close

Вход

Забыли?

вход по аккаунту

?

6.1989-3392

код для вставкиСкачать
89-3392-0
Deficiencies of Long-Term Dynamics Requirements
and New Perspectives
Gottfried Sachs
*
Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1989-3392
Technische Universitat Miinchen
Miinchen, Germany
Abstract
=aC~/da
= aC~/a6,
The long-term dynamics of aircraft in supersonic flight at
a Mach number M 2 are considered and it is shown how
their characteristics compare with existing flying qualities
requirements and flight standards.
>
One topic dealt with is concerned with the correlation between airspeed and pitch attitude for commanded pitch
attitude changes. It is shown that the airspeed/pitch attitude correlation in supersonic flight is not unequivocal
and substantially differs from the relationship for subsonic
and transonic flight. The flight mechanics factors underlying are identified and it is shown that the height mode
which is an additional mode existing in supersonic flight
exerts a significant influence.
lift coefficient
=~ C L / ~ ( U / ~ O )
= aCL/acu
= BCL/B6,
pitching moment coefficient
pitch damping, Cmq = 2aCm/a(q~/Vo)
speed dependent pitching moment,
Cmu = a c m l a ( u / v ~ )
= 8Cm/da
=aacm/a(~qvo)
= dCm/a6,
Another topic considered is concerned with aperiodic airspeed divergence. It is shown that there are deficiencies
in regard to the connection considered to be existent between the requirement concerning aperiodic airspeed divergence and related control gradients. There may be
aperiodic instability despite the fact that the control gradients would indicate stability. The opposite can also be
true. It is shown which are the reasons for these characteristics. Here again, the height mode exerts an influence.
altitude dependent pitching moment,
cmh
= (l/~h)acm/ah
mean aerodynamic chord
acceleration due to gravity
altitude
radius of gyration
Mach number
factor describing thrust speed dependence
Nomenclature
factor describing thrust altitude dependence
dynamic pressure, ij = (p/2)V2
A(s)
coefficient matrix of the homogeneous system
reference area
B
scaling matrix for control inputs
Laplace operator
CD
drag coefficient
height mode root
cDU = a c ~ / a ( ~ / v ~ )
*
thrust
Professor Dr.-Ing. Gottfried Sachs, Director of the Institute of Flight Mechanics and Flight Control.
Associate Fellow AIAA.
Copyright O American Institute of Aeronautics and
Astronautics, Inc., 1989. All rights reserved.
time
speed pertubation
are adequate for subsonic and transonic flight. It will be
shown which are the stability and control characteristics
that lead to the differences existing for supersonic flight.
airspeed
It may be of interest to note that existing flight standards
variable vector
thrust-line offset
angle of attack
pitch attitude angle
especially developed for supersonic aircraft (Refs. 4, 5 )
show similarities to the requirements described. Therefore, the discussion and findings in the present paper may
also be of interest for these standards.
relative mass parameter, p = 2 m / ( e S ~ )
Flight Dynamics Considerations
Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1989-3392
pitch control surface deflection
air density
density gradient, ph = (l/po)de/dh
phugoid damping
reference time, r = c/Vo
phugoid frequency
In supersonic flight at a Mach number M = 2 or beyond,
forces and moments due to altitude perturbations have a
significant effect on dynamic stability of horizontal flight
(Refs. 6 - 10). Accordingly, these forces and moments
must be taken into account in the mathematical model as
expressed in Eqs. ( l a - d ) for the linearized equations of
the longitudinal motion for horizontal flight
where
Introduction
X(S) = [u/&
a
A ~ / F ] ~
(1~)
Recently, flying qualities have been proposed as MILPrime Standard and Handbook "Flying Qualities of Piloted Vehicles", Refs. 1, 2. In the meantime, the requirements are published in their final form, Ref. 3.
Some of the requirements are concerned with the longterm dynamics of aircraft. They address flying qualities
for longitudinal (speed) axis and for long-term pitch response. The requirements relate control and response
characteristics to the long-term stability of the aircraft.
Particular attention is given to aperiodically divergent
modes of motion. The requirement for longitudinal (speed)
axis is intended to assure that there is no aperiodic instability in attitude and airspeed.
It is the purpose of this paper to show that discrepencies
exist between the flying qualities requirements under consideration and the dynamics of aircraft in supersonic flight
(at a Mach number M = 2 or beyond). The requirements
In particular, the long-term dynamics of the aircraft are
substantially influenced by forces and moments due to altitude perturbations when flying at supersonic speed (see
Fig. 1). Firstly, altitude dependent effects result in the
occurence of a third mode of motion (not existent in subsonic and transonic flight). This mode called height mode
shows an aperiodic behavior which may be stable or unstable. It can be expressed as (Cm, = 0, Cmh = 0)
where
Correlation between Airspeed
and Pitch Attitude
Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1989-3392
denote the dependence of thrust on speed and altitude,
respectively.
From Fig. 1 it follows that the height mode appears
much more pronouncedly in airspeed than in altitude response characteristics. It is even predominating airspeed
response practically after a time interval no longer than
one phugoid period.
As to the second item concerning long-term dynamics,
the well-known phugoid is dominated by altitude effects
in supersonic flight. This becomes evident by the following expressions for frequency and damping (emu
= 0,
Cmh = 0)
From the expressions presented it follows that altitude
perturbations exert a significant influence. This particularly holds for phugoid frequency which no longer shows a
decrease with airspeed (as in subsonic flight) but is even
independent of it and solely determined by the change
of density with altitude e h = (l/e)dp/dh. In addition,
phugoid damping is significantly influenced by the change
of thrust with altitude whereas the effect of thrust/speed
dependence can be neglected (Ref. 10).
A quantity needed in the following is the steady-state correlation between airspeed and pitch attitude. It may be
derived from Eqs. ( l a - d) by using the relation
and accounting for s = 0 (steady-state). Thus, the steadystate correlation between airspeed and pitch attitude may
be expressed as (with C L and
~ C D neglected)
~
One of the flying qualities requirements mentioned is concerned with the correlation between airspeed and pitch
attitude. In this requirement, it is required that airspeed
tracks pitch attitude in the conventional way both in the
long and short term when pitch attitude changes are commanded. This means that an increase in pitch attitude is
correlated with a steady-state decrease in airspeed and
vice versa.
The airspeedlattitude consonance as prescribed by this
requirement is generally existent for stable aircraft in subsonic and transonic flight (leaving out forms of direct force
control such as direct lift control or autothrottle the effects of which are not considered here). This is because
there are relationships with unequivocal correlation between airspeed and pitch attitude both for short-term response as well as for steady-state response. The relationship for the latter may be expressed as
In supersonic flight (M 2 2), this no longer holds. Whereas the short-term response characteristics basically show
the same behavior as in subsonic flight, the steady-state
correlation is substantially altered. It may be expressed
as (for stratospheric flight where nh = 1, see Eq. (6))
This expression is no longer unequivocal in regard to the
correlation between airspeed and pitch attitude. This
is because the correlation can change its sign. A dominant effect is due to the change of thrust with speed
nu = (Vo/T)dT/du.
The reason why the steady-state correlation between airspeed and pitch attitude is no longer unequivocal and substantially altered (when compared with the classical relationship according to Eq. (7)) is due to the fact that the
height mode is existing in supersonic flight as an additional
mode of motion. This mode has a substantial effect on the
long-term response characterisics of aircraft.
Further insight is provided by Figs. 2 and 3 which show the
responsein pitch attitude and airspeed to a step pitch control deflection input. Fig. 2 shows the classical response
characteristics corresponding to subsonic flight. In this
Fig., a distinction has been made between a time region
called intermediate which corresponds to the time region
Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1989-3392
of the &st phugoid periods and the long-term response (or
steady-state response). In regard to the classical behavior
shown in Fig. 2, the long-term response may be considered
as a kind of average value during the intermediate time region. This no longer holds for supersonic flight as shown in
Fig. 3. In supersonic flight, the long-term response characteristics completely differ from the intermediate response
characteristics, even showing a sign change in pitch attitude response. This behavior is due to the existence of the
height mode. From Fig. 3 it also follows that airspeed response characteristics are dominated by the height mode
whereas the phugoid is of minor influence. This already
holds for the intermediate time region despite the fact that
the height mode develops more slowly than the phugoid
(i.e., despite Ish[ < wp).
The results shown in Fig. 3 represent basic response
characteristics of an aircraft where the long-term modes
of motion are not significantly changed or even suppressed
by an augmentation system. Such a system which may
utilize moment feedback loops for improving stability and
control can have a substantial effect on the mode shapes.
Nevertheless, the correlation between airspeed and pitch
attitude may not be changed. This is because Eq. (6) or (8)
still holds for all type of moment feedback loops (neglecting
the effect of CDg and CLs). An example is presented in
Figs. 4 and 5 which show the characteristics of an aircraft
with a combined speedlaltitude feedback loop. From
Fig. 4 it follows that such a feedback loop provides an
effective means for stabilizing the long-term dynamics of
aircraft. The response characteristics are shown in Fig. 5.
As may be seen, the response is significantly altered as
far as the time history is concerned. However, the correlation between airspeed and pitch attitude still shows the
same sign as for the unaugmented aircraft considered in
Fig. 3.
In Fig. 6, a changed correlation between airspeed and pitch
attitude is shown. This change is due to altered thrust
characteristics which are such that thrust dependence on
airspeed is now considered as zero (aTli3u = 0). From the
results presented in Fig. 6, it also follows that the change in
the correlation is due to a change in pitch attitude response
characterisics.
In regard to the modes of motion however, the results of
Fig. 6 again demonstrate that the height mode is the
mode exerting a substantial influence on the correlation
between airspeed and pitch attitude. By contrast, phugoid
characteristics appear to be of minor influence. Thus, the
results of Fig. 6 confirm the results discussed before that the
existence of the height mode is the reason for the difference
in airspeed/pitch attitude correlation existing between the
subsonic and supersonic flight regimes.
It may be emphasized that the correlation between airspeed and pitch attitude as shown in Figs. 3 and 5 belongs to aircraft with usual characteristics and that this
correlation is not caused by a special feature like a form
of direct force control which is generally considered as a
possible reason for a non-consonanceof airspeed and pitch
attitude. In addition, it may be of interest to note that
there is no aperiodic instability in attitude and airspeed as
regards the aircraft behavior shown in Figs. 3 and 5.
Usually, aircraft that meet the equivalent phugoid and
short-period requirements are considered to automatically
meet the requirement concerning the correlation between
airspeed and pitch attitude. However, such a firm connection between phugoid and short-period requirements
and airspeedlpitch attitude correlation is not existent in
supersonic flight.
Prohibition of Aperiodic
Airspeed Divergence
The long-term stability of aircraft in supersonic flight is
addressed in various flight standards and flying qualities
requirements (e.g., see Refs. 1, 4, 5). They show basically similarities but may have different levels of details in
the individual formulation of the requirements and in their
supporting documents and background information.
According to Ref. 1, the requirement concerning airspeed
divergence is intended to assure that there is no aperiodic
instability in attitude and airspeed. It will be shownfor supersonic flight that there are deficiencies in regard to the
connection considered to be existent between the requirement and related control gradients. These control gradients are the gradients of pitch control force and deflection
with airspeed. According to the requirement guidance, the
requirement is considered satisfied if the gradients of pitch
control force and deflection with airspeed are stable (i.e.,
an airplane-nose-down pitch control yields a steady-state
airspeed increase and vice versa).
Basically, an aperiodic instability may (or may not) exist
in supersonic flight due to the existence of the height mode.
This is illustrated in Fig. 7 which shows an aperiodic airspeed divergence as well as a stable characteristic. There
is no connection between this kind of aperiodic instability
and the pitch control gradients as used in the requirement
on the prohibition of aperiodic airspeed divergence. This
is because altitude dependent effects on dynamic stability
Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1989-3392
are not accounted for in the pitch control gradients under
consideration. For flight testing at high speed, a measurement technique for the control gradients is recommended
where the altitude is considered as constant. This means
that such a control gradient cannot account for forces and
moments due to altitude changes because altitude changes
do not occur during the test. Referring to Fig. 7, the pitch
control gradient with airspeed is stable for all aircraft considered.
Usually, pitching moments due to speed (or Mach number)
are considered to have a significant influence on aperiodic
airspeed divergence and on the pitch control gradients with
airspeed. Such moments may be caused by compressibility
effects, thrust-line offset (zr) or aeroelastic deformations
due to dynamic pressure changes, thus resulting in an effective pitching moment stability derivative
In addition, C,, may also be considered as a moment
artificially introduced by feeding back velocity to the pitch
control surface. Such a means may be used for improving
stability and control characteristics.
Fig. 8 shows the effect of C,, in supersonic flight. For
the pitch control gradient, substantial changes exist. In
particular, the negative C,, value considered results in
unstable gradient. However, the dynamic characteristics
show only minor changes. This particularly concerns the
case where the control gradient is unstable. For such a
case, the phugoid is generally considered to be aperiodically divergent. However, no such aperiodic phugoid instability exists. Rather, the phugoid still is an oscillation
the characteristics of which show only slight differences
compared to the stable pitch control gradient cases.
Further insight is provided by Fig. 9 which shows an evaluation of the effect of C,, on phugoid and height mode
eigenvalues. It confirms the rather insignificant influence
of C m UParticular
.
attention is given to the value
of pitching moments due to speed or Mach number on dynamic stability in subsonic and transonic flight, e.g., the
well-known tuck-under effect.
An effect not considered in existing requirements and flight
standards concerns pitching moments due to altitude perturbations. These moments may be caused by thrust-line
offset or aeroelastic deformations due to dynamic pressure
changes, thus resulting in an effective pitching moment stastability derivative
In addition, Cmhmay also be considered as an artificial moment resulting from an altitude feedback loop for
improving the stability and control characteristics of aircraft.
The stability derivative Cmhis of great significance on
dynamic stability in supersonic flight. By contrast, its
effect in subsonic and transonic flight can be neglected.
The significance of this stability derivative for supersonic
flight has been recently reported, using YF-12 aircraft
data (Refs. 9, 11).
In regard to the requirement concerning aperiodic airspeed divergence, no such effects are taken into account.
Fig. 10 shows that Cmhhas a substantial effect on dynamic stability, particularly in regard to aperiodic instability. However, the pitch control gradient does not show
a change. Even for the aperiodic instability introduced
by C m hthe
, control gradient would still indicate a stable
characteristic.
Further insight is provided by Fig. 11 which shows the effect of Cmhon phugoid and height mode eigenvalues. Positive Cmhvalues have primarily an influence on phugoid
frequency. The main effect of negative values concerns
aperiodic instability which can reach a severe level. Aperiodic instability exists when Cmhis more negative than
This critical value is also indicated in Fig. 11.
This value denotes the boundary between a stable and an
unstable pitch control gradient. According to Fig. 9, the
change in dynamic stability appears to be rather insignificant. In particular, the phugoid still exists as an oscillation and its stability characteristics are not significantly
changed even if Cmutakes on much more negative values
usually considered as introducing severe dynamic instability (particularly aperiodic instablility). The results shown
in Fig. 8 and 9 are in contrast with the significant effects
Further Considerations
From the results presented it follows that new perspectives
exist for flying qualities in supersonic flight. In particular,
the existence of the height mode introduces such a new
perspective. According to Eq. (2), this mode is primarily
determined by thrust characteristics, with thrust dependence on speed as well as on altitude exerting an influence.
The height mode which is an aperiodic mode of motion
Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1989-3392
may be stable or unstable. Thrust increase with speed
(aT/du > 0, a characteristic existing in supersonic flight)
contributes to instability, whereas thrust decrease with altitude ( a T / a h < 0, typical for airbreathing engines) has
an effect increasing stability. The height mode is a slow
mode of motion. Nevertheless, it is significant for response
characteristics. This holds even for the time scale of the
phugoid in terms of one phugoid period. This is particularly true for airspeed response characteristics for which
the height mode may be qualified as the dominant mode
of motion (refer to Fig. 1).
Another perspective concerns the phugoid because of the
dominating influence of forces and moments due to altitude changes. By contrast, speed effects are insignificant.
This is illustrated in Fig. 12 which shows the mode shape
of the phugoid and a time vector presentation identifying
the individual contributions of the forces perpendicular to
the flight path. The latter makes evident the dominant
effect of forces due to altitude changes as compared with
speed influence for supersonic flight. It may be of interest
to note that the opposite is true in subsonic and transonic
flight. This is also illustrated in Fig. 12.
A particular topic in regard to flying qualities of the longterm dynamics is concerned with the combination of a divergent phugoid and height mode. The case of such a combined instability (not known for subsonic and transonic
flight) would seem to be definitely worse than a divergence in only one mode of motion. It would seem reasonable that the boundary acceptable for such a combination
from a flying qualities standpoint should be more stringent than the boundary for the individual modes. This
problem may be aggravated by the lack of tactile cues.
Therefore, longer minimum times may be required to adequately control such combinations of divergent modes of
motion.
which is an additional mode of motion in supersonic flight
exerts a significant effect on this correlation. A factor of
particular relevance is the change of thrust with airspeed.
Other flying qualities requirements or flight standards are
concerned with the long-term stability, with particular attention given to the prohibition of aperiodic airspeed divergence. It, is shown that there are deficiencies in regard
to the connection considered to be existent betwen this requirement and related control gradients. The height mode
which may be aperiodically stable or unstable shows stability characteristics which are not related to the control
gradients under consideration
Pitching moment stability derivatives due to speed are usually considered to have a strong influence on the long-term
stability. It is shown that this influence is significantly
reduced in supersonic flight. As a consequence, the control gradients may indicate severe aperiodic airspeed divergence whereas the long-term stability characteristics may
show minor changes only. In particular, the phugoid can
still exist as an oscillatory mode of motion despite the fact
that there is an unstable control gradient which is usually
associated with aperiodic phugoid instability.
It is shown that pitching moment stability derivatives due
to altitude have a significant effect on long-term dynamics
in supersonic flight. However, they do not have any influence on the pitch cont,rol gradients as used in the requirements. Thus, severe aperiodic instability can be caused
by such pitching moments, and the pitch control gradient
would still be stable. The critical derivative value beyond
which aperiodic instability exists is described.
Further considerations concern additional characteristics
of the height mode and the phugoid in regard to their time
behavior and their mode shapes.
Conclusions
On the basis of a dynamic stability analysis, the long-term
dynamics of aircraft in supersonic flight at a Mach number M = 2 or beyond are considered and it is shown how
their characteristics compare with existing flying qualities
requirements and flight standards.
In regard to the requirement concerning the correlation between airspeed and pitch attitude, it is shown that the supersonic flight regime shows substantial differences to the
subsonic and transonic regime. In particular, the classical
airspeed/pitch attitude consonance in the short-term and
long-term response characteristics no longer exists. This
is because the correlation between airspeed and pitch attitude is not unequivocal. It is shown that the height mode
References
[I] Flying Qualities of Piloted Vehicles - MIL Standard
and Handbook. Draft prepared by AFWALIFIGC,
Wright Patterson AFB, 1986.
[2] Woodcock, R. J., Browne, J. T., "The MILPrime Standard for Aircraft Flying Qualities,"
AIAA Atmospheric Flight Mechanics Conference
Proceedings, pp. 6-1 - 6-15, 1988
13) Woodcock, R. J., "A Second Look at MIL Prime
Flying Qualities Requirements," in: AGARD-LS-157
"Advances in Flying Qualities," pp. 6-1 - 6-15, 1988.
[4] Special Conditions for the SNIASIBAC Concorde
Model Airplane, FAA, Washington, D.C., 1972.
[5] TSS Standard No. 3-5 "Handling Qualities," Civil
Aviation Authority, London, 1976.
1 I
I [LONG-TERM]
I
/INTERMEDIATE
I
[6] Etkin, B., "Dynamics of Atmospheric Flight," Wiley,
New York, 1972.
I
I t w t p
I
I
Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1989-3392
[7] Stengel, R. F., "Altitude Stability in Supersonic
Cruising Flight," Journal of Aircraft, Vol. 7, Sept.
1970, pp. 464 - 473.
[8] Sachs, G., "A New Concept of Static Stability and
Its Flight Testing in Supersonic Flight," Journal of
Aircraft, Vol. 14, Sept. 1977, pp. 874 - 880.
[9] Berry, D.T., "Longitudinal Long-Period Dynamics
of Aerospace Craft," AIAA Atmospheric Flight
Mechanics Conference Proceedings, pp. 254 - 264,
1988.
[lo] Sachs, G., "The Effect of Thrust/Speed Dependance
on Long Period Dynamics in Supersonic Flight,"
Journal of Guidance, Control, and Dynamics
(accepted for publication 1989/90).
[ll]Powers, B. G., "Phugoid Characteristics of a YF-12
Airplane with Variable-Geometry Inlets Obtained in
Flight Tests at a Mach Number of 2.9," NASA TP
1107.1977.
Fig. 2 Airspeed and pitch attitude response characteristics in subsonic flight (step pitch control deflection, t p : phugoid period)
/ ALTITUDE 1
TOTAL RESPONSE
500
PHUGOID
.
Ah
[ml
u-t b l
-5
TOTAL
RESPONSE
I
'-------
HEIGHT MODE
PHUGOID
\
1
1
I
t/b
1 AIRSPEED I
200
' ['I
MODE
roo
TOTAL
RESPONSE
I .! I
PHUGOID
1
TOTAL RESPONSE
'
Fig. 1 Long-term dynamics in supersonic flight (altitude and speed response to a step pitch control
deflection input, Mo = 3.0)
*)INCLUSIVE SHORT PERIOD
CONTRIBUTION
Fig. 3 Airspeed and pitch attitude response characteristics in supersonic flight, Mo = 3.0 (step pitch
control deflection input, t,: phugoid period)
Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1989-3392
NOMINAL
VALUE
FOR €3
a-
.l
['/'I
u/Vo0
1
VALUE
FOR FIG. 5
-x
-. 1
.o
0
'7
-4:.
..? - - - - - .-.
-....4 - - - - - I
HEIGHT
I
[l/~]
*)INCLUSIVE SHORT PERIOD
CONTRIBUTION
Fig. 4 Effect of speed and altitude feedback on phugoid
and height mode eigenvalues in supersonic flight
(from Ref. 9)
Fig. 6 Changed correlation between airspeed and pitch
attitude response characteristics in supersonic
flight, Mo = 3.0 (step pitch control input,
t p : phugoid period)
PITCH CONTROL GRADIENT
(STABLE)
0.5
1
V Im/sl
VALID FOR
AIRCRAFT 1, 2, 3
AIRSPEED
Fig. 5 Airspeed and pitch attitude response characteristics of an aircraft with speed and altitude feedback in supersonic flight (from Ref. 9)
RESPONSE (STEP 6, INPUT)]
Fig. 7 Pitch control gradient and height mode stability
characteristics in supersonic flight ( M o = 3.0)
Aircraft 1: height mode stable
Aircraft 2: aperiodic instability of height mode
Aircraft 3: increased aperiodic instability of
height mode
1 PITCH CONTROL GRADIENT I
PITCH CONTROL GRADIENT
(STABLE)
0.5 1
Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1989-3392
-0.5
1
/
(AIRSPEED RESPONSE (STEP 6e INPUT)]
Fig. 8 Effect of C m Uon pitch control gradient and dynamic stability in supersonic flight ( M o = 3.0)
FREQUENCY
Fig. 10 Effect of C,,, on pitch control gradient and dynamic stability in supersonic flight (Mo = 3.0)
FREQUENCY
DAMPING
DAMPING
. .I
ap [ l
/sI
1 HEIGHT
Fig. 9 Effect of C,, on phugoid and height mode eigenvalues in supersonic flight ( M o = 3.0)
C k U : boundary value for stablelunstable control gradient
MODEI
Fig. 11 Effect of C,,, on phugoid and height modeeigenvalues in supersonic flight (Mo = 3.0)
(Cmh)crrt:boundary value for aperiodic instability
I SUPERSONIC
FLIGHT (Mo = 3 )
1
ALTITUDE Ah [krn]
pp!5tp
SPEED u/V0
Downloaded by UNIVERSITY OF NEW SOUTH WALES (UNSW) on October 27, 2017 | http://arc.aiaa.org | DOI: 10.2514/6.1989-3392
TIME VECTOR
I SUBSONIC
Ah [krn]
FLIGHT (Mo=0.25)
SPEED u/V0
1
I
Irn
Fig. 12 Phugoid characteristics in supersonic and subsonic flight
Документ
Категория
Без категории
Просмотров
0
Размер файла
663 Кб
Теги
3392, 1989
1/--страниц
Пожаловаться на содержимое документа