are
well
predicted
by
the
theory.
For
reference,
in
Fig.13
the
stabilit
y
boundary
results
from
linear
theory
are
shown
and
compa~ed
to
experimenta
l
data
obtained
w1th
the
nonl
inear
dampers removed. The
agreement
bet
ween
theory
and
experime
nt
in
this
case
is
good.
Fi
g.14
shows a
phase
plane
p
lot
which
is
dire
ctly
plotted
by
a
X-Y
recorder.
The
measurement
point
of
displacement
and
ve
locity
is
located
at
the
same
position.
Th
eir
phase
difference
approximates
to
9 if,
the
pattern
displays
an
ellipse
approxima
tely,
and
the
higher
order
harmonic
components
are
included
in
the
ellip
se
curve
. From
this
pattern,
we
also
can
see
the
transient
process
leading
to
the
steady
state
limit
cycle
amplitude.
Different
disturbances
have
different
un
stable
processes.
In
this
case
the
roll
os
cillation
of
the
fuselage
tends
to
a
li
mit
after
about
7
pseudo
cycles,
and
the
tra
nsient
process
includes
more
higher
order
harmonic
components.
Fig.lS
shows
the
results
of a
frequency
spectra
l
analysis
of
the
lagging
motion
of
th
e
individual
blades
at
a
rotor
speed
of
2.
04(HZ).
It
is
found
that
in
the
in
stability
region
the
individua
l
blades
have
an
identical
oscillation
frequency,
p,
which
is
equal
to
the
difference
between
the
rotor
speed
and
the
oscillation
frequency,
",
of
the
fuselage.
This
result
is
in
agreement
with
49
conclusion
which
is
derived
by
claasical
linear
theory.
However,
the
individual
b
lades
have
different
amplitudes
and
phase
angles
due
to
the
actual
small
different
blade
damping
characteristics.
The
equation
of
motion
of
the
k-th
blade
can
be
written
as
follows:
f.
+2n;'
+w
2
f =
~
(x
sin~k
-zcos
;k)
k
~k
5 k L
v
•
h
The
periodic
term
can
be removed by
co-ordinates.
in
the
above
equation
using
the
COleman
n
'7
=
L:
3"k
s
in
}v,k
k=1
~
=
t
~cos~
kozl k
If
it
is
represented
oscillation
defined
as:
expected
that
~
and
t
are
well
by a
fundamental
harmonic
with
f
req
uency,
A,
then
t k
may
be
j k =
~c
o
s
(Pt+211
k i n
i.e
the
follOWing
conditions
satisfied:
must
be
*p=Q-A
*
the
amplitudes
of
blades
are
identica
l.
*
the
phase
angl e
of
the
equal
to
2k"/n.
In
the
present
experiment,
condition
is
satisfied.
violations
of
the
latter
are
the
main
reasons
for
the
theoretical
and
experimental
the
individual
k-th
blade
is
only
the
first
The
modest
two
conditions
error
between
resul
ts.
In
order
to
provide
a
better
phYSical
understanding
of
the
unstable
motion,
we
discuss
the
relationship
between
motion
of
the
hub
center
and
the
center
of
gravity
(C.G)
of
the
blades.
In
the
fixed
coordinate
system
XOZ,
the
motion
of
the
common
center
of
gravity
of
the
blade
system
is
expressed
by
X
=X-
Pc
t..
fksin'f,k
c
~=I
Z
=Z+
Pc
trk
c osl!,k
c
nk=l
X=
ate
c
osA
. t
i=1
xO
i 1
Z=
ate
O
.
cos(.A
.
t-~
.
)
i-I
Z 1 1
Zl
Fig.l~
shows
the
time
history
of
~
and
S •
The
test
curves
include
components
with
frequent;}'
(2S2-
....
),
and
A,
in
addition,
a
cos
....
t
component.
This
also
can
be
clearly
found
from
Fig.17.
The
motion
locus
of
the
C.G
of
the
blades
is
not
exactly
an
ellipse,
but
a somewhat more
complicated
curve.
The
starting
point
at
taO
does
not
~
.