C.P. No. 476
C.P. No. 476
(2 I ,607)
A.R.C. Technical Report
MINISTRY
OF
AERONAUTICAL
RESEARCH
CURRENT
AVIATION
COUNCIL
PAPERS
The Longitudinal
Frequency
to Elevator of an Aircraft
Short Period Frequency
Response
over the
Range b
bY
D. M. Rid/and
LONDON:
HER MAJESTY’S STATIONERY
I960
PRICE 7s. 6d.‘NET
OFFICE
C.P. wo. 476
U,D,C, No. 533.6.013.~+12:533.6.0~3.417
Technical
Note No. Aero.
September,
ROYAL
52
AIRCRA3'T
2647
"rY59
GSTABLISHN?ZNT
LONGITUDINAL TX!%cJUENCYRESPONSE TO ELEVATOR OF AN AIRCRAFT OVER THE
SHORT PERIOD F,REQUXlKY RANGE
D, lX. Ridland
Formulae for the evaluation
of aircraft
longitudinal
frequency
response characteristics
are presented.
The simple short period frequency
response is illustrated
for three different
types of aircraft
and the
effects
of neglecting
elevator
lift
and speed variations
associated with
the phugoid motion are separately
determined
and discussed,
l
.
LIST OF CONTEY?TS
Page
4
INTRODUCTION
1
4
3
4
SHOW PZRIOD FREqUE$JCYRESPONSE
3J::
3.3
3.4
Assumptions and equations
of motion
Transfer
functions
Frequency response formukae
lkan@es of short period frequency response
curves
9
EFFECT 0.7 ELEVATOR LIFT
4
Ikl
Asswptions
and equations
of motion
Transfer
functions
Frequency response formulae
Corrections
for elevator
lift
EWITipleS
of corrections
for elevator
t:
4.4
k5
9
IO
-IO
12
lift
14
15
Assumptions and equations
of motion
'Transfer functions
Frequency response formulae
Corrections
for the phuyoid
Example s of three degrees of freedom
response curves
;::
5.3
5.4
5.5
6
16
16
-I8
21
frequency
CONCLUDING REP&'&KS
24
2.5
LIST OF SYXBOLS
25
LIST OF REFEREXXS
29
APP?XJXXXS I ARD 2
TABLE - Values
ILLUS~~rTIONS
31-36
of aircraft
- Figs.~-I
parameter3
6
LIST OF A..PPF~WICES
&en&x
1
37
-
2 -
Tran-,fer
case
functions
for
the general
longitudinal
31
Range of validity
of the approximate values of the
short period sta?eility
parameters B and i:
-2-
36
LIST 03 ILLUSTRATIOKS
The characteristics
of the non-dimensional
short
response of incidence
and normal acceleration
period
frequency
The characteristics
response of rate
period
frequency
of the non-dimensional
of pitch
short
2
An example of the non-dimensional
short
of incidence and normal acceleration
period
An example of the non-dimensional
of rate of pitch
period
frequency
response
3
short
frequency
response
4
An example of the effect
of incidence
of elevator
lift
on the frequency
response
An example of the effect
of normal acceleration
of elevator
lift
on the fkeq.uency
response
An example of the effect
of rate of pitch
of elevator
lift
on the frequency
response
5
7
An example of the effect
of elevator
lift
frequency response of normal acceleration
on the modulus
of the
8
An example of phugoid
frequency
response
curves
for
incidence
An example of phugoid
acceleration
frequency
response
curves
for
normal
An example of phugoid
frequency
ratio
J-
for
different
curves
for
oscillation
rate
of pitch
on short
11
period
12
The percentage
error incurred
neglect of the phugoid
period
response
the phugoid
The frequency at which neglect
error in modulus
of short
9
IO
An example of the effect
of
frequency response mcduli
Ratio
6
of the phugoid
13
at short
period
frequency
by
14
frequency
lift
causes 5 per cent
to phugoid
frequency,
$ , and the
1
coefficients
R2
Comparison of the true values
of the short pried
stability
B',
Cl and approximate
parameters
-3-
values,
3, C
16
I
INTRODUCTION
techniques
the actual aircraft
is
Vith modern stability
analysis
frequently
treated as part of a system which may also include such items
To apply these techniques
the response
as servocontrols
a,nd a human pilot.
characteristics
of the aircraft
may be expressed in general terms, by the
or specifically,
by frequency response CUTV~S.
aircraft
transfer
functions,
The object of this report is to summarise the formulae most frequently
used
in assessments of aircraft
longitudinal
response to elevator
at frequencies
near the natural
frequency of the aircraft
short period pitching
oscillation.
The high frequency band which will be affected
by structural
modes is outside the scope of the paper.
The emphasis is generally
on simplicity
and tne principal
formulae are
The effects
of
derived
for the basic two de,grees of freedom approximation.
neglecting
elevator
lift
and the phugoid freedom on this simplified
short
period frequency response are expressed as correction
factors
and discussed
in detail,
':xamples of the correction
factors
are given for typical
aircraft types.
The responses to elevator
are derived
for aircraft
acceleration
at the centre of gravity
and rate of pitch,
principal
variables
normally considered.
2
that
incidence,
normal
to cover the
RE'lYTDDOF MWXSIS
The method by which the formulae
used in this work1,2,3;
~omorily
in the report have been obtained is
it is described briefly
below* ,
The equations
of motion are written
in the usual D operator
notation
This form is adopted as the equations
of motion for most cases will already
They are then solved for the ratio
have been stated elsewhere in this way,
The resulting
of the relevant
output variable
to the input variable.
of the expression
expression
is a transfer
function 4. The Laplace transform
function 5
obtained simply by substituting
p for D, gives the Laplace transfer
transform,
or simply the 'transfer
function'
of the system, while the Fourier
into modulus (ampliobtained by substituting
iw for D, gives on resolution
The
the frequency response formulae.
tude) and argument (phase angle),
output variables
normally considered
in aircraft
stability
work are incidence
+ , normal acceleration
n and rate of pitch q, while the i-nput is negative
0
elevator
angle (-n).
Negative elevator
angle has been used because it
produces positive
pitching
moments.
Pitching
moment could equally well have
but elevator
angle is more easily measured
been chosen as the input variable,
in practice.
The transfer
functions
in this report have been expressed 'n terms of
the usual aerodynamic parameters
in the concise Neumark notation- ii whereas the
frequency response functions
are given in terms of damping and frequency
parameters.
3
SHORT PERIOD FREQUENCY%%POmE
Transfer
Appendix 1.
functions
for the general
They can be simplified
for
longitudinal
motion are given in
the short period frequency range
*An excellent
discussion
of the underlying
theory and the practical
aspects of analysing
and programning
the results
of response experiments
given in Convair Report FZA-36-195 (P72631) entitled
"Dynamic response
Presentation
and theoretical
programme on the B-36 airplane:
Part III,
considerations
of the transient
analysis method employed for obtaining
frequency response functions
from flight
data", by T.P. Breaux and
E.L. Zeiller
(1952).
-4-
is
by neglecting
both elevator
lift
and the phugoid freedom.
Uhen the motion
of the aircraft
is represented
in this manner the drag equation is redundant.
% anal kt are omitted;
terms -i;kr = - $- CI, tan ye ,
e
ye being the angle of climb.
$ is commorily neglected for both conventional
and tailless
aircraft
because of the large values of I-I appropriate
to modern
aircraft
and k' is zero for level flight.
The restriction
to level flight
is, however, not stringent;
even at large angles of climb the values of the
short period stability
coefficients
B and C are hardly changed by the term k'.
As will be shown later,
responses computed on this simple basis are reasonably
correct
in the case of tailed aircraft
for frequencies
near the short period
frequency,
but for tailless
aircraft
it would be advisable
to include elevator
lift,
Further,
the small valued
The simplified
equations of motion from which the short period transfer
functions
are derived are stated below, together with the assumptions
governing
them; the relevant
response formulae follow.
3.1
Assumptions
It
stability
and equations
of motion
is assumed that
1
speed is constant
(no phugoid
2
the aircraft
3
there
A-
the aircraft
5
elevator
6
the system is linear.
is trimmed
is no coupling
mode),
to level
with
flight,
lateral
modes,
is rigid,
lift
is neglected
With the above assumptions
axes are
and
the equations
of motion with
respect
to the
(1)
3,2
Transfer
functions
The resulting
transfer
functions
and rate of pitch are respectively
-5-
for
incidence,
normal
acceleration
(2)
6
a@$
2m
Sp+$
(
>
ii- ~~ /
(4)
(3)
9
p2+Bp+C
p2+Bp+C
3
(4)
where
c
z
6l+p
lJ “w
=
aflh
-‘-2iB*LU
period
6 is the elevator
moment coef'ficient,
stability
parameters.
3.3
Frequency
Incidence,
given
response
while
B and C are the short
formulae
7WI
0
The aircraft
by:
incidence
frequency
response
obtained
from Equation
(2) is
Modulus
(5)
Phase lag
(of
incidence
behind
negative
elevator)
(6)
-6-
In these expressions,
dimensional
frequency
is
x is tne ratio
of the frequency,
of the short
period
is
of actual
the ratio
which have been non-dimensionalised,
oscillation.
f,
to the natural
the non-
- JC
2x%'
frequency,
The damping ratio
to critical
damping.
It is apparent from equations
5 and 6 that the incidence frequency
response is of exactly the same form as that of a simple, one degree of
freedom, mass spring and damper system.
The frequency
response curves
(one for modulus and one for phase lag) plotted
against non-dimensional
frequency,
x, are uniquely defined by the damping ratio,
z, and the main
features
of these curves are shown in Fig.1.
It can be seen that the phase
lag is always 90' at the natural
undamped frequency,
x = 1.
Hence, from
measured frequency responses the value of C can be estimated and the value
of B can be determined from the damping ratio,
G.
With B and C known, 6,
the elevator
moment coefficient,
can then be evaluate&
Normal acceleration,
n
The aircraft
frequency response in terms of normal acceleration
the centre of gravity,
corresponding
to Equation (3) is:
at
(7)
Phase lag
*
9,
=
2z%x
tan -1
1
-
x2
0%
l
.
apSV2
, in the modulus expression
Apart Ifrom the constant,
2m
equations are identical
to those for incidence
response,
Rate of pitch,
The rate
these
q,
of pitch
frequency
response
by:
-7-
derived
from Equation
(4) is given
Modulus
1+(&-y
!
=
-r
PT
*g;
/
(9)
i (1-x2)2
+ (2
G
x)2
\
l
Phase lag
qsq= tan-’
(4 ;:
za + x2 - 1)x
2 ca(l-x2)
4. 2 g x2
.
In this case the frequency response curves are again defined
non-dimensional
frequency x, the damping ratio
Z, and an additional
The term Za, defined by
'a*
the ratio
is the damping ratio due to the heaving motion alone;
the ratio of damping due to heaving to the total damping.
by the
parameter
Ga% is thus
The main characteristics
of the rate of pitch frequency response curves
It will be seen that the value of ca can be obtained
are given in Fig.2.
from measured rate of pitch responses and, since B is known from the acceleraslope, a, can be detertion or incidence response, the value of the lift
mined.
From frequency response measurements, therefore,
it is fezsible
to
evaluate
B, C, a and 6,
It can be seen from the foregoing
that the incidence
and acceleration
The rate of pitch response
responses depend only on the damping ratio G.
depends not only on C, but also on the distribution
of damping between that
derived
from heavir.3 (a) and that derived
from rotation
(me) as defined by
Za/L
3.4
Examples of' short
period
frequency
response
curves
Examples of short period frequency response curves for normal acceleration and rate of pitch are given in Figs.3 and lb for three different
types of
The data used in the calculations
are given in Table I for each
aircraft,
of the aircraft,
Aircraft
A, a tailed aircraft
with a ratio of restoring
margin6% to
elevator
lift
arm of l/20 (7 = 0.05)) has high total damping (i: = 0.6) with
reasonably
normal distribution
of the contributions
to damping from heaving,
rate of pitch and rate of change of incidence
(za = 0.3, g,, = 0.2 and
This could be a typical
tailed
aircraft
in flight
cr = 0.1 respectively).
medium altitude.
<<Restoring > margin
are negligible,
is equal
to static
-o-
margin
when Mach number effects
a
at
Aircraft
B, a tai less aircraft
with a ratio of restoring
elevator
lift
arm of 3&J (7 = 0.15) has low total damping (Z
consists of equal contributions
from heaving and pitching
(Ga
cV = 0.1 respectively);
there is no damping contribution
from
of incidence
(PY$ = 0).
This could be regarded as a tailless
supersonic
flight
margin to
= 0.2) and this
= 0.1 and
rate of change
aircraft
in
at high altitude,
Aircraft
C is again tailless
with the same ratio between restoring
margin and elevator
lift
arm and the s,ame total damping as the second aircraft.
The difference
between the second and third aeroplanes
is in the
It is assumed that the effective
damping is
rotary damping contributions.
entirely
supplied by the heaving motion (Ga = G = 0.2), the damping contributions due to rate of pitch and rate of change of incidence being numerically
equal but of opposite sign (ZY = -ZX = 0.05).
This could represent
aircraft
B at transonic
speeds.
Figs.3 and ,!+ give a general impre ssion of short period frequency
response curves and show the effects
of different
values of G and c a'
4
EFPECT OF ELEVATOE1LIFT
The effect
of elevator
lift,
z q, has been determined by including
elevator
lift
in the equations of mo&on (Equation
(1))
otherwise
the
approximations
made here are identical
to those made in'section
3.
The
effect
of neglecting
elevator
lift
has been represented
by corrections,
for
both modulus and phase angle, to the simplified
short period frequency
responses derived
in the previous
section.
r-0
An in the previous
case the simplified
assumptions are stated below and the various
4. 1
Assumptions
It
stability
and equations
equations of motion
formulae follow.
and governing
of motion
is assumed that
I
speed is constant
2
the aircraft
3
there
4
the aircral"t
5
the system is linear,
(no phugoid
is trirrmed
is no coupling
With the above assumptions
axes are
to level
with
is rigid
mode),
flight,
lateral
modes,
and
B
the equations
of motion
with
respect
to the
.
6 - zw)
(Fj
-q^t
-z$J
= 0
I
-P-
4wL
Transfer
functions
pitch
normal acceleration
The transfer
functions
for incidence,
obtained from tlle above equations are respectively
w
77
-=-0
Y
p2+Bp+C
Y
2w
p*+Bp+C
.... ..
(6 c z
Ai-.
of
zv p - (6 - zr V)
G)E
-=' ii
G>E
and rate
x)p+~G+wz
=gJ+p
Y
p*+Bp+C
(-i&7
(13)
(14)
where
, the rate
V
=
of change of incidence
the rotary
-. “s
damping
damping coefficient
coefficient,
and
'By
w
=
- -fJ “w , the static
iB
It can be seen
equations
of motion
equations
2 to 4).
quite different
and
damper system as in
4.3
t
Frequency
stability
coefficient.
that retaining
the elevator
lift
term, - zV ?I, in the
results
in complicated
transfer
functions
(compare with
The expressions
for incidence
and acceleration
are now
no longer strictly
resenlble a simple mass, spring and
Section 3.3.
response
formulae
Incidence,
The aircraft
is given by:
incidence
frequency
response
ModuJus
-
10 -
corresponding
to Equation
(12)
Pl-zse lag
-1 2 m + 4 ca zv y) - 2 za y(1- x2)
95 = tan
0 + 4 za q) r> (1 -x2) + 4
0 E
(16)
w
;:
L;,
y
x2
x
l
T
As in Section 3.3 x is non-dimensional
frequency,
Z;, Ga and z,, are
damping parameters,
while y is a non-dimensional
coefficient
defining
the
ratio of elevator
lift
to elevator
moment derivatives.
Y = ?.lc
zw
Normal acceleration,
gravity
F
l
n
The normal acceleration
derived
from Equation
frequency
(13) is:
response
at the aircraft
centre
of
m
Modulus
7%
(I + Y x2 - r)2 f
= “;
E
4(Z,
-I-q2 Y2x2
x2)2 -t (2 ;= x)2
(I-
t
l
07)
Phase lag
=
2(q)
tan"'
#nE
+ $)
Y(l-
x2> -I- 2 Z(1 e y x2 " 7)
(1 + Y x2 - 7) (1 -x2>--
4 yJ($)+
";y) y x2
I,
xy
(18)
where
r =zdth*
2
W
.
mr
y is the ratio of restoring
margin to elevator
lift
arm and Gx is the
component of the damping ratio due to rate of change of incidence.
Rate of pitch,
The rate
q
of pitch
frequency
response
- 11 -
obtained
from Equation
(14) is:
.
r)2
-+
*
2
(1
”
Ga
z;x
r12
0
(
>
q%3$
zi
zxl2
4
a
1
(I - xy2
.
(19)
-I- (2
Phase lag
#
QE
4.4
2
Corrections
for
z: r;,(l- F) - (I - 4 ca Gxr> (1 -x2,
x.
~a(l-y)(l- x2>+ 2 z x2(1 - 4 ga t;X Y)
4
= tan-’
elevator
Lii?t
The effect of elevator
lift
on the simplified
short period frequency
responses of Section 3 can be represented
by correction
factors which are
defined
as
0$
FjJ-, ?
+
-t--l7
ZZ
, etc.
I
5
(>I
xFiJ-i, 9
, .-.
are defined
by Equations
(5) to (IO)
arki
0+
are defined
These correction
factors
by Equations
are discussed
-
12 -
(15)
below.
to (20).
(20)
Incidence,
$
0
Modulus correction
l?hase angle
factor
correction
= -tan-',+4g
A$
22;
yx
y
a
Qv
0
Y
l
(22)
?
E
For a tailed aircraft
all
normally positive.
to increase the amplitude ratio
the parameters
in these expressions
are
, the effect
of elevator
lift
is
over the whole frequency range
and to decrease the lag of incidence behind elevator
deflection,
frequency
(x = 0) the modulus correction
factor
is (I + 4 Ga GV y)
phase angle correction
is zero;
when frequency
approaches infinity
the phase lag approaches 90' and not 180' as when elevator
lift
is
No such general observations
can be made for unorthodox
aircraft;
be considered
on its merits.
Normal
acceleration,
At zero
and the
(x '00)
neglected.
each must
n
Modulus correction
factor
(23)
i?hase angle
correction
A#
nE
-1 2 Y x(q,! + $1
= tan
1+yx2+
'
(24)
Here when frequency
is zero the modulus and phase angle corrections
are
(I- 7) and zero respectively.
At very high frequencies
the modulus correction factor becomes very large and the phase angle correction
becomes
negligible.
Rate of pitch,
q
Modulus correction
factor
-
13 -
F
K
q3
=1
(1 - 4 ca zx yy xz c 4 zia(l- ^ry
I
x2 + 4 cjy
(25)
.
Phase angle correction
A@% =, tan-'
may be noted
2
za x(4
za
r;,Y-3
(1 - 4 Ga 3 y) x2 f 4
Ga*(l
-
4)
l
that
restoring
manoeuvre
margin
margin
l
The effect
amplitude
ratio,
V
of elevator
lift
in this case is to reduce slightly
the
At zero frequenoy
, and to decrease the phase lag.
lprl-v
(x = 0) the amplitude reduction
factor
is (I- T) and the phase angle correcThese
corrections
are
identical
to those fore-the acceleration is zero.
tion at zero frequency and the static
relationship
- n g zzw- q V is
aircraft
and 0.85
The fat-tar
(I- 7) is roughly 0.95 for tailed
retained,
for tailless
aircraft.
f
4 .5
Examples 0f corrections
for
elevator
lift
The effects
of elevator
lift
are shown in Figs.5,
6 and 7, for inciThe three airdence, normal acceleration
and rate of pitch respectively.
craft,
A, B and. C, which were described-in
Section 3.4 have again been used
to indicate
the variation
in elevator
lift
effect with aircraft
type.
Incidence
In Pig.5, which illustrates
the corrections
to incidence
response,
elevator
lift
is seen to incrtiase the modulus and to reduce the phase lag
The increase in modulus is, however, less
throughout
the frequency
range.
than two per cent in all the cases considered
at the aircraft
short period
natural
frequency
and, as it increases only slowly with frequency,
it could
reasonably
be neglected.
The phase lag reduction,
on the other hand,
At the
increases rapidly
with frequency asymptotically
approaching
pO".
only 4O in the worst example
short period frequency,
however, it is still
of Fig.5 and in comparison with experimental
errors this is not a large
amount.
When elevator
is applied to
These effects
have a simple explanation,
This lift
accelerates
the
increase
incidence
it produces a negative
lift.
in an additional
increase in incidence.
aircraft
bo&ly dovrnvtards, resulting
stability,
The aircraft
then pitches nose down, because of its inherent
the net result
is an
partial1.y
alleviating
this increase in incidence;
increase in incidence.
The correction
factor expresses basically
the ratio between the aircraft response to the combined effects
of elevator
lift
plus moment and the
response to elevator
moment only.
Changes in the motion of the aircraft
.
14-e
are resisted
by aircraft
inertia
and, as with increasing
response to elevator
moment approaches zero more rapidly
to elevator
lift,
the correction
factor,
being a ratio,
as frequency is increased.
frequency
the
than the response
tends to infinity
Acceleration
The effect
of elevator
lift
on normal acceleration
frequency response
is shown in Fig. 6,
The acceleration
modulus is decreased at frequencies
below approximately
the short period frequency
and increased above this
frequency,
while the phase lag is increased at all frequencies.
The
increase in phase lag in the worst case, however, is only about 5' which, as
previously
suggested, is not a very large amount from the experimental
point
of view.
This value decreases with decreasing
rotary damping, (GV + zx),
being smaller for the second, tailless
aircraft,
13, and zero for the third
aircraft,
C, for which the rotary damping is zero (c$, + G = 0).
x
The physical
interpretation
of these effects
is similar
to that given
above for incidence,
but it can be seen that the acceleration
correction
(Equation
(23))
is more complex, having a frequency component in each term.
At low frequencies
the primary effect
of elevator
lift
is to produce a
negative
lift
on the aircraft,
There
thereby reducing the total acceleration.
will also be a secondary increase in acceleration,
due to the incremental
incidence
induced by elevator
lift,
and this till
be alleviated
by the
pitching
nIOt ion of the aircraft
as in the incidence case.
correction
factor
tends to
very high frequencies
the acceleration
factor,
but,
infinity
in much the same manner as the incidence correction
whereas the absolute value of the incidence
response modulus including
elevator
Lift tends to zero with increasing
frequency,
the corresponding
acceleration
modulus tends to a finite
value, I?, as the elevator
lift
will
always be apparent even though the aircraft
can no longer respond in pitch.
This effect
of elevator
lift
on acceleration
response at higher frequencies
is shown clearly
by plotting
the moduli of the two acceleration
responses,
on a logarithmic
scale as in Pig.&
it
Rate of pitch
The effect
of elevator
lift
on the rate of pitch frequency
response is
shown in Fig. 7.
Both modulus and phase lag are, in general,
decreased by
elevator
lift,
but the corrections
are appreciable
only at frequencies
below
the short period frequency,
Physically,
the effe.ct of elevator
lift
on rate of pitch response
follow's directly
of the incidence an2 acceleration
from the explanations
by linear
motion alone,
corrections.
As the rate of pitch is unaffected
only the pitching
motion in response to the incidence
induced by elevator
lift
contributes
to the correction
factor.
This, as previously
indicated,
is only significant
at low frequencies
because of aircraft
inertia,
and as
it is opposite in sense to the basic short period pitching
motion, results
in a decrease in rate of pitch response.
Unlike the assessment of the effect
of elevator
lift
on the short
period response characteristics,
which*was made simply by retaining
the
term zT r in the equations of motion, consideration
of the effect
of the
phugoid oscillation
involves
an additional
degree of freedom.
The drag
-
15 -
,
equation must be included and the system becomes one of fourth
instead of
second order.
As, however, the aim is to examine the errors incurred
at
low frequencies
by neglecting
the phugoid, rather
than to study the
Elevator
lift
is
phugoid itself,
several simplifications
are made.
neglected
(z = 0) and it is assumed that there is no Mach nun&er effect
rl
X
Z
$,
$
and x7 are neglected and,
Further,
the small quantities
(1% = 0).
in conformity
with the two previous
cases, zw is defined as - t instead of
-;. (a + CD) when deriving
This leads to
the frequency response formulae.
some useful simplifications
but for p'ractical
purposes leaves the value of
zw unchanged.
k' too is omitted assuming as in Section 3 that the aircraft
is trimmed for level flight.
As with elevator
lift,
the effects
represented
by correction
factorti e which
short period responses of Section 3.
of the phugoid oscillation
are
can be applied to the simplified
The assumptions a nd equations of motion are stated
by the transfer
functions
and response formulae.
5.1
stability
Assumptions
and equations
It
is assumed that:
1
the aircraft
2
there
3
the aircraft
4
elevator
5
the system is linear.
of motion
is trimmed
is no coupling
with
lift
is neglected
( >
u
lateral
flight,
modes,
v
and
the equations
U
-x
to level
is rigid,
With these assumptions
axes are
D
below and followed
of motion
W
-
xw
0
with
respect
‘QC,@
‘fT
- zu+ (D- zwj(;j
Transfer
=0
1
-9%
-q%+D0
5.2
to the
= 0
J
functions
The transfer
incidence,
normal
functions
acceleration
obtained from the above equations
and rate of pitch respectively
-
16 -
are,
for
($)
qp2
e s
p + $.)
?
A
(-rl)F=
where
2
cL
+ B CD c 2
A = p4 + (B + CD) p3 -b
>
+
2
CL pi-wc CD i- fu + x3 -Tjy>
(
(31)
*
p2
2
CL
2'
A better physical
insi,3ht into the short period-phugoid
relationship
may be gained by re-vlriting
Equations (J-1) to (33) in the following
form
(f)
'x3$
6(p2 + s
p + $-)
(34)
I
A
=
2
-= ii
6-r' lp
apl,jlsc. T 2 VJ
(
p2 -+ CL, p f ZL2
2
2$
,)
(35)
A
2
a 'L
-F;-2-
p+t
>(
A
>
9
(36)
*
l
where
c2
A =
(p2 f s p + c) - ;+-
(p - u).
(37)
reduce to the simple
It can be seen that when CL = 0 these expressions
At
short period transfer
functions
of Section 3 (Equations
(2) to (4)).
CL z 0 the phugoid frequency
is zero and the short period responses should
2
'
5
not be affected.
The last term in Equation (37), - t --& (p - v), can
-
-17 -
bc regarded
oscillations;
as a coupling
termbetween
the short period and phugoid
when CL = 0 there is obviously
no coupling.
Frequency
5.3
response
formulae
To simplify
the derivation
of the frsquency
response2formulae,
Equation (37) is considered
in the form (p + B' p + Cl)(p
+ bp + c) and
the expressions
for incidence,
acceleration
and rate of pitch are treated
as products of two complex numbers.
Approximate values of the coefficients,
obtained by approximate
factorisation7
of the quartic
(Equation
(37)) are
B'c!B and. C' r C, where B and C ere the short period stability
parameters as
defined
in Section 3.1, and
b=
-(
2
cL
- (B + CD) "2
2
cL
C-dP2.p~
and
To further
simplify
error in each case being
Appendix 2).
matters it is assumed that By = B and C' = C, the
extremely small (less than one per cent - see
Equations
(31) to (33), describe two oscillaThe transfer
functions,
The previously
considered
each
with
its
own
natural
frequency.
tory modes,
involving
only
the
short
period
oscillation,
were,
on
transfer
functions,
non-dimensionalised
by
dividing
conversion
to frequency response functions,
This is again done in
C.
through by th-t: short period frequency parameter,
the present case, because it is the short period response which is of primary
concern.
Incidence,
The incidence
$
0
frequency
response
formulae
obtained
from Equation
(31)
are:
?Kodulus
(38)
- 18 -
Phase angle
=
$
tan-'
%
W
07
P
2x [4rT,5,x2+~(R,2-x2)~~-x2)+~(R~-x2)(,nx2)-~(,-x2)(R,2-x2)]
(~-X2)(R~-x2)(R-f-x2)+4x2[Q?&-x2)+Z
,
c,,(R:-x~)-~:z;,(R$~~)]
. . . . . (39)
The symbols, R,,
defined
R2,
Zb and $,
represent
phugoid
parameters
and are
below,
R
as the ratio of phugoid frequency to short period frequency,
is
1'
the most important
parameter in the assessment of phugoid effects
on short
period response.
The value of l/k, , as obtained by the approximations
given
a,bove, is plotted
against CL in Fig.15 for aircraft
A.
The values given are
for the present purpose, reasonably
representative
of conventional
aircraft
and show that R, may be taken as being proportional
to CL.
R2
=
It can be shown that R2 c? RI , an3 R2 is therefore
an approximate
ratio
of phugoid frequency to short period frequency,
Values of l/k2 are compared
with l/R, in Fig.15 for aircraft
A.
At low CL the absolute values of R, and
R2 are small with respect to unity (in Pig.l5,
R, = 0.0367 at CL = 0.2);
as
in the response equations R
R2 occur only in the form (a2 - x2), at
1 '
frequencies
near the short period frequency,
when x -N 1, R, and R2 may be
neglected.
$, is a phugoid damping parameter
period frequency,
flC,
Phugoid damping
second order
term.
and is related
is very small
As CL -+ 0, b + CD and at all
low CL therefore,
the order
of $, is given
which can usually
be neglected,
-
19 -
by
here to the short
and r;D is clearly
a
2
cL
CL, c = 5 2'
At
a small
quantity
P
,
is an approxin-ate
be neglected at low CL
form of t;, and for
%
Briefly
then,
for
conventional
similar
reasons
can usually
aircraft
R2>R4
and
$-,'$.
At low lift
coefficients
and for frequencies
frequency R,, R2, ';, and $, may be neglected,
well
above the phugoid
It can now be seen that the combined phugoid and short period incidence
simple short
response (Equations
(38) and (39)) is, like the corresponding
period incidence response (Squations
(5) and (G)), completely
defined by
frequencies
and damping ratios.
Assuming x = 1 and low values of CL, and
thence that R, = pcL = 'i, = s E 0, the phugoid terms disappear
and the
expressions
become identical
to the corresponding
short period equations of
Section 3.
It can be expected therefore
that, with low CL, phugoid
effects
dn incidence response will be negligible
at short period frequencies.
Normal acceleration,
The frequency
Equation (32) are:
response
n
formulae
for
normal
acceleration
obtained
from
ixodulus
x2(x2
+
4 5-f)
I-
x2)2 + (2 2; x)"][(R,~-x~)~
-I- (2 Gb x)~]
l
xc4
2:y”bX2+4 ‘i3 $,(1-x2)+
4 g $,(R,2-
x2)-
(,-
IC~)(R,~-~~}]
. .
'
.
.
.
(40)
l
. . . . . . (41)
In the simplified
short Period case both incidence
ard acceleration
responses were identical,
apart from a constant,
but in the Present case they
However, making, as before,
the approximation
R, = R2 = Zb = s = 0,
differ.
the corresponding
short period acceleration
equations are obtained (Equations
It
can
again
be
expected
therefore
that the phugoid will have
(7) ad (8)),
negligible
effect
on the short period acceleration
response at short period
fre+encies.
- 20 -
Rate of pitch,
given
The rate
by:
q
of pitch
frequency
response
derived
from Equation
(33)
is
Modulus
4
P
X2(ca-I-$,)2 + (4
[(I -x2)2+
(2
ti x)~][(R
Ga 5
.
+ R22 - x2)2
I 2 -x2)2+
(2
“b
x)2]
l
. . . . . . (42)
Phase angle
. ...*.
(43)
On making the approximation
R, = R2 = 5 = ';, = 0, Equations
(4-Z) and
reduce to the short period rate of pitch expressions,
Equations (9) and
Only negligible
interference
by the phugoid need therefore
be
anticipated
in the rate of pitch response at short period frequencies.
5.4
Corrections
for
the phugoid
As in the elevator
lift
case, the above phugoid frequency response
expressions
can be used to derive correction
factors
for the effect
of the
phugoid on frequency response in the short period range.
The phugoid
response expressions
are cumbersome, however, and as the object here is to
determine
the frequency above which the phugoid can be neglected without
significant
error rather than to enable corrections
to be made, only the
modulus correction
factors
are examined.
The examples of phugoid frequency
responses given in Figs.9 to 11 (which are discussed in the next section)
show that it is reasonable
to assume that where modulus errors are small,
phase angle errors are also small.
An individual
assessment can of course
be made in any particular
case.
Incidence,
Modulus
At zero
F
0
correction
factor
expression
-
reduces
21 -
to
i
2
cL
C+BCD+2
w
R2
E-y=
L’ cw
II
manoeuvre
restoring
=e=
margin
margin
(45)
'
At the short period frequency,
x = 1, the effect
of the phugoid as
expressed by the correction
factor is very small and at higher frequencies
it will tend to diminish altogether.
Differentiation
of Equation (44) shows that maximum and minimum values
of the correction
factor
occur at frequencies
given by the smaller and
larger positive
values of x respectively
obtained from
-b + "Jb'2a
x2=
4ac
(46)
where
8
gives
a
= R,2 - %2 + 2(52
b
= R24-R,4,
C
= R, ' Rz2(R,'
- R~')
- q,
,
+ 2(G2
Rz4
-
s2
R,4)-
Vhen R, = 0.06, R, = 0.07, ;;b = 0.0036 and 5 = 0.004-4,
x = 0.0601 and 0,;701
(in fact, R, and R2 approximately)
and minimum values respectively
in agreement with the turning
of incidence modulus.
points in Fig. 12.
If y is defined as the proportional
error
neglecting
the phugoid freedom in the incidence
Equation (46)
for maximum
These figures
are
in the modulus incurred
by
response, it follows
that
(47)
Choosing as the maximum permissible
error a value y = lyl, Equation
(47) can be solved to give the limiting
frequencies
outside which neglect of
In the case of the incidence
the phugoid freedom is no longer permissible.
response y is negative
(i.e. K, , < 1.0) near the short period natural
0
W
fi:
frequency
of y.
and in consequence
Equation
(47) must be solved
- 22 -
for
negative
values
Equation (47) has been used to obtain the incidence curves of Pigs. 13
and 14 for aircraft
A.
As aircraft
A is considered
representative
of
conventional
aircraft,
the curves will indicate
the magnitude of the errors
to be expected with conventional
aircraft,
Fig.13
shows, for ranges of lift
coefficient
and phugoid/short
period frequency ratios,
the frequency at which
neglect of the phugoid causes a 5% error-in
modulus,
Fig. 14 similarly
shows
the error,
incurred
by neglecting
to take account of the phugoid, at short
period frequency.
Normaal acceleration,
Modulus correction
n
factor
.
period
This factor becomes zero at zero frequency,
almost unity at the short
increased.
frequency and tends nearer to one as frequency is further
The maxkm.~n value
obtained at a frequency
Rl4
of the acceleration
modulus correction
x given by the positive
root of
i- 1R,8
I
+ 4(R,*
- 2 7%* + 2 s*)(*
s2
factor
is
R14)
X
2(R,*
- 2 $,*
R, is the phugoid natural
frequency,
period frequency,
and one would physically
have its maximum value at this frequency.
which R-I = 0.06.
(49
+ 2 r;D*)
expressed
in terms of the short
expect the correction
factor
This can be seen in Fig.12
to
for
By putting
Kn
= *l+y,
P
where, as before,
y is the maximum acceptable
percentage
error in the
acceleration
response modulus and solving for x, the highest value obtained
will be the frequency ratio at which neglect of the phugoid causes this
maximum error;
as frequency is increased beyond this value the error
becomes smaller.
Curves of acceleration
errors assessed in tnis way are
given in Figs. 13 and 14 along with the incidence
error curves,
Rate of pitch,
q
IGodulus correction
factor
- 23 -
Y
response tests are made at R < 0.07 and simple short period theory is
I
applied,
the maximum possible
error at the short period frequency due to
neglect of the phugoid is 0.69, and tnis occurs in the acceleration
response.
c2
Generalising,
as the coupling
term of Equation (37), - t +
(p - v), is
mainly a function
of CL and thus of frequency,
the interaction
between the
short period and phugoid oscillations
is governed by the frequency ratio,
R
I'
(l/R, is more convenient
in practice.)
If the frequency ratio l/R, is more
than 10 (i.e. with a short period frequency
of say 6 c.p.s.,
for a phugoid
period of more than 20 seconds) the phugoid can be neglected with less than
5$ error down to x = 0.5, and less than 1;s error at the short period frequency, x = 1.
error
should be greater
down to x = 0.2.
6
CONCLUDING RZ&fARKS
l/R,
than 30 if
the analysis
is to have less than 5%
The formulae derived
in this report can be used either
to estimate
aircraft
response characteristics
or to obtain information
from measured
aircraft
responses,
The first
application
is straightforward
and the simplified short period formulae should be adequate for the relevant
range of
It has been shown that in
frequencies;
the second needs a little
care.
interpreting
experimental
results
the phugoid effects
can be neglected
providing
that tests are made at a high value of the ratio of short period
frequency
to phugoid frequency.
This separates the two frequencies
and will
generally
be accomplished
in flight
tests if these are made at low lift
coefficients,
say CL < 0.2.
The effects
of elevator
lift
can also be
neglected
if errors of 5% in amplitude
and 5' in phase angle can be tolerated.
These errors can, however, be m&h reduced if the-analysis
is restricted
to a
small frequency range around the short period frequency of the aircraft.
Complete transfer
functions
to elevator
are given in Appendix
for
1,
the three
.
degrees
of freedom
P
a
response
LIST OF SYMBOLS
a per rad
lift
a2 per rad
lift
slope of the elevator
(referred
tailed aircraft
and to wing area for
aT per rad
lift
B slug ft2
pitching
B++u+x
short
=--a
2
b
slope of the aircraft
slope of the tailplane
(aT = a2 for
moment of inertia
period
damping
stability
coefficient
"s _ "t;
iB
iB
phugoid
to tail
tailless
damping
stability
-2 5-
coefficient
all
r
arez for
aircraft)
moving
tail)
c
LIST OF SYMBOLS (Contd.)
c=w+
short
period
stiffness
stability
coefficient
pmw
=,-,a?3
iB
2 ig
5
natural
undamped short
in aerodynamic time
.
CD
drag coefficient
cL
lift
c
phugoid
=c ft
mean aerodynamic
D =d3-d
operator
f c.p.s.
frequency
of longitudinal
manoeuvre
margin
%-l
=Km--
ei
s 1-1
kg 2
5
0
ig=
0t
Y $ YK
P
of the aircraft
stiffness
stability
ccefficient
choti
of pitching
oscillations
moment of inertia
restoring
margin (Restoring
margin is equal to static
margin when there are no Ivbch number effects)
K ,K K
nE: '
qE
F
0 E
K
frequency
coefficient
coefficient
a 'rn
L
Km=-r
period
Qp
correction
factors
to allow for the effect
lift
on short period incidence,
acceleration
pitch response moduli respectively
of elevator
and rate
correction
factors
to allow for the effect
of the phugoid
on short period incidence,
acceleration
and rate of pitch
response moduli respectively
P
kB =
radius
of gyration
in pitch
k' = -Q CL tan ye
e ft
mU &---
reference
z
of
length,
usually
tail
arm length
acm
pitching
moment derivative
a$
0
-
26 -
due to forward
velocity,
u
LIST OF SYMBOLS (Contd.)
pitching
moment derivative
due to heaving
velocity,
ac m
pitching
moment derivative
due to heaving
acceleration,
ac m
pitching
moment derivative
due to rate
z. acm
mrl =zx
pitching
moment derivative
due to elevator
mb =m q +m*w
full
n 'gl
normal acceleration
q rad/sec
rate
R,=II;;
ratio
22
m*
W
=$
E
mq
d-
rotary
of pitch,
w
q
deflection,
damping derivative
increment
above lg
of pitch
of aerodynamic
stiffnesses
2
R2=
cL
2-E
approximate
s ft*
wing area
STfY2
tail
t = Z 1: set
w
t =gpsv
time
unit
ratio
of aerodynamic
stiffnesses
area
of aerod,ynamic
u ft/sec
increment
V ft/sec
forward
velocity
W lb
weight
of aircraft
w ft/sec
increment
vu’*
0V rad
incremental
2KfE
x=-YE--
non-dimensional
time
of velocity
along
x axis
of aircraft
of velocity
aircraft
in undisturbed
along
flight
flight
z axis in disturbed
flight
incidence
frequency
- 27 -
in disturbed
of short
period
&
oscillation
rl
LIST OF SS'MBOI.3 (Con&)
longitudinal
force
derivative
due to forward
longitudinal
force
derivative
due to incidence,
longitudinal
force
derivative
due to rate
1 aCx
=ET--Tyxr
longitudinal
force
derivative
due to elevator
2u =- 21 a 54
normal
force
derivative
due to forward
velocity,
u
zw =- 21 acz
normal
force
derivative
due to heaving
velocity,
w
1 a %
normal
force
derivative
due to rate
1 a %i
% =73-q-
normal
force
derivative
due to elevator
Y, rad
angle
X
1 a cx
w
Yt
6 = -- p *v
velocity,
F
of pitch,
of pitch,
u
q
deflection,
q
deflection,
V
of climb
elevator
moment coefficient
iB
E rad
downwash angle
short period
damping
'a = ZE
$ =ikf
damping ratio,
lift
contribution
rate
of pitch
rate
of change of incidence
ratio
of actual
to critical
to damping ratio
contribution
- 28 -
to damping ratio
contribution
to damping ratio
77
LIST OF SYNB0I.S (Contd.)
%=&
phugoid
damping term
phugoid
damping term
q rad
elevator
deflection
0 rad
angular
displacement
W
P= gpse
aircraft
V=-
3
rotary
density
in pitch
from equilibrium
position
parameter
damping coefficient
iB
slugs/f$
p
air
density
T=-t
e
aerodynamic
9 degs
phase angle
A# I 4
0? E
corrections
to short period phase angle for
acceleration
and rate of pitch respectively,
for the effect
of elevator
lift
xzmrn$
rate
time
of change of incidence
incidence,
to allow
damping coefficient
iB
Pm
cd=--
W
static
stability
coefficient
iB
y=t?l c
“w F
non-dimensional
elevator
lift
ratio
coefficient
defining
derivative
to elevator
of restoring
margin
the ratio of the
moment derivative
to elevator
lift
arm
LIST OF ijXZ%RENCES
-’No
1
2
Author(s)
Title,
etc.
James, 13.X.
Nichols, N.B.
Phillips,
R.S.
Theory of servo-mechanisms.
McGraw-Hill
Book Co. Inc., New York.
(X.1.T. Radiation
Lab. Series No. 25)
Schetzer,
Notes on dynamics for aerodynamicists.
Douglas Report No. SN-14077.
J.D.
1955.
- 29 -
1947.
LIST OF 'RZFE2ENCSS (Con%%)
&.
etc.
3
Jaeger,
J.C.
An introduction
to the Laplace
Methuen ard Co. Ltd., London
1953.
4-
Porter,
A.
An introduction
to servo-mechanisms,
Methuen and Co. Ltd., London,
1954-O
Lawden,
D.F.
Mathematics of engineering
systems.
Methuen and Co. Ltd., London.
1954-O
7
E
Title,
Authw(s)
transformation
oscillaof flight
Neumark, S.
Analysis of short-period
longitudinal
tions of an aircraft
- Interpretation
tests.
2. & i% ql+o.
1952.
Duncan,
The principles
of the control
and stability
aircraft.
Cambridge University
FYess, Cambridge.
1952.
W.J.
-
30 -
of
.
c
.
TlUXXER
PUXCTICNS FCR TIB CZXER&L XJCNGITUDINAL CASE
functions
are given
Bithin
the assumptions stated complete transfer
below for each of the dependent variables
in the commoriLy.accepted equations
of motion in the longitudinal
plane (apart from that for 67~ which is simply
Additionally,
the transfer
the integral
of the expression
for -CT).
functions
for incidence
and rate of pitch have been combined to give the
corresponding
functions
for normal acceleration.
Assumptions
It
is assumed that
1
there
is no coupling
2
the aircraft
3
the system is linear.
Equations
is rigid
(D >
U
u
Transfer
lateral
modes,
and
of motion
With these assumptions
axes are
-X
with
-
the equations
of motion
written
in stability
xw$
T
functions
The following
transfer
functions
for forward speed, incidence
and- rate
of pitch are obtained from the above equations by the methods of Section 2.
-
31 -
- t
C-3
B2 p3 + C2 p2 + D2 p + E2
A
= -
P3+
B3
5
vi
(-3
The corresponding
expression
equations
(55), (56) and
-ng
p2 c D3 p + E3
A
(55)
PCC& P2 + D4 P + E4)
.
A
g () 3 v
-=3
for
5
.
C3
(-5)
(54)
normal acceleration,
(56)
obtained
.
= "-qv,
from
(57)
iS
c
- ii
(-5)
~(3~
= -
p3 -I- C5 p2 + D5 p + E )
.
A
A is the characteristic
In these equations
expression
A, p4 + 2, p3 + C, p2 + D, p + E,
and the coefficients
A,
=
B,
=
A
1'
Bl
'
B2 etc.
are as follow:
1
- 32 -
(58)
f
+ k' xu)
m
+ MT x&J
33 c
+ zq) -x
z - &t k'
?I. u
I
m
c- u P 4 cI; - xtp(P + "(J + x z
9 wI
iB c
p mu
-t k’ xu) - (& CL zlv + k' xw)
33
B2
= x,,,
'2
= - fliB
CJ
=
(xw
CL zv + k' xv)
'q _ xrl zw)
(xv zu - xu ZJ
- 2
arl -4 2
-
(I
33-
+ $)
D3 =
E
3
lJ “u (-2 CL zr + k' xrl) - T
= --q
($ CL zu + k' xu>
Pm
D4 = .---x u
lB
E4 =
+
'
-lJ*u (xwzrl - xr zw)+ Ek..,,
( ru
iB
iB
+3(xziB
uw
xwYJ
B5 =
c5
=
m*w
4-7
("7,
=B
zu "
-
34 -
uq 25>
E5
=
p m*
--q-
-I----lJ mr xu(k'
53 c
c E$ + z,($
CL -
t zw) i- z,($
CL - xw)
”
35
Xiv)
-
3
@mw( x
+ - ig
ru 2
- xu %j)
RANGE 0.b VALIDITY OZ' TIE AFl?.iTOXIM.A'lSVALUES
03' THE SHORT -PEJRIODSTABILITY I'ARAXETERS B AND C
In Section
5.3 the quartic
equation
(37))
(Equation
C2
(p2 + B P + C> - gwas expressed
(59)
(p - u)
as
n
=
(p2 -k b p e c)(p2
-t- B'p
+ Cl)
and in the subsequent analysis
it was assumed that B1 = B and C'
B and C were the tvfo degrees of freedom short period damping and
parameters respectively.
The divergence
of the true values,
B*,
the approximate
values, B, C, of these parameters with increasing
shown in Pig.1 6.
(60)
= C, where
stiffness
C', from
CL is
To obtain these true values, Equation (59), with the selected
coefficients,
B = 5, C = 16, a = 5, u r- 2 and C,., = 0.02 -+ 0.08 CL2,
expressed in the form of Equation (60) by iteration,
was
It can be seen from Fig,? 6 that the errors
incurred by making the
assumption are negligible
at CL = 0.33 (i.e. in the examples given
Section S;), and increases with C!L'
At high lift
coefficients,
therefore,
correct
values of the coefficients
must be used,
above
F
-
36
-
in
Values
of aircraft
A
Tailed,
medium
altitude,
subsonic
IFor
aircrai?
Tailless,
high
altitude,
supersonic
Tailless,
high
altitude,
transonic
0.1
0.2
0.2
0.2
0.1
0.05
0, I
0
0.066
0.156
o. 4%
0.05
0.15
0.15
A only:
-I-
B
c
4
4
II,?
and
-
in England
9
0.06gz
R, = 0.06,
0.0036
-0.05
a
(as in Figs.9
cD = 0.029,
- Printed
C
0.2
when CL = 0.33
WZ'.x)78.C.P,U7F;.K3
B
0.6
0.3
CD = 0.022
+
parametera
37
”
cLL
to
12)
R2 = 0,07,
SD = 0.0044
L
ZERO
SLOPE
I
I
0
v
V
3
2
---&P
5
’
NON-01
MENSIONAL
FREQUENCY
X
= 27rft/c
f
FIG. I. THE CHARACTERISTICS
OF THE NONDIMENSIONAL
SHORT PERIOD FREQUENCY RESPONSE
OF INCIDENCE
AND NORMAL
ACCELERATION
I t 83’,2 +
6
ZERO
SLOPE
0
2
I
NON-01MENSlONAL
FREQUENCY,
X=
3
2zrrF $?/!!
LEAD
T
I
I
TAN-’
-2d
Al
LAG
ASYMPTOTE
FlG.2.THE
CHARACTERISTICS
OF THE NONDIMENSIONAL
SHORT PERIOD FREQUENCY
RESPONSE OF RATE OF PITCH
.
0
1
\
\
AIRCRAFT
l
8 AND
Al RCRAFT
I
I
A
0
C
\
C
4
086
0.2
o-2
\
\
\
ACCELERATlON
MODULUS
Al R&AFT
A
0
0
NON
I
- DtMENSlONAL
3
2
FREQUENCY
X
MODULUS
2
I
RAFT
i
INCIDENCE
AND
-to8
ACCELERATIt
PHASE
LAG
6
AND
C
I
A
AIRCRAFT
NON - DlMENSlONAL
PHASE
FREQUENCY
X
LAG
FIG. 3. AN EXAMPLE OF THE NON-DIMENSIONAL
SHORT PERIOD FREQUENCY
RESPONSE OF
INCIDENCE
AND NORMAL
ACCELERATION
AIRCRAFT
I3
I
\
\
T
RATE
A’
B
C
\
\
‘\\ \ \
\ \
OF
PITCH
MODULUS
0.6
o-2
0.2
AIRCRAFT
0*3
0’1
0.2
c
C
\
\
4
2
.
I
t
0
NON-
DIMENSIONAL
FREQUENCY
3c
MODULUS
60”
~~,~‘-+,~/W&A~T
AtRCdAFT
B
?
f
A
\\I
KEH°F
PHASE
ANGLE
AtRCRAFT
C
‘>
-----
\
--
-loo”NON-DIMENSIONAL
FREQUENCY
PHASE
FIG.4.
-
-
X
ANGLE
AN EXAMPLE OF THE NON-DIMENSIONAL
SHORT PERIOD FREQUENCY
RESPONSE
OF RATE OF PITCH
Z-I
K
0YE
YE
A
A
AIRCRAFT
l-02
INCIDENCE
MODULUS
CORRECTION
AIRCRAFT
0 0 0
FACTOR
i*or
_
C
-
do)
----
-S-
C--
me-
/--
__
-
-
---
-IAIRCRAFT
6
I-00
I*00
0
NON-
MODULUS
no
no
FREQUENCY
CORRECTION
X
FACTOR
2
I
0
3
2
I
DlMENSlONAL
I
3
AIRCRAFT
8
INCIDENCE
PHASE
LAG
CORRECTlOb
AIRCRAFT
C
\
-6’
\
\
\
\
AIRCRAFT
\
\
\
r \ \
-0’
AIRCRAFT
A
%
O-6
s,d&
0+3
0*2
0.1
B
C
O-2
0.1
o-2
0-I
o-05
0
-0.05
F
O-2
r
0.066
0.156
0.156
A
&
0.05
0815
0~15
\
‘\
\
\
\
\
\
-IO0
PHASE
FIG.5
ELEVATOR
LAG
CORRECTION
AN EXAMPLE
OF THE EFFECT OF
LIFT ON THE FREQUENCY
RESPONSE
OF INCIDENCE
TO ELEVATOR
AIRCRAFT
I.6
A
8
C
O-3
0.1
0.2
0.2
0-I
0.1
0
0#05-0.05
0~156
0456
0~15
O*l5
/
K
“E
AIRCRAFT
NORMAL"4~
ACCELERATION
MODULUS
1
CORRECTION
FACTOR
(ALMOST
B AND
1
+
C.
z
IDENTICA
I*2
/
I-0
\
2
NON-
DlMENSlON~‘
3
FREQUENCY
X
0.8
MODULUS
CORRECTION
FACTOR
NORMAL
ACCELERATION
ACCELERATIC
PHASE
LAG
CORRECTION
2O
AIRCRAFT
B
f3
AIRCRAFT
AIRCRAFT
C
C
/
2
I
NON
- DIMENSIONAL
PHASE
LAG
FREQUENCY
CORRECTION
FIG. 6. AN EXAMPLE
OF THE EFFECT OF
ELEVATOR
LIFT ON THE FREQUENCY
RESPONSE
OF NORMAL
ACCELERATION
TO ELEVATOR
I*05
AIRCRAFT
c
8
I.00
095
RATE OF
PITCH
MODULUS
CORRECTION
AIRCRAFT
AIRCRAFT
I
FACTOR 0190
I
1
I
AIRCRAFT
AIRCRAFT
MODULUS
I
I
A
0
Aa’eE
-c
AIRCRAFT
-\
-*O\\
\
/
/’
-*O\\
RATE
OF
PITCH
PHASE ANGL
\
4 1
-4”\\
CORRECTION
/
‘\\
\ \
\
/
/
/
-/
0
0
PHASE
4’
0
/
ANC
FIG. 7 AN EXAMPI
ELEVATOR LIFT ON TH
OF RATE OF PI
4
AIRCRAFT
A
4
O-6
%
O-3
/f
0.2
9,
0.1
‘F
0.066
F
O-05
8
0.2
0.1
0-J
0
0.156
0.15
5 6 78910
4CY x
20
30
‘ATOR LIFT ON THE MODULUS
ACCELERATION
TO ELEVATOR
.
OF THE
33 0.020
4
4
II*1
0.6
O-3
Od-
0*2
0.06
O-07
O-0036
0.004r
3
AIRCRAFT
A
0*3
0*4
INCIDENCE
MODULUS
PHUGOID
0
0-I
FREQUENCY
0.2
NON-
DIMENSIONAL
o-5
FRCWENCY
X
MODULUS
PHUGOID
FREQUENCY
0.2
0.1
o-3
0.4
o-5
INCIDENCE
PHASE
LA6
-Id
B--w
SHORT
FREQUENCY
PHASE
PERIOD
FREQUENCY
RESPONSE
RESPONSE
INCLUDING
PHUGOID
LAG
FIG.%AN EXAMPLE OF PHUGOID
FREQUENCY
RESPONSE
CURVES FOR INCIDENCE
0*6
I
CL
co
’ a
8
c
I f
0.33
0~029
4
4
11-f
006
RI
0.06
R2
0.07
Jf$
OeOO36
90
0,0044
Ai
‘24
I*
0.3
0.2
0.1
20.066
F
0.05
i
‘ACCELERATI
MODULUS
PHUGOID
FREQUENCY
I
I
0.6
NON-
DIMENSiONAL
FREQUENCY
SC
MODULUS
t 160'
----
SHORT
PEiRlOD
FREQUENCY
+8d
FREQUENCY
RESPONSE
RESPONSE
INCLUOING
PHUGOIO
+4d
ACCELERATI
PHASE
ANGLE
0
FREQU
PHASE
ANGLE
FIG.10. AN EXAMPLE
PHUGOID
RESPONSE CURVES FOR NORMAL
FREQUENCY
ACCELERATION
I
CL c,
8
0.33
O-029
RI
a 16
c
4
II.1
4
Jlo
O-3
096
R2
6
RATE
OF
PITCH
MODULUS
o-2
NON
-
O-3
DIMENSIONAL
0.4
x
FREQUENCY
MODULUS
SHORT
-m--w
PERIOD
FREQUENCY
FREQUENCY
RESPONSE:
RESPONSE
INCLUDING
PHUGOID
+4cf
i
RATE OF
PITCH
PHASE
?
LEAD
ANGLI
---,-I
0”
0-I
It
PHUGOl5
-----b-d--
o-2
0*3
FREQUENCY
LAG
PHASE
ANGLE
FIG. II- AN EXAMPLE
OF PHUGOID
FREQUENCY
RESPONSE CURVES FOR RATE OF PITCH
9
8
7
t
I
c
I
AIRCRAFT
A
I’
I
i
I’
I’
6
NORMAL
kCCELERA~ON
AND
RATE
OF PltCH
;I
I’/’
ll
K
2
0 “P
Il
5
K
CL
co
O-33
6
c
4
ltdl
0029
“4
R!2
O-07
96
O-0036
1
Oa6
I 1
“P
AND
I’
II
II
I I
4
I I
I I
II I1I
’ I
1 I
1 I
RI
0 l 06
30
000044
lr
0.066
4,
083
9v
0*2
s$
O*f
F
O-05
t
3
G
MODULUS
CORRECT/ON
FACTORS
2
-
PHUGOID
0-I
FREQUENC:Y
0.2
NON
- OIMENSIONAL
0.4
O-3
FREQUENCY
FIG.12. AN EXAMPLE
OF THE EFFECT OF THE
PHUGOID
OSCILLATION
ON SHORT
PERIOD
FREQUENCY
RESPONSE
MODULI
O-6
T
6
O-6
t
IO
FREQUENCY
RATIO
C‘
I
R,
30
NON-DIMENSIONAL
0 74
FIG. 13. FREQUENCY
PHUGOID
CAUSES
AT WHICH
5O/o ERROR
FREQUENCY,
X
I
Oj6
0
,,8
I* 0
NEGLECT
OF
IN MODULUS
6
7
IDENCE
ACCELERATION
0.6
\
RAT7 3F PITCH
IO
FREQUENCY,!
RATIO
CL
R
I
0.4
NOTE
15
NEGLECT
?
o-2
3c
-
0
OF THE
P HUGOID
GIVES
COMPUTED
VALUES
OF ACCELERATION
-1
WHICH
ARE LOW BY
THE AMOUNT
SHOWN
/
INCIDENCE
VALUES
ARE
HIGH
PERCENTAGE
-4
-2
ERROR
FIG. 14. PE6 ,ENTAGE ERROR IN MODULUS
a iNCURRL
.
AT SHORT PER OD FREQUENCY
(X=
I) BY
NEGLU :T OF PHUGOID
.
T
i%
k
0.6
4C
FREQUENCY
RATIOS
I
RI
I
AIRCRAFT
30
I
2c
I/
R2
I’0
0.1 08066
I
Z
AND
0.3 0.2
A
I
I
0*05
4
4
11-1
I
SHORT
PERIOD
FkEQUENCY
PHUGOlb
FREWJENCY
‘
c
0
0
FIG.15 RATIO
0.2
OF SHORT
0.6
PERIOD
FREQUENCY
TO PHUGOID FREQUENCY,
& AND THE RATIO
I
I
&FOR DIFFERENT
LIFT COEFFICIENTS
(AIRCRAFT
A)
7,
8’
40
30
C’
l
----Ms--
c-----(SPEED
AS
l
cL
3
ASSUMED
------
IN PHUGOID
C+NST
5
4
$
---
EXAMPLES)
6
1
AND
I
PERIDD
NATURAL
FREQUENCY
Rz IO
0
THE
1
ASSUMED
VALUES
2
CL
3
64,
C-16,
r=
c1)=0*02+
WERE
USED
IN EQ.37
TO OBTAIN
cp2+
( p2t6p
c,,
+ C
X
+ p2)
p2+
B’p
(p *t&
+C’)
4
6
2.5, V= 2,
O-08
cf
+c)-;
BY
5 ?(p-v)=A
ITERATION
FIG.16. COMPARISON
OF TRUE VALUES B;C’ AND
APPROXIMATE
VALUES
B,C OF THE SHORT
PERIOD STABILITY
PARAMETERS
C.P. No. 476
(21,607)
_
A.R.C. Technical Report
0 Crow Copy&# 1960
Published by
HER MAJESTY’S
STATIONERY
OFFICE
To be purchasedfrom
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or through any bookseller
Printed in Engtand
S.O. Code No. 23-9011~76
C.P. No. 476