Effects of structural flexibility on a reaction jet satellite attitude control system

December 1975

EFFECTS OF STRUCTURAL FLEXIBILITY

ON A REACTION JET SATELLITE ATTITUDE CONTROL SYSTEM

1 SEP.

1978

by

Gunter Ma1ich

UTIAS Technical Note No. 199 CN ISSN 0082-5263

EFFECTS OF STRUCTURAL FLEXIBILITY

ON A REACTION

JEr

SATELLITE ATTITUDE CONTROL SYSTEM

by

Gunter Ma1ich

Deceniber, 1975

UTIAS Technical Note No. 199

"

(

'

.

Acknowledgements

The author is indehted to Dr. P. C. Hughes for suggesting this topic of investigation, and for his continued interest and helpful discussion.

The work described herein was made possible through the financial assistance of the Federal Department of Communications (Communications Research

Centre), Contract No. OSR3-0017.

•

'.0(

Sunnnary

The perf'ormance of' a satellite reaction jet attitude control system cau be severely degraded by structural f'lexibility. Using a computer-modelled spacecraf't wi th a pseudo-rate controller, a quantif'ication of' the perf'ormance loss is presented. Flexibility has been introduced into the simulation in a very general way by reducing elastic inter act ion to a series of' modal f'requency and gain parameters. The modelled system has been f'ound to remain stable under all conditions studied, although perf'ormance may suf'f'er various degrees of degradation.

TABLE OF CONTENTS Page Acknowledgements i i Sununary iii , "" Notation v l . INTRODUCTION

1

2.

TEE GENERAL MODEL

1

3.

TEE CONTROLLER

2

3.1

Pseudo-Rate Control

₃

3.2

Controller Characteristic Frequency

₃

3.3

Test of Wc and t*

₇

4.

TEE DYNAMICS BLOCK

₇

4.1

Modes

₇

4.2

Flexi bility Parameter s (k and

w)

8

4.3

Constrained and Unconstrained Parameters

₉

5.

TEE FINAL COMPillER MODEL

10

5.1

Values of Controller Parameters

10

5.2

Values of Flexibility Parameters

11

5.3

Nondimensionalization of Variables

11

5.4

Initial Conditions

12

6.

COMPillATION PROCEDURE

12

6.1

Integration Method

12

6.2

Operating Procedure

13

7.

DATA PRESENT AT ION AND ANALYSIS

13

7.1

Simulation Plots

13

7.2

Performance Quality Numbers

15

7.3

Plots of Performance Quality Regions

15

la

8.

RESULTS FROM PERFORMANCE ANALYSIS

16

...

8.1

Concluding Remarks

17

REFERENCES

18

FIGURES APPENDICES

Algebraic

DB

H I R r

t

t on 0: . Computer

DB

H IFLEX

IRIG

ITOT

KF KJET KlU KlC KTHETA R TC TD TNET TIME TON TSTAR ALPHA BETA EPSl EPS2 EPS3 NOTATION Definition Dead-band of controller PSR hysteresis

Inertia of flexible part Inertia of rigid part Total inertia

PSR feedback gain Thruster torque gain

Mode 1 unconstrained gain Mode 1 constrained gain Sensor gain

PSR controller output Thruster duty cycle Control torque Disturbance torque net torque Effective torque Real time Thruster !lon-time!l Thruster !loff-time!l

System characteristic period Acceleration

Sensor output PSR feedback error PSR controller input

NOTATION - Continued

Alr:iebraic C0!S2uter Definition

9

TH Body angle

8

E THE Error angle to sensor

8

r

TRI Control angle

8 I

R THR Body angle of rigid part

el THl Body angle of mode 1

~ DAMP Appendage damping factor

'T

f TAUF PSR feedback time constant

<I> PHIC

cp

PRIU

Wc WN Control loop natural frequency

n~ WlC Mode 1 constrained frequency

w~ WlU Mode 1 unconstrained frequency

x

XN Nondimensional form of "X"

.

X DX Time derivative of "X"

X XO Initial condition of "X"

1. INTRODUCTION

The study of structural flexibility and its effect on control system performance has become of crucial importance in the design of modern space vehicles. This was poignantly brought to light in Explorer I, as previously unknown effects of whip antenna motion led to dynamic instability. A few years later, the success of 000 111 was seriously hampered by excessive oscillations created by control system interactions with flexible booms. There are many other examples of space missions being hampered by these problems. The inter-ested reader is directed to Ref. 1 for an outline. Reference

8

also provides an informative overview of flexibility effects on control systems.

structural flexibility has been, and will probably continue to be, an important area of study. With spacecraft power consumption and sophistica-tion on the increase, antennae and solar arrays tend towards greater prominence. If costly weight penalties are to be avoided, this isbound to result in less appendage rigidity. There are limits to the degree of flexibility which can be tolerated. Excessive appendage motion can feed back into the body of the

satellite, and hamper pointing accuracy. Furthermore, the added accelerations created by appendage oscillations are bound to increase the stress and fatigue levels on the spacecraft.

Particular problems can be created for those vehicles wi th an acti ve control 'system, such as the previously mentioned 000 111. Acti ve systems incorporate sensors to check satellite attitude and a controller to supply terques to maintain some desired attitude. Excessi ve motions of the vehicle, as a result of flexibility-induced oscillations, will tend te trigger the attitude control system more often than for a rigid satellite. The control torques applied can produce an added fuel consumption, a degraded control response, and even instability.

Investigating flexibility interactions through ground testing is both costiy and dynamically awkward. structures designed for the weightless

state do not lenÇi themselves well to a one-g field. structural engineers thereforè ,try .ta design appendages stiff enough so that interaction problems are unlike.ly t·o occur. This generally entails arranging for the natural

vibratión frequencies of the structure to be much higher than the passband of the att:i.;~ude controller. This results in a trade-off between costly stiff-ness, and design confidence.

It becomes imperative to estimate the amount of performance lost for a given loss in rigidity. An attempt is made in this Note to contribute towards thi s e stimate • Wi th the aid of a computer model, a simple satelli te with a nonlinear attitude control system is simulated. The analysis is arranged so that the ratio of 'flexible' inertia to tbtal inertia can be

I

varied. In this way, dynamic effects are modelled from a fully rigid to a fully flexible satellite. The structural model, in conjunction with the attitude control system model, thus allows an investigation into the inter-actions of structural flexi bili ty, control performance, and stability.

2. THE GENERAL MODEL

In order to help isolate the effects of flexibility on attit ude control, i t is desirable to investigate a reasonably simple satellite model.

Though this entails a loss of accuracy for individual spacecraft, results can be of a more general nature.

A diagram demonstrating the vehicle studied is shown in Fig. 1. The model is assumed to possess a synnnetric structure , wi th a central rigid body. All flexibility is contained in two diametrically opposed appendages affixed to the central body. Furthermore, the appendages are considered to behave as rod-like members, such as booms or antennae.

Attitude perturbations from a desired reference position are coun ter-acted by control jets supplying a pure torque, Tc, about the satellite centre. Tc can assume a positive or negative value, depending on the sense of the satellite' s attitude angle. I t is not necessary to consider any addi tional torques or forces for our purposes. Internal torques, due to fuel sloshing, friction, etc., are also considered beyond the scope of this analysis (Refs. 2 and 3 give an indication of the magnitudes of these additional torques.) Attitude information for the rigid main body is limited to one rotational degree of freedom, measured by 8. The lack of translational motion for the centre of mass innnediately implies that synnnetric modes of appendage flexure are being ignored. This results in no loss of generality for our purposes, since the symmetric modes do not affect 8.

A general control loop may be drawn for the system, as shown in Fig. 2. Attitude error, SE, is sensed bya controller which, in turn, applies a correcting torque to the satellite. The resulting motion of the body, which may pe written as a superposition of rigid and flexible components, provides input to an attitude sensor feedback loop. In this study, it is assumed that the sensor processes the angle 8 instantaneously, and with perfect accuracy. It is recognized th at this latter assumption is quite idealistic and likely eliminates important instability possibilities; it is planned to remove this assumption in a subsequent study.

3.

THE CONTROLLER

Though a simple controller model would be mathematically desirable, some sophistïcation is required in the simulation, Tt would be unrealistic, for example, to employ a basic relay-type controller. Such a system would provide a torque to counteract only the sense of SE. The jets, being non-throttling, would be constantly firing and expending fuel and thus an

undesir-able limit cycle would also be exhibited, as shown in Fig. 3(a), for a rigid system. Some improvement would be possible by providing a deadband region in the controller. No jet thrust would be applied while the attitude error was

wi thin certain bounds. However, fuel consumption would still be almos·t as high, and the limit cycle would also remain, see Fig. 3(b).

In order to achieve realistic performance in the simulation, it was decided to employ the slightly more complicated pseudo-rate (PSR) con-troller. A view of Fig. 3(c) illustrates typical PSR performance on a phase-plane plot. Velocities are quickly reduced, avoiiding limit cycle instability.

..

3.1 Pseudo-Rate Control

The main feature of this nonlinear system is torque control through

a form of pulse modulation. The spacecraft attitude control jets fire for brd.:ef intervals , reducing on-time, a..n.d therefore fue1 requirements 0 Reference

4

provides the reader with more PSR in:formation. Additional descriptions of ON-OFF controllers can be found in Ref.5. Thè width and frequency of the

control pulses are determined by the attitude error angle, ~E, arrd an

artifi-cially produced estimate of

é

_E •

Figure

4

shows a schematic for the PSR controller. E::r.. is a funC'tion

of the angle 9E. The output, R (either -1, 0, or +1) act s as a switch for the

torque jets. Some expla..n.ation of E:s is required. If the satellite were

l'erfect1y rigid, and had no initial angular velocity , then

ê

wou1d be calculabIe from the satelli te' s torque history.

é

~

f

T dt

c

If we further assume the presence of ideal control jets, R will be related to

Tc by a constant. E:s, therefore, roughly approximates the time integral of R by virtue of the PSR feedback lag system. Tt is apparant that Es can never be identically equal to velocity , due to the saturating effect of the lag

network.

The feedback time constant, T f' of ten assumed two values, depending on whether the jets are on or off. This gives the control-system designer

extra freedom for performance optimization. For our purposes, however, it

wil1 suffice to fix T f at one value only.

The limits of attitude error are defined by the bounds of the

controller's deadband region. When 9E exceedsthis region, correcting torques

may be applied. At either end of the de adband, a small area of hysteresis is found. These are particularly useful in reducing fuel consumption during limit

cyc1e operation. This can occur, for example, under the influence of an

external disturbance torque.

3.2 Controller CharaC'teristic Frequency

It is possible to define a convenient Ucharacteristic frequEncylt

for the pseudo-rate control system. This derivation follows closely that

shown in Ref.

4,

and provides a useful reference base with which to reduce

later data.

For sma1l angular velocities about one axis, a linearized rigid

equation of motion may be written:

I 9

=

T_eff

where I and Teff are the tota1 inertia and effective control torque, respect-ively.

The e:f:fective torque wi11 be a :funct,ion o:f reaction jet torque, and the :fraction o:f thruster on-time (duty cyc1e):

Thus:

I

e

=

T

r

c

Consider again Fig.

4.

Let us assume , :for the moment, that ~ is a constant. This will be valid i:f the sate11ite dynamics respond slow1y compared to thruster on-time. We may then write:

where E20 is the initial va1ue o:f E2. The switch turns ,on when Es

=

1, and turns o:f:f when Es

=

1 - H. Substituting (3.3) into Es = E1 - E2, we have:

After rearranging:

During steady state operation, E2

=

El - 1, at the time o:f pu1se turn on. o

Thus in the steady state:

(

K:f

+

1 - El ) ton = -r:f,en K:f

+

1 - H ... El When the switch turns o:f:f,

The :feedback circuit wi11 decay according to the equation:

whereE2 _o

=

El. - (1 - H).

The switch will turn on again when:

Af

ter some rearranging:

E;s

=

1

=

E,. - Ez _o

If we now allow E~ to change slowly, we have

E.l. (t + 6t)

=

El. ( t) +.6;t

~i

( t) + •••

Substitute into

(3.5)

.

Since H and E~ are small, and provided that Ej,

>

1, the logarithm may be

expanded to gi ve :

If we take t off« 1" f' then

1"fH

t off

=

--.:----.-El. - 1 + 1" f E;l.

Frem

(3.4),

the on-time may be similarly approximated:

(3.6)

provided that El.

<

Kf + (1 - H), that i's, less tha.ll the saturation level.

The duty cycle was defined as:

t

r

=

-:---.:o;.:n~_

t

+

t .p.p

In a we11 designed controller, t on ~ t off" Thus, Substituting

(3.6)

and

(3.7):

t _ on r -toff

.

€J,. - 1 + '-f el. r

=

~---~---~---Kf + 1 - H - €l.

Let us assume that the input is much greater than the de adband, and we11 be10w the saturation level. (This approximation would be inva1id for limit cyc1e operation.)

1

«

el.

«

Kf

+

(1 - H)

The H may be rem::>ved if the deadband is very sma11, Le., H « 1 . The fo11owing simp1ifications resu1t:

(i)

(ii)

Substituting into the duty cyc1e equation, we find,

We may now back substitute into (3.2):

T

I

ë· -

I C {6l.

+

1" ~) (

3 .8)

- Kf f

el. is re1ated to S through an~amp1ifier of gain KS' Th~s:

Substitute into

(3.8),

and rearrange:

s

=

0

"'

.

~.

The undamped natural frequency of the satellite system thus becomes:

W

=

c

The characteristic frequency above allows us to define also a characteristic periodof the form:

M

Kf

t* = 27T T K

c

e

Test of W and t*

c---The natural frequency and period, Wc andt*, form basic measuring

tools with which . later data are reduced. It is, therefore, imperative to

discover just how universal these characteristic values really are. A munber

of computer simulations were undertaken of the control system in Fig.

4.

All

parameters in the system were individually varied, and a plot was made of response vs. t/t* (Le., multiples of the characteristic period). For t* to truly be a natural period, all plots should have similar period with respect to t/t*.

Sample plots are shown in Fig.

5,

for a variation in the feedback

time constant, Tf. It is seen that the first quarter periods cluster about

t/t* ::;

0.25,

as desired, although subsequent period fractions tend to deviate

from their predicted values. Plots investigating other control loop

para-meters showed very similar patterns. The consistency of these results, though only for the first quarter period, demonstrates th at our definition of t* (and

therefore wc) does indeed produce

a

characteristic parameter.

4.

THE DYNAMICS BLOCK

The vehicle dynamics portion of the control loop describes the structural response of the satellite model. The transfer functions therein

al,low for rigid and flexible contributions to the total motion. (Ref.

9

provides an informative overview.) The flexible motionsare initially

con-fined to linear and nondissipative elastic effects • Linearity is

mathe-matically preserved by restricting motions to small scale deflections. This rel;rtriction is not considered serious, since modern satellite control systems are designed for small attitude excursions. The assumption of linearity in the structural response considerably simplifies the mathematical formulation of the model since the problem is then amenable to a convenient IOOdal analysis.

4.1

Modes

The equations of motion for our satellite model may be derived either

through classical continuum mechanics theory (e.g., Ref.

7),

or more modern

equations may initially be written in space and time variables. Separation of variables is then employed to isolate the (sinusoidal) time-dependent portion from the space-dependent (modal) portion. Results resolve into an eigenvalue problem, with the eigenvalues and eigensolutions indicating individual modal frequencies and shapes. The solution is typically of the form:

where 5

(x)

n ~(t) 00 Y(x,t)

=

I

5n(x)~(t)

n:::l

is the n th normalized mode of the complete system, and

is the time dependent generalized displacement coordinate associated with the nth mode.

For a general discussion of these matters, the reader is referred to Ref.

6.

It is apparent that the deflection Y(x, t) can be related to the

attitude angle

e

of the rigid core. I t can also be seen that

e

will be composed

of contributions due to the various modes. We may write:

00

e

:::e+Ien~(t)

n:::l

where

e

is the main body angle due to rigid motions;

th

e

n ~ is the attitude angle contribution from the n . mode.

The modal deflections are typically of the form shown in Fig.

6.

In

theory, there are an infinite number of shapes, corresponding to the infinite

number of eigenfunctions. Half of these will be symmetric motions, involving

no angular displacement of the central body. These, as mentioned previously,

are of no direct interest in the present context.

Of the remaining infini te number of antisynnnetric modes, only the

first few would be of apy importance. I t is unlikely that the higher frequency

modes would be excited by disturbances th at a real spacecraft would encounter.

Furthermore, higher order Iflotions would tend to be transparent to the control circuit, due to the filtering effect of the attitude sensor.

4.2

Flexibility Parameters (k and

w)

Each characteristic motion will contribute its own dynamic effects to the control loop. For the purposes of a mathematical simulation, it is advantageous to describe these effects in terms of two basic parameters. The

first, 'w, is the previously mentioned modal frequency. The second may be termed

the modal gain, k. Loosely speaking, the gains will indicate the fraction ef

attitude acceleration attributable to individual modes.

,

..

Each flexible mode, as represented by W and k, contributes to the

transfer function acting upon a satellite's ideal rigid metion. The larger the

valuè of k for any individual mode, the larger will be the influence of that

oode. Reference

7

develops a number of formulae to estimate va1ues for W and

k. However, before introducing these results, some further background is

necessary.

4.3

Constrained and Unconstrained Parameters

Spacecraft designers have the option of defining satellite modes

from two vantage points (Ref. 10). For satelli tes wi th a rigid main body anG!.

flexible appendages, i t is possible to consider motions of the elastic members

separately. This is equivalent to assuming a fixed central body. Actual main

body motion may subsequently be modelled as driving forces to the appendages.

Modes of this form are termed "constrained", and give rise to constrained

frequencies and gains, n and K. Conversely, mode shapes of the complete

satel-lite may be considered, with the main body free to rotate. These modes result

in "unconstrained" frequencies and gains, W and k.

The relationship between the two systems can be further appreciated

through Figs.

7

and

8,

where block diagrams are shown of constrained and

un-constrained vehicle dynamics. The former involves a feedback mechanism to alter

the rigid response, whi1e the latter has a feed forward summation of moda1

contributions. Both formulations provide similar solutions provided that the

number of modes considered is made sufficient1y large.

According to Ref.

7,

constrained frequencies for 'rod-like' f1exible

members follow the approximation:

(p

=

1, 2,

3 ••. )

(4.1)

where prepresents the mode number. It is apparent that an equality can be

made if any mode' s natural frequency is known.

Once the np are found, the constrained gains for the rod-1ike

append-ages are approximated by:

~

n

E p

~

=

2.084

(4.2)

where ~

=

If/I is the rati.o of flexible inertia tothe satellite' s tota1 inertia.

The parameter ~ is a measure of the degree of flexibility in the vehic1e. It is

apparent that 0 ~ ~ :::. 1.

To relate the constrained parameters to their unconstrained counter-parts, we may make use of the f0110wing identities:

(Xl

I

)?

1

_(4.3)

=

-WE _

n

E _WE

00

:

-I

1

(4.4)

P=l

Vehicle dynamics computed using constrained parameters are found to have lesser accuracy neat resonance than those calculated using an equal number

of unconstrained modes. However, experience has shown that accuracies near

resonance will be comparable if the number of constrained modes is made suffi-ciently larger than the number required in the uncorl3 trained format.

5.

THE F INA!, C OMPurER MODEL

The full simulation block diagram, complete with controller and

structural dynamics sections, is shown in Fig.

9.

.An unconstrained format was

adopted for the body dynamics block, with modal parameters derived from a

constrained system. This allowed greater accuracy tban would have been possible with the constrained method, given a similar number of modes.

Only one flexible unconstrained mode was included with the rigid mode,

in the interests of computational economy. I t was not felt that this

simp")..ifica-tion would alter the basic character ofthe results.

5.1

Values of Controller Parameters

The choice of values for PSR parameters req uires a detailed analysis

by the satelli'te designer. The system must be optimized for performance, fuel

economy, cost, etc.

In order to present arealistic system, it was deemed best to employ

values designed for a practical spacecraft. In this regard, we were fortunate

to have available an early design study (Ref.

4)

of the back-up pitch controller

for Canada f s CTS satelli te. The following 'parameter s, originating in that

report, were used in our model.

Parameter Value

13.3

0.0188

8.8

sec. 0.12 ft-lb

68

slug-ft2

In the simulation, a time integral was taken of the PSR relay output. Termed ton' the value provided a measure of thruster on-time, and therefore an

indication of fuel consumption.

5.2 Values of Flexibility Parameters

The modal frequency and gain for the unconstrained flexibility block were derived from an equivalent system of three constrained modes.

A value of

~ and

n).

would be set for the first constrained mode. Equations (4.1) and (4.2) would then be employed to estimate the values of

n

and

K

for the first three

constrained modes. These, in conjunction wi th the transform equations

(4.3)

and (4.4) then provided w). and kJ. for the first unconstrained flexible mode. Thus,

the two parameters ~ and

n).

could be thought of as defining a 'condition of flexibility' for the unconstrained dynamics block.

In this study, interest centres on appendage frequencies close to the satelli te' s control loop natural frequency. A parameter was defined to measure this feature. Expressed in terms of constrained or unconstrained frequencies it becomes, respectively,

or

cp

=

w).

w c

Values of

n).

were restricted so that <I> would not exceed the range

0.1 to 10. Figure 10 shows the relationship of

cp

to <I> over the entire range of fij. It is seen that

cp

approaches <I> as the value of ~ diminishes.

The presence of a damping parameter, ~, in the flexibility bleck, requires some explanation. It is a standard, though mathematically

non-rigorous, practice to include this energy dissipative term in dynamic simula-tions. This parameter has a small value, in practice. For our model, a value of 0.001 was chosen.

5.3

Nondimensionalization of Variables

Simulation variables which would be of later interest were made dimensionless, and denoted -by the symbol (h). , The following chart lists the nondimensionalizing factors: Nondimensional Variables Factor time: t, t _on (l/t*) angles: 8, 8~<u (l/DB)

.

0 (t*/DB) angle rate: 8, 8).g,.

5.4 Initial Conditions

A number of initial conditions nrust be set for the integrating blocks. Conditions were made cOlJ!latible wi th a step change from a motionless attitude.

That is, q,

q,

é

_{o ' were zero, while} 8

0 was given an arbitrary rotation of

10 DB.

6. ,GOMPUTATION PROCEDURE

An IBM-packaged cOlJ!luter language named GSMP (Gontinuous System

Modelling Program) was available to numerically solve the control loop equations. CSM!? has the great advantage of being a digi tal dynamic simulation program, while

offering many advantages of analog computation. In use, the programmer is simply

requi:rrd to list system transfer functions, and set the various numerical con-stants and initial conditions.

The GSM!? user nrust pay the price, however, for the programming ease. Since the program is supplied as a prepackaged language, the programmer must

arrange his problem to suit the program, and not vice-versa. Input and output

formats are limited, as are the variables available as output. Another dis-advantage is the largeamount of cOlJ!liling time required to process a GSMP simulation.

The particular program written to solve the system of Fig. 9 is found

in Appendix A. Most of the statements are self-explanatory. Two subprograms

are added; one simulates the PSR relay with deadband and hysteresis, while the

other provides punched cards of required output varia1:iL es •

6.1 Integration Method

A GSMP-supplied fifth-order Milne method was selected for integration. step size was allowed to vary, being decreased until prescribed error criteria we re met.

Predictor and corrector calculations were applied by GSMP, using the following formulae: Predictor : Corrector: ? ( t + t.t) :::: y(t - t.t) + (t.t/3)[8X(t) -5X(t - t.t) + 4x(t - 26t) - X(t - 3t.t)] yC (t + L:.t) =

~

[y (t ) Pi> 7Y (t - L:.t)] +

~~2

[65X (t + t.t) (6.1) + 243X(t) + 51X(t - t.t) + X(t - 2t.t)J (6.2) "

The integration interval was then adjusted, such that one of the following equations would be satisfied:

or

where A is the allowable error.

<A--

,

<A--

;

(6.3)

(6.4)

Once these criteria were' satisf'ied, the integration estimate became:

6.2

Operating Procedure

The CS:MP program was run to provide output f'rom t

=

0.0 to t

=

20.0.

(This corresponds to t ~ 855 seconds of' real time.) Wi th each run, new values were set f'or the f'letibility parameters wJ. and kJ. corresponding to a predeter-mined <1> and~. ~ was varied f'rom 0.1 to 1.0 in steps of' 0.1. <1> took on :five

values in the range 0.1 to 10.0. This resulted in a total of' 50 simulations with f'lexibility, in addition to one rigid ref'erence case.

A

number of' simulations were undertaken to determine a reasonable error criterion. Tt was f'ound that an error of'

5

x 10-6 was required at large

~ and <1> to f'ind output approachi~~ a limit. At lower values of' ~ and <1>, the

error could be relaxed to 1 x 10 ,allowing better computational economy. Computer CPU times on the University of' Torontol _{s IBM 370 system were generally}

in the,range of' 0.25 to 0.70 minutes per simulation.

7

° DATA PRESENTATION AND ANALYSIS

A A Punched card output f'rom the CSMP program provided val ue s of' t,

ê,

é,

8J.,

él.,

and ton at regular small intervals of' time. These provided the data base f'rom which simulations were analyzed. From here on, we write simply 81 f'or 81qJ. •

7.1 Simulation Plots

To allow a qualitative overview of' perf'ormance trends, a number of' computer plots were drawn of' each simulation run. The attitude angle and rate, êAand ~ were plotted versus

t,

as was the measure of' f'uel consumption,

€

n0 (ton was divided by the value of' ton f'or the rigid spacecra:ft tE> give a \1f'uel f'actor" 0) A steE-by-step calculation of' satellite :m=chanical energy was also plotted against t. (The f'ormula used is presented in Appendix B.) This display is particularly valuable, since it provides some insight into stability. The more rapidly the vehicle loses its mechanical energy, the more quickly it approaches its ideal ultimatestate of' zero

é

and zero 8. Clearly if' the

energy were to steadily increase, the satellite conf'iguration could be labelled unstable. This would indicate that energy f'rom control jet pulses was being added to attitude oscillations, instead of' being subtracted, as required.

Figures 11 through 31 show some of the simulation plots. The first

.-of the series, Fig. 11, presents the performance of the fully rigid spacecraf't

(t3

= 0). Following are graphs at

t3

=

0.1,0.4,0.7, and 1.0 for the full range

of~. A number of trends are evident.

" "

9 vs t

All simulations show a rapid initial reduction of 9 to values hovering

about the deadband. There are pattern changes, though, that relate to ranges of

ep. At low ep, the attitude angle tends to ride the edge of the deadband. Tt

appears that the control jets have the power to hold the main body in place, but

must constantly fight the slowly but surely moving appendages. When ep is greater

than 1.0 (above controller resonance), the response tends to follow the rigid simulation pattern. The flexible vibration of the booms seems to add only a

high-frequency ripple to a steady motion, as shown for the case ~ = 0.1, ep

=

3.16.

Unlike the lower frequency examples, the response of the appendages does not dominate the pattern.

"

.

9 vs t

The oscillatory frequency is readily apparent in these plots. Like

the displacement vs time series, the response pattern is more a func·tion of ep

than it is of~. At low values of CP, the pattern follows the form of alternating

spikes about a relatively low velocity. The steady low velocity corresponds to

the intervals where 9 hugs the deadband edge. The large spikes show the effect

of the appendages intermittently swinging the core to the opposite side of the

deadband. Velocities will be high until the thruster fires repeatedly, again

bringing the attitude angle t 0 the deadband edge. For the larger CP'!3, there is

a ripple of varying magnitude about steady coasting velocities.

t

vs

t

-on~--Fuel usage at larger values of cP follows clos~ly the figure for the

rigid satellite. The thruster uses little power af ter t = 2.0. This contrasts

sharply to the requirement of satellites with cP

=

0.1. Af'ter the initial spurt

of fuel,the usage figure slowly, but steadily, increases. The total flow at

t

=

20, however, remains somewhat lower. The largest figures for fuel usage

are found when the satellite' s natural frequency is approached, and especially

for large

t3.

Similar to the low frequency case, a large portion is spent af'ter

the initial j~.

Energy vs

t

There seem to be three patterns for the energy plots. At low CP, there is a tendency to drop rapidly to a high and steady value. This indicates that

the attitude jets have l i·ttle effect on the vibrations of the low frequency

appendages, once those appendages have been set in motion •

At the slightly larger frequency of cP

=

0.316, the energy follows a

slow steady decrease, showing some controller effectiveness in reducing vibration.

For cP = 10.0 and for cP

=

3.16 at high

t3,

there is a rapid decrease to

a low energy value. Being co~arable to the rigid case, it provides an extra

7.2 Performance Quality Numbers

The simulation data was subjected to a more quantitative analysis

through a program called "SIGVALS". This program, found in Appendix A, isolated

or calculated values from the simulations which were considered significant in terms of showing satellite performance. The items of interest were:

(i) Total Fuel Expended

The total thruster-on time for each flexible simulation was divided

by the corresponding ton for the rigid satellite. This provided ~ fuel factor

corresponding to the fuel used in a particular run, divided by the fuel usage for the reference rigid run.

(ii)

Time in Deadband

The primary function of the control system is to maintain the attitude

angle,

e,

within the deadband region. A measure of the system' s success in

meeti~ this requirement is shown by the percent of time that it actually

satis-fies

Ie I

<

l.O.

(iii) Initial OVershooting

When the control loop is first excited, there is a tendency for state variables to overshoot. A satellite with less overshoot is better capable of

coping with attitude perturbations. Program SIGYALS, therefore, was designed to

find the maximum overshoot of the attitude angle and rate

(ê

and

ê),

along with

the component of angle and rate due to flexibility (ê~ and ê~).

(iv) Secondary Maximurns of Variables

~

In the time span of t ::: 2.0 to the final t ::: 20.0, it can be assumed

that initial overshooting hastapered away. The maximum magnitudes of the state

variables now provide some indication of deviation over an extended period.

(v)

Energy

Ultimately, the energy represented by the state variables should

become zero. How closely a given simulation approaches this goal indicates

system stability. By calculating the energy at €

= 20.0, a powerful indicator

of performance quality was measured.

7.3

Plots of Performance Quality Regions

The "significant numbers" found above for each simulation were assessed. Grades were assigned, ranging in value from "A" to "E", with an A identifying the best level. Generally , an "A" signified performance equal to, or better than, the performance of the rigid reference case. Each performance criteria of Section 7.2 received a quality grading for all flexible simulations.

This allowed quality region plots to be drawn, as shown in Fig.

32

to

38.

(The

computer program which calculated the quality grading, and drew the plots is

found in Appendix A.) Each criterion has a display of ~ vs

t3.

Quality gradings

are entered in the position corresponding to each simulation's flexibility parameters.

Figure 32 shows plots for criterion 1 using both the constrained and

unconstrained format (i. e. , <I> vs

t3

and

cp

vs

t3).

Subsequently, only the

uncon-strained type is displayed. The latter can easily be converted, using the

transformation of Fig. 10.

8.

RESULTS FROM PERFORMANCE ANALYSIS

Figures 32 through 38 show many interesting performance trends for the

flexible satellite model. Each criterion of quality will be reviewed individually.

Criterion 1. Fuel Expended

There is a definite increase in fuel consumption as the appendage

modal frequency approaches the satellite 's natural frequency (i.e.,

cp

~ 1.0) •

In addition, extra fl,lel is required at l~ge (3. The excellent economy shown at

low

cp

may be misleading, however, due to the finite interval of~our simulations.

The fuel usage plots show little sign of easing consumption at t = 20.0, indicating

th at larger long term figures can be expected. Criterion 2. Time in Deadband

A definite trend is shown of decreasing deadband time as

cp

decreases,

and as

t3

increases. Only those frequencies above Wc show good results.

~

Criterion

3.

9 Overshoot

The 'attitude angle is shown to be less likely to overshoot at large

t3

and small

cp.

ThiS

t3

trend is due, no doubt, to the decreased main body inertia

which the attitude jets must control. The lower frequency of vibration allows

plenty of time for the jets to assert authority.

Criterion

4.

~

.

9 Overshoot

The area of highest vel 0 city overshoot is found at low cp and large

t3.

Most of the performance degradation is caused by the worsening contribution of

é~ (se~ Criterion

6

below).

Criterion

5.

ê~ Overshoot

The trend in this criterion is exactly opposite to that observed in

'Criterion

3.

The implication is apparent. The proportion of appendage inertia

is largest when the rigid main body inerlia is least, rnaking the high

t3,

low

CP,

zone most subject to avershoot.

~

Criterion

6. 81

Overshoot

~

.

The region of high 9~ overshoot is also the region of largest 91.

The larger flexible displacements at given frequencies result in correspondingly

larger modal velocities.

Criterion

7.

Secondary

B

The long-term attitude angle perturbations become considerably

degraded around the resonance frequency. Good performance is found only at the

•

~

Criterion

8.

Secondary 8

The trends of the velocity excursions follow closely the pattern established above for the angular displacements.

Criterion

9.

Secondary 8;t,

The simulation plots of Figs. 11 to 31 showed how the appendage motion became more violent at low

cp

and high

t3.

This is distinctly underlined bythe larger long term ê~ in this regime, as shown in the performance quali ty plo·t.

A

.

Criterion 10. Secondary 8;

~

A Areas of highest appendage velocity , 8~, coincide wi th the regions of

maximum 8~. This is compatible1with the results of Criterion

9.

A

Criterion 11. Energy at t

=

20

The quality pattern found for the energy criterion is not as clear as some others. However, one result is readily apparent. At

cp

above the resonant frequency, energy drops very well, with somewhat less improvement at smaller ~.

The low frequency, high flexibility region (low CP, high

t3)

retains a large amount of residual energy.

8.1

Concluding Remarks

The plots of the preceding section show definite variations in satel-lite performance as a function of appendage flexibility.

The ability of the pseudo-rate controller to operate well is very much a function of the natural frequency of vibration of the appendages. At higher frequencies, the elastic modes are not as likely to become excited by control inputs. At lower ranges, however, elastic oscillations can become a dominant motion, particularly when large boom inertias are present. In add.ition, these modes tend to persist much longer, as shown by the energy plots.

It is of marked interest that energy levels for all simulations show a decreasing trend wi th time. This indicates that no unstable behaviour has been found for any flexible condition of our spacecraft model •

1. Likins, P.

w.

Bouvier, H.

K.

2. Greensite, A. L. 3. Deutsch, R. 4. Sta1ey, D. A.

5.

Greensite, A. L. 6. Likins, P.

w.

7. Hughes, P. C. Garg, S. C. 8. NASA 9. Gevarter,

w.

B. 10. Hughes, P. C. REFERENCES

"Attitude Control of Nonrigid Spacecraft", Astro-nautics and AeroAstro-nautics, May 1971.

"Ana1ysis and Design of Space Vehicle F1ight Control Systems", Vol. XII, NASA CR-831.

"Orbital Dynamics of Space Vehicles", Prentice-Hall, Inc., New Jersey, 1963.

SPAR Interoffice Memo, oct. 12, 1971.

"Analysis and Design of Space Vehicle Flight Control Systems", Vol. IV, NASA CR-823.

"Dynamics and Control of Flexible Space Vehicles", JPL Tech. Report No. 32-1329, Rev. 1, Jan. 1970.

"Dynamics of Large F1exible Solar Arrays and Appli-cation to Spacecraft Atti tude Control System Design" , urIAS Report 179, Feb. 1973.

"Effects of Structural F1exibility on Spacecraft Control Systems", NASA SP-8016.

"Basic Relations for Control of Flexible Vehicles",

AIAA Journal, April 1970.

"Dynamics of Flexible Space Vehicles with Active Attitude Control", Celestial Mechanics Journal, Vol. 9, pp. 21-39, 1974.

..

-y

Actual Attitude

=

8

--Flexible

Appendage,~ +=---+--~---~x

Oesired Attitude

=

81

-

--Rigid Core

FIG. I SATELLITE

-

GEOMETRY

8

1

+

8

E

..

Control

Tc

..

Structura I

8

--,

_-

.

-.,

System

Dynamics

Attitude

Sensor

-FIG. 2

GENER AL CONTROL LOOP

..

-81 +

r

~

J-8

fJ

8

FIG. 30 IDEAL RELAY CONTROLLER

FIG. 3b RELAY WITH DEADBAND

FIG. 3c PS R CONTROL

FIG. 3 PHASE PLANE PORTRAlTS OF SEVERAL CONTROLLERS (RIGID SATELLITE)

-:t

1

EI +.-,. €2

_fT

{ KJET

1-K8 R

I

-

~

IJ

-elHI-E3 Kf Tfs +1 BODY DYNAMICS Tc FIG. 4 PSR CONTROLLER

o

ru

LL

=:J

«

r-LL

0

Z

0

H

r-«

H

CL.

«

>

In

,.,

H 0 I Z 0 Z liJ ::;:! H i- -0 (IVNOISN3~IO-NON)

V13H

o

~ ! liJ

U1

Z

0

D..

.

en

liJ

lY

:2

LU

~.

U1

>-lil

0

f-J

CJ

H

c:r

L[) (9 LL

Rigid Mode

- - ~ Anti-Symmetrie

FIG. 6 NATURAl FlEXIBlE MODES (SCHEMATIC)

Te _I I _...

8

.

-r-

,~

_ST

KI 52 + 2 ZI .al 5 + .0.₁2 K2 52 + 2Z2 .0_{2 5} + .0.₂2 I I I I

Tc

8

•

8

•

8

1 I

-1-18

1

I

I S2

Additional Higher Modes

FIG. 8 BLOCK DIAGRAM OF UNCONSTRAINED DYNAMICS

P SR Controller I---R~

-l

I

EO _,

+I!

1I:

R

I

=0

_I I

I

€2

I

I

TF~+I

I

L _ _ _ _ _ _ _ _

~

Body Dynamics

-

- I

I S2

a

I I

I

1

L ___ _

_J

I

FIG. 9 SIMULATION BLOCK DIAGRAM

B

+ ) 4

0 0 rJ 0 LD 0 ~« 0 LD 0 '.-1 0

I--a

.

.. -I

.

0 lT1 -:-i c •

.

W

0 (Tl

...,

_{0 0} ~

rn

11 11 11 11 11 _m ( , ( , ,

.

, I U

·

1....0 I...J L.J I....J ₀ H H H H H I I I I I EL EL IL EL EL. CD < 0

U

!'- H

·

I

0

D-

I--LO

Z

·

«

0

I--U1

Z

0

L'l

_U

·

0

I--«

-q-

...,

I--·

(Tl 1\

0

0 1\ Ul

-..J

W

D-UlD WCJ

=:J

f!i O~

·

~

[J H 0 _L.i.l

_I

() z

D-WH

Z<

HO:::

0

ru

<I-

_O:::Ul

·

f - Z 0 _Ul[) (9 ZU

oz

LL u

::J

' .. -l _[.J(J

u...

!.J_ " 0 0 0 ZZ 0

·

0

.

0

.

c-l 0 r . 0

...,

0 ,-I

8~~ 1iI~ 8~~ _{ö ...} Iil~ H à'"'" H ';!I I-';!I I-<'u:> ~~~ ~n ~ ~ § ~

..

_:I à

~

~

~

~

~ _H I-~ ~ _~ >- _~ >-

I-~

H Ol

~

'" W UI UI 0 r

-D

11 U ... ( :c 0... 5\ Ol lD ru 0 _{'l' 'I 'f cp ~ 'l'} r;> "'f 'I' 'f t;- cp f'l-ilD (A~3N3)OD- ₀

.

0 11 ex: r-w c:a r-ex: z 0 . ~~ Iil~ ... 8~~ r-H ex: ö ... H ';!I I- - I ';!I I-::> <,uil _:::E: ~h ... ~ ~ (/) r -§

.

r -8 :I

..

à

~

c.!J ~ ~ ~ ... ~ u.. !!! ...J ~ U. ~

I-~

~

\'"

U _al u.

~

§ H .:. UI lD 0

)

§ " 8 .~ à'" 11 11 u.. <,uê ~if

~

ru ru ~ UI

..

cu 0 _{'l' 'I 'f cp} 0 0 0 ~ _{'" ru .:. à} ~l 30VSfl 131.:1

w s ~ f-6 2 o

tI I

-2 -4 -6 -a 2·0

~

~

1·0 [JN.f' = o· 001.00 2~ 6 a CJISPLACEM:NT RESPONSE SETA • o·w PHIC. 0·100 fLEI.. FACl[Jl • 0.gT7 BET'" = a·w PHIC = 0·100 A-4!U = 001.04 0.0~1~f--+-~~--r-~-+~~+-~~--r-~-+~--+--f--+--f--~ 8 10 6 2 12 14 16 18 20 TIMEN FLEL FLOW 10

gat

n

6 2 cw.P • 0·001 fETA = 0·10 ~IC = 0·100 PHru· o·w.

0II'IT:;

2 -4

~

,

6

,~~q

8 10 18 r==, 20 TIMEN -2 -4 -6 -8 -w -2

~

-3

~

-5 -6 -7 -8 2 6 VELOCITY RESPONSE a 10 12 ENERGY VS· TIMEN 14 16 SETA' o·w PHIC' 0·100 PHJU· o·w. 18 20 TItJEN

w+

(t

6 A

2~

:IV

-8+ 2·0

~

~

1·0 \./ 8ET~ = O.W PHIC = 0·316 ON.P = 0·001.00 ~I5PL.,A.C8.€NT RE5F'C:NSE FLEI. FJICTlJ'! = 1·005 BET ... = 0·10 Pl<IC = 0·316 PHIU = 0·331 1.LIW:.r ... 0·0 I I I I I I I I I I I 2 • 6 8 W 12 1.4 16 18 al TIM::N FlEL FLOW W

t

OI\W = 0·001 ~8 0 6 2 0 -2 -4 -6 -8 -W VELOCITY RE9"CJ'l5E -2

~

-3 ~ B -. -1 -s -6 -7 -8 +---+--+-2 6 8 W 12 1.4 El'ERGY VS· TIM::N 16 8ET~ = O·W Pl<IC = 0·316 Pl<IU = 0·33:1 EETI> :: 0·10 Pl<IC = 0·316 PHIU = 0·33:1 18 al TIM::N

10+

~

8t

6

•

₁₁

~l

-6

-8

2-0

~

~

i-a (lMf' = 0-00100 QISPLACEMENT RESPONSE IETA = 0-10 PHIC = 1-000 FUl. F ACTlR = 1-208 IETA = 0-10 PHIC = 1-000 PHIU = 1-049 0·0 I t I I I I I I I I I 2 • 6 B 10 ~ ~ ~ ~ ro TIMEN FLEL FLOW 10 ~8 0 6 2 0 -2

-.

-6 -8 -10 -2 r

01-3

~

-s -6 -7 -8

~

6 2 6 (lMf' = 0-001 h A V ~ V ~ V VELOCITY RESPONSE B 10 ~

,.

ENERGY VS· TIMEN

FIG. 14 SIMULATION AT BETA

=

0.1, PHIC

=

1.0

A ij ~ IETA = 0-10 PHIC = 1-000 A-UU = 1·().4S \-E

E~

V V

,

EETA = 0.10 PHIC = 1-000 PHIU = 1.049 ~ ro TIMEN

10+ B

f

I-6 4 2 0

~:tV

]

2-0

~

~

[)I.f.P :a< 0-00100 .or~caENT RE9"CNSE BET" • 0-10 Pl<IC. 3-1&! A..EL F"1CTtR • 1-062 EET" = 0-10 PHIC:= 3·162 A-4IU = 3 • 31.9 20 TIM::N

"I(

0-0 I I I I I I I I I I I 2 4 6 8 10 12 U 16 18 20 TIM::N FLEL FLOW

flO

t

lul

[ w p . 0-001 1- 9 0 6 4 2 0 -2

-.

-6

I

I

-8 -10 VEl..1X:ITY RE9"CNSE -2 ):

!ï

-3

~

B

-4 ...J -5 -6 -7 -B +--+----+-_{2 6 8 10 12} 14 El-ERGY VS· TIM::N 16 BET" • 0-10 Pl<IC • 3-1&! Pl<IU' 3-31.!1 BET" • 0-10 Pl<IC. 3-1&! PI<IU' 3-31.!1 18 20 TIM::N

W

B

~

₆

2

[Wf' = o· OOWO !ETA = O·W

A-4IC : 1Q·CXXJ PHIU = W.0497 0 "

~

~~---V

2~6 l.6 19 20 -2+ -A -6 -8 2·0

~

ê

1·0 B 10 12 :::JISPLACEMENT RE5PCJ'.JSE EETII = 0·10 PHIC = W·OOO F\..EL F"~ = 0·999 iA TIIIoEN 0·0 I , , I I I I I I , I 2 4 6 8 W 12 U l.6 19 20 TIMEN FlJEL FLOW 10 ~B 0 6 4 2 0+ 1 -2 -4 -6 -e -10 -2

~

-3 W B -4 ...J -5 -6 -7

~

4 [lN.f' = 0·001 6 8 VELOCITY RESF'O'JSE !ETA = O·W PHIC = W·OOO PHIU = W·0497 TIMEN EETA = 0·10 PHIC = W.OOO RiIU = 10· 497

-

8t

I~ 2 4 6 8 ENERGY V5. TIMEN

10 8

~

₆ .4 2 [W.f' = 0-00100 OCTA = 0-.40 A-<IC = 0-100 PHIU = 0-125

oF

2 .., 6

-2i

/

~ 1.A ' \ . 16

~

___ - - - 9 / 1 0 12 -.4 -6 -B 2-0

~

~

1-0

'r

EETA = 0·«> A-<IC = 0-100 QISPL,A.CD.ENT RE5F'CN5E F\.EL FIICTCR = 0- BB6 0·0 I I I I I I I I I , 1 2 .4 6 9 1 0 1 2 U 16 19 20 TIMEN FlEL FLOW 10

~8

o 6 2 [W.f' = 0- 001 EET'" =: Q·AO PHIC = 0-100 PHIU = 0-125 o I ,1)0 .... , ~ ,_ ,.c, ob '"' ... '. '{'4Y' .r= -2 -4 -6 -9 -10 -2

~

-3

~

B -4 ~ -5 -6 -7 -9 2 4 6 VELOCITY RE5F'O'6E 9 10 12 ENERGY VS· TIMEN 14 16 EET" == 0 • .40 A-<IC = 0-100 PHIU = 0-125 19 20 TIt.EN

10+

fl

6 ~

2~

-

'I

-6 -8 2·0

~

~

~·o ON*' = 0.001.00 CJISPLAcaENT RESPCN;E BETA : o·~ PHIC: 0·316 Fln. F ACTCR : ~. ~~ EETA = a-AIO A-lIC = 0·316 f'HIU: o· 3!3B ~~~ TI~ 0·0 I I I I I I I I I I I 2 • 6 9 10 ~ U ~ W ~ TIMEN FlEL FLOW

lOH

~8 0 6 2 0 -2 -4 -6 -8 -10 -2

~

-3

~

-9 2 CW,fl = 0·001 VELOCITY RESPONSE 6 9 10 ~ u Er-ERGY VS· TIMEN

FIG. 18 SIMULATION AT BETA

=

0.4, PHIC

=

0.316

~ EET1-= 0·40 fIt.(!C = 0·316 f'HIU: O· 3!3B EETA = 0-410 PHIC: 0·3~ PHIU: o· 3!3B w ~ TI~

10+ 10+

I

tw.P : 0-001 tw.P : 0-00100 SET'" ;: 0·,,:) PHIC.; 1'OCXJ ~B

(\

R-«u;: 1· 259 0 6 4 2 2~

A

0+

II1

~

\4f\r 6

'è V\Aà.JV}é V V U o 11 11 11 /1 I' '\: I 1_ J«I .1_ "'"Ef:c::=::!: 1A U; 18 ~ TIMEN -2

~:l v

V -4 -6 -6

1

_I

_I

-B -B -10

QISPLACEMENT RESPONSE VELOCITY RESPrnSE

lo--2 2-0 EET~ : 0-"" r ~ -3 W Z w

8

-4 ~ -5 -6 PHIC;: 1· (XX) Flll. F!oCTCR : 1-3M

~

~,,(

-7 -8 +--+--+-0·0 I I I I I I I I I I I 2 4 6 8 1 0 1 2 U 16 18 20 TIMEN 2 6 8 10 12

FLEL FLOW ENERGY VS· TIMEN

16 16 EET~ : 0-"" Pl<IC: 1- 000 Pl<IU: 1- 259 18 20 TIMEN EET ... ;: 0.40 A-tIC;: 1·(x)() PHIU;: 1·259 18 20 TIMEN

10 9

~

₆ 2 ON*' = 0- 00100 BETA = 0-40 PHIC = 3-162 PHIU = 3-!Il1 o 11 /, \< , / , ,=+--... , ,1 iA .\; .'--2 -A -6 -9 2-0

~

~

1-0 EET1-= Q • .4() PHIC = 3-162 CJI5PL.AC8iENT RESPONSE F1..EL F N:TCR = 1·005 TII'.EN 0-0 I , , , , , 2 6 9 10 12 1.4 16 18 20 TII'.EN FUEL FLOW ON*' = 0-001 2 o I 11 1111111\ -2

-.

-6 -9 -10 VELOCITY RESPONSE -2 >-~ -3

~

-5 -6 -7 -9 2 6 9 10 12 l ' ENERGY VS· TIM::N

FIG. 20 SIMULATION AT BETA

=

0.4, PHIC

=

3.16

16 BETA = 0-40 PHIC = 3-162 PHIU = 3 - _ BETA = 0-40 PHIC = 3-162 PHIU = 3·931 19 20 TIMEN

10 a

~

₆ 2 0Mf' • O· 001D0 EET~ • 0·"" A-He::: jD·OXl R-4IU = 12·591 0"

~

- - - :

V

2~6 8 10 12 lA 16 18 20 -2+ -4 -6 -8 2·0

~

~

1·0 EET~ • 0·4() PHIC • 10·000 CJISPLACEIllENT RESPONSE Fln.. FAC~ ::: O·!E7 TIMEN 0·0 I I I I I I I I I I I 2 4 6 8 10 12 U 16 18 20 TIMEN FlJEL FLOW 10

g8

6 -4 -6 -8 -10 -2 >-!:E -3 W Z w B -4 -1 -5 -6 -7 -8 +---+----+-2 0Mf' • 0·001 6 VELOCITY RE5PCNöE 6 8 10 12 14 ENERGY VS· TIMEN 16 EET~ • 0·"" PHIC • 10·000 PHIU • 12·591 TIMEN EET" ::: 0·40 PHIC • 10·000 PHIU • 12·591 18 20 TIMEN

~o B

~

₆

"

DMf' ~ o· 001.00 EETA = 0·70 R-4IC = 0·100 R-tIU = 0.167 12

'r'

-2-'-f

'\. I-.: 6 'ij w .1i\~ ~~ -~ -6 -8 2-0

~

~

1-0 L..-' EET" = 0·70 F'HIC - o-wo OISPLACEIvENT RESPCNSE FI.EL r ICTLR - 0- 796 0-0 I t t t t t t I I t I " • 6 B W 12 ~ ~ ~ ffi TIt-.EN FLEL FLOW w ~B 0 6 ~

"

0 -2 -~ -6 -B -w -2 >-Éi1-3

~

8 -.

--' -6 -7 -B (]NoP - 0-001 2 ~ VELOCITY RESPCNSE EETA = 0·70 F'HIC- O·WO f'l<IU - 0·167 ~ ffi TIt-.EN EETA = 0·70 f'I<IC - o· WO f'I<IU - 0 ·167 I I I I t t I t I I I " • 6 8 W 12 ~ ~ ~ ffi Tlt-.EN El'ERGY VS· TIt-.EN

10+ 9 ~ f-6 4 2 0 -2 -4

:1

2·0

~

~"l

.

Cl.'M' • 0·001.00 OI~T RE5PCN5E BET" = 0·70 I'HIC = 0·316 F\E.. FKTCR = 1· 31.6 EETA = 0·70 I'HIC: = 0·31.6 Pl<IIJ = 0·529 0.0 r i l I I I 1 I 1 I I 2 4 6 8 10 ~ U ~ ~ ~ TIf.EN Fl.EL FLOW

I

I

10

t

l

t::w,f) = 0·001 ~9 0 6 4 2 0 -2 -4 -6 -9 -10 VELOCITY RE5PCN5E -2 r ~ -3

~

B -4 ...J -5 -6 -7 -8 +---+----+-_{2 6 8 10 ~ 14} Et-ERGy VS· TIf.EN ~ BET" = 0·70 A-IIC = 0·31.6 f'I-UU = O· 529 BET" = 0·70 Pl<IC = 0·316 f'I-UU = 0·529 18 ~ TIt.€N

lOT

~

Bl

6 4 2tl o I I

JV

-9+ 2·0

~

~

1·0 f\ I1 D.'Of' = 0·00100 I1 1 TI ,::loet I""",~I 1 14 DISPL"CEMENT RE~ BETA = 0·70 f'HIC = 1·000 F\.EL F ACTl:R = 1· 369 EETA = 0·70 f'HIC = 1·000 PHIU = 1·672 1 1 1 16 18 20 TIM::N 0·0 I I I I I I I I I I I 2 4 6 B 10 12 U 16 18 20 TIMEN FLEL FLOW lOf

~B

0 6 2 0

t

I -2 -4 -6 -B -10 -2

i

-3

8

-4 ...J -5 -6 -7

I

\«

c:w.p :: 0·001. \ A I 6 \MIW 12 U VELOCITY RESPCNSE EET" :: 0.70 f'HIC = 1.000 f'HIU = 1.~ 16- - -iS" ~ -20 TIM::N BETA = 0·70 f'HIC = 1·000 f'HIU = 1·~ ~ I 1 1 1 1 1 1 1 1 1 1 2 4 6 B 10 12 U 16 18 20 TIMEN ENERGY VS· TIM::N

lOt 10+

M

0Mf' = 0·001 BET" = 0·70

0Mf' = 0-001.00 EET" = 0.70 PI-IIC = 3·1.62

PHIC = 3.162 ~ I A-4IU = S·2Ee

9 PHIU = 5.2BEI r:: 9 f 0 ~ 6 6 4 4 2 2

'reG,

6 8 10

, , "

12 U 16 18

, -.;

,

TIMEN -2 -2+ -4 -4

1

-6 ~ .~ -8+ -10

DISPLACEMENT RESPCNSE VELOCITY RESPCNSE

2·0

~

~

1·0

(

EET1-= 0·70 Pl<IC = 3·162 F\..EL F~ = 0·957 ~ol _~ 4 _{6 B} ~ FLEL FLOW 12 14 16 -2 ;: }E -3 ~ W

B

-4 ..J -5 -6 -7 -8 18 20 +--+--+-TIMEN 2 6 B 10 12 ENERGY VS· TIMEN 14 16 EET" = 0·70 Pl<IC = 3·162 18 20 TIi'JEN

10 B

~

6 A 2·0 ~

~

~

i·a CW.f' = 0·00100 12 ~A OISPLACEMENT RESPONSE EET" :; 0·70 A-UC = 1O.()(X) FtEL FACTrn = 0·999 ~ EET~ = 0·70 PHIC :; 1D·CXXJ A·nu = 16·723 1B eo TIMEN 0·0 I I , I I I I I I I I 2 A 6 8 10 12 U ~ 1B eo TIMEN FUEL FLOW 10 ~8 o 6 A 2

DIM" = a· 001 EET" = 0·70

PHIC = 10.ClCX) PHIU = 16·723 o 11,1 ~ 'JjII!~';\'Iw'~~~d.~~~ -2 -A -6 -8 -10 -2

~

-3

~

-5 -6 -7 -8 2 • 6 VELOCITY RESPONSE 8 ia 12 l ' Er-.ERGY VS· TIMEN ~ TUIEN EETA = 0·70 A-4IC = iQ.1XX) PHIU = 16· 723 iS 20 TIMEN

10+

0H4I' • 0·00100 SElA lil: 1.00

A-lIC z 0.100

~

st

PHIU = O·2Be 6 ~ 2

oF?\'

,

J

,

,\

,

'd'

~T~

~ B _{10 \ 1Z} .. ~ i6 -2

:1

-8+ 2·0

~

ê

.. ·0 BETA = 1·00 PHIC = 0.100 DISPLACEl\€NT RE5PCNSE na f'ACTCR = 0·907

---0-0 I I I I 1 I I 1 , I t 2 4 6 8 2.0 1Z U i6 ~ ro TIf,EN FLEL FLOW 10+ ~B ~ 0 6 -6 -8 -2.0 -2

~

-3

B

-4 -1 -5 -6 -7 DI'W = 0·001 VELOCITY RE5PCNSE EET" 11: 1AD PHIC = 0·100 fll-4IU = 0 .. 282 BETA = "·00 PHIC = 0·100 f'HIU = 0·292 -8 I 1 I I I I I I I I I 2 4 6 8 2.0 1Z U i6 ~ ro TIf,EN EN::RGY VS. TIM::N

10 B

~

₆ -4 -6 -9 2-0

~

ê

1-0 [lN.f' : 0-00100 DISPL"CEMENT RESPCN5E EETA = 1-00 PHIC.: 0-316 F\.E... FACTI:R = 1.710 EETA == l·CI) Pl<IC: 0-316 PHIU == 0-994 0-0 I I I I I I I I I I I 2 4 6 9 10 ~ " ~. w ro TIr-.EN FLEL FLOW 10 [lN.f' : 0-001

~9

6 -4 -6 -9 -10 ·/ELOCITY RESPONSE -2

~

-3 ~ B _.J -. -s -6 -7 -9 +---+--+-_{2 6 9 10 ~ 14} ENERGY VS· Tlr-.EN

FIG. 28 SIMULATION AT BETA

=

1.0, PHIC

=

0.316

~ EETA == 1-00 PHIC:. 0.316 ""'JU: 0-994 !ETA : 1-00 PHIC : _0-316 ""'JU: 0-894 w ro TIr-.EN

10+ B

~

₆ ~ 2 0

::j~

-8-1-2·0

~

~

1·0 [lH.P .. 0.00100 U DISPLACEMENT RESPCNSE BET/\ = 1·00 PHIC = 1·000 F\.EI. ,1\CTCJl = 1· 360 0.0' ~ I~ ~ B 10 12 u FLEL FLOW BETA = 1-00 PHIC = 1·000 PHlU • 2·SëEI 16 19 a:J TI~ 16 19 a:J TIM::N lO

t

~9

0 6 ~ 2 0 -2 -4 -6 -8 -10 -2

~

-3 W

8

-4 .J -s -6 -7 -8 IJIM:' s 0-001 VELOCITY RESPCNSE 2 ~ 6 B 10 12 JA Ef'.ERGY VS· TIM::N 16 SETA = 1,..(JO Pl-4IC = 1·000 PHlU = 2·SëEI BET/\ = 1·00 PHIC = 1·(XX} PHlU = 2·828 19 a:J TIM::N

10 8 ~ I-6 4 2. 0 -4 -6 -8 2-0

~

ê

1-0 ~ = 0-00100 6 8 10 <2 '4 OISPLACEtvENT RESPONSE IJZT~ = 1-00 R-UC = 3·162 FI.EL FACTlJ'l = 1- 020 <G EET ... = 1·00 ~C = 3·.1S2 PHIU = 8·944 18 ~ TI~ 0·0 I I I I I I I I I I , 2 4 6 8 10 <2 U <G ~ ro TIMEN FUEL FLOW 10 DMf' = 0·001

~8

6 2 0

+1

l:

6 -2 -4 -6 -8 -10 VELOCITY RESPONSE -2 ~ ~ -3 W ~ -4 -5 -6 -7 -8 +---+---+-2 6 8 10 <2 14 ENERGY VS- TIMEN

FIG. 30 SIMULATION AT BETA

=

1.0, PHIC

=

3.16

<G EET~ :: 1·00 PHIC:: 3·162 PHIU:: 9·944 EET" :: 1.00 PHIC = 3-162 R<IU = 8 -5144 ~ ro TIMEN