The Stability of Aircraft under Automatic Throttle Control and the Cross-Coupling Effects with Elevator Control

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MINISTRY OF AVIATION
R. & M. No. 3314
¢
AERONAUTICAL' RESEARCH COUNCIL
REPORTS AND MEMORANDA
The Stability of
al l
Aircraft under Automatic
Throttle Control and the Cross-Coupling
Effects with Elevator Control
By D. E. FRY and M. R. WATTS
LONDON HER MAJESTY'S STATIONERY OFFICE
I963
. NINE SHILLINGS NET
The Stability of an Aircraft under Automatic
Throttle Control and the Cross-Coupling
Effects with Elevator Control
By D. E. FRY and M. R. WATTS
COMMUNICATED BY THE DEPUTY CONTROLLER AIRCRAFT (RESEARCH AND DEVELOPMENT),
MINISTRY OF SUPPLY
Reports and Memoranda No. 33±4 *
October, z958
Summary.
The stabifity of an aircraft with automatic throttle and elevator controls has been investigated theoretically
using an analogue computer. Throttle application proportional to change in airspeed,, incidence, or rate of
pitch may provide damping of the long-period motion, but speed is shown to be the most suitable variable.
The control of an aircraft having negative static margin is considered and shown to require either the
addition of an integral control on the throttle or a combination of throttle and elevator controls.
Where it is required to control height by means of the elevator, some aircraft flying under certain conditions
can only be adequately stabilised by means of an automatic throttle control.
1. Introduction.
The longitudinal stability of various aircraft with automatic throttle control has been investigated
theoretically. Several types of throttle control have been studied.
The main effects of elevator controls are known 1, but are summarised in this report. When a
height lock is added to an elevator control an unstable mode is often introduced. Thi s instability
may be removed by the addition of a speed control on the throttle. Each type of throttle control
has been shown to have a different effect on the longitudinal stability, when operated in conjunction
with a typical elevator control. The cross-coupling effects of the two types of control are not always
beneficial.
Use has been made of a Shorts' analogue comput er tO substantiate the theoretical analysis~
results being given as actual responses in pitch, height, etc. to-a horizonthl.step gust~ ~' ,
Replaces R.A.E. Tech. Note No. I.A.P. 1079--A.R.C. 21,0401
2. Ai rcraf t and Control Equations.
The aircraft eqmitions of motion, in non-dimensional form are:'
A-I .t A
Du = x~u -~ x~w = kO + T
~( ~- 0) = z~,a + z~v~
A . , A A. .
Dq = - Ka xDw- ~o~ - vq - 3~7
where
~) d t"
m dO
- ; q=~
pSV
09 = -- ~1- -
= -- lZl
mz~ o
ZB
mw°
IB
/f l y.
z B
and T is a small change of thrust produced by the au(omatm
The elevator-control equation used was:
m~b
X : - - l gl =-
ZB
mq
1,, -
k= ½cL
control.
0)
(2)
, (3)
v = GoO + Gq + G~ + G~f,~& (4)
where Go, Gq, GI~, G~ are control gearings, and the height deviation tz is reiated to 0 and v3 by the
equation
b~ : 0 - ~. (5)
The throttle equation was of the form:
T = A,, a + d,~ + doO.., etc. (6)
Au, dw, A o etc. are control gearings.
The meaning of all symbols other than' those defined can be found in R. & M. 18012.
Since all equations in this report are in non-dimensional form, a, a3, etc. will be written as u, w, etc.
The stability polynomial for the aircraft with controls fixed is a quartic. With the addition of
control laws, the order of the equation may be raised.
Table 2 gives the contributions from the control terms to the coefficients K¢ of the stability
equation
Y, Ki A i = O.
Thus for the control law ~7 = GoO + Gift," the coefficients would be
K4=Ko~
K 3 -- Ko3
K 2 = Ko~ + 3G o
G = Koi + N~3Go
Ko= Koo + P13Go - zw~Gn ,
K 1 = (PI.+R1)3Gh.,
where Koi are the stability coefficients for the basic aircraft.
2
The contributions from any derivative or integral of the control variable are obtained by moving
the value in the table up or down one square. For example, the control law ~1 = G d would add
3G~, N13G ~ and P13Gq to the coefficients Ks, K 2, K 1 respectively.
It should be noted that the coefficient K s is equal to minus the sum of the roots of Y~ K~A i = 0,
and since negative real parts of roots correspond to positive damping it is also equal to the sum of
the damping terms. Thus from Table 2 it can be seen that examples of controls that actually increase
the total damping of the system are: elevator/rate of pitch, (SG~); throttle/speed, (A~); throttle/
vertical acceleration, (A.x). A control which does not add to K 8 cannot affect the total system
damping, but will re-distribute the damping between the several modes of motion.
3. Basic Principles of Elevator Control.
The main features of automatic elevator control are outlined below, but a more complete
description has been given by Hopkin and Dunn 1.
3.1. ~7 = Go O.
This control does not add any damping to the system. The effect of a G o gearing is to increase the
frequency of the short-period motion, and decrease the frequency of the long-period motion,
damping being transferred from the short to the long-period oscillation (Fig. 1).
3.2. "q= G~q.
This control adds damping to the system mainly to the short-period motion.
3.3. = CoO + C,,h = CoO + ;
This height control has the effect of increasing the order of the stability equation and thus
introducing a new mode of motion. The damping of the long-period oscillation is reduced and the
frequency increased (Fig. 2). There is hardly arty effect on the short-period motion. The new mode
may be stable or unstat~le depending on the sign of the combination of aircraft derivatives,
(XuZ~-X~z~) + z~CL/2 , i.e., P1- Rr In Fig. 2, P1- R1 is negative and therefore there is a
divergence. The stability criterion just quoted is only strictly true if z,; = 0.
4. Throttle Control.
The three basic throttle controls are throttle proportional to speed, incidence and attitude or any
derivative of the above three variables. Throttle proportional to errors in speed, incidence, or rate
of pitch s increases the damping of the long-period motion (Figs. 3, 4), but only speed control actually
adds damping to the system; throttle control has little effect on the short-period motion.
Lag on the throttle control of up to about five seconds time constant also has very little effect on
the stability of the aircraft. This is shown in Fig. 5 which gives the effect of different lags on a
speed-controlled aircraft.
5. Comparison of Various Throttle Controls.
The damping power of.the three long-period throttle controls A~,, Aw, Aq depends approximately
on certain combinations of aircraft derivatives. The analysis is simplified if we assume that the
long-period damping is determined mainly by the ratio of the two coefficients/£1, K o of the stability
equation. That this is reasonable can be seen from the following examples.
3
(86574) A*
The stability polynomial
)t 4 + Ka;~ a + K~A ~ + K1;~ +/'2o = 0
usually factorises into two quadratics (As+ AA + B)(A2+ aA + b) where B is large, A is often large,
and a, b are small. The first factor corresponds to a well-damped short-period motion and the
second to a long-period motion. Comparing coefficients of the stability polynomial with those
obtained from the product of the quadratic factors, we have
Ka=A+a
K~ = B+b+Aa
K~ = Ba + Ab
Ko = Bb.
It is seen that K 0 and Kj will be small compared with K~ and Ka, and a first approximation to the
factors will be given by K a = At, K~ = B1, K1 = Bl al, Ko = Bt bl. The coefficient K 2 is said to
be pivotaP because the approximate factors are
(
(z, + + K.) + +
A better approximation for a is obtained from the relation
i.e.,
K 1 = Bl as + At b t ~ K~a~ + K3Ko/Kz,
1
and still better values obtained by writing
A~ = K3- as,
Bz = Ks - bt - A~a~, etc.
However, it is not necessary to proceed any further in order to see that the major contribution to
the long-period damping coefficient a is equal to Kt/K ~ provided K 2 is sufficiently large. It is not
essential that K a be large as well. Two examples are given to illustrate this:
Example 1 Example 2
I£~ 3. 2564 2.6815
Ks 113. 696 (highly pivotal) 9. 5469 (pivotal)
K t 2.5133 1.2253
K 0 - 0. 1467 1" 0742
The approximate and accurate values of a and b are compared in the following table:
Example 1
Example 2
a 1
0.0221
0-1283
12" 2
0.0221
0.0967
a
0.0222
0.1007
bl
--0.00129
0-1125
-0.00130
0.1171
4
Throttle-control terms are important in K 1 (Table 2) but are relatively unimportant in K~, so that
the effects on damping of the three types of control are almost directly proportional to the K 1
contributions, which are - A.~M1, AwM ~ and - AqS. ]1/I1, M~ and S are functions of the aircraft
derivatives: M 1 is proportional to the standard manoeuvre margin, and S to the static margin,
while M S is proportional to another kind of manoeuvre margin suggested by Hopkin. These
three quantities are
M 1 = (~o- vzw) ,
M~ = ( ~- ~.),
S = (~zw-~oz~3.
Thus, by using the known values of the aircraft derivatives, the relative values of A~,: Aw: dq can
be determined to give the same amount of long-period damping. Examples are given in Figs. 3 and 4
for different aircraft and flight conditions.
With 'normal' values of derivatives the two manoeuvre margins and the static margin are all
positive, and therefore the coefficient K 1 and the damping Will be increased by negative values for
A~, and dq, and positive A~.
M~, the manoeuvre margin, is invariably positive for subsonic aircraft, but it is possible for it to
be negative for supersonic aircraft in the subsonic condition.
Ma is usually positive, but could possibly be negative for large negative K(= -tz~mJiB).
The static margin S is also usually positive, but like 3//1 may be negative for supersonic aircraft
in the subsonic condition. This is because rn~ may have to be made positive (oJ < 0) in order to
avoid large negative values at supersonic speed. Quite apart from this the static margin may be
negative because m~, is large (negative); so that S will change sign if z~K becomes larger than zu~o
(Aircraft 2 and 3).
This may happen at transonic speeds (see diagram).
- Cll tl
\
I
1.0 M ~-
Typical relation between m~ and Mach number.
Thus to keep the increment to K 1 of the same sign over the flight range, the sign of the throttle-
control gearing Aq may have to change. Also, since the amount of long-period damping depends
approximately on - Aq(a:z w- ~ozu) and the value in the bracket goes from positive to zero to negative,
and probably to zero and positive again, the value of the Aq gearing would have to vary considerably
and also change in sign, over a comparatively short flight range. Thus it would appear that a rate of
pitch control on the throttle would not be suitable for continuous long-period damping.
( 86574) A* 2
Examples of the relative strengths of the three control gearings for different flight conditions of
the same aircraft are:
Aircraft A~ : A w : A~
(1) 1 : 13.13 : 2-26
(2) 1 : 4.85 : - 198.6
(3) 1 : 13.92 : - 61.61
In order to get the approximate equivalent damping contributions to the long-period motion it
can be seen from the above Table that it is necessary to have an extremely large gearing A t
{Aircraft (2), (3)}. This means that we not only affect coefficient K 1 but also make a large contribution
to K m since - RAp is now comparable with M 1 (Table 2). This will increase the frequency of the
short-period motion and thus the damping per cycle is reduced. The resulting high-frequency
oscillation can be seen in Fig. 6.
The derivatives for the aircraft 1, 2, 3, etc. are given in Table 1.
For aircraft 2, 3 with negative static margin, i.e., coefficient K 0 negative, the uncontrolled
long-period motion is either an unstable oscillation or consists of two exponential modes one of
which must be a divergence. None of the three throttle damping controls will make the system stable,
since no increments of A~, A~o or Aq are added to coefficient K0, which must always remain negative.
One remedy would be to use the integral of the control variable in combination with the basic
control itself, e.g.,
T= A~u + A~ f u d-r.
The strength of the integral term necessary will depend on how negative the static margin is.
For the cases illustrated in Figs. 6 and 7 the static margins are - 0. 0398 and - 0. 1521 respectively,
and the curves show that a compromise must be made between effective control over the divergent
tendency caused by the negative static margin and a conflicting tendency for the integral term to
introduce a long-period oscillation.
If A~ is kept constant and A; increased to a very large value, the stability quartic has two quadratic
factors:
+ + M1) ( & + - = O,
i.e., the basic highly damped short-period oscillation (A ~ + LI~ + M1), and a barely stable very-high-
frequency oscillation.
If, on the other hand, the throttle gearings A,~ and A m are both made large together, but having
a fixed ratio (time constant), ~'1 = A~,/A~, the stability quartic splits up approximately as follows:
( 0
+
i.e., the basic short period again, and two subsidences, one heavily damped and the other lightly
damped. The above two approximations indicate that too large an increase in ~/~ without a
corresponding increase in A** will lead to a poorly damped oscillation.
An alternative suggestion for counteracting the negative components of coefficient K 0 is given
in the next paragraph.
6. Cross-coupling Effects of Various Throttle Controls with Basic Elevator Control.
An elevator control proportional to change in angle of pitch 07 = GoO) could overcome the
negative static-margin effect, but with the addition of a height lock (~ = Ghh), and with z~ assumed
zero, (P~- R~) still determines the sign of the last coefficient of the stability equation (see Table 2).
eL
P1- R1 = x~z ~- x~z~ + z~ 2 "
This combination of derivatives is often positive but is usually very small, and with relatively
little change in derivatives it can become negative, e.g., aircraft (1) and (2). Since one of the
conditions for positive stability is that the last coefficient be positive, an elevator control with
height lock will always produce an unstable mode if Px - R1 is negative (Fig. 2). The condition
P1 --- R1 i s almost identical with the condition for minimum drag in steady straight and level
flight ~,%w,~*. :! ~=5 Hc;'_~'z~_
To check the cross-coupling effects of a basic elevator-pitch attitude control and a throttle-speed
control, the computer was used to give responses in speed and pitch for various values of G o and
A~ (Fig. 8).
= GoO
T = d,~u.
Increasing A~ damped the long-period motion without materially affecting the short, while increasing
Go improved the long-period mode by subtracting damping from the quick oscillation. The cross-
coupling terms due to the product At, G o were beneficial, adding increments to coefficients K 1 and K 0.
The cross-coupling of the throttle-spee~t control with any of the elevator controls appears to be
beneficial. Fig. 9b gives the aircraft response in pitch with an elevator control only. Fig. 9c shows the
effect of a height lock, giving a divergence (P1- R1 being negative), i.e., the final coefficient K_ 2 of
the stability equation being negative. Immediately a throttle-speed control is introduced, the
cross-coupling contribution A~G~ swamps the small negative component and makes the coefficient
K_~ positive. This has the effect of making the extra mode introduced by the integral height control
a subsidence instead of a divergence. By inspecting Table 2 it can be seen that all the cross-products
of elevator-control terms with A u (speed control) are positive and associated with the derivative z w-
To show the variations in the cross-coupling effects of the three long-period damping throttle
controls, d u, d w, dq, with elevator control and height lock, 'equivalent' values of the throttle
gearings were used, i.e.,
= GoO+Gah+G~ t-h&"
~7
d
with
or
or
T = Auu
T = i wW
T = Aqq.
The cross-coupling terms are shown in Table 2 and examples are given in Figs. 10, 11 for which
aircraft the value of (P1- R1) is negative. Thus the speed control gives a damped response in height
and pitch, the incidence control (A~v) a damped response, but less damped, while the rate of pitch
control gives a damped oscillation and a divergence. The explanation for these variations in stability
7
comes from the cross-couplings terms A,,G; etc. A~,G~ is positive and cancels out the negative P1 - R1
of K_~, associated with the integral height gearing. The A~G; cross-coupling term is beneficial
(positive), but is associated with the relatively small derivative z~,, as opposed to the A,Gg term
associated with z,,. Finally, the AqGr, term is negative, reducing the coefficient K_~, but still leaving
the final coefficient K_~ negative.
The above controls do not add damping to the short-period motion, the remedy for any lack of
short-period damping being an elevator control proportional to rate of pitch. Fig. 11 gives the
aircraft and pitch response with the addition of a Gq control as compared with Fig. 10.
Although elevator proportional to rate of pitch is necessary for short-period damping, it also has
an effect on the frequency and damping of the long period. This is apparent for the large q control
gearing (G~ = 2.0) of Fig. 12. In the practical case of an autostabiliser which incorporates a 'high-
pass filter' the elevator signal will be proportional to {~D/(1 + ;1D)} q. Comparison of Figs. 12 and 13
shows that an unsuitable choice of time constant ~1 may introduce instability. There is scope for
further investigation into this particular problem, but it is not intended to deal with it in this report.
Two other flight conditions with both elevator and throttle controls are shown in Fig. 14:
(a) with negative static margin and negative P1 - R1
(b) with negative static margin and positive P1 - R1-
Both are completely stable when a combination of elevator and speed-throttle controls are used.
Condition (a) is unstable if either set of controls is used separately, while condition (b) is unstable
with throttle control alone.
7. Conclusions.
Automatic throttle controls mainly affect the long-period motion of the aircraft. Moving the
throttle proportional to the speed of the aircraft seems to be the most suitable of the various throttle
controls, the other types sometimes needing large changes in gearing to keep the same amount of
damping over the flight range. The unstable mode introduced by negative static margin cannot be
stabilised by basic throttle controls themselves, but addition of an integral term will remove the
instability. However, elevator-pitch attitude control seems a better method of counteracting negative
static margin.
Analogous results are found when automatic height control is obtained by elevator action alone.
A new mode of motion is introduced which may or may not be stable depending only on the sign of a
certain combination of aircraft derivatives (P1- R1). When this mode is unstable it cannot be made
stable by changing the normal elevator-control gearings, but adding an integral term can be effective.
Alternatively the instability can be overcome by throttle-speed control.
The combination of a throttle-speed control and an elevator control (elevator proportional to
pitch attitude, rate of pitch and height deviation) has an appreciable stabilising effect on the aircraft
motion even for aircraft with both negative static margin and negative (P1- R1). The other throttle
controls (proportional to rate of pitch or incidence) with elevator control are not so powerful in
their stabilising effect and may even be de-stabilising in some flight conditions.
8
Au, Aw, A o etc.
D(b) -
Go, G h etc.
k =
K~, K a etc.
T
t
t =
"7"
.(a)
0
x
8
/xl
LI ST OF SYMBOLS
Non-dimensional throttle-control gearings
d
dr
Non-dimensional elevator-control gearings
Non-dimensional height deviation
½-C~
Coefficients of stability equation
Incremental change in thrust
Time in true seconds
m/pVS, unit of time in non-dimensional form
t
~, time in air-seconds
Non-dimensional lag time constant
Non-dimensional speed error along x axis
Non-dimensional component of speed along z axis
Pitch deviation of aircraft
Elevator angle from equilibrium position
- ~l mwl i.
- mq/i.
- ~irno/is3 Portmanteau functions of pitching-moment derivatives
- - [Zl mu/i B
- ~l m~/i B
m
pSl
No. Author
1 H.R. Hopkin and R. W. Dunn ..
2 L.W. Bryant and S. B. Gates ..
3 K.H. Doetsch .......
4 H.R. Hopkin ......
REFERENCES
Title, etc.
Theory and development of automatic pilots.
A.R.C. 13,825. August, 1947.
Nomenclature for stability coefficients.
A.R.C.R. & M. 1801. October, 1937.
The time vector method for stability investigation.
A.R.C.R. & M. 2945. August, 1953.
Routine computing methods for stability and response investiga-
tions on linear systems.
A.R.C.R. & M. 2392. August, 1946.
10
TABLE 1
Aerodynamic Dat a and Derivatives
Aircraft
Altitude
gi (kts)
M
CL
X u
X w
z**
z~,
mz,
m w
mw
m a
i.

OA
X
Y
8
P1- R1
(1)
0
172
0.26
0.55
40-5
2-44
-0-0585
+0-0578
- 0.55
- 1.403
0
- 0- 061
0
- 0.428
(2)
40,000
256
0.9
0.242
164
3.31
-0.0115
+0.0085
- 0.327
- 1.61
- 0.0276
- 0.134
0
- 0.508
(3)
40,000
313
1-1
O. 165
164
2.71
- O. 0284
- O. 0065
- 0.1
- 1.618
- 0.017
- 0.237
0
- 0.565
(4)
40,000
214
0.75
0.274
69
3.09
- 0.020
+o.oil
- 0.365
- 2.56
+0.00123
- 0.0282
-0.00457
- 0.
O.
0
7.
0
1.
24
- - 0"
204 - 0.233
35 0.35
13
1 63
0
22 1.45
- 0.244
0.35
8
111
0
1.61
0377
109
- 0.0182
114
0.0373
- 0- 45
- 0.24
0.1
O. 849
19.5
3.15
4.5
165.6
O. 1052
11
TABLE 2
Coefficients of St abi l i t y Equation Y, Ki h i = 0
Basic
Aircraft
1
g~ K0~
K~ Ko~
K~ Ko~
K1 Kol
Ko Koo
K_I
Automatic-Control Terms
3G o 3G h - A u A w - Aq - 3GoA~ 8GoA w - 8GhA u 3GhA w - 8GhA q
1
N,
- - Z w
P, - R1
1
L1
M~
- - Z u
M~
R
S
1
- z ~
- z ~
- - Z u
"~u
Control equations
r] = GoO + Gqq + Ghh + G~Sh dr
T = A.~u + Aww + Aqq
Aircraft equations
Du= xuU + Xww- kO + T
D( w- O) = ZuU + zww
D20 = - u- xDw- wW- vq- 87
Li= v+ X-Z w
MI=~- vz~
M2 ~ K- v~ u
NI : -- xu -- zw
R1 : - kzu
R = - ( K+%X)
S = Kz w - ~o%
Ko4 ~ 1
//7o8 = L 1 -- x u
Ko~ = M, - xuL 1 - xwz u
Kol = - xuM l + xwM ~ + kR
Koo = kS
To illustrate the use of the Table, we write
K 2 = (Ko~) + 8G o - L1A u - zuA w - RAq
K_I = ( Pz - Rt ) SGh + zwSGt ~Au- zu8Gt~Awetc.
12
NO TE
The following figures have all been reduced to half linear
size. The various scales should be adjusted accordingly.
13
SPEED REISPONSE
UNCONTROLLED
RATE OF PITCH RE6PONSE
~-'k x.//-'x /-'-x
AIBCRAFT
k
o .
Ere = O'P-
C-a= O-'I-
%.o-a
t
o~
E
V v
ld
R
Q
6
u
¢~e-!
TIME SCALE: {C~= [4-~.5EC
w"
Fro. 1. Response of Aircraft (I) to a horizontal gust; pitch control
on elevator ( 7/= GoO ).
PITCH RESPONSE
RATE OF PITCH RESPONS~
i
n
L~ U
6
UNCONTROLLED AI RCRAFT
,~ HEIGHT RESPONSE
,L=e
-/- e ÷ K + o.o4efKan-
"rime 5CAL~- JCM ° a*.*SEC
t-
~r
to
{L
i
E
B
d
o
o
6
Fie. 2. Aircraft (1), effect of height lock on elevator-pitch control.
PITCH RESPONSE
UNcoNTROLLED AIRCRAFT
I--a
t.9"
L
iz:
ILl
0_
:1:
":t"
0

A r -. "
V"-
L,
SPEED CONTROL oN THROTTLE
T ~ - oq.u.
(~ 55"8 Le, THRUST pEP-, KNOT ESROR)
P~&TE OF PITCH CONTROL ON
THROTTLE
T = - O-~Z.ro ~ v
(=- 888 LB, THRUST PER DEG/SEC)
t NCI DFNCE CONTROL ON THROTTLE,
T: 1"31W
(-= a~zo L~ THRU~T P~F~ ~Ea~E~)
TIMF-. SCALE: I cM ~48.8 5EC
Fla. 3. Pitch response of Aircraft (1) for three different
types of throttle control.
ad
:I:
F_
S
c~
/
14 L~ THRUST ~ER KNOT ~EED
ERROR ( A~ =- O.OZ)
A /X /-', .--. .._..
V V V ---
&PEED CONTROL
T = Au.,u,-
14-0 I..B THRUST PER KNOT SPEED
(E~ROR-a~--o-z)
lO,~,O LB THRU.ST PER DEG/5EC
(A~,o -o-o~7)
RATE OF
FITCH CONTROL
T :A,t, !,
ID, GO0 LB THRUST PER DEG/SEC
4~o40 LB THRUST PER DEGREE
(Aw = 0-784)
f ~/""~ v " iNCIDENCE T=AwW CONTROL
TIME SO'ALE: IcM =GI 6EC
40,400 L5 THRUST PEP, DEGREE
(Aw ° 7"8*)
FIG. 4. Pitch responses for three different types of throttle control [Aircraft (4)].
PITCH RESPONSE
No LA~
~a
~k
i-
o
~o
b
u
LA~ TIME CONSTANT ('Jp) o~
,A~RSEC @a44SEC )
-~-~V~V/~/-', J -'-
LAC~ TIME CONgTANT OF
3 AIRSEC (= 7"3~ ~EC )
LAC T TI ME CON6TANT OF
to A~SEc (-a4.÷ SEC )
TIME 5CAL~ : I C.M = 48"8 EEC
FIG. 5. Effect of lag on throttle-speed and integral
speed control.
A~ - 53- 81b/knot~
5PEED RE.SPON 5E~
F
(5
uJ
a
#_
¢/
u
5PEED CONTROL: ID-8 L5 THP, UST SPEED & }NTEG'~,AL SPEED CONTROL 1$.5LB/KNOT~
FER KNOT TIME CON6TANT 40 SECOND5
}f
RATE OF PITCH CONTI~OL:
116,ooo LEE/DEG/SEC
RATE Of PITCH & PITCH ATTITUDE CONTROL:
116,ooo LIE/DEG/EEC, TIME CONbTANT 40 5EC
INCIDENCE. CONTROL :
870 L8/DEG
iNCIDENCE & INTEGRAL INCIDENCE CONTROL :
87OLB/DE~, TIME CONSTANT 4O 5ECONDb
TIME- 5CALE: ICM = 8~'~ 5EC
Fro. 6. Effect of addition of integral controls to the three types of
throttle control [Aircraft (2)].
~3
g)
L
0
Fro. 7.
SPEED RE5PONSE5
SPEED CONTROL: ?-4-£ L5 SPEED AND INTEGRAL SPEED
THRUST PEP-. KNOT CONTROL: ?..4-Z LB/KNOT
TFME CONSTANT 40 SECOND5
RATE OF PITCH CONTROL:
44,600 LB /DEG/.SEC
RATE OF PITCH AND PITCH
ATTITUDE CONTROL:44,GOOLO,/DZG/SEC
TIME CONSTANT 40 SECOND5
Y
INCIDENCE CONTROL ;
~,72.O LB / DEC~
INCIDENCE AND INTEGRAL INCIDENCE
CONTROL : 5,720 LB/DEG
TIME CONSTANT 40 6ECOND5
TIME ~CALE: i cM=54.2..SEC
Effect of addition of integral controls to the three
types of throttle control [Aircraft (3)].
SPEED RESPONSE
.RATE 0~' pITCH RE6PONSE
A~_=-O,I
(s3.s LE,
qe " q'I
O.
Ig
6
i
I
TIME SCALE :
C-Te ~ 0'[ -~f "- - -"
Au= - 0-4
Ge " o.I
Aa" - 0'+
Ge= 06
A~ ~ - 0.4
Ga = 0-8,
I CH= 14'6 5EC
t2r
to
(1.
F_
[k
u/
gL
u
L3
O
6
FIG. 8. Response of Aircraft (1), pitch control on elevator,
speed control on throttle (~7 = GoO; T = A~U).
PITCH RESPON6E F~ATE OF PITCH RE6~ON6E
A~x=O
HEIGHT RESPONSE
~e = I
vaR~ 1
~K= 0-049
v
1
o_
~E
t2T
w
:x
o=
Ct $= I
~-~ = o.o48
T~
q$: I
A~ = -0.5
FIG. 9.
u
t~
&
I
G
i11
8
o
6
u
T-~_ : o.o4~
cTq: I
A~L = -t
TildE SCALE ICM = ~4"~ 8EC
Response of Aircraft (1) showing effect of throttle-speed
control on elevator height lock.
(1;) I
~] = GoO + Geq + GT~ h+ ~ h dr ; T= Auu
HE1QHT P, ESPON sE
R,.hgg OF FITCH RES,PONSC
Au. =- 0"£
+
Ksor)
L
A~I,.=- 0--151
-
m
- - 6
" A w = E-GZG
"~ (4£40 LD
INCIDENCE)
71M~ aCALE : I cM = ~4' 4 5Ec
FIo. 10. Aircraft (1), comparison of three types of throttle
control combined with elevator-pitch attitude control and
height lock.
(~7 = 0+h+ 0"049J'hdr; T= A,uorAqqorA~vw )
29
L
0~
0
H EIC~HT P, E6PON~K
RPII-E OF YITCH RE.~poN~E
TH~U~T PER
~NOT)
A~ =-o.451 .d~
(l??e I-a
THRUST P~R
os~/a~c)
Aw = #_-~,£G
u (4a4q L~
DE.CaREE
li"CII~ENC E)
}-
g
uJ
c~
O
6
u
TIM~ OCALE: IcM ~ a4.4 5Kc
Fro. 11. Aircraft (1), addition of elevator-rate of pitch control
to Fig. 10 examples.
L
~'ATE OF PITCH R~'6pON$ ~"
HEIC~HT RE6PONSE
'CT¢ = a 1
ELEVATOR
PER OEGI~EC)
t
b
L~
TIME 5C#,LE: (Cl~ = P-4'~ 5EC.
FIG. 12. Aircraft (1), effect of elevator-rate of pitch control
with throttle control.
( :0+ o +h+0049 I d T---0:ul
-,.i
bl
I-
5
E
g
o
o
6
5
RATE OF PITCH RE6PONSE
/'- X
NO FII-T E N
T~ = I AIRSEC
"C~ = 0- 3 AIRSEC
HEI~MT RE6PONSE
AAA AA
v vvvv/
'1: t : O'l AIRSEC
_/V~ ~o o ~,~ ~ ) _
W
Ib"
&
- - 4~ "gl = O" 08 AIRSEC
(~ o.o~ ~ ) ....
TIME SCALE: ICM = 2- 4"4 ,.~C
Fro. 13. Effect of 'high-pass filter' on q signal.
= 0+h+0.049 AdT+I- _i:~D q; T- - - 0- 2u
K
I
2.
Z
~r
o
HEI~T RESPONSE RATE OF PITCH I~ESPONSE
Or"
v
6 
TIME SCALE: ICM= P-.4'~ 8EC ~ <~ F
Fro. 14a. Response of elevator and throttle controlled aircraft,
with height lock [Aircraft (2)].
~to
5~
i,J
TIME SCALE.; ICM = P-4-4 5EC
tO
a N
m
FIG. 14b. Response of elevator and throttle controlled aircraft,
with height lock [Aircraft (3)].
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