Analysis of the errors of the dynamic analysis and control laboratory flight simulator

-ANhALYSiS Ok ThE EilR O F '.-ThE DYNAMIC ANALYSIS AND CONTROL LABORATORY FLIGHT SIMULATOR

by

Herbert Jaoobs, Jr. B.S. University of Arizona

(1944)

SUBMITTED IN PARTIAL FULFILLMENT

OF THE REQUIREMETS FOR THE DEGREE OF MASTER OF SCIENCE

at the

MASSACHUSETTS INSTITUTE OF TECHNOLOGY

(1949) Signature of Author Certified by =

Signat

Di

ure redacted

ptA

Elect.

Jn 108 1949

:ure redacted

/., 62 T,

~Supe

rvi s or

Signature redacted

INST. r

APR 18 1949

Date Acknowledgement

To Thomas F. Jones, Jr. I wish to express my sincere appreciation for his supervision and guidance throughout the performance and writing of this thesis.

To Dr. Albert C. Hall I wish to express my sincere

appreciation for providing the opportunity for me to undertake this thesis and for his making available to me the full facili-ties of the Dynamic Analysis and Control Laboratory, without which this particular thesis could not have been undertaken.

Page

Acknowledgement . . . . . . . . . . . . 11

Abstract . . . . . . . . I Introduction General . . . . 1

Flight Simulator Equations.

. . - - . . . . 3

3

Solving the Flight Simulator Equations . . . - -. . 13

Summary of Objectives . .0... .0.. . . . . . .*. 17

II Analytical Analysis of the Flight Simulator General . . . . . . . 19

Solution by the Classical Approach . . . . 22

Solution by the Servo Approach . . . . . . . . . . 29

Solution by Gain Considerations . . . . . . . . . . 32

Conclusions . . . . . . . . . . . . . . . . . . . .. 37

III Experimental and Analytical Analysis of Integrator Servos Introduction . . . . . . . . . . . . . . . . . . . . 40

Electro-mechanical Integrators . . . . . . . . . . . 42

Translational Rate Servo . . . . . . . . . . . . . . 51

Experimental Equipment for Testing Integrator Servos 63 Results of Tests on Integrator Servos . . . . . . .. 66

Conclusions . . . . . . . . . .. . . . - . . . . .. 76

IV Conclusions General. . . . . . . . . . . . . . . . . .. . . .. 79

(Table of Contents)

Standard Types of Tests for Accuracy Evaluation . -. 85 Summary . . . . - . . . . - - . . - - - 90 Bibliography - - - . - *

.92

This paper deals with the analytical and experimental

analysis of the Dynamic Analysis and Control Laboratory Flight Simulator. The simulator analyzed is a computer which solves the aircraft equations of flight by means of two types of instru-ment servos. The objectives of the analysis of the errors are: first, to determine and evaluate criteria for the choice of the time scale to be used in solving any given problem so that the simulator will be used in an accurate region of operation; and second, to determine standard types of tests by which a continuous evaluation of the accuracy of the solutions by the simulator can be maintained.

It is concluded in this thesis that there are two impor-tant criteria for a change of time scale. These criteria are: First, the order of magnitude of the frequency components which the simulator can handle accurately is at present 2 cycles per second. Therefore the time scale must be changed until the highest frequency components of importance in the problem are

of this order of magnitude. Second, the per-second gain of each integrator servo is limited in value. Therefore the time scale must be changed until each estimated per-second gain required fr the problem is equal to or less than the available per-second

gain for each integrator servo.

vi

on the results of this thesis. The first type of test is

a check of the individual loop time constants with the "prob-lem at hand" set up in the simulator. The second type of test is a check on the calibration of the non-linear terms. The third type of test is to record the error signal of each integrator servo as a simple and direct measure of the

accuracy of the solution on an instant to instant basis. In general, the most effective results are obtained when the standard types of tests are used in conjunction with the

important time scale criteria for setting up a given problem.

Chapter I Introduction General

The DACL Flight Simulator is an analogue computer particularly adapted to solving two general types of dynamic problems of air-craft flight. The first general type of problem is the dynamic testing of the response of a motion sensitive automatic control device of an aircraft whose aerodynamic stability characteristics

are well known. The second general type of problem is the determin-ation of the degree of aerodynamic stability of a given airframe

either with locked controls or controlled by a mathematical ideali-zation of an autopilot. The solution of these dynamic problems is obtained from three moment equations and three force equations based on the three orthogonal body axes of the aircraft.

The solution of the first general type of problem requires a flight table whose angular motion is equivalent to the actual angular motion of the aircraft. The DACL Flight Simulator has a flight table with the angular motion being supplied by a system of four gimbals. The three inside gimbals are acceleration-servo con-trolled. The body-axis angular accelerations, as given at the output of the computer, are resolved into the gimbal axis accelerations which drive the respective gimbal servos. The outer gimbal is position servo controlled in such a way as to prevent gimbal lock. In order for the

outputs of the motion-sensitive automatic control devices to have meaning the motion of the flight table and therefore the gimbals must

be on a one to one time scale. This further requires that the

acceler-ation servos be sufficiently fast to drive the gimbals on a one to one time scale.

The other general problem of determining the aerodynamic sta-bility of a given airframe requires no physically equivalent motion and therefore can be solved on any time scale. Only a mathematical representation of the motion is required. A system of

electro-mechanical integrators has been built to be used interchangeably with the gimbal system and to fulfill their mathematical function. This thesis analysis will be carried out on the simulator with the system of electro-meohanical integrators replacing the gimbals.

The electro-mechanical integrators were not designed as high speed servos because the time scale can be slowed down in all prob-lems where they are used. However many considerations are necessary in choosing the time scale to be used and the method by which the time scale changes will be accomplished. One objective of this thesis

is therefore to determine and evaluate criteria for the choice of the time scale and the method by which the scale change will be made so

that the electro-mechanical integrators will be used in an accurate region of operation. The other objective of this thesis is the development and evaluation of a system for testing the simulator which will be indicative of the accuracy, precision and sources of

error of the simulator with electro-mechanical integrators replacing the gimbals.

Flight Simulator Equations

The equations of motion which the flight Simulator solve were first formulated by Michael Witunski1 ,2. These equations are the summation of the moments about each of the three orthogonal air-craft body axes and the summation of forces along each of the three orthogonal aircraft body axes. Figure I-1 defines the body axes and surface deflections with polarities of each indicated. The origin

of the axis system is the center of gravity of the plane. The X-axis is directed out through the nose of the ship with right wing down being a positive rotation. The Y-axis is directed out to the right with nose up being a positive rotation. The Z-axisis directed down-ward with right wing back being a positive rotation. The surface deflections are designated AFV, meaning the angle of the front-vertical surface deflection, of which the conventional aircraft has nones ANx, representing the aileron deflections A2 and A4 which

pro-duoe the roll of an aircraft; ARV, the angle of the rear-vertical surface deflection, the vertical tail surface-deflection of the con-ventional aircraft, and A , the angle of the rear-horizontal surface deflection, the stabilizer of the conventional aircraft.

The large variations in design, size, altitude and speed of the conventional aircraft present an impractical limit on the kinds of aircraft which can be simulated unless an effective method of equation

'Report #1 - DIC-6387, DACL - MIT January 2, 1946 Airframe Eqiations of Motion (Unclassified) Author: Michael Witunski

2_Report

_#13

_{- DIC-6387, DACL - MIT May 15, 1946}

Introduction of Elementary Aerodynamic Concepts (Unclassified) Author: Michael Witunski

YZ PL

ANE

(so-AoiI.

+ 1+) (-) At

D

REF. LEMPT REF. AREA

S

NOTATION FOR AIRPLANES

normalization is adopted.

As a result, the equations used for the

flight simulator have been normalized with respect to the aircraft

forward velocity, V; the reference distance D, and the reference

area S, shown in Figure 1-1; and the density of the air,,p. In order

to normalize the equations with respect to the forward velocity it

was necessary to assume that the propelling force was essentially equal

and opposite at all times to any drag forces along the X-body axis.

Therefore the acceleration along the X-body axis may be considered

zero and the velocity considered a constant parameter of the

equa-tions. This makes one of the three translational force equations

unnecessary.

The aerodynamic equations used for the Flight Simulator as

analyzed in this thesis, with the electromechanical integrators

re-placing the gimbals are given in Figure I-.2 .

A short discussion

of the physical meaning

of the various terms gives a better

under-standing of the equations.

It is known that a surface deflection,

the rudder of a conventional aircraft for example, causes

three

gen-eral effects when deflected. First it causes a drag to the forward

direction of flight, second it causes a force on the aircraft in a

horizontal direction, and third

it produces a moment about the

Z-body axis. Since the forward velocity, V , is considered constant,

the drag effect is assumed overcome by the driving power of the

air-craft thus producing no resultant force. The force in the horizontal

direction is given in the equations by

f An

(Eq. 1-3). The

A

'Memorandum 3.4-2 - DIC-6387, DACL - MIT August 4, 1948

Revised Plan B Equation Sheets for Integrator Bank Synthetic Gimbal

System (Unclassified)

q

F V

e s

Ys( a) + YFV(AFV)

1

CV2S 0 Za(a-0) ZFH( FII) Fy vi F Y s Fz Fz

-

2

?-+ fY) + fZRH (ARH)

!vs

T!evs

m N

eVb S

xo +

(nX

_D)

_A

8

a

+

(n

)(

)

2 +(bXb ) WX + +

(bX

D)

(

Z

N y N _z e~ VD S V

(nYaDY )a

Sf+ (n~₈ V)

As

+ _{) AH} + (n )+

ARV

+

(ba) w

Figure I-2a SIMUIATOR EQUATIONS (1-2) (I-3)

(I-s)

(I-6)

(b

1

b

_(A1)

(ba) W

(I-8) (1-9)

V) A

ZFV 17

FV + (n

ZRV IF

V)

a

., N

a

2

.CTh S

a z _Z2

(I. tVD S

WX faidt WY - faydt Wz - fa&Zt

Vy

Fy

Y

Y

V FZ

z

z-2 CV 2 2 V

(yms

VD2-C'Czs

-W

+

W

+

WY - WX

)

v

vy

0

vpPrdt

Aa

V

V

V v

A--

mY+ YY

8a

T

_{SIMULATOR EQUATIONS}Figure I-2b

(I-10) (I-12) (I-13) (1-14) (1-15)

(1-17)

(I-18)

(I-19)

(I-20)

(1I-21)

a

Acceleration (angular)

A

Angle

b

Dimensionless damping moment coefficient

b2

Wing span squared

B

Damping moment coefficient

D

*Reference dimension

E

Error signal indication

f

Dimensionless static force coefficient

F

Force

g

*Acceleration due to gravity

G

Transfer function indication

J *Moment of inertia

m

*M&sS

N

Dimensionless static moment coefficient

p

Time-derivative indication

*Constant for Plan B System

a

B

0

(R)

R

5

(T)

W

I

Y

z

Attack

Referred to missile (body)

Constant for a particular case

Referred to axis of measured signal

Referred to reference axes

Sideslip

Reference to measured quantity

Wind

Referred to X-body axis

Referred to Y-body axis

Referred to Z-body axis

Figure

I-2o

SIMULATOR EQUATIONS

p Second time-derivative indication q *Dynamic pressure

C

*Density S *Reference area

t

Time

V Velocity (Translational) w Frequency- (angular) W Velocity (angular)

x,y,z Displacement along particular reference axis (translational)

XYZ Displacement about particular gimbal axis (angular)

Y Displacement (angular) of outside gimbal frame

V

Two letters

(XX, UY, Yy, etc.)

Reference to effect of rotation about axis repre-sented by outer letter upon axis reprerepre-sented by inner letter.

(e.g. b a dimensianless damping moment

coefficient about X-body axis due to rotation about Y-body axis.) FH a forward horizontal surface

FV n forward vertical surface RH n rear horizontal surface RV a rear vertical surface NX - roll control surfaces

*Constant for Plan B System

Figure I-2d SIMLATOR EQUATIONS

0

is the magnitude of the surface deflection in radians. The force produced is assumed to be proportional to the magnitude of the sur-face deflection and this proportionality constant is a function of the aircraft construction and is given by f .

The direction of a positive deflection is defined as that

direction which causes a positive force. This is the basis for deter-mining the positive direction of all surface deflections. As a

re-sult of this definition, the moment caused by a positive AV defleo-tion is a negative one about the Z-body axis as is given by

n V(AR) Eq. (1-9). -The sign of .the constant of proportionality is therefore inherently negative. These same three effects are

caused by the other surface deflections of the aircraft (see Equation (I-3), (1-4), (I-8) and (1-9). There is one exception and that is when one of the pairs of surface defleotions is operated differentially

rather than in unison, for example the ailerons of a conventional air-craft. In this case a rolling moment is caused but no resultant force. It is expressed in the equations as n (A ) Equation (1-7).

The angle of attack and angle of sideslip are defined by Equations (1-20) and (1-21). If it is assumed that the air is still, i.e. the wind velocities V and V are zero, the angle of attack becomes

VV

equal to and the angle of sideslip becomes equal to '-"

.

The angle

V

-_

V

of attack, a, produces a lift on an aircraft and also produces a moment about its Y-body axis. The lift force is expressed in Equation (1-4) by fZa(Aa) where fZa is inherently negative since a lift is in the negative Z direction. The moment is expressed by Ya A) and

ny V is again inherently negative to express the negative moment

produced by a positive A. The angle of sideslip causes a similar

effect in the Y force, Equation (1-3) and in 'the Z moment, Equation

(1-9). Dihedral in the wings of an aircraft produces a rolling

moment which

is

proportional to the angle of'sidealip and is expressed

in Equation (1-7) by the term

n

₁

!.Ly(A)0

b

An angular rate about Y or Z is directly a rate of change of Aa

and Aa respectively, thus they are included in Equations (1-16) and

.

(1-17). But angular rates also have other effects. For example when

the aircraft is rolling at a given rate, there is a resistance to

that roll due to the damping effects of the wings. This damping

moment is also present about the Y and Z axes and is expressed by

b D2

(M.

), b

),bf

(W).

and bZZ

d(WIz)(7)

() respectively in Equations

(I-7),

b

(1-8) and (1-9). This moment opposes the direction of the forcing

moment, therefore the coefficients b D

2

b

and

b

are inherently

b

negative.

The terms of the equations which have been discussed up to this

point are called linear functions, i.e., the forces and moments

pro-duced are directly proportional to the rates and angles which produce

them. Mathematically speaking the differential equations formed by

the inclusion of only these 'terms are linear and can be solved by

classical methods. However in aircrafts there are certain

inter-action forces and moments which are referred to here as cross-product

terms. These cross-product terms in the equations are non-linearities

12

which prevent the equations from being solved by classical means. The aerodynamic interaction cross-product effects are now discussed.

When an aircraft is flying with a given angle of attack, and a rate about the Z-axis occurs simultaneously, there is a difference in the relative wind velocities at the two wing tips of the aircraft. This difference produces a difference in lift of the two wings which results in a moment about the X-axis. This moment is included in

(1-) b th tem bzaD

2

Equation (-7) by the term b -(Aa) (WZ). A similar moment is

produced about the 1-axis when the aircraft has an angle of sideslip ocouring simultaneously with a rate about the Y-axis. This moment is given by b D2

(As

)(W)

Equation (I-7). Also due to the

differ-Xrs- 8s1

b

ence of the relative wind velocities at the wing tips of the aircraft, but produced by a velocity along the Z-axis occurring simultaneously with a rate about the I-axis, a force along the Y-axis is produced.

A similar effect occurs with a velocity along the Y-axis and a rate about the X-axis. These forces are expressed as(WX) and (W ) Y in Equations (1-16) and (1-17).

These are the only cross-product interaction terms considered in this thesis. In the final layout of the simulator there will be many cross-proauct terms for the Euler inertia interaction terms, for

example. It will also have products of many sine and cosine terms for the resolutimn of the force of gravity, for example. It has been estimated that it would require three and one-half times the capacity of the MIT differential analyzer #2 to solve the equations to be solved

Solving the Flight Simulator Equations:

The philosophy used in the Flight Simulator for solving these

equations is very straight forward and shown schematically in Figure 1-3. It is divided into three basic parts, the electro-mechanical

integrators , the translational rate servos, and the aerodynamic com-puter. The electromechanical integrators are used to compute the rates from the values of acceleration fed in from the aerodynamic computer. The translational rate servos give the velocities from the rates of change of velocity as fed in from the aerodynamic computer. The outputs are then fed back into the computer or used to excite multiplying potentiometers as shown in the figure. It is pointed out that _{the translational rate servos must work with the electro-mechanical} integrators on a reduced time scale and also be fast enough to work with the gimbal acceleration servos on a one to one time scale. There-fore the translational rate servos cannot be replaced by the electro-mechanical integrators since the latter are too slow to work with the gimbals.

The method of signal transfer in the computer is 400 cycle suppressed carrier. There are five primary signal input potentiometers as shown in Figure 1-3. These pots are each fed from the 400 cycle supply via step-down transformers' and the outputs from each is a 400 cycle sig-nal whose amplitude is proportiosig-nal to the displacement of the servo

1_{Thesis - E.E. Department MIT, January 1948} A High-Precision Electro-Mechanical Integrator Author: William James Deerhake

2

_Memorandum

_{- 1106-2 DIC-6387 - DACL - MIT April 20, 1948}

Translational Rate Servo (Unclassified) Author: T. M. McCaw

K COMPUTER p R V. SDx COMPUTER l.a. y WY w INTERGRATOR I. Z Wa W INTERGRATOR K -E

______J

I

r~I

I NOTES: DEFINITIONS

j.D.=AERODYNAMIC

CONTROL SURFACE DEFLECTION - - ANGULAR ACCELERATION (BODY AXES)

W = ANGULAR RATE (BODY AXE3)

Z = SUMMING POINT

V - TRANSLATIONAL VELOCITY

(

R.V - REFERENCE VOLTAGE

SIMULATOR

BLOCK

DIAGRAM

BODY AXES)

FI GURE I-3

POT.

R.v. MT 3E

UNHEAR UNEAR Yn TRAMBLAIOn A vS

D-P~r'. OT. RATE 3ERYO

shaft from the zero position. These signals are then fed around

through the computer as specified by the equations to be solved.

The computer is made up of various standard components which,

like building blocks, can be put together in various ways to solve the necessary equations. The basic computer component is the

ampli-fier rack shown in Figure 1-4. Each rack holds six pair standard amplifiers1 which plug in from the rear, and the standard plug-in units which plug into the front of the panel as shown. The standard plug-in units consist of phase-correctors, standard attenuators, coefficient potentiometers, amplifier termination networks, and summing networks.

The specification on maximum error for each component of the computer is .1 per oant of full scale. In order to maintain this accuracy in the amplifiers, they are calibrated at a fixed gain with a given load of 10 K, for exampit. In order to maintain this load-ing, special plug-in amplifier termination networks are used. These furnish the proper loading for the amplifier and also the 5 K source impedance for feeding the summing circuits where necessary. The

suming circuits are designed for 2, 3, 4, 6 and 8 inputs and each in-put is assumed to have a 5 K source impedance. The coefficient

potenticmeters are used for multiplying the input variable by an aero-dynamic coefficient and for calibrating a signal level. They are dual pots with a constant 10 K input impedance and a constant 5 K output

'Report +47 - DIC-6387,

DACL

- MIT November 25, 1947

A 400-Cycle Multiple-loop Feedback Amplifier (Unclassified) Author: Max T. Weiss

16

11 ~L w6 w6 a~** m6 * (* '~ask :.* 0 - g a6 C *so .0' . SPe be* o* -,

F

I

t

I:

AMPLIFIER RACKS

Figure

I -4

Oft:%~

046 * to ea $ ~r4

.11

II

-I 4 I i

i

17

source impedance.

The phase-corrector units are

to correct the

phase at the output

of

a resolver, for example, and the

attenu-ators are for changing the scale level of the signal at any point.

Summary of Objectives:

There are a number of possibilities for error in

the

solu-tion of problems with the simulator. The

two principal sources

encountered are those introduced by operating the mechanical

units beyond their dynamic limits of linearity and those

intro-duced by inaccuracies in the various components in normal operation.

Therefore the two principal objectives of this thesis are first

to determine criteria for a proper choice of time scale to

con-trol the errors of the first type, and second to set up and

evalu-ate a system for testing the simulator to determine the errors

of the second type.

There are a number

_{of'methods for changing the time scale}

of operation of the simulator, each of which affects the accuracy

of simulation in a different way. It will be necessary, therefore,

to investigate the possible methods of time scale change, and

their affects upon the accuracy in order to determine a criteria

for a proper choice of time scale.

The accuracy of the simulator is to be checked by comparing

its solution to a given problem with an accurate solution of the

same problem. The complexity of the problem to be used may vary

from a simplified linearized form to the complete set of equations

18

for the simulator. An accurate solution of the complete set of

aircraft equations is both difficult and expensive to obtain. Such a solution may be desirable ultimately, but until the

simu-lator is more highly perfected it appears more desirable to use a simplified system of testing which can indicate how the in-accuracies of the various components affect the accuracy of the

**

overall system.

+

It has been reported that I.A.M.

Company

of New York has given

a general estimate of 20 hours at $500 per hour for the solution

of a set of aircraft equations.

Analytical Analysis of the Flight Simulator

General

A complete analytical analysis of the flight simulator is,

in the strictest sense, impossible by non-machine methods, for

it would require an explicit solution of fi"e simultaneous, second

order, non-linear, differential equations.

Therefore, in order

to give meaning to the statement, "analytical analysis of the

flight simulator" certain qualifications must be made.

These

qualifications include simplifying assumptions which will reduce

the equations to a linear form. When a linearized form for the

equations is obtained there will be only one "system function"

1

for each variable, however there may be a number of approaches

for obtaining the system function.

It is necessary to

investi-gate a number of approaches in order to become thoroughly

familiar with the computing technique of the simulator.

The objective of making an analytical analysis is to obtain

an exact solution against which the simulator solution can be

compared. Any simplifying assumptions which reduce the

cross-product terms to linear terms are in general not sufficiently

accurate to give a solution which can be used as a stapdard

against which the simulator solution can be checked. The

cross-product terms are the only terms which cause the equations as

used in this thesis to

be non-linear, therefore

an analytical

analysis can be made if they are made zero.

1

_{Transients in Linear Systems: Gardner and Barnes, John Wiley &}

K_{2 0 A + fYFV(AFV)} K 20 _{(a) + fZFH(AFH)}

F

1 2 _K2 1

F

Fz

F

_Fy

Vm F z - U +fYR(RV)

K

₂₁

N

1PVDZS0 a S n Tf YDZS (+K 3 1(A)+ (nF ,)A.,,+ + 31 a+(ZFV D +

tOVD

S Wy n a

dt

W _z

V

y

V

z a

dt

Fy v K1 1 -WZ

Fz

Z _{+ WY} V py

d

V

p

v

d7

Linearized Equations Figure II-la

F

vzs

(uI-i)

(11-2)

(II-3)

(11-4) (nRD v K 0 Y.

(I1-5)

ZRV IF1V+10-(II-6)

(II-8)

(1I-9)

(II-10)

(II-12)

(II-13)

S V p dt AV VY a v Vs.-Symbolism Transormation 2 K1 0 a b '_b b _bzz 1

2

Tv IVD 3 K_{1 1} -J K20 *Ys Za 1 K 7RVS

K

_{2 1}

m

K

_{3 1}

Ya

-nZs Linearized Equations Figure II-lb ,~- ~ (11-14) (1II-15) (II-16) 1

₂

7toVD S (I1-17) (11-18) (11-19) (II-20)

If it is assumed that the aircraft being tested is per-feotly stable about its roll axis, i.e., a. and W are zero, all

x

cross-product terms are eliminated. The computer equivalent of this assumption is to make the W integrator inoperative and looked in its zero position. The equations which result are given in Figure II-1. A study of these equations indicates that there are two independent sets. One set of equations involve the variables A and W and the other set involve the variablesA₅ and

a

Y

W . Therefore any disturbance in Aa or WZ should have no affect on either Aa or WY and visa versa. The solution of these equa-tions can be used to indicate the accuracy of operation of each of these two sections of the simulator and to indicate any erroneous interactions. The two sections should be perfectly independent with the roll axis looked in its zero output position.

It is also necessary to check the accuracy of each of the integrators and each of the translational servos as it is used in the computer. This can be done by assuming all of the co-efficients zero except those involving a single variable. These equations are given in Figure 11-2. The accuracy with which each of these sets of equations is solved is a measure of the accuracy of integration of each of these units.

Solution By the Classical Approach

The classical approach for getting the solution of a linear differential equation can now be applied to any of these equations

Y axis Translational Equations

F

-K 20 (A8) (II-22)

F

F

.22'

(K

₂₁

)

(II-23)

V F

y

y

V-

W

(11-24) y

vfp'rdt

(11-25)

VY

A* -- -

(11-28)

Z axis Translational Equations

F

91-Z. K20

a

(11-27)

F

F

F

Fz

(K

₂

)(11-28)

Vm

nT21

Z

FZ,(11-29)

V V p z dt (11-30)

V

Aa (11-31)

X axis Rotational Equations

NX

X 2 3 K10 ]) (11-32)

.p

₁

Vb 3

axa

(T-

N

)

(K

₁₁

)

(11-33)

AgVb S

w

1

a dt (II-34)

Individual Axis Equations Fig. II-2a

4

Y axis Rotational Equations

N

1

y

0

(KY10)

w.

(11-35) -ZI VDZS.

SY M 1NY2')

K11

(11-36)

jVD

S

Wy f Jaydt (11-37)

Z axis Rotational Equations

1-z (K)10wZ (11-38) "r VD

S

a 1

(

2 ) (K ) (11-39)

~pVD

3

W

-

a dt (11-40)

Individual Axis Equations

Fig. II-2b

and a complete solution can be obtained without considering the simulator. The equations of Figure II-1 involving the variables As and W will be solved. The lower case p will be used as the LaPlace transform 6perator in order to be consistent with the nomenclature of the simulator equations. Combining Equations

(II-1) (II-3, (I- 1),(I-13) and (II-15) l fig9UrI-gives:

--

TAFV(fZFV)

+ AV(fZRV3 (K2 1) (II-41)

A(K )()

- WZ+ pA

20 21)

combining Equations (11-6), (II-8), and (II-10) gives:

[nZO . + V(nZF V A RV (K 1)

PW +1- (K3 1 + WZ (K1 0)] (K11) (-42)

All of the coefficients in these two equations are assumed positive which is the reason for the reversal of sign of the coefficients (n ), (K3 1 and (K10). In the setup of the

simulator as considered in this thesis the terms on the left of the two equations are independent of the variables and merely constitute forcing functions. In testing the actual computer it is found to be more practical to consider these various input forcing functions zero and insert a single disturbance in the form of a force or a moment applied directly to the force or moment summing circuits. Aerodynamically this is equivalent to assuming the control surfaces looked in the zero position with a fictitious

r&)

force or moment applied to the aircraft. For this particular

solution a disturbing moment,

M

₂, is assumed.

Applying these assumptions, the equations can be written in the following form:

W - As P + (K2 0) (K2 1

J

(11-43)

-A

(K

₃₁

)(K

₁₁

)

K(1-4)

WZ 0

[p+(Kl1(

j p+(Kl0)kK 11)

As(p)

K

₁₁ p 2_{+p (K} 2 0)(K21)+(K1 0)(Kll I + (K1 1) [K3 1+X 2 1)(K1 0)(K2 0}

1

(K1 0)(K2 1)(K2 0) +KY 3 1 (11-45) 1

1

₂

_(K

2 0

)(K

2,)+(K

)(K-T)

Kl (K110)(K21)(K20)

_~~F

_io

_~

_z+K

_i~K

+ K311 p ₁₁

[(Kl

+ ₀

)(K

Ki ₂₁

)(K

20(

)+K 311

10 1 1 2) + which is a quadratic response function with

S- K _{1 1} PK ( 10)(K21)(K)20 + K3 1 (11-46)

and

1

(Kl

₁

)(K

₁₀

)+(K

_{2 1}

)(K

20

)

(11-47)

K 1 (K1 0)(K2 1)(K2 0 )+43

The simulator should give a response for Aa exactly as

specified by Equation (11-45) for any given forcing function MZ(p). Also there should be no interaction into the other portions of

the simulator as a result of this disturbance. The equations of Figure II-1 involving the variable A. and WY are of the same form and therefore have the same response function with an interchanging

of the subscripts. This is also an independent loop and should

cause no interaction with the rest of the simulator.

The form of the equation for the response of any one of

the five sets of independent axis equations is the same.

Con-sidering the equations for the X axis (see Figure 11-2) give

W (P)

K0

'()

₁₀

(11-48)

1

)

p

+

l

(K

11)(K

₁₀₎

This is the response of a simple

lag

whose time constant is

given by

7"

1

(11-49)

The exact time solution for any kind of analytically

ex-pressible input signal can be obtained from Equations (11-45)

and (11-48). However it may be desirable to change the time

scale of the solution. The following theorem is used for

chang-ing the time soale .

-

KTF(KTp)

(11-50)

applying this theorem first to Equation (11-48) gives

KT

W - K

(II-51)

(K

10

p)(1

p

+

1

1

_{Transients in Linear}

_Systems.

_{Gardner and Barnes, John Wiley}

If the input is a unit step, then

1(p)

K

(11-52)

in the new time scale

K _T 10

T

W

1

(p)

KT

P

K1 ) (K

10)

p +

K . 10 _(II-53) p K1 K KT 10 p +1

In a similar manner the time scale change theorem can be applied to equation (11-45) giving:

1 (K K21 )( +K31

(10)(K

) ( 20

K+T(II-54)

K2111 2)(K 2 0) ) 1 + 10 p p + 0T p+

[(K1)

() )

(K

2 0

)+

K10)(

)(K

?

2

By investigating the dimensions of the various aerodyrmmic coefficients of Equations (11-13) and (11-14) it is found that all are dimensionless except those which have the dimension of seo. They are:

In the two forms of Equation (1I-53) the KT was manipulated in

order to be associated with one of the coefficients having the

dimensions of sec

a,

by

similar manipulations in the derivation

of Equation (11-54), each KT was associated with such an

aero-dynamic coefficient. It is always possible to make such an

association since there must be a

term

(coefficient) with the

proper dimensions of time associated with each power of p for

the equation to be diwensionally correct. The KT appears in

the

numerator of Equation (II-54) because A8

and the input moment

IL.

are not of the same time dimension.

In conclusion, therefore,

it can be said that the time scale of a problem can be changed

by changing the magnitude of each aerodynamic coefficients which

has the dimension

seoc

1

by the desired factor T'*

All the above solutions were obtained with no considerations

of the simulator or its technique for obtaining a solution.

At

times this has its advantages but generally more insight into the

simulator is obtained by having the solution or the manner of

obtaining it, a direct

result of considerations of computer

aom-ponents themselves.

Solution By the Servo Approach

The equations are solved in the simulator by using a servo

to generate each of the five variables. Each servo is controlled

by the necessary feedback loops as indicated by the equations.

30

a zero rate error instead of the usual zero position error, and therefore are called rate servos. However they can also be called integrator servos if the feedback term proportional to the rate is considered to be fed back within the servo. As a result the three servos which generate the value of W , W and W are called integrator servos, and the two that generate A5

and Aa are called rate servos.

The basic servo loop set up for generating W is given in Figure 11-3. In this form it solves only the X-axis roll equa-tions as given in Figure 11-2. There are other input terms

there-fore which are not shown. The boxes showing the coefficients are simply gains equal to the magnitude of the coefficient in each case. The transfer function of the integrator-servo box is ideally ., though in actuality it could be represented more

p1

accurately y p+ and even more accurately by higher order terms in the denominator. But at this time the ideal solution is desired, so the transfer function of 1 will be used. However,

p

the inability of the integrator-servo to be a perfect integrator, as is indicated by these higher order terms, is one indication

for the possibility of error in the system. This will be dis-cussed in more detail in Chapter III.

The closed loop system shown in Figure 11-3 can be solved by servo analysis methods from a knowledge of the various transfer

kt -i

input (Mx)

K

Integrator

Servo

L

_{K 10}

Integrator Servo Loop Block Diagram

FIGURE HI

3-functions

.

The solution is based on the general expression

outputK(p)

lG(p)

(11-55)

Input (p)

1+K

_{lG (p)K2G2(p}

where KG (p) is the forward loop transfer function and K

2

G

2

(p)

is the backward loop transfer

function.

Therefore for Figure 11-3

11

W

(

NX 1 + (K l)(1)(K10)

or

IX

K10

_(11-57)

I p + k~(K lo

'Principles of Servcmechanisms, Brown and Campbell

John Wiley &

Sons, Inc., Page 225

This approach for determining the ideal response to be

expected from a given portion of the computer is equally appli-cable to each of the other integratbr servos. In the case of the translational rate servos there is no coefficient in the forward loop and two in the backward loop. The solution in this case is given in Equation (II-58).

A8(p) (K2 0)(K2 1)

r

_{p + 1}

(K_{2 0)(K}

2 1)

A comparison of Equation (II-57) and (11-58) shows that as long as the magnitude of the coefficients remain the same the time con-stant of the response does not change. But a variation of the location of the coefficients, i.e., the location of the gain around the loop, varies the magnitude of the response.

Solution by Gain Considerations

More familiarity with the computer, as well as a better understanding of the effects of gain variations around the loop

can be obtained by approaching the ideal solution by means of gain considerations only. First consider more closely what is inside the box marked Integrator Servo. It consists primarily of a high gain amplifier whose output furnishes the control field voltage of a two-phase servomotor. A tachometer is connected directly to the shaft of the motor to feed back a signal

propor-33

tional to velocity and a potentiometer is also connected to the

motor shaft through a variable gear box to feed back a signal

proportional to position. There is a slight delay in the electronic amplifier and a little larger delay in the control field of the motor, and ideally this delay should be zero in both cases. There-fore in order to get the ideal solution for the response, the trans-fer function- from the input rate error to the output potentiometer shaft position is assumed to be a gain K approaching infinity in the ideal case. Figure 11-4 shows the servo loop arrangement shown in Figure 11-3 in terms of equivalent loop gains instead of aero-dynamic coeffibients and an integrator.

Input

(Mx) _{Position Servo} WX Output

Gain

Gain

Cp

K

Ta ch

Taock

Gea

r

Gain""" Constant

Raotio

CT

phI

KgI

PositionPo

GL

""""p"Constoant

-Definit ions:

k

Volts output per r.p.s. of tach shaft.

k p

Volts output per revolution of pot shaft.

Integrator Servo Loop Gain Schematic

34

The transfer function of the inner loop is given by

W (p)

_X)

_K

(11-59)

KKgKTCTP

+

1

Now considering the overall loop

K

W

₁

(p)

KKgKTCTP+1

_(11-60)

+C

K

K

)(K C

pl

KKgKTCTP+1

P p

C K

1+C Cp KP

~

~2

p(II-61)

g T T

~

+1

1+ p

2

p

but as

K--+oo,

1(<

CP

₁

Cp

₂

KKp.

Therefore

1

W (p) K C

(p)

K

_(11-62)

p1Cp

2

p

A

similar expression can

be

derived for the translational rate

servo loop. Figure

II-5

shows a block diagram of the loop in terms

of equivalent gains. Here the block of gain K which is assumed to

approach infinity, is a simplification which includes the servo

amplifier with its various compensating networks, the

hydraulic

flow valve and motor, and an internal feedback of the power piston

position. This is an idealized simplification, but the accuracy

ks

k4.-part of the simulator can solve its equations.

Input

(Wz)

Servo

Amplif ier

Output

Gain

Tooh

Trach

Gear

-Gaoin

--

onstant

R

atio-Gt

_{P Kt}

_Kg

Position-

Position-

Pot

Loop

__Loop

-Constant-G ain

Gain

Surf ace

Deflections

Translational Rate- Servo Loop Gain Schematic

FIGURE

1

- 5

The transfer function for Figure 1-5 in terms of

equiva-lent loop gains is

A(p)

C1

C63)

wzp

K K

_{g T T}

d

C C p + 1 p p2Kp

Considerable information can be obtained fram these transfer functions expressed in terms of loop gain. The time constant of

36

a given integrator loop is given directly by the ratio of the

equivalent tachometer feedback gain to the equivalent position

feedback gain.

The gain constants C

and C

in the equations

p1

.p

2

(11-62) and (II-63) are K

11

, K10 and K

21

, K

20

respectively.

Therefore their magnitudes can not be changed at will. The

tach-ometer constant, i.e. volts out per revolution per second of the

output shaft in fixed by the hardware used and can not be varied.

The same is also true of the potentiometer constant. There

re-mains, therefore, only three places where the gain can be changed.

Changing the time constant of each integrator loop is

equivalent to changing the time scale of the whole simulator.

Therefore these three places where the gain of a loop can be

changed can be interpreted as ways in which the time scale of

the simulator can be changed.

They are first, a change in the

gear box ratio; second, a change in the tachometer feedback

gain, and third, a change in the aerodynamic coefficients.

A change in the gear box ratio not only has the effect of

changing the time scale but also

it changes the effective

reso-lution of the output potentiometer. These potentiometers are

wire wound and the more turns that are used to represent a given

quantity the better the resolution of the quantity represented.

The gear

box change is usually used

in conjunction with one of

the other methods to

obtain optimum resolution of the output

signal.

IL)

two possible ways. The tachometer output is fed back as one of two inputs to a summing circuit. The other input to the summing

circuit is the signal to be integrated. Therefore a gain of C in the tachometer input to the smming circuit is equivalent to a gain of in the other input to the summing circuit. The gain

1

of 1 of the integrator input is more easily obtained and is

C

usually used.

It was found in the classical approach that the time scale could be changed by changing the magnitude of each aerodynamic coefficient which contained the dimension of time. Since the aerodynamic coefficients can not be changed indiscriminately, this specifies a special way in which they can be changed to

cause a change of time scale throughout the simulator.

Conclusions

The solutions of the equations as obtained by the various approaches suggest that the linear portions of the simulator can be tested by a response technique. For example a step function applied at the input, , of the W servo produces an exponential

rise of the output. W , whose time constant should compare exactly with a computed value. Also a sinusoidally modulated input can be used. In this test the frequency is varied until a phase

shift of 450 from input to output is produced. The magnitude of the output at this frequency should be .707 of the maximum of

t , C.

(in radians per sec.) should be the time constant. This forms an accuracy check of the system. If it is desired, a complete frequency response of the servo loop can be made and plotted in the normalized log modulus form and compared with an exact plot of the simple lag desired. From such a comparison the frequency limits can be determined within which the loop simulates a simple lag to a specified accuracy. It can be concluded that if each

of the input signals contain no frequency components outside these limiting frequencies, the servo can position the output shaft to the accuracy specified.

The inputs to a given servo are driving signal inputs,

signals directly generated by one of the five servos in the simu-lator, or are non-linear cross-product terms which originate from an output of one servo being multiplied via a potentiometer by an output from another servo. These cross-product terms are simi-lar to a sine squared term and thus introduce a d.o. displace-ment and a frequency doubling effect. These non-linear terms

therefore introduce higher frequency components into the input of the respective servos. If these frequencies are above the previously determined upper critical frequency, additional errors will be introduced into the computation, if on the other hand

they are not, they will introduce no new sources of error. By considerations similar to those, a first order approximation of the overall accuracy of the simulator can be made with the

cross-product terms included.

Therefore it is concluded that the accuracy of the simu-lator as a whole need not be determined from a study of the over-all simulator with its many non-linearities, but can be deter-mined from a detailed study of the sources of error in each of the five integrator servos, and evaluating the affect of these errors on computations in the overall simulator.

Experimental and Analytical Analysis of the Integrator Servos

Introduction

The objective of the experimental and analytical analysis

of the two types of integrator servos is to determine their

limita-tions in being able to solve a given form of differential equation.

The form of equation to be solved by each integrator servo

con-sidered independently is a linear, first-order, differential

equa-tion. The system function for such an equation was found in

Chapter II to be a simple lag with a time

constant 7

.

The time

constant was found to be a function of the various gains around

the loops and these gains were assumed to be constant and

inde-pendent of the frequency components of the excitation function.

In the physical equipment there are, however, certain points in

the frequency spectrum near which certain of these gain factors

become functions of frequency.

At the points where such

fre-quencies are involved, the integrator servos

no longer simulate

the calculated time response function.

Therefore from the

analy-sis

of

the integrator

servos

the

range of time-constant which

can

accurately be simulated must be determined.

In Chapter II it was pointed out that there was only one

basic way to change the time constant of an integrator servo, and

that was by a relative change of gain in either the tachometer

feedback loop or the position feedback loop, or both.

This can

(W

ELECTRO - MECHANICAL INTEGRATOR

FIGURE

M

-1

be done by three principal methods, each of which affects the

accuracy of simulation in a different way. It is necessary

there-fore to determine from

the analysis of

the integrator servos to

what extent each of the methods

of time constant change affects

the accuracy of simulation.

There are two types of

integrator servos in the simulator,

the electro-mechanical integrators which generate W ,

_{X WY}

, and W

_z

and the translational rate servos which generate Y and VZ

The

VT

V_

first to be considered is the electro-mechanical integrators.

Electro-mechanical Integrators

1

The

electro-mechanical integrator can be divided into three

sections, the mechanical unit, the servo amplifier and the

integrator

control panel. The mechanical

unit consists of a servo motor, a

tachometer, a gear train, and one or more potenticmeters. The

servo amplifier is of conventional a.c. amplifier design with

preamplifier, filter, and power amplifier sections. The

inte-grator control panel is used to introduce initial values of rate

into the integrator or change its function from a position servo

to an integrator.

The servo amplifier and mechanical unit

oper-ate together in the rack and are shown in Figure III-1.

The servo motor on the mechanical unit is an Eclipse-Pioneer,

400 cycle, two phase, twelve pole motor.

The reference field is

Thesis Electrical Engineering Department, MIT, January 1948,

High Precision Electro-Mechanical Integrator

excited by 110 v /90* 400 cycle per second voltage and the maxi-mum control field voltage is 220 /00* 400 cycle voltage. In

the conditions of actual operation in the integrator the maxi-mum acceleration was found by measurement to be 1750 radians

per second per second. The viscous friction load was determined from the voltage speed curves to be essentially constant over the normal ranges of speed and equal to 1 in. oz.

The tachometer which is directly coupled to the motor shaft is an Arma 1B-400 precision induction generator. The primary exciting voltage is 110 x /0* 400 cycle voltage. The output of the tachometer is linear with speed within + 0.1 per cent up to speeds of 3000 rpm. The calibration of the output varies from one tachometer to the next but is of the order of 5.5 v per 3000 rpm. There is a small output from the tachometer at zero speed. Its magnitude is a function of the rotor position and has a variation of as much as 5 mv. from maximum to minimum. The quadrature component of the output is + 0.10 for either direction of rotation.

On the opposite end of the servo-motor shaft from the tach-ometer there is an arrangement for inserting gear boxes of

various ratios. As was pointed out in Chapter II, this is one of the methods for changing the time constant of response for the integrator. These gear boxes are to be made available in speed reducing ratios of 2:1 to 128:1 in steps of powers of two, including a 1:1 drive. On the output shaft of the gear box -is

44

a fixed gear train with a speed reduction of 30:1. The fixed

speed reduction is necessary to prevent the potentiometers which are driven by the output shaft from excessively loading the servo motor. There are five output positions available in which either potentiometers or resolvers can be mounted. The motor inertia load change from no potentiometers to five potentiometers was com-puted to be less than 10 per cent of the motor inertia.

Two kinds of potentiometers are used. A. Fairchild, one-revolution, wire wound, 10 Kilo-ohm potentiometer is used on the W integrator and two, ten-revolution, wire wound, 10 Kilo-ohm micropots are used on each of the W and W integrators. The

Y

Z

resolution of the Fairchild potentiometer is .07 per cent of full scale and that of the Micropot is .01 per cent of full scale.

The servo amplifier has a gain available from zero to 15,000, however, all of this gain can not be used in the closed loop for two reasons. First because of saturation of the output stage by the quadrature in the input signal, and second because of

satur-ation in the output stage due to high frequency oscillations when the loop is closed.

The quadrature which produces the saturation is a phase shift of the carrier frequency of the input signal. The output from the tachometer is a 400 cycle voltage of the reference phase; therefore it can cancel out only the in-phase component of the carrier in the input signal. This leaves a small error signal of in-phase carrier and a large unattenuated component of

out-of-4 0lL

phase or quadrature carrier as the input to the servo amplifier. The servo-motor control field responds only to the in-phase com-ponent but the quadrature present saturates the amplifier and

limits the effective gain of the in-phase component. By compen-sating the phase shift in the various computer components it was possible to reduce the quadrature at the input to the inte-grator from as much as 500 millivolts to the order of 20 milli-volts. The oscillations in the amplifier saturated the output

stage and limited the effective gain of the in-phase component and in general have a similar effect on the operation of the ampli-fier as the quadrature in the input signal. As a result of these two effects the operating gain of the servo-amplifier was limited

to about 1800.

The characteristics of each of the component parts of the inte-grator affects its accuracy in various ways. A detailed analysis

of the integrator with a position loop closed around it as it is used in the simulator is necessary to determine the extent of the effects of the various physical characteristics. Figure 111-2

shows the detail block diagram upon which the following analysis is based.

The constants K 1 and K10 are the aerodynamic-constants as

defined in Chapter II. The gain of the tachometer CT is unity and there are no provisions in the servo-amplifier for varying it,