Echnical Paper 21 65 E -2 June 1984 STABCAR-A Program for Finding Characteristic Roots of Systems I Having Transcendental Stability Matrices

NASA Technical Paper 21 65 e -2 June 1984 STABCAR-A Program for Finding Characteristic Roots of Systems I Having Transcendental Stability Matrices

WfimM. Adams, Jr., Sherwood H. Tiffan?, Jerry R. Newsom, and Ellwood L. Peele

NASA

t NASA Tech nica I Paper 21 65

1984 STABCAR-A Program for Finding Characteristic Roots of Systems Having Transcendenta1 Stability Matrices

William M. Adams, Jr., Shenvood H. Tiffany, Jerry R. Newsom, and Ellwood L. Peele Langley Research Center Hampton, Virginia

N a t toiid I At.:or7,iil t ic 5 and Spa c c Ad nii fi1 L t r J t to ri

Scientific and Technical Information Branch CONTENTS

MATHEMATICAL REPRESENTATION AND SOLUTION TECHNIQUES ...... 3 Equations of Motion ...... 3 Unsteady aerodynamic forces ...... 4 Sensor input definition ...... 5 Control-System Representation ...... 8 Sensor dynamics options ...... 8 Compensation options ...... 8 Actuator dynamics options ...... 10 Combination of sensor. compensator. and actuator matrices ...... 10 Characteristic Root Determination ...... 10 Determinant iteration ...... 11 Initial estimates for roots ...... 12 Prediction of Gil and gi2 at a new value of independent variable ...... 13 Logic during convergence difficulties ...... 13 Precise determination of point at which change in stability occurs ...... 13 Matched-point computations ...... 13 PROGRAM DESCRIPTION ...... 14 Features for Enhancement of Resource and User Efficiency ...... 14 Overlay structure ...... 14 Variable dimensions ...... 14 DITIZIE~Cstzrage allocation ...... 14 Free-field reads ...... 15 Model selection ...... 15 Automted parameter variations ...... 15 Graphical display of output ...... 15 Interactive execution ...... 15 Restart capability ...... 15 Functional Description of Overlays ...... 15 STABCAR Flowchart ...... 17 INPUT REQUIREMENTS ...... 17 Description of hput Files ...... 17 Input Aids ...... 17

PROGRAM CAPABILITY DEMONSTRATION ...... 18 Aircraft Mathematical Model ...... 18 Graphical chtput mom Sample Cases ...... 19 Case 1 .Aerodynamic forces and p-plane approximation ...... 19 Case 2 .Open-loop characteristic roots as function of altitude ...... 19 Case 3 .Open-loop characteristic roots as function of density ...... 19 Case 4 .Open-loop characteristic roots as function of dynamic pressure ...... 19 Case 5 .Characteristic roots as function of velocity (truncated model) ...... 19 Case 6 .Elastic mode gain root loci ...... 20

iii Case 7 .Effect of phase errors upon elastic mode gain root loci ...... 20 Case 8 .Gain root loci of control system poles ...... 20 Case 9 .Short-period altitude root locus ...... 20

CONCLUDING REMARKS ...... 21

APPENDIX A .p-PLANE APPROXIMATION TO THE UNSTEADY AERODYNAMIC FORCES ...... 22

APPENDIX B .EXAMPLE OF INCLUDING SCHEDULED COMPONENTS IN COMPENSATOR ...... 26

APPENDIX C .OPTIONS FOR HANDLING DENOMINATOR DYNAMICS ARISING FROM ACTUATORSr FILTERS. AND SENSORS ...... 28

APPENDIX D .ALGORITHM FOR MATRIX ITERATION FOR CHARACTERISTIC ROOTS ...... 30

APPENDIX E .GRAPKICS ROUTINES NOT INCLUDED IN STABCAR ...... 33

APPENDIX F .DETAILED FUNCTIONAL DESCRIPTION OF OVERLAYSr MAJOR SUBROUTINES, AND IDENTIFICATION OF KEY PARAMETERS ...... 46

APPENDIX G .INPUTS FOR SAMPLE CASES ...... 68

TABLES ...... 95

112

iv I I SYMBOLS

~A matrix whose eigenvalues are found by using matrix iteration approach 1 (eqs. (18)) column matrices in p-plane approximation to jth column of I AO;'"=IA (nRj+2) J j generalized aerodynamic forces (eq. (3) )

aAd ,...,a denominator polynomial coefficients in transfer function for i Ad (nAdi )i ith actuator (eq. (1 2))

,...,a numerator polynomial coefficients in transfer function for i An (nAni )i ith actuator (eq. (12))

denominator polynomial coefficients in ith filter transfer 1 i a Fd (nFdi )i function shown in equation (IO)

aFnli ,...,a numerator polynomial coefficient in ith filter transfer Fh (nFni )i function shown in equation (1 0)

aSd lg~gla denominator polynomial coefficients in transfer function for 11 Sd (nSd. ) j 7 jth sensor (eq. (8))

aSn ,...,aSn numerator polynomial coefficients in transfer function €or 1j (nSn, ) j 3 jth sensor (eq. (8))

Bn matrix defined by equation (A61

b reference length used in computation of generalized aerodynamic forces

column vector (n x 1) of constants in denominator terms of p-plane fit bj Aj (eq. (3))

c,co matrices defining constraining relationships imposed on p-plane coefficients (eq* (A41 1

C mean aerodynamic chord of wing

D complex determinant of characteristic matrix of homogeneous equations (eq. (1) or (2))

D' modified determinant (eqs. (1 4)

d common denominator of nonscheduled elements of transfer matrix relating actuator outputs to sensor inputs (see eq. (C3))

e ,e ,e error functions associated with least-squares p-plane fit to ij c n ij unsteady aerodynamic forces (eqs. (A51, (A71, and (A81 1

FD matrix of modal coefficients converting hinge moment outputs produced by actuator to generalized forces (eq. (1))

vector of natural frequencies in vacuum, Hz fn

V G gain matrix (eq. (9)

9s vector of structural dampinq coefficients

H matrix relatinq sensor input to qeneralized coordinates (eq. (7))

matrix relatinq actuator elongation to qeneralized coordinates (eq. (1) ) HE h altitude

J. inteqer; J. = 0, 1, or 2 defines sensor j as position, rate, or 7 1 acceleratlon sensor, respectively (eq. (8)

K generalized stiffness matrix

KO' K1lK2 gain for inteqral, proportional, or derivative filter, respectively

k reduced frequency, wb/U

M generalized mass matrix

N N. ./d is transfer function relatinq output of actuator i to input ':ensor j (see eq. (~4))

Mach number Ma nAni ,nAdi number of numerator and denominator polynomial coefficients, respectively, in transfer function for ith actuator

nFni ,nFdi number of numerator and denominator polynomial coefficients, respectively, in transfer function associated with filter i (eq. (10) 1

nSni ,nSdi number of numerator and denominator polynomial coefficients, respectively, in transfer function for sensor i (nSni contains Ji in addition to basic sensor dynamics)

nkk. number of values of k for which Qj(k) will be employed in determininq 3 least-squares fit for p-plane approximation to jth column of aerodynamic forces (eq. (A7))

number of high pass terms in p-plane approximation for jth column nlj of unsteady aerodynamic forces

n numher of subintervals to divide interval (Vi,Vi+,) into when havinq Av '"AV C S convergence difficulties or locatinq a point of chanqe in stability, I respectively ,n ,n number of qeneralized coordinates, control surfaces, riqid-body nt;rnb,nr s k degrees of freedom, Sensors, and reduced frequencies for which aerodynamic forces have been computed, respectively

vi difference in pressure between upper and lower surface of lifting body at point (x,y) due to motion in jth node

nondimensional Laplace variable, sb/U

matrix of generalized aerodynamic forces

p-plane approximation of Q (eq. ( 3) 1

tabular value in unsteady aerodynamic force matrix at a reduced frequency of ki Q(ki

obtained by interpolation given Q' (k. ) = Im ~ , i = 1, ..., '5~ 1 (if k, = 0, Q'(kl) Z Q'(k2)) ki

dynamic pressure , pU2/2

1 RxlIRy' angular rotations about x' and y' axes, respectively (eqs. (4) and (6))

r column vector of sensor inputs

r' column vector of sensor outputs

rD column vector of linear displacement sensor inputs r e column vector of angular displacement sensor inputs S surface area

S Laplace variable

n Sin nth estimate of ith characteristic root

diagonal matrix of actuator transfer functions

transfer matrix relating actuator hinge moment output to actuator elongation

THS transfer matrix relating hinge moments produced by actuator to input to J sensors (s j is also included in THM, see definition of J.) 3

TL transfer matrix relating actuator inputs to sensor outputs

TL excluding gain, phase and scheduling components (eq. (9)) T;, error,

scheduled portion of compensation between jth sensor and ith control (eq. (9))

J diagonal transfer matrix relating sensor outputs to sensor inputs (s j is also included in Tm, see definition of J.) 3 J T transfer matrix relating control surface position to sensor inputs (s J is 6 also included in T see definition of 6' Jj)

vi i t time variable

U aircraft speed tJ-u aircraft design dive speed

open-loop flutter velocity UF

(x, y, z) position triple defining three-dimensional coordinate system

(x' ,y ' ,z ' coordinate system rotated and translated from (x, y, z (eq. (5))

X x coordinate of center of gravity c9

(XOI YO'O) origin of (x',y',z') coordinate system with respect to (x,y,z) zi(x,y) displacement in ith mode at (x,y,O)

scalar defining step from si to si for which n n+l

column vector of control rotations, x1 n6

actuator and control logic output, respectively, n6 x 1

criterion for converqence upon characteristic root (eqs. (1 6) )

damping ratio associated with complex conjugate pair of roots

damping ratio associated with notch filter

normalizing angle in definition of rigid-body pitch mode shape (see section entitled "Sensor input definition"

sweep angle of elastic axis, positive clockwise (eq. (5)1

jth characteristic root

column vector of Lagrange multipliers associated with constraints imposed upon coefficients in p-plane approximation to jth column of aerodynamic forces (eq. (A8))

column vector of generalized coordinates, nF density

mass surface density of f lat-plate idealization of airplane

time constants in first-order lead-lag (lag-lead) filter

phase error introduced in compensation between jth sensor and ith actuator

frequency of oscillation

viii natural frequency associated with notch filter I F natural frequency of ith mode in vacuum I i

~ Subscripts:

IF f i lter

S sensor

Abbreviations:

€ req. frequency

nos. numbers

Red. reduced

Specia 1 ma thema ti ca 1 s ymbo 1:

€J diagonal matrix

ix SUMMARY

STABCAR is a computer proqram that can be used to determine the characteristic roots of flexible, actively controlled aircraft, including the effects of unsteady aerodynamics. A modal formulation is employed to characterize the aircraft. me control system representation is input as a matrix of transfer functions. Roots are determined through the use of determinant iteration utilizing either oscillatory or approximate fully unsteady aerodynamic forces. Provisions are included to allow automatic variation of velocity, density, altitude, or feedback gains.

The mathematical model employed for the aircraft is described, a flowchart is provided, detailed description is given of the function of key program elements and parameters that direct program operation, proqram inputs are defined, and sample inputs and outputs are presented.

STABCAR can he executed in either a batch or an interactive mode. It is written specifically for use on CDC@ CYBER 175 equipment. Modification would be required for operation on other equipment.

INTRODUCTION

The equations of motion of flexible aircraft contain aerodynamic force terms which, when expressed in the Laplace domain, are transcendental functions of the Laplace variable. Determination of the characteristic roots of the system, therefore, requires the use of iterative or graphical procedures. Alternatively, one may approximate the aerodynamic force terms with rational functions of the Laplace variable and convert the equations of motion to a standard eigenvalue problem (refs. 1 and 2).

This paper describes a computer program which utilizes the method of determinant iteration to solve for the characteristic roots of flexible, actively controlled aircraft. References 3 through 6 describe other programs for iterative determination of characteristic roots. Reference 3, which was closely followed in the development of STABCAR, describes results obtained with a code developed by a major airframe manufacturer; reference 4 describes methods employed in mqland. Neither code is readily available and no reference is made to a user's guide for the programs. Ref- erences 5 and 6, however, describe NASTRAN@, a publically available program. The NASTRAN program was developed for batch-type operations. In contrast, STABCAR, described herein, is tailored for interactive execution and contains a number of features which facilitate the interactive study of the effects upon stability of variations in control-system parameters.

STABCAR is a module of the ISAC (Interaction of Structures, Aerodynamics, and Controls) program (ref. 7). me present paper is the first which documents a major segment of the ISAC program. E'uture papers which document other seqments may be found by employing ISAC as a key word in a library search.

The equations of motion are outlined by employing a modal representation of the aircraft. Aerodynamic forces for either purely oscillatory or approximate fully unsteady motion can be considered. General linear control laws can be analyzed either by direct input of the elements of a transfer matrix or by construction utilizing selectable bui 1t -i n € i lter types.

Considerable effort has been directed toward the incorporation of featiires that allow the zser to .-L-u~.L~lze 1: his time efficiently. Interactive program execution is an option. Characteristic root variation with altitude, density, velocity, or control system gains can be performed in an automated manner with the results presented graphically.

The STABCAR program and all user options are described in detail in this report. The functions of major program elements are specified, and key parameters are identified that direct the input to, the computations in, and the results from each major subroutine. Flowcharts and a detailed description of input parameters are also p r o vide d .

Results are presented and discussed which illustrate major program options. Characteristic root variations are shown with and without feedback control as a function of parameters dependent on flight conditions. Root loci, as a function of variation in control parameters, are also shown; comparisons of characteristic roots obtained by using approximate fully unsteady and oscillatory aerodynamics are presented.

The inputs for a series of sample cases are presented. The cases correspond to the graphical results that are shown. These sample cases provide explicit examples of the inputs required to exercise the major capabilities of STABCAR.

STABCAR has been written specifically for efficient operation on CDC@ CYBER 175 equipnent. Revisions are required to run this proqram on other equipment.

OVERVIEW

In STABCAR, stability characteristics are investigated. me dynamic elements that may be included are shown in figure 1. Mathematical descriptions of each of the dynamic elements are presented in the next section. me element representing the aircraft includes riqid, elastic, and control deqrees of freedom as well as unsteady aerodynamic forces. 'Ihe unsteady aerodynamic forces cause the equations of motion in the Laplace domain to be transcendental functions of the Laplace variable. Multiple- input/multiple-output control laws can he studied; these can include dynamic compensation expressible in terms of a transfer matrix.

STABCAR has been developed to solve for characteristic roots of the stability matrix. Typically, a search is made for roots as a function of the variation of a parameter of interest such as altitude, density, velocity, or a feedback gain. This procedure is illustrated in figure 2. Since the stability matrix is a transcendental function of the Laplace variable, the characteristic roots are determined by an iterative procedure. Consequently, initial estimates of the characteristic roots are required. Methods employed in STABCAR for determining initial estimates are presented in a later section.

MATHEMATICAL REPRESENTATION AND SOLUTION TECHNIQUES

In this section, the mathematical description of STABCAR is outlined. ?he equations of motion are discussed as well as alternate options as to the form that can be

i ~~ ~ ~ 1 chosen for the aerodynamic forces. A capability for interpolation of beam- or plate- type mode shapes to obtain sensor inputs is described. The manner in which the control system is included is described, and the techniques for characteristic root I determination and major options for output of information are detailed. ,I

Equations of Motion

I I The equations of motion consider only perturbations from a level equilibrium flight condition. ?he perturbed aircraft motion is represented in terms of a truncated set of riqid and elastic modes. The resultinq equations, expressed in terms of generalized coordinates, masses, stiffnesses, dampings , and aerodynamic forces, are essentially the same as those developed in reference 8. The homoqeneous part of these equations, expressed in the Laplace domain, are given in the followinq equation:

where

Mij element of generalized mass matrix, zi(x,y) o(x,y) z. (x,y) dS S 3 '%j element of generalized stiffness matrix; K(x,y;u,v) is influence coefficient giving force at (x,y) due to unit displacement at (u,v), zi(x,y) K(x,y;u,v) z .(u,v) dS 3 dS 12 S s1 2

element of generalized aerodynamic force matrix with hp. (x,y;Nmlp) being the pressure difference due to 3 aPi rr (xIy;NMalp) notion In jth nede, J j z. !x:y! - dS S 1 9

gi(s) = \J-1 9s.t -Is\S gsil or g -; user chooses desired form. 1 wn i If w = 0, gi set to zero n i

9s structural damping coefficient associated with ith mode

"6 matrix of transfer functions relating actuator hinge moment"' outputs to sensor inputs

THE(S) ng x n diagonal matrix of transfer functions relating actuator hinge moment6 outputs to actuator elongation

n x n matrix of modal coefficients converting hinge moment outputs to FD 'generalized6 force

5 vector, n X 1, of rigid and elastic generalized coordinates 5 6 vector, n6 x 1 , of control-surface generalized coordinates

3 "5 zi(x,y) ith mode shape, z(x,y,t) = zi(x,y) ti(t) i=l

Hij (XI Y) linear or angular modal deflection contrihiiting t~?inp~t to seiis~r i resulting from unit deflection in jth generalized coordinate, r = HE where H is n X n s5 matrix of modal coefficients relating actuator elongation to HE n6 X nt ge ne ra li z e d coo r dina t es

An alternate form of the equations neglects aerodynamic hinge moments and hinge moments due to inertial couplinq in which case the control surface deflections are

6 = T (s) HE 6

?he dimension of 6 is n X 1 and T~ is a matrix of transfer functions relating control deflections to sensor6 inputs. This expression replaces the control rows of equation (1 ); the resulting expression is

I- 1

Unsteady aerodynamic forces. - In equations (1 ) and ( 2) , the aerodynamic forces are expressed as functions of Mach number and .the complex variable p = sb/U. The programs currently available for production generation of aerodynamic forces can only compute these forces for p = bw/U (e.g. , refs. 9 through 12). Two approaches are taken in STABCAR to circumvent this difficulty.

The first approach, known as the p-k approximation (refs. 3 through 5) is valid for lowly damped modes. TWO options are implemented for this approximation:

(a) Form of reference 3

(b) British form (refs. 4 and 5)

Q'(k) is obtained by interpolation of the matrix of functions {Q'(k,), Q'(k2), ..., Q'(k "k )} where Q'(ki) = Im-r:i)). If kl = 0, Q' (kl) E Q' (k2). The approximation becomes exact at the flutter point. The second approach, known as the p-p method (ref. 31, requires the aerodynamic forces as a

4 function of p. In STABCAR, use is made of the concept of analytic continuation to develop an approximation to the aerodynamic forces for arbitrary motion in terms of knwn aerodynamic forces for oscillatory motion (p = J-1k) . The form assumed for the jth column of the aerodynamic force matrix is

R= 1

This expression has the same form as that employed in references 1 and 2. Determina- tion of the polynomial coefficients is initiated by computation of Qj(F k) for a number of values of reduced frequency. This computation must be done by an external program such as those described in references 9 through 12. The user specifies the vector bj and may impose linear constraints upon the coefficients in equation (3). For example, the A. columns associated with riqid-body linear displacements should be zero. The coefficients of each element of 4 (p) are determined, subject to any imposed constraints, such that the errors between the curve fit (eq. (3)) and the tabular data are minimized in a least-squares sense. The least-squares-fit procedure and allmable constraints are described in appendix A.

References 13 and 14 present a theoretical basis for the curve-fit procedure and describe the asymptotic form Gj(p) should take. ?he asymptotic limits are not enforced in STABCAR because constraininq the curves to fit hiqh frequency limits tends to degrade the €it in the limited frequency band where tabular data are available. Reference 1 5 describes the application of a constrained optimization procedure to optimally choose in equation (3). 'j References 16 and 17 describe a program nw under development which will allow computation of aerodynamic forces for transient motion where p has a nonzero real part. STARCAR can be readily modified to utilize such data. ?he required modifica- tiOn is to provide for interpolation of tabuiar deicjdynaiik;z force arrays as a function of two independent variables (the real and imaginary parts of p) rather than as a function of only the imaginary part of p.

Sensor input definition.- Sensor inputs in STABCAR are assumed to be linear combinations of the generalized coordinates. For linear displacements ,

u i j=l

For angular disp acements,

re (XfY i j =I

5 These expressions would be multiplied by s for a velocity type sensor and by sL for an accelerometer. The multiplication by the appropriate power of s is included within the computation of the effects of sensor dynamics. (See eq. (8).) Thus, in computing sensor inputs, the assumption is made that inertial axes are employed. Ihe .-- - ~b~iiiiay override this assumption by appropriateLy modifying the rigid-body portions of the mass, damping, and stiffness matrices.

The mode shapes or their derivatives at the sensor locations must be specified before the sensor inputs can be determined. Two options are included for defining the modal data for input to sensors. Ihe first option is by direct input of the modal data required at the sensor locations. me second option is to input all the modal data (mode shapes and node-point locations) and interpolate for the modal amplitudes at the sensor locations. This option is attractive when performinq s ensor-1 oca ti on studies.

The remainder of this section describes the interpolation procedure. Vibration analyses, which use modelings ranginq from simple beams to f initeelement representa- tions, calculate modal data at a number of grid points. In order to accommodate output from a variety of vibration analyses, it is necessary, in STARCAR, to arranqe the modal data into subset divisions called structural sections. For aircraft configurations, the structural sections follow naturally as the wing, fuselaqe, horizontal tail, and so forth.

There are two types of interpolation methods in STABCAR correspondinq to the two types of structural sections available in the program. The first is a displacement- twist interpolation method which is most commonly used when the mode shapes are qenerated from an elastic-axis beam model. !he second is a surface-spline interpolation method (ref. 18) which is used when the mode shapes are generated from a finite- element "plate-type" model. me entire modal representation may consist of a combination of both types of structural sections.

For a zero-thickness idealization of a surface and a beam structural representation, the displacement in the ith mode normal to the surface can be written as

where

(x,Y,z) riqht-hand coordinate system with x positive aft and z positive up

(x',Y',z') right-hand coordinate system with z' parallel to z and X' and y' swept from (x,y) by angle A

zi(x,y) deflection in ith mode at (x,y) normal to surface

zi(y') vertical bendinq in ith mode at (x' = 0,y')

R (y' ) rotation about y' axis in ith mode at (x' = 0,y' Y; Figure 3 shms the geometrical relationships for a flat surface parallel to the xy plane. The origin of the swept, translated axis system is at (xolyo); A is the sweep angle (positive clockwise); and positive z is out of the page. %e transfor- mation from (x,y) to (x',y') is given by

cos A -sin A (5) j;] = sin A cos A

For angular displacements, the partial derivative of equation (4) is required and is given as f 0 1lows :

a I (Y') &i 'i - = (y') - x' sin A - R '(y' ) cos A [.X! ax dY 1 ' 'i

where R is rotation about x' (positive counterclockwise). X! 1 For "piate'! type Inodes the iiiodal data from a structural inalysis zre detin~dat points distributed over the structural section. Thus, there are two independent variables (x,y) and the interpolation is done by using the surface spline methods described in reference 18. Slope information (e.9. , az. (x,y)/ax) is qenerated by 1 interpolating for the derivative of the surface spline.

Rigid-body mode shapes may be defined by input by the user. In this case, the rigid-modal data must be treated as if they are elastic modes (i.e., nr = 0 where nr is the number of rigid modes). Alternatively, if the symmetric rigid-body mode shapes of vertical translation and pitch about the aircraft center of mass are desired, the user may set nr = 2. Then the following built-in expressions are employed to compute the rigid-body contributions to sensor inputs:

Linear displacements :

7 An gu lar dis place me n t s :

dz 2 - ,,(x,y) = 0; -dx - -'n

Con t r o 1- S y s t e m Re p r e senta ti on

?he control system is divided into three parts: sensor, compensation (filtering) , and actuator dynamics. 'Ihe input-output relationships between these dynamic elements and their coupling with the aircraft dynamics are indicated schematically in fiqure 1 and explicitly in equations (1 ) and (2). In a subsequent discussion of these dynamic elements, the term "block" is used interchanqeably with "transfer matri x. 'I

Sensor dynamics options.- Input to the sensor dynamics can be either linear or angular position information expressed in terms of the qeneralized coordinates as

r = HC (7)

The dimension of r is n x 1. Within the sensor dynamics block each ri can be S transformed to a position, velocity, or acceleration input by multiplication by the appropriate power of the Laplace variahle s. A qiven sensor output will he of the form

nSn,-1 a +a s + + a s' Sn Sn ... Sn 11 21 (nSn .Ij J r', = 3 sjr =T (s)r 3 nSd, -1 i Sj j a +a s + ... + a s' Sd Sd , Sd 11 21 (nSd.)j 1 where Jj = 0, 1, or 2 corresponds to a displacement, rate, or acceleration sensor, respectively. Default values are built into the program so that only J must be j input if sensor dynamics are neglected.

Compensation options.- The transfer matrix TL, of dimension ng x n specify- SI ing compensation of the signals from the sensors prior to sending them to the actuators can be full with distinct dynamics in each element. The transfer matrix relationship between compensator outputs and compensator inputs (sensor outputs) is

6 = T (s) r' I, L where

8 l is a gain, each [TL(s)]i may contain dynamic compensation elements but has a efault value of 1. Alternale values for [T~(s)]ij may be defined either by direct input of polynomial coefficients OK by utilization of the following I bui It-in f i lter types :

1. Notch filter:

where and are the natural frequency and damping ratio of the notch filter w”F CF 2. Integral filter:

3. Proportional plus derivative filter:

K1 + K2s

4. Lead-lag (lag-lead) filter:

1 +zs 1 1+TS 2

5. Polynomial filter:

nFn, -1 I a +a s + t a 5 Fn ... Fn li 2i (nh, )i I nFdz -1 1 a +a s + ... + a S Fd~i Fd 2i Fd ( nFd i

Any combination of these filter types may be utilized including repetitive use of the same filter type with the same or distinct constants. Additional filter options are as follcws:

6. Scheduling:

The capability for providing scheduled compensation elements TLS(~,NMa,q)i is also included. For each control-sensor pair, it may be desirable to allow the control law to vary as a function of Mach number and dynamic pressure. mis variation is accomplished by providing a call to a subroutine which has available for use control-sensor pair identification, the value of the Laplace variable s, Mach number, and dynamic pressure. ?he subroutine, which defines the desired scheduling, must be supplied by the user. This procedure is illustrated in appendix B for a specific scheduling law.

9 7. Phase error:

An additional feature included is the following option of introducing a phase error Qi4 into the compensation for one controlsensor pafr: -J

4-1 Qij e (11)

This option allows one to study sensitivity to phase errors.

Actuator dynamics options.- The transfer-function matrix relating inputs to and outputs from each actuator is diagonal. Its default value is the identity matrix. I If actuator dynamics are to be considered, the following numerator and denominator polynomial coefficients are input:

nAn, -1 1 a +a s + + a S ... An Anli An 2i

a +a s + ... + a S Adli Ad 2i Ad (nAdi )i

(L)= PA] 6L

where the dimension of gA is n6 x I.

Combination of sensor, compensator, and actuator matrices .- The relationship between dA and r is

If aerodynamic and mass coupling hinge moments are considered, the resulting transfer matrix corresponds to THs in equation (1); otherwise it corresponds to T 6 in equation (2). The polynomial d(s) in equation (1 3) is the common denominator of Ti(s) (see eq. (9)). This factor can be cleared from the denominator in equation (1) or (2). Appendix C describes this option.

Characteristic Root De te rmination

The primary objective of STABCAR is to determine characteristic roots of an aircraft represented mathematically by the transcendental matrix equation (eq. (1 or (2)). Roots can be found as a function of variations in velocity, density, altitude, or control parameters.

10 Determinant iteration.- Characteristic root determination is accomplished mode by mode with determinant iteration given initial estimates for the roots. The roots to be traced are specified by input. Methods for obtaining the initial estimates are described after the followinq discussion of the procedure for determining a characteristic root.

An attempt is made to find the ith characteristic root in a manner which avoids convergence upon a previously determined root, j < i. This is accomplished hy determining the zeros of

(1 4a) or

(i > 1)

n (S - h.1 j=l 3 where

D(s 1 value of complex determinant of stability matrix in equation (1) or (2)

h previously determined characteristic root j

A Let sin+l denote the desired n + 1 estimate of the zero of equations (1 4) correspondinq to the ith characteristic root and Gin denote a previously determined

A h or is the ..1>.- -6 r)' (SI ;It P. estimate; therefore, i~: D! -."CZLUC "I nr "L : i 1 "In+l n n+ 1 n respectively. ?here exists an a > 0 such that, for n

a D! ni h - A n 'in+l - sin- dDf /d~ n

n+l n

In the search for hi, is constrained to be CO.l in order to reduce the an likelihood of jumping to a point sufficiently far away from h as to cause converq- i ence up2n another root. Tne general procedure is to solve equation (15) repetitively until differs from zin by less than a tolerence E specified by the user.

11 I That is, if

- ;i I : E (1 6a) I:ln+lIC. nl

A converqence has occurred and Ai Z ssi in+ln+l . Initial estimates for roots.- The alqorithm for determininq characteristic roots described in the previous section requires both function (eqs. (1 4)) and slope (eq. (1 5) ) information. Thus, two initial estimates are required for each root that is to be traced. Three options are available for specifying the initial estimates:

1. Direct user input:

2. Computations based upon system natural frequencies of vibration in vacuum:

gil =o+$Ty,i (1 7)

A si2 = I

where dl and d2 are input parameters havinq default values of 0.01 and 1.0, respectively. This approximation works well for the elastic degrees of f reedom at sufficiently low dynamic pressures.

3. Initial estimates by matrix iteration: In this approach the matrix equation (1) or (2) is expressed in the form

12 Appendix C shows how A(s) is obtained, and appendix D describes the matrix iteration method.

Prediction of and gi2 at a new value of independent variable.- Subse- G:L1 quent predictions for gil and zi2 at each new value of the independent variable are obtained by using an extrapolation of either a linear fit to the two preceding converged roots (after second iteration), a quadratic €it to the precedinq three converged roots (after third iteration), or a cubic fit to the preceding four converged roots (after the fourth iteration). The type of extrapolation to be performed (linear, quadratic, or cubic), after the requisite number of iterations, is specified by input.

Logic during convergence difficulties.- Failure to converge upon a characteristic root occurs periodically when the root is changing rapidly as a function of the parameter being varied. When such a failure occurs, the matrix iteration approach is employed in an attempt to recover. The matrix iteration approach is started with the last successful estimate for the root. If this attempt is successful, normal determinant iteration is then' resumed. If the attempted recovery by using matrix iteration fails, the following scheme is utilized:

1. Let vi denote the value of the parameter beinq varied (independent variable) at the point the root was last successfully obtained. Let Vi+l denote the value of the independent variable at which convergence failed.

2. Divide the interval (V.,Vi+l) into a user specified number of increments 1 n av C 3. Solve for the root by using determinant iteration at each intermediate point and the end pdnt Vi+l.

If convergence is achieved at Vi+ll normal execution is resumed. If not, that particular root is deleted from the set being souqht and normal execution is resumed for the remaining roots.

Precise determination of point at which change in stability occurs.- When the first change in stability is noted, the value of the parameter being varied which corresponds to zero for the real part of the root is automatically determined. me determination is accomplished by

1. Dividing the interval between Vi+ll where a chanqe in stability is noted, and Vi into a user specified number of increments n AVS 2. Evaluating the characteristic root at all intermediate points

3. Performing an interpolation with these data to find the precise value of V at which the real part of the characteristic root is zero

Matched-point computations .- Two options €or automatic matched-point computations are provided. Each seeks matched points at a particular input Mach number. A matched point is one for which the density, Mach number, and velocity are consistent with actual physical conditions (e .g. , atmospheric conditions or tunnel operatinq conditions). Characteristic roots are computed and their corresponding KOOt loci or damping ratios and natural frequencies can be plotted as a function of the parameter

13 being varied. The point at which the real part of a root becomes zero denotes a change in system stability. For an elastic mode, this corresponds to a matched flutter point. The two automated types of matched-point computations are

1. Wind-tunnel matched point (density variation at fixed Mach number):

(a1 Characteristic roots are examined to determine stability characteristics with velocity held fixed.

(b) Outputs for this type run include damping ratios and natural frequencies or root loci as a function of either density or dynamic pressure.

Density variation analyses correspond to the method of operation of some wind tunnels and are useful for correlating analytical and experimental results for scaled flutter models.

2. Atmospheric matched point (altitude variation at fixed Mach number) :

(a) Reginning at a hiqh-altitude (low-q) condition, characteristic roots are determined at specific altitudes with the data of reference 19 to determine the correspondinq density and speed of sound. This type run yields a matched flutter point, if one exists, for the specified Mach number.

(h) Cutputs for this type of run include damping ratios and natural fr~q~ienciesor root loci as a function of either altitude or dynamic pressure.

PROGRAM DESCRIPTION

In this section, proqram features are outlined and the primary functions of major program modules are identified.

Features €or Enhancement of Resource and User Efficiency

Considerable effort was expended to develop a code that makes efficient use of resources and allows the user to efficiently study the stability characteristics Of flexible aircraft. The following features are a result of the effort.

Overlay structure.- STABCAR is an overlay proqram with two levels of OVerhYS. The overlay structure is indicated in fiqure 4. Overlaying allowed modularization of the code which facilitates modifications affecting only one overlay. It also reduced the amount of code required in core at one time.

Variable dimensions.- All large arrays are variably dimensioned, with the dimension specified by input. This enables the user to either study a new ldrqer problem or make efficient use of central memory when a smaller problem is to be investiqated without makinq proqram modifications.

Dynamic storaqe allocation.- Within each overlay, the required computer memory is computed and updated at critical points during job execution. This dynamic allocation of storage results in more efficient utilization of central memory. Further description of this technique is given in reference 20.

14 Free-field reads.- Free-field reads are employed to input most of the array data (e.g. , mode shapes , generalized masses , stiffnesses and dampings , and unsteady aerodynamic forces ) .

Model selection.- A subset of the model that has been input can be selected for study. This option allows one to consider specified degrees of freedom while iqnor- ing others that are represented in a given set of data. The appropriate generalized masses, stiffnesses, aerodynamic forces, and so forth are automatically extracted.

Automated parameter variations .- Stability characteristics are typically studied as a function of variation of dynamic pressure or control-law parameters. Automatic variation of dynamic pressure can be specified by the user in three ways: (1) altitude variation, (2) density variation, or (3) velocity variation. Gain variations can also be performed in an automated fashion. The effects of variations of other parameters - for example, sensor locations - upon system stability can be examined by running several cases, each having a different value of the chosen parameter.

Graphical display of output.- Stability characteristics can be displayed graphically in two forms. The' damping ratio and natural frequency corresponding to each characteristic root can be plotted, and characteristic root loci can be plotted as a function of any parameter which can be varied in an automated manner. In addition, plots can be made which show the variation of the unsteady aerodynamic force with reduced frequency. The aerodynamic-force plots can exhibit the data points input into STABCAR, curves resulting from interpolation, and curves resultinq from a p-plane approximation to the input data. The Langley Graphics package employed to obtain graphics output is not included in STABCAR. A description is given in appendix E of the function of the routines in this package to facilitate substitutions of equ i va lent s of twar e.

Interactive execution.- Interactive execution of STABCAR is an option. Some advantaqes of interactive execution are the more rapid receipt of output, the immedi- ate visibility of input errors, and the capability of early termination.

Restart capability.- Data that define the mathematical model being employed in the current case are stored on TAPE1, a random access file. The data include selected modal information and corresponding aerodynamic forces includinq a p-plane fit if it has been generated. In addition, if the plot control parameter is nonzero, Characteristic roots are stored as a function of the parameter beinq varied along with the user-supplied case title for identification. Consequently, if TAPE1 is saved, it may be used in a subsequent case if the model selected is a subset of the previous case. Thus, for example, the p-plane coefficients need not be recomputed. The new case might be to extend the range of variation of a parameter, to delete redundant or erroneous characteristic root data, or to combine sets of characteristic root data into one composite plot. Examples are given in the sample cases which illustrate several of these possibilities.

Functional Description of Overlays

The program is structured with two levels of overlays. me first level (overlay MAIN) is an executive. A brief summary of the major functions of each overlay will be presented here. A more detailed description including the function of major subroutines and definition of key parameters which direct the computations is given in appendix F.

15 MAIN: This executive overlay directs the computations by calling other overlays based upon input parameters. Namelist data are input which define the type of studies to be performed.

INTERP: INTERP performs geometry-related computations needed for a surface spline fit if modeshape data are for a plate. The data required in these computations are input here. The results of the computations are subsequently used in SENSOR to generate Surface Spline coefficients that fit the plate modal data.

SENSOR: This overlay determines surface spline coefficients for fitting plate- type modes. Beam-type modal data are also input here. Interpolation of beam and plate modal data is performed to obtain modal coefficients defining either deflections or slopes that would be measured by sensors at specified locations. These modal coefficients are stored for subsequent use and define the matrix H in equation (1) or (2).

MATINPT: Large data arrays such as the aerodynamic data, qeneralized masses, generalized stiffnesses, and control-system dynamics definitions are input. In addition, the selection feature is contained, which enables one to choose a mathematical model that is a subset of the model defined by a qiven set of data.

PPLANE: This overlay is called if a p-plane approximation to the aerodynamic forces is desired. ?he best coefficients are determined, in a least-squares sense, to employ in equation (3), qiven a set of aerodyamic force data defined over a range of reduced frequencies. These coefficients are stored for subsequent use in overlays PKFLUT and AEROPLT.

CONTROL: The matrix of transfer functions T TIT is constructed. (See eqs. (9) and (13).) 'he ( )ij element relates tReLo?tput of actuator i to the input of sensor j excluding (G)ij and any schedulinq or phase error. ?he polynomial coefficients for sensor, compensation, and actuator dynamics are constructed separately and then combined. Fach element of the resultinq matrix can then be multiplied by a distinct feedback gain and schedulinq component in overlay PKFLUT.

PKFLUT: The matrix in equation (1) or (2) is formed. Characteristic roots are determined as a function of variation in altitude, density, velocity, or feedback qains and the resultinq output is stored for possible subsequent plottinq. By operatinq in a multicase mode, the effect of variation in other parameters such as actuator or compensator dynamics or sensor location can also be studied.

AEROPLT: Plots are constructed which show hcw the oscillatory aerodynamic forces vary with reduced frequency and depict how the p-plane approximation fits the data.

PKPLOT: Plots showing root loci resultinq from variation in velocity, density, altitude, dynamic pressure, or feedback qains can be qenerated. Alternatively, dampinq ratios and natural frequencies can be plotted for specified modes as a function of any of these parameters. Namelist inputs are accepted which define the types of plots desired; scales; plot titles and which plot data files, amonq those available, should be combined to make a composite plot. STABCAR Flowchart

A flowchart is presented in figure 5, which describes the computational flow in STABCAR. This flowchart, although relatively detailed, is not intended to describe either the effects of all input options or the function of all program subroutines.

INPUT REQUIREMENTS

Description of Input Files

In this section STABCAR input requirements are defined. The documentation of the input requirements is presented in tables I through V. Therein, the types of data are outlined; a detailed description of the inputs is given which defines the individual input variables along with case-dependent conditions that determine whether certain of the data are required; and the input files that contain the data are identified. The correspondence between the FORTRAN names and the alqebraic names of the variables is also indicated in these tables.

TAPE2 contains data which define problem size, user options, starting estimates, control system definition, and so forth. These data, primarily namelist, are outlined in table I. Table I1 contains a detailed description of the TAPE2 inputs. If ICASE is positive, the run is to be an interactive one where changes in data are made on a remote terminal during program execution. In this case, all such changes in input are made interactively from file INPUT during program execution.

TAPE5 is a file containinq the bulk of the array data required for proqram execution. All such data are read free field. The data contained on TAPE5 and the array sizes are shown in table 111.

TAPE1 0 contains mode-shape data to be employed in determining sensor deflections for computing feedback siqnals as indicated in table IV.

TAPE1 is a random-access binary file created by the proqram. !kta on T.APE1 have been either input or generated in a particular overlay of STABCAR. 'Ihese data may be required in other sections of the proqram during the current case or may be needed in a subsequent case. For example, the user may have selected a subset of the mathematical model. This file could be saved by the user for future runs. The option of using the TAPE1 data as input is accomplished by setting ICASE = fl and the option of selecting a subset of the model data stored on TAPE 1 is exercised by setting ISELECT = -1 as illustrated in case 6 in appendix G. The contents of TAPE1 are indicated in table Iv.

Input Aids

Many cases are run which employ a previously generated TAPE1 €or the bulk of the input. ?he user may be in doubt as to precisely what values are stored on TAPE1. TAPE2 variables and selected sensor data stored on TAPE1 may be determined by executing STABCAR with IFLUT = 0, ICASE = +I, ICONSYS = 1, and TAPE1 local. If IAEROWR is set equal to 1, the aerodynamic force tabular arrays will also be output. If IMODEWR is set equal to 1, generalized masses, structural dampings, and either natural frequencies or the generalized stiffness matrix will be output.

17 A local scratch file called STABIN is produced by STABCAR which reflects TAPE2 or INPUT file data generated during a series of batch or interactive STABCAR runs. When ICASE = t2 (i.e. , an initialization run) , STABIN contains the values of all variables includinq default values, except for array variables where only the first value and any nonzero values are exhibited. When ICASE = +1, STABIN contains the values actually input as well as any default values pertinent to the analysis being attempted. STABIN is not rewound by STABCAR; therefore, it can provide the user with a record of what was done durinq a multicase session.

When IPKLT = +1, the user has a number of options pertaininq to any sets of plot data stored on TAPE1. If the user sets IDELPLT = +1, he may delete sets of bad plot data by specifying which of the stored plot sets he wishes to keep. The user can specify which sets he wishes to keep in any order. They will then be resequenced in the order selected. If IDELPLT = 0, the user may combine a series of plot sets to form a composite plot. (See appendix G, table GVII, for example). Here too, the plot sets may be chosen in any convenient order. 'Ihe choice of order is qenerally made so as to maximize ISAME (table II), which minimizes the number of times namelist SPLOTP has to be input (table GVII).

PROGRAM CAPABILITY DEMONSTRATION

In this section some of the capabilities of STABCAR are demonstrated. A mathematical model of an aircraft is defined and STABCAR is employed to examine its stability characteristics as a function of several parameters. The inputs required to qenerate each sample case are shown and a qraphical display of the results from each case is presented.

Aircraft Mathematical Model

Results obtained by using STABCAR are shown for a mathematical model of an early version of the DAST ARW-2 (Drones for Aerodynamic and Structural Testing, Aeroelastic Research Wing Number 2). The DAST ARW-2 has a high-aspect-ratio (10.3) supercritical wing having 25' sweep at midchord mounted on a Firebee drone fuselage. The structural design of the wing was performed by the Boeing Military Airplane Company (refs. 21 and 22) under the assumption that an active flutter suppression system would be operational. The wing will flutter within the flight envelope with the flutter suppression system turned off. The structural model was supplied to NASA by the Boeing Military Airplane Company. The structural model was input into the NASTRAN program (ref. 6) to obtain the modal characteristics of the model. Two rigid-body modes (plunge and pitch), 10 symmetric elastic modes, and 1 control mode were retained for computation of generalized aerodynamic forces. The resulting modal information together with geometric and paneling data were input into the ISAC (Interaction of Structures, Aerodynamics, and Controls) program (ref. 7) where unsteady aerodynamic forces were generated by using a doublet lattice method. The aerodynamic paneling employed, as well as the location of the control surface and sensors to be employed for a flutter suppression system, is indicated schematically in figure 6. The flutter suppression control law that is used is shown in figure 7. In this control law, the difference in signals from two accelerometers located at (xs,ys)l = (7.144,2.083) m and (X~,Y,)~= (7.336,2.134) m is fed back to an actuator driving an outboard aileron. Note that there is a scheduled component in the filter which varies with Mach number and dynamic pressure to provide appropriate phasing between sensor output and actuator input at varying flight conditions. Appendix B shows how the scheduled component was incorporated into STABCAR.

18 Graphical Output From Sample Cases

lhe amount of information to be output by STABCAR is selectable by the user. The basic file for printed output is TAPE6 which can be sent to the printer for either batch or interactive runs. This lengthy output is well identified by Hollerith descriptors and is not shown here. When interactive runs are being made, the user will typically minimize the information displayed to the screen. The variables employed to designate the output desired have been defined in table I1 in namelist SINPUT. In the results which follow, only graphical outputs to the screen are presented. The inputs required to generate each figure are given in appendix G.

Case 7 - Mrodynamic forces and p-plane approximation.- In case 1 (aerodynamic forces and p-plane approximation) , a p-plane approximation was made to the aerodynamic forces. Figure 8 shms the selected aeroiiynamic force data input into STABCAR, the curves resulting from a quadratic interpolation for points between the data, and a p-plane fit. The elements shown are those for the three elastic modes that are most critical in the flutter mechanism and those for the corresponding control surface effectiveness elements. lhe inputs required to qenerate these figures are presented in appendix G.

Case 2 - Open-loop characteristic roots as function of altitude.- Figure 9(a) shows the root loci of the aircraft without controls as the altitude is varied. Case 2 (open-loop characteristic roots as function of altitude) is for a Mach number of 0.86; each point is matched by solvinq for the atmospheric density at the altitude as well as the speed at that altitude corresponding to the 0.86 Mach number. The open-loop flutter occurs at h = 6.705 km which corresponds to a velocity of 269.6 m/sec and a density of 0.6101 kq/m 3 . Note that the roots correspondinq to modes 3, 5, and 8 are varying with altitude, whereas the other roots considered are relatively unaffected. It will be shown, in a subsequent plot, that essentially the same flutter point is predicted with a mathematical model that retains only the 3d, Sth, and 8th modes in the analysis. Figure 9(b) presents essentially the same information as figure 9(a) except damping ratios and natural frequencies are plotted as explicit functions of the altitude.

Case 3 - Open-loop characteristic roots as function of density.- Results obtained by running case 3 are shown in figure 10. Figures 10(a) and (b) show root loci and damping ratio and natural frequency variations, respectively, as a function of density for NMa = 0.86 and U = 269.6 m/sec for no controls. The flutter point occurred at p = 0.6101 kg/m3 which is in perfect agreement with the altitude variation case.

Case 4 - Open-loop characteristic roots as function of dynamic pressure.- Fig- ures ll(a) and (b) for case 4 (open-loop characteristic roots as function of dynamic pressure) correspond precisely to figures 10(a) and (b). They are density variation runs but the independent variable has been changed to dynamic pressure to show the characteristic roots as a function of q. Plots of root variations as a function of dynamic pressure can also be made in altitude and velocity variation runs.

Case 5 - Characteristic roots as function of velocity (truncated model).- Case 5 shows results from using a truncated model. Figure 12(a) is an open-loop plot of damping ratios and natural frequencies versus velocity. The aerodynamic forces for Nm = 0.86 and the matched-point density found in figures 9 and 10 are employed. Thus, only the point at which U = 269.6 is a matched point. In this run, the select feature has been employed to consider only the effects of the modes that are

19 most important to the onset of flutter. These are modes 3, 5, and 8 which are the lst, 3d, and 6th elastic modes. Tne predicted flutter speed of 271.2 m/sec is within 1 percent of that found earlier with two rigid-body and six elastic modes. In figure 12(b) damping ratios and natural frequencies are plotted as a function of velocity under the same conditions and assumptions as for figure 12(a) except that the control loop is closed. The control-sytem representation is that shown in figure 7. The aircraft does not flutter for the range of velocities shown.

Case 6 - Elastic mode gain root loci.- Fiqure 13 shows gain root loci (case 6) for elastic mode roots at Nm = 0.86 and h = 4.572 km which correspond to UD. For this condition the open-loop ARW-2 confiquration is unstable as indicated by the root in the right half-plane in figure 13. The root loci are shown for both a p-k (solid lines) and a p-p (dashed lines) analysis. The unstable mode is stabilized at a gain of approximately 0.7. The slight difference between the p-k and p-p analyses for this root at the ja-axis crossinq is an indication that the least-squares fit of the aerodynamic forces is not precise at the flutter frequency. Note that the mode number identification does not correspond to the numberinq of the earlier fiqures. The matrix iteration technique was employed here to obtain initial estimates for the open-loop roots. The roots are numbered in ascendinq order of maqnitude of the open- loop roots. Since the two rigid-body roots are not shown, the lowest number is 3.

Case 7 - Effect of phase errors upon elastic mode qain root loci.- Fiqure 14 illustrates the use of STABCAR to determine whether specified phase and gain marqins are met by a candidate control law. This figure presents the gain root loci of the elastic mode roots (case 7) at the same fliqht condition as those for figure 13 for the p-p analysis only. Additional root loci are shown where the control signal has been modified by phase errors of -45O and 45". The fiqure indicates that the candidate control law has less than a -45O phase marqin at the nominal gain and does not meet a -6 dR qain marqin at nominal phase.

Case 8 - Gain root loci of control system poles.- Figure 15 shows the gain root loci for the control system poles (case 8) at NMa = 0.86 and h = 6.705 km, which is the open-loop flutter point. The dynamic pressure at this condition is 25 percent less than that at UD (the fliqht condition of fiqs. 13 and 14). For this dynamic pressure, root loci are shown for both p-k (solid lines) and p-p (dashed lines) analyses. Loci on the neqative real axis were determined for the p-k analysis only because of the complications occurring in the p-p analysis due to isolated singular points and poles introduced by the p-plane approximation to the aerodynamic forces. Note in particular the branch where two real poles merqe to form a complex pair. This type of locus can best be handled in an interactive run by tryinq different initial estimates havinq nonzero imaqinary parts when the branch point is approached. Another curve of interest is the locus of the filter root which has an open loop pole at (-50,196.5) rad/sec. Its locus is similar to that of the 3d elastic mode in fiqure 13 which emphasizes the variation in root loci patterns that can occur as a result of different zero and open-loop pole locations between two fliqht conditions.

Case 9 - Short-period altitude root locus.- A short-period root locus (Case 9) with altitude is presented in figure 16 for p-k and p-p analyses. The riqid-body degrees of freedom considered in this study are plunge and pitch about the center of mass with respect to an inertial set of axes. Draq contributions to the perturbation equations are neqlected. The resultinq equations for the rigid-body roots correspond approximately to those for the short-period approximation with additional effects of unsteady aerodynamic force contributions due to both rigid and elastic motion.

20 CONCLUDING REMARKS

STABCAR is a computer program designed for the study of the stability characteristics of flexible, actively controlled aircraft. Analyses can be performed that include either oscillatory or approximate fully unsteady aerodynamic forces. Deter- minant iteration is employed for characteristic root determination. Consequently, sufficiently accurate initial estimates and sufficiently small steps in the quanti- ties being varied are required to converge upon and follow the characteristic roots.

STABCAR can be executed interactively with a remote terminal which allows the user to rapidly assess the results he is obtaining and conveniently change the input data. Changes such as deleting modes from the model, altering transfer functions or sensor types, or executing different types of analyses may be performed quickly and easily.

In an effort to aid potential users, a description of the mathematical techniques employed has been outlined, a flowchart has been provided, key variables have been defined, and a description of the input requirements has been given. In addition, a series of sample cases have been presented which provide explicit examples of the inputs required to utilize the major program capabilities.

STABCAR has been written specifically for efficient operation on Control Data equipment. Revisions are required to run this program on other equipment.

Langley Research Center National Aeronautics and Space Administration Hampton, VA 23665 March 7, 1984

21 APPENDIX A

p-PLANE APPROXIMATION TO THE UNSTEADY AERODYNAMIC FORCES

Least-Squares Solution

It is desired to approximate the jth column of aerodynamic forces with the following function:

The form of this expression is the same as that found in references 1 and 2. HOW- ever, constraints would have to be imposed upon the coefficients to have full equiv- alence between the fit described here and that of references 1 and 2.

Assume tabular data for oscillatory motion have been computed:

The objective is to determine the coefficients in equation (Al) such that $, (p = fi k) best fits the tabular data in a least-squares sense subject to a set of linear equality constraints which may be imposed upon the coefficients.

A A Consider a particular element Qij Of Qj* Let

denote the coefficients to be determined. Suppose that there are linear equality constraints which the coefficients must satisfy denoted by

CTA = C I ij Oi j

The error in the fit of equation (A1 ) to the data at k = k n is

22 APPENDIX A or

where

and - - k2 2 n 10-k n 2 *-- 2 b2 + k b2 + k lj n nnj j n B= n bk b .k n jn 0 11 n O kn 2 *** 2 b2 + k b2 + k lj n nnj j n -

Define the square of the magnitude of the error over all data points included in the least-squares €it to be

nkk . J leij l2 =EeT nn e (A7 1 ij ij n=l where nkk < nk. Then the coefficients which provide the least-squares error sub- j ject to the imposed constraints are found by determining the Aij and - hi,, which minimizes

23 APPENDIX A

Performing the indicated minimization yields

Selectable Constraints

The user may select constraints to impose upon the p-plane coefficients for column j from the followinq list:

1. Force agreement with tabular value at k = 0 (asumes that kl = 0):

2. Constrain slope at k = 0 based upon tahular data for column j (assumes that kl = 0 and k2 is small):

3. Constrain slope at k = 0 for column j to be related to the tabular data in the mth column at k = 0 (assumes that kl = 0):

This constraint can he used to relate plunqe and pitch columns at k = 0 (ref. 23)

4. Constrain slope at k = 0 to be zero:

24 APPENDIX A

5. Null out % : j For example, let k = 2,

where 1 occurs in column k + 1.

6. Require a precise fit at a specific, nonzero frequency k which need not be a tabular point:

2 k2 k2 1 0 -k Aij = Re [Q. . (k)I 2 *** 2 b2 + k +k 17 11 bn,,j

bl .k kO 2 *** b2 + k L 11

If this constraint is selected, the tabular data are interpolated to obtain Qi (k).

Instructions for preparing program input to include one or more of these constraints are given in table 11. See specifically the variables ICOF, THETAN, and SPKO.

25 APPENDIX B

STABCAR contains a provision for includinq compensator elements that vary as functions of Mach number and dynamic pressure. This capability is provided in function subproqram SCHEDUL where control/sensor pair identification, Mach number, and dynamic pressure are available. A phase error Qij introduced into the compensation for one control/sensor pair is also available. mere are block data statements in SCHEDUL. ?he user inputs data required to define his schedulinq laws by usinq the block data statements. He then provides code in SCHEDUL which utilizes the data and the previously mentioned parameters to define the desired schedulinq laws. An example i llustratinq the procedure for a sinqle-input/sinqle-output control law follows. The schedulinq law implemented is that shown in fiqure 7.

~ ~~ APPENDIX B

COMPLEX FUNCTION SCHEDUL(IC,JC,XMACH,QBAR,S) COMMON/GAIN/IGAIN,DELGAIN~l~,2O~,GN~lO,2O~,IPH,JP!~,PHASE,PHERR ,ISCHDUL(10,20) COMPLEX PHERR COMMON/AEROP/NMSAVE,NCSAVE,NRSAVE C ,NKSAVE,NMNCS COMMON /CNSCHC/ COF1(10,2O),COF2(10,20),COF3(10,2O),COF4(10,20) ,COF5(10,2O),COF6(10,20),COF7(10,20),COF8(10,20),COF9(10,20) ,COF10(10,20, COMPLEX S COMMON/SELECT/NMODES,NOC(25),NKNEW,NCNEW,NRNEW C ,NOK(15),ISELECT DATA COFl/O.,O.,61,197*0/ C ,COF2/0.,O. ,2. 3924807E+04, 19710. / C ,COF3/0.10. ,3.147319E-05,197*0./ C ,COF4/0.,0,1034.21,197*0./ C ,COF5/0 ,O.,l ,197*0./ C ,COF6/0.,0.,735.,197*0./ C ,COF7/0 ,0.,833.3,197*0./ C ,COF8/0.,0.,.0033257154,197*0/ C ,COF9/0.,0.,300,197*0./ C ,COFlO/O ,0.,75.,197*0./ NMNC=NMODES-NCNEW CFl=COFl(IC,JC) CFP=COFP(IC,JC) CF3=COF3(IC,JC) CF4=COF4(IC,JC) CFS=COFS(IC,JC) CF6=COF6(1C,JC) CF7=COF7!IC,JC) CFB=COFB(IC,JC) CF9=COF9(IC,JC) CFlO=COFlO(IC,JC) c*--) PHASE ERROR CAN BE INTRODUCED BETWEEN CONTROL IPH AND SENSOR JPH C* EXP(I*PHERR) SCHEDUL=(l ,O.) IF(IC.EQ.IPH.AND.JC.EQ.JPH)SCHEDUL=SCHEDUL*PHERR iFIiSCHDUL(1c,jc) .NE.I)RET~~RN c*--) THIS CODE IMPLEMENTS THE SCHEDULED COMPONENTS OF THE CONTROL LAW C* DEFINED IN FIGURE 7. XM= XMACH c*--) IF(M ( 0.61) THEN M=.61 IF(XM.LT.CFI)XM=CFl QB=QBAR c*--) IF(Q ( 2.3924+04) THEN c1 = 2.3924+04 IF(QB.LT.CFZ)QB=CFZ

C*--) -1 (= KS = (3.1417-05)*(1.61-M)*(Q - 1034.21) (= 1 XKS=CF3*(1+CFl-XM)*(QB-CF4) IF(XKS CT.CFS)XKS=CFS c*--) -75 <= DS = 735.-.333.33*~+ 0.0033267*~(= 300. DS=CF6-CF7*XM+CFE*QB IF(DS.CT.CF9)DS=CF9 I F ( DS . LT . CF 10 ) DS=CF 10 c*--) (KS/(S+DS) SCHEDUL=CMPLX(XKS,O.)I(S+DS>*SCHEDUL RETURN END

27 APPENDIX C

OPTIONS FOR HANDT.TNC, DENOMINATOR DYKAMICS .W.ISI?!'G FROM ACTUATORS, FILTERS, AND SENSORS

This appendix describes options contained in STABCAR which allow denominator dynamics arising from actuators, filters, and sensors to be cleared from the denominators of the equations. This option is employed when control roots are to be found with qains near zero to avoid division by zero.

Consider the followinq transfer function between actuator i and sensor j (see eqs. (9) and (13)):

Let

and

Define

d = d + dls + d2s2 + ... + d s "d 0 "d

as the common denominator of {iij}, i = 1, ..., "6, j = 1, ..., ns. Define

An option is provided to multiply the control rows of either equation (1) or (2) throuqh by d. The multiplication is performed if the variable IDMULT is input as 1. If IIIMULT is input as 0, the multiplication is not performed. If equation (1) 1s to be employed, the variable ICSACT is input as 0. If equation (2) is to be PmPloYed, ICSACT is input as 1.

When characteristic roots are determined with the determinant iteration approach, the modifications to equations (1 and (2) that occur for IDMULT = 1 are

L APPENDIX C evident. ?he modifications employed for the matrix iteration option are defined below (dr is the first nonzero coefficient in the polynomial d(s) (see eq. (C3) ):

1. Let ICSACT = 1. Partition the matrix (see eqs. (2) and (18)) rag< A=

Then A and A have the following definitions: 6< 66

(a) For IDMULT = 0,

(b) For IDMULT = 1,

2. Let ICSACT = 0 (see eqs. (1) and (18) ).

(a) For IDMULT = 0,

(b) For IDMULT = 1,

r (d(s) - drs ) - F T (s) HHE d(s) - FDN(S) D HE ] d sr r

29 APPENDIX D

liLGORiTHM FOK MATRIX ITERATION FOR CHARACTERISTIC ROOTS

The matrix iteration algorithm for characteristic root determination employed in STABCAR during convergence difficulties and for obtaining initial estimates for the characteristic roots is presented in this appendix.

Let

0 option type for which matrix iteration routine is being employed. P = 1 during converqence difficulties; 0 = 2 if initial OP . estimates are being souqht and all gi2 = 0: 0 = 3 if initial P estimates are being sought and some # 0. 12

"h number of characteristic roots to be found by matrix iteration. nh = 1 if 0 = 1; if 0 = 2 and ICSACT = 0; nh = nE + n6 P = "5 P if 0 = 2 and = 1; nh = Number of nonzero if 0 = 3 P ICSACT gi2 P i = counter for number of roots beinq found. i 1, ..., "h i root number in determinant iteration for which converqence difficulties n were incurred

counter determininq number of iterations for a root

eiqenvalue number (from set of eiqenvalues of which \ corresponds to si,,*I the initial estimate of current root being

determined (see step 2) si best initial estimate for ith characteristic root

h Si second initial estimate for ith root (see step 6) 1 A estimate from determinant iteration routine for root for which convergence difficulties were incurred s ! ith root being determined 1

AI si. jth estimate for si 3 square root of__ eiqenvalue of which corresponds to s'.1' i.e., s! = (Ab 1. 7

frequency used in estimatinq 0 =2 P see step 2 APPENDIX D

M-l inverse of generalized mass matrix JAR square root with positive imaginary part Step 1: Initialize:

i=l

"I A s. = s if 0 =1 11 (in)2 P

= (-0.5~,w ) if 0 = 2 rr P

C - if 3 =3 'i P

Determine M-l

Step 2: Set j = 1:

Determine eigenvalues } of A("l) (see eqs. (18)) {A'r

Determine index R such that sil = IJhk

R=nA- (i - 1) if o =2 P

AI Or R is defined such that s' is closest in magnitude to sil if il op # 2

Step 3: Determine convergence:

Step 4: Set sij+l"1 if j = 1 = tij +

Step 5: Set j = j + 1

If j > 30, print out nonconvergence, and go to step 7

Determine eigenvalues {hkj} of (see eqs. (1 8))

Repeat step 3

31 APPENDIX D

A Step 6: Set s(i - si and return if 0 = 1 n2 j P c Set si2 = sij and gil = 0.99Ei2 if Op # 2

Step 7: Set i = i + 1

If i > nh return

if op =

Repeat step 2

32 APPENDIX E

GRAPHICS ROUTINES NOT INCLUDED IN STABCAR

This appendix is devoted to the description of elements of the Langley Graphics package that have been employed by STABCAR but are not included as a part of STABCAR. The description is included to facilitate a substitution of equivalent routines. Alternatively, the Langley Graphics package can be purchased separately from Computer Software Management Information Center (COSMIC), Suite 112, Barrow Hall, University of Georgia, Athens, GA 30602, as LAR 12722 entitled LRCGOS (Langley Research Center Graphics Output System).

SUBROUTINE PSEUDO

PURPOSE: To create and write an appropriately named Plot Vector file. Through link- ages set up by an initial call to PSEUDO, all subsequent graphics data generated by the user will be routed through one of the PSEUDO entry points and written on the Plot Vector File. The PSEUDO processor is designed for use with the frame dependent postprocessors described in Section 1.3, Volume IV, of the Computer Programing Manual.

-USE: CALL PSEUDO

This will establish a Plot Vector File named SAVPLT.

CALL PSEUDO ( 6LMYFILE)

This will establish a Plot Vector File named MYFILE.

NOTE: The plot Vector File (or Files) will usually be written to disk (as opposed to tape) and may be postprocessed following user program termination v5a appropriate specification of one or more PLOT control cards. (See Section 1.3, Volume IV, Computer Programing Manual).

RESTRICTIONS: An initializing call to PSEUDO (with or without a file name argument) must be made prior to any calls to CALPLT or any other graphics output routine.

OTHER CODING INFORMATION: CALPLT and NFRAME are entry points in PSEUDO. PSEUDO checks for indefinite and infinite value and provides a traceback.

ENTRY NFRAME

PURPOSE: To provide users specific means of executing frame advance movements on any plotter device via an appropriate f rame-oriented device postprocessor. Frame advance distances are generally defined to be incremental from current frame origin. A frame is defined as a finitely bounded plotting area; it is an imaginary boundary around the usable plotting area. CALL NFRAME is intended to be used as a frame advance mechanism, =as a plot origin offset.

-USE: CALL NFRAME

33 APPENDIX E

ENTRY CALPLT

PURPOSE: To move the plotter pen to a new location with pen up or down. This is the basic plotting routine that is used by all other graphics routines; therefore; a check is made on the parameters for indefinite and infinite. If any of the parameters are indefinite or infinite, the program will abort and a traceback to the

routine is provided. (See “OTHER CODING INFORMATION. ‘I )

-USE: CALL CALPLT(X,Y,IPEN)

where

XI Y are the floating-point values €or pen movement.

IPEN = 2 pen down

= 3 pen up

Negative IPEN will assign X = 0, Y = 0 as the location of the pen after moving the X,Y (create a new reference point).

RESTRICTIONS: All X and Y coordinates must be expressed as floating-point inches (actual page dimensions ) in deflection from the origin.

SUBROUTINE LEROY/BALLPT

PURPOSE: To set up the parameters necessary to accommodate plotting with the liquid ink pen. Once set, this mode will remain.

-USE: CALL LEROY

RESTRICTIONS: The CALL LEROY is only recognized by the CalComp postprocessor; it is ignored by the other postprocessors. In addition to reducing the speed of the plotter for all plotting movements, the number of plot vectors in any annotation is considerably increased.

The CALL LEROY must be made prior to any plotting calls, but after the CALL PSEUDO.

34 APPENDIX E

SUBROUTINE NUMBER

PURPOSE: To convert a floating-point number to BCD (expressed in F format) and to draw the resulting alphanumeric characters.

-USE: CAtL NUMBER(X,Y,HEIGHT, FPN,THETA,NODIGIT)

where

X,Y are the coordinates in floating-point inches of the left lower corner of the first digit of output.

HEIGHT is the height of the plotted number in floating-point inches. (See NOTATE routine. 1

FPN is the floating-point number to be drawn.

THETA is the angle in floating-point degrees at which the number is to be drawn. (See NOTATE routine. )

NODIGIT is the number of decimal digits to the right of the decimal point for output .

NODIGIT=-I or NODIGITzO both specify no decimal places; however, -1 sup- presses the decimal point.

35 APPENDIX E

SUBROUTINE NOTATE

PURPOSE: To draw alphanumeric information for annotation and labeling and provide special centered symbols for annotation of data points.

USE: CALL NOTATE(X, Y, HEIGHT, BCD,THETA,NOCHAR)

where

XI y are the floating-point coordinates of the first character.

For alphanumeric characters, the coordinates of the lower left-hand corner of the characters are specified.

HEIGHT specifies character size and spacing in floating-point inches for a full-size character.

BCD is the string of characters to be drawn and is usually written in the form: nHXXXX--- (the same way an alpha message is written using FORTRAN format statements). Instead of specifying alpha information as above, the beginning storage location of an array containing alphanumeric information may be given.

THETA is the angle in floating-point degrees at which the information is to be drawn. Zero degrees will print horizontally reading from left to right, 90° will print the line vertically reading from bottom to top, 180' will print the line horizontally reading from right to left (i.e., upside down), and 270° will print vertically reading from top to bottom.

NOCHAR is the number of characters, including blanks, in the label. NOCHAR is limited to 400 characters in one call.

36 APPENDIX E

SUBROUTINE AXES

PURPOSE: To draw a line, to annotate the value of the variable at specified inter- vals with or without tic marks, and to provide an axis identification label.

-USE: CALL AXES(X, Y,THETA,S,ORG,SPX,TMAJ,TMIN,BCD,HEIGHT,NOCHAR,NDEC)

where

x, y are the coordinates in floating-point inches of the starting point of the axis with reference to the plotting area origin as established by CALPLT.

THETA is the angle of rotation measured counterclockwise from the X-axis in floating-point degrees. Normally, THETA is Oo for an X-axis and 90° for a Y-axis.

S is the length of the axis in floating-point inches. Should be a multiple of TMAJ.

+S will generate tic marks. -S will eliminate tic marks.

ORG is the functional value to be assigned to the origin (i.e., the value of the first scale) in floating point.

SEX is the adjusted scale factor for the array to be plotted (change in value per inch). NOTE: Values of ORG and SEX which will produce a reasonable scale may be calculated using subroutine ASCALE or BSCALE.

TMAJ is the distance in floatinq-point inches for major tic marks (0.25 inch high). If the values are integer multiples, the decimal point and decimal places are eliminated. A riegative TI.tnYJ will cause the actual value to be written instead of the adjusted value.

TMI N is the number of divisions per inch in floating point for minor tic marks (0.125 inch high). To eliminate minor tic marks the following may be used:

TMIN = 0.

BCD is the character label for the axis (see NOTATE routine).

HEIGHT is the height of the full-size characters in the BCD title. Numbers at the tic marks will be (0.75 *HEIGHT) high. HEIGHT is in floating- point inches.

If HEIGHT = O., all annotation will be eliminated.

37 APPENDIX E

NOCHAR is an integer specifying the number of characters in BCD title. A negative NOCHAR places the annotation on the clockwise side of the axis and a positive NOCHAR places the annotation on the counterclockwise side of the axiq. N0CFTA-P. = c! Is not diiowed. ~t it is desired to have no label, then the RCD parameter should be lH, and NOCHAR either +1 or -1. Normally NOCHAR = + for Y-axis and NOCHAR = - for X-axis.

NDEC is the number of decimal places after the decimal point for the numbers placed on the axis at the major tic marks. NDEC may be omitted. If NDEC is omitted, the default value of 2 is set.

ENTRY ASCALT?

PURPOSE: To compute a scaling factor for an array of numbers to be plotted over a certain area and find the minimum data value with the array.

-USE: CALL ASCALE(ARRAY,S,N,K,DV)

where

ARRAY is the name of the array containing the floating-point values to be scaled. (See RESTRICTIONS. )

S is the lenqth (floating-point inches) over which the data are to he plotted (usually the length of one of the axes).

N is the number of data values in ARRAY from which points are to be plotted in accordance with K.

K is the interleave factor which specifies the sequence in which data are stored.

= 1 indicates the values are stored sequentially.

= 2 indicates that values are stored in every other location in the array, etc.

DV is the number of divisions per inch of the plotting paper to be used (should be: 10.0 or 20.0).

RESTRICTIONS: The array must be dimensioned to include storage space for two extra elements per interleave factor. For example: N = 100, K = 1, DIMENSION ARRAY (1 02) ; N = 75, K = 3, DIMENSION ARRAY (231).

METHOD: This routine scans the elements starting at location L of the array to find the minimum and maximum. (If the scaling heqins with first location of array, L = 1; if scaling begins with third location of array, L = 3.) ASCALE computes an adjusted minimum (origin) and stores it in ARRAY(L+(N*K)) and computes a scale factor and stores it in ARRAY(L+(N*K)+K). The scale factor will be (A*lO**J) where A is 1, 2, 4, or 5 and J is integer power. The data in the array may be scaled to floating-point inches by:

SV = (AE - AMV)/SF

38 APPENDIX E where

SV = scaled value AE = present value of array element AMV = adjusted minimum value computed by ASCALE SF = scale factor computed by ASCALE

39 APPENDIX E

SUBROUTINE BSCALE

PURPOSE: To compute a scaling factor for an array of numbers to be plotted over a certain area and to find. sfi ~cceptc.bleii.Liilmum data value for the array.

-USE: 1. Stores the adjusted minimum value (origin) in ARRAY(L+(N*K)) and scale factor in ARRAY(L+(N*K)+K). If the scaling begins with first location of array, L = 1; if scaling begins with second location of array, L = 2, etc.

CALL BSCALE( ARRAY, S ,N ,K, TMAJ, M, ORG)

where

ARRAY is the name of the array containing the floating-point values to be scaled. (See RESTRICTIONS. )

S is the length (floating-point inches) over which the data are to be plotted (usually the length of one of the axes).

N is the number of data values in ARRAY from which points are to be plotted in accordance with K.

K is the interleave factor which specifies the sequence in which data are stored.

= 1 indicates that values are stored sequentially.

= 2 indicates that values are stored in every other locaiion in the a r ray, e tc.

TMAJ is the distance in floating-point inches for major tic marks along the axis (length of S).

= 0. the routine scales in a manner similar to routine ASCALE.

> 0. the routine will adjust the origin and scale factor in order to give regular multiples of the delta number that will be written at the tic marks by the AXES routine.

M is an integer that has multiple uses.

= 0 if M is zero and TMAJ is zero, the routine scales in a manner similar to routine ASCALE.

> 0 if M is positive and TMAJ is zero, the data will be scaled over the full range of S, with no attempt to adjust the scale factor for appearance.

< 0 if M is negative, the valid value contained in parameter ORG will be used for a forced origin.

40 APPENDIX E

0RG is a floating-point value which will be used for a forced origin if parameter M is negative. The value in ORG may -not be exactly the same on return from subroutine.

ORG may be omitted if M is positive or zero.

41 APPENDIX E

SUBROUTINE PNTPLT

PURPOSE: To draw NASA Standard Plot symbols centered on a given coordinate value.

-USE: CALT, PNTPLT(X,Y, XSYFi,iSj where

X is the X coordinate €or the centered symbol in floating-point inches.

Y is the Y coordinate for the centered symbol in floatinq-point inches.

ISYM is an integer specifying the symbol to be used. (See figs. El and E2.1

= 21 for a point.

= 22 for a plus sign +.

IS is an integer value specifying the size symbol to be used.

= 1 small

= 2 medium

= 3 large

NASA STANDARD PLOT SYMBOLS

INTEGER SIZE INTEGER SIZE REFERENCE SMALL MEDIUM LRRGE REFERENCE SiiRLL nEDIUfl LARGE

1 11

2 12

3 13

Li 111

5 15

6 16

7 17

8 18

9 19

10 20

Figure El .- Plot symbols.

42 APPENDIX E

SUBROUTINE LINPLT

PURPOSE: To draw a line between and/or draw a NASA standard symbol at each successive data point (stored in an array), or to draw a continuous line through the data points and draw a specified symbol at the end of the line.

-USE: CALL LINPLT( XARRAY, YARRAY, N, K, J, ISYM, IS, NG)

where

XARRAY, are the names of arrays containing the X values and Y values, respec- YARRAY tively, to be plotted. Values must be in floating point. (See RESTRICTIONS.)

N is the number of points to be plotted.

K is the interleave factor which specifies the sequence in which data are stored.

= 1 indicates that values are stored sequentially.

= 2 indicates that values are stored in every other location in the array, etc.

J is positive for line and symbol plot, negative for symbol-only plot. The magnitude specifies the alternate number of data points at which to plot a symbol.

= 0 for line plot with a specified symbol at the end of the line.

= 1 for symbol for every data point.

= 2 for symbol for every other data point, etc.

ISYM is an integer describing symbol to be used, see PNTPLT routine for list (fig. El).

= 0 no symbol will be drawn.

IS 1s an integer specifying the size of the symbol (see PNTPLT routine).

NG is an integer used to specify data points which will be enclosed by a I-inch square grid, 20 x 20 lines to the inch. NG may be omitted. If NG is omitted, the default value of 0 is set.

= 0 no grids.

= 1 for every data point.

= 2 for every other data point, etc.

RESTRICTIONS: LINPLT expects the adjusted minimums and scale factors as described in ASCALE.

43 APPENDIX E

SUBROUTINE DLINPLT

PURPOSE: To draw a dashed line between and/or draw a NASA standard symbol at each successive data point (stored in an array) , or to draw a continuous c?ashed line through the data pnl.n.ts mi: draw a specified symbol at the end of the line.

-USE: CALL DLINPLT(XARRAY,YARRAY, N,K, J, ISYM,IS,LP)

where

XARRAY, are the names of arrays containing the X values and Y values, respec- YARRAY tively, to be plotted. Values must be in floating point. (See RESTRICTIONS.)

N is the number of points to be plotted.

K is the interleave factor which specifies the sequence in which data are stored.

= 1 indicates that values are stored sequentially.

= 2 indicates that values are stored in every other location in the array, e tc.

J is positive for line and symbol plot, negative for symbol only plot. The magnitude specifies the alternate number of data points at which to plot a symbol.

= 0 for line plot with a specified symbol at the end of the line.

= 1 for symbol for every data point.

= 2 for symbol for every other data point, etc.

Ism is an integer describing symbol to be used, see PNTPLT routine for list (fig. El).

= 0 no symbol will be drawn.

IS is an integer specifying the size of the symbol (see PNTPLT routine).

LP is an integer used to specify the line pattern.

=2 --_------_---

=3 -

=4 -_-______

=6 ~____~~

44 APPENDIX E

RESTRICTIONS: DLINPLT expects the adjusted minimums and scale factors as described in ASCALE.

45 APPENDIX F

DETAILED FUNCTIONAL DESCRIPTION OF OVERLAYS , MAJOR SURROrJTINES , AND TDEN'I'IFICATION OF KEY PARAMETERS

In this appendix, a description of the functions of each overlay and its major subroutines is given including definitions of key input and internally generated parameters that control program flow.

Overlay MAIN

Overlay MAIN is the STABCAR executive. Data are input here which determine the type of case to run, where to find array data, and which overlays are required. Key parameters are defined as follows but are defined more completely in table 11:

I CAS E Integer indicatinq type of case (interactive or batch) to run and where to get data.

I SPLANE Integer indicating whether to make p-plane approximation to aerodynamic forces .

ISENSE Integer indicating whether to perform interpolation for modal deflection or slopes at specified sensor locations.

I NTERP Integer indicating whether to compute geometric coefficients required for surface spline interpolation.

IPLATE( I) Integer defininq whether Ith structural section is modeled as beam or as plate.

ICSREAD Integer defining whether sensor data are to be read from TAPES.

ICONSYS Integer defining whether control system is to be included.

ICHSEN, ICHFI L, Inteqers defining whether changes to previously defined control and ICHACT system are to be made in current case; if not, overlay CONTROL is not needed.

I SELECT Integer defininq whether subset of current model is to be selected and where data for current model can be found.

IFLUT Integer defininq whether to perform stability analysis in current case.

IAPLT Integer defining whether to make plots of aerodynamic forces.

I PKPLT Integer defininq whether to save data for makinq plots of stability characteristics.

IDELPLT Inteqer defining whether to delete unwanted characteristic root data from data storaqe file TAPE1.

46 APPENDIX F

Overlay INTERP

Geometry-related, mode-independent coefficients which are needed for a surface spline fit for later interpolation purposes in overlay SENSOR for plate-type sections are computed by overlay INTERP. Primary subroutines called are INTEXT, SYMSURF, NONSURF, and MATINV. Key parameters are defined as follows:

NNODES (or NNODS) Total number of nodes at which modal data are available.

NPMX Maximum number of nodes in one structural section.

NSECTNS Number of structural sections vehicle is divided into.

I PLATE NSECTNS X 1 array.

= 1 Ith section is modeled as a plate. IPLATE (I) = 0 Ith section is modeled as a beam.

TAB NNODS X 4 array which, for plate-type sections, defines x,y coordinates (global) and z-deflection; fourth column is not employed for plate sections. For beam sections, array elements are distances along axis of rotation at which data are available, z-deflection, rotation about

swept Y-axis (y' ) and rotation about swept X-axis (XI ). (See fig. 3. ) There is a TAB array for each elastic mode shape.

ISS 2 x NSECTNS array. ISS(1,I) is node number of first node of Ith structural section; ISS(2,I) is node number of last node of Ith structural section.

RO,XO, and YO NSECTN X 1 arrays. For plate-type modes, RO = 1/B, where B is one-half root chord of Ith structural section; XO(1) is x-coordinate of Ith section root semichord; and YO(1) is y-c~crdinateat root semichord. For Seam-type sections, IPLATE = 0, (XO(I),YO(I)) is elastic axis intercept with Ith structural section root chord; RO is sweep anqle of elastic axis of Ith structural section in radians.

H+ Array of geometry coefficients.

Subroutine INTEXT.- Subroutine INTMT is an extension of INTERP and it is here that the TAB arrays are actually read and geometry coefficients are obtained. Sub- routines called are SYMSURF and NONSURF. Key parameters used are all those previously defined for overlay INTERP.

Subroutines SYMSURF and N0NSURF.- Subroutines SYMSURF and NONSURF compute the mode-independent geometry coefficients for symmetric and antisymmetric or nonsymmetric plate sections, respectively. Subroutine called is MATINV. Key parameters used are TAB and H.

___ ~~~ 'Internally generated parameters.

47 APPENDIX F

Subroutine MAT1NV.- Subroutine MATINV performs the matrix inversion needed in computing H, the geometry coefficients.

Overlay SENSOR

Overlay SENSOR determines surface spline coefficients for fitting mode shapes over plate-type sections. Beam-type modal data are also input here. Interpolation of beam and plate modal data is performed to obtain modal coefficients defining either linear or angular displacements that would be measured by sensors at specified locations. These modal coefficients are stored for subsequent use and define the H matrix in equation (1) or (2). Primary subroutines are SENCTL, DEFLECT, CSIUNI, IUNI, SPLDER, ZFUN2, AFUN2, ZFUN, AFUN, ZDER, and ADER. Key parameters are defined as follows:

NS Number of sensors.

XS and YS NS X 1 array of sensor locations.

= 1 Linear measurement. ITYPE = 2 Angular measurement.

NSS NS X 1 array identifying structural section where each sensor is located.

ISS 2 X NSECTNS array defining beginning and ending node numbers for each structural section; see section "Overlay INTERP" for additional i n f or mat i on.

= 1 Ith structural section is modeled as plate. IPLATE(1) = 0 Ith structural section is modeled as beam.

TAB,RO,XO, and YO Defined in section "Overlay INTERP."

NR Number of rigid-body modes. In this overlay NR must be either 0, 1, or 2. If NR = 1, the rigid-body mode mst be the negative of plunge; the user may input his own rigid-body modes by setting NR = 0 and treating his riqid modes as if they were elastic.

XCG x-coordinate of airplane center of mass. Since first rigid-body mode is assumed to be the negative of plunge, and second, pitch about center of mass, this is only required if NR = 2.

U and V Sensor location with respect to swept axis.

cs array containinq modal participation of each to input of each sensor.

I W Geometry coefficients computed in overlay INTERP for plate-type sections.

48 APPENDIX F

Subroutine SENCTL.- In subroutine SENCTL, the data required for modal interpolation are read in from file TAPEl0, and after linear or angular displacements are calculated, resulting modal coefficients are written to data storage file TAPE1 . Primary subroutine called is DEFLECT. Key parameters used are all those defined for overlay SENSOR.

Subroutine DEFLECT.- In subroutine DEFLECT, the actual modal coefficients defining linear or angular displacement inputs to sensor are obtained. Primary subroutines called are CSIUNI, IUNI, SPLDER, ZFUN2, AFUN2, ZFUN, AFUN, ZDER, and ADER. Key parameters used are all those defined for SENSOR except NS and NSS.

Subroutines IUNI and CSIUN1.- Subroutines IUNI and CSIUNI are used to perform interpolation of modal data used in calculating both linear and angular displacements for beam-type sections (IPLATE = 0). IUNI performs linear and quadratic interpolation when there are less than four tabular values and CSIUNI performs a cubic spline interpolation when there are four or more tabular values. Key parameters used are TAB and V.

Subroutine SPLDER.-.Subroutine SPLDER interpolates for Ry' and estimates dRy'/dy' for beam-type sections (IPLATE = 0) used in computing angular displacements (ITYPE = 2). Key parameters used are TAB and V.

Subroutines ZFUN2 and AFUN2.- In subroutines ZFUN2 and AFUN2, spline coefficients are determined for symmetric (ISYM = l), nonsymmetric or antisymmetric (ISYM # 1) plate-type sections (IPLATE = 1). Key parameters used are TAB and W.

Subroutines ZFUN and AWN.- In subroutines ZFUN and AFUN, modal coefficients for linear displacement-type sensors are determined for symmetric (ISYM = 1) and nonsymmetric or antisymmetric (ISYM # 1) plate-type sections (ITYPE = 1 and IPLATE = 1). Key parameters used are TAB, W, XS, and YS.

Subroutines ZDER and ADER.- In subroutines ZDER and ADER, modal coefficients for angular displacement or rotational-type sensors are determined for symmetric (IS\?.? = 1) and nonsymmetric or antisymmetric (Ism # 1) plate-type sections (ITYPE = 2 and IPLATE = 1). Key parameters used are TAB, W, XS, YS, and RO.

Over lay MATINPT

Large data arrays such as the aerodynamic data, generalized masses, generalized stiffnesses, and control system dynamics definitions are input in overlay MATINPT. In addition, the selection feature is available which enables one to choose a mathematical model that is a subset of a previously defined model. Primary subroutines called are MATEXD, CSINPUT, and SELECTS. Key parameters are defined as follows:

= 1 ,2 Read p-plane coefficients. ISPLANE = 0,3 Read aerodynamic coefficients.

49 APPENDIX F

< 0 Read modal data from data storage file TAPE1 (select from previously used data).

> 0 Read modal data as formatted input from file TAPE5 (select from ISELECT 1 nriyLlr*IILIL-4 m - 1 data;. 0 No selection of array data; for initial case, data from file TAPE5 are used; in subsequent cases, array data from preceding case are used for current case.

= 1 Call subroutine CSINPUT to input sensor and control system data.

ICONSYS

(= o NO control system input.

KASEt Number of cases run during current program execution.

= 0 Read sensor data from data storage file TAPEI, record 16 as computed and stored in overlay sensor. ICSREAD = 1 Read sensor data as input from file TAPE5.

1 Interactive run is being made (additional input after initial ization case is requested from keyboard). INTERAC 0 Run is in batch mode (additional input after initialization case must also be on TAPE2.

= 1 Change in sensor dynamics. ICHSEN = 0 No change in sensor dynamics.

= 0 No change in filter dynamics.

= 1 Change in filter numerator. ICHFIL = 2 Change in filter denominator.

= 3 Change in both numerator and denominator.

= 0 No change in actuator dynamics.

= 1 Change in actuator numerator. ICHACT = 2 Change in actuator denominator.

= 3 Change in both numerator and denominator.

NM and Number of modes employed in model before selection and number to be NMODES selected (ISELECT # 0).

tInternally generated parameter.

50 APPENDIX F

NR and Number of rigid-body modes employed in model before selection and NRNEW number to be selected (ISELECT # 0).

NC and Number of controls employed in model before selection and number to be NCNEW selected (ISELECT # 0).

NK and Number of reduced frequencies at which aerodynamic tabular values NKNEW are employed in model before selection and number to be selected (ISELECT # 0). If NKNEW = 0 (default), all will be used and no selection per frequency will be performed.

NOC Array of mode numbers indicating which modes to select (ISELECT # 0).

NOK Array of frequency indices indicating which reduced frequency data to select (ISELECT f 0 and NKNEW f 0).

NS Number of sensors.

Subroutine MATEXD.- .All noncontrol array data are read in subroutine MATEXD and all array data are rewritten to data storage file, TAPE1, in form to be used by other overlays. Primary subroutines called are CSINPUT and SELECTS. Key parameters used are ISPLANE, ISELECT, ICONSYS, KASE, INTERAC, NM, NR, NC, NS, and all arrays defining aeroelastic model and control system.

Subroutine CS1NPUT.- All control system data are read in subroutine CSINPUT, or default values set, including sensor deflection coefficients. Key parameters used are ICSREAD, ICHSEN, ICHFIL, ICHACT, NS, NC, and arrays defining control system.

Subroutine SELECTS.- All modal and frequency selection of data is performed in subroutine SELECTS. Key parameters used are ISPLANE, INTERAC, KASE, NM, NMODES, NR, NRNEW, NC, NCNEW, NK, NKNEW, NOC, NOK, NS, and all arrays defining basic aeroelastic mode 1.

Overlay PPLANE

The p-plane curve approximations to the oscillatory aerodynamic forces are computed in overlay PPLANE corresponding to a specified set of denominator constants ({b,} (eq. (3))). The best coefficients to employ in equation (3) for a given set of oscillatory aerodynamic force data at a number of reduced frequencies are determined in a least-squares sense. The resulting coefficients and the corresponding denominator constants are stored for subsequent use in overlays PKE'LUT and AEROPLT. Pri- mary subroutines called are SET and AERCOEF. Key parameters are defined as follows:

NK Number of reduced frequencies at which aerodynamic data are available.

NKK ( J) Jth element of NM x 1 array indicating that only first NKK reduced frequencies of aerodynamic data are to be employed in least-squares fit for Jth column of aerodynamic forces.

NCOEF Number of coefficients desired in p-plane fit, includinq polynomial and denominator (lag) terms (Max = 10).

NPOLYC (J) The number of Dolvnomial coefficients for column J.

51 APPENDIX F

Integer indicatinq whether to include nth constraint (as defined below) when fitting Jth column of aerodynamic forces. (See appendix A.) It is assumed that kl = 0 and k2 is small.

efault) Nth constraint nct ezplcie3 i11 Jth coiumn ot aerodynamic forces. ICOF(N, J) > 0 Employ Nth constraint.

ICOF (1 ,J) > 2 Force agreement with first tabular value at kl = 0; Q. . (0) = Q. . (0) 13 13 ICOF( 2, J) > 0 For kl = 0, force slope,

ICOF(3, J) > 0 For kl = 0, Force slope = -Re{Qi2(O)}/(ben).

ICOF (4,J) > 0 For kl = 0, Force slope = 0.

ICOF(5,J) = !t + 1 Force AI, = 0. 3 ICOF( 6, J) > 0 Force 4(ko) = Q(k0) for some kor where Qj (kO) is interpolated value of Jth aerodynamic force at kor obtained with a piecewise quadratic fit to Qj(ki), i = 1, ..., "k' BN(I,J) Constant in Ith denominator term of form P/(BN + P) correspondinq to Jth column (pressure mode); if same set of BN is used for several columns they need only be input once. (See NCOL.)

THETAN Normalizing factor employed when applying constraint number 3, ICOF(3,J) = 1; Default = 1.

SPKO Value of reduced frequency for which p-plane approximation mst precisely fit data.

NM Number of modes being employed in model.

NC Number of controls being employed in model.

NOC Array of mode numbers indicating which modes are included in current mode 1.

= N Integer indicating preceding column number, if any, for which BN(1,J) = BN(1,N) (default is 1). NCOL( J) i (= 0 BN terms for Jth column are to be input.

52 APPENDIX F

NBN ( J ) t Number of nonzero BN for column J.

C(IrJ,K) Ith p-plane coefficient for (J,K) element of generalized aerodynamic force matrix.

= 1 Write percent errors for p-plane fit at tabular values to TAPEG. I ERPRT = 0 Do not write.

Subroutine SET.- The number of lag terms is determined for each column of the aerodynamic forces based upon input parameters. The logic in subroutine SET was developed to reduce the input required of the user. ?he parameter NCOL(J) is employed to indicate whether lag terms for column J were read in (NCOL(J) = 0) or are the same as those for column JJ = NCOL(J). Key parameters used are NCOEF, BN, NM, NC, NOC, NCOL, and NBN.

Subroutine AERC0EF.- Subroutine AERCOEF solves for the coefficients that yield the minimum error in the least-squares sense in the p-plane fit to the aerodynamics. A library subroutine MATOPS is employed to solve the set of linear equations which result. (See appendix A. ) Key parameters used are all for overlay PPLANE.

Over lay CONTROL

For overlay CONTROL, a matrix of transfer functions TATATs is constructed. (See eq. (9).) %e ( )i element relates the output of actuator i to the input of sensor j, excluding iG)ij and any scheduling or phase error. The polynomial coefficients for sensor, compensation, and actuator dynamics are constructed separately and then combined and a common denominator is found. mch element of the resulting matrix can then be multiplied by a distinct feedback gain and scheduling component in overlay PKFLUT. Primary subroutines called are CONTEXT, SENDYN, CLOGIC , FILDYN, ACTDYN, PMULT, NCOF, and EQUATEP. Key parameters are defined as follows:

NC Number of controls.

NS Number of sensors.

= 0 Perfect sensor (default). ISDYN ( I ) i = 1 Sensor dynamics included. (= o Position sensor.

IORD(1) = 1 Rate sensor.

= 2 Acceleration sensor.

XKs (J,1) Jth coefficient of numerator polynomial for Ith sensor.

Interna 1ly gene rated parame te r.

53 APPENDIX F

= 1 If J = IORD(1) + 1 and ISDYN(1) = 0.

ASN(J,I)+ = 0 If J # IORD(1) + 1 and ISDYN(1) = 0.

L= x~s

1 If ISDYN(1) = 0 and J = 1.

ASD( J, I 1 = 0 If ISDYN(1) = 0 and J > 1.

Jth coefficient of denominator polynomial for Ith sensor if ISDYN(1) = 1.

ISN( I) and Number of coefficients in numerator and denominator, respectively, ISD( I) of transfer function modelinq Ith sensor dynamics.

= 0 Ith actuator is modeled as perfect actuator. IACT(I) = 1 Actuator dynamics are included for Ith actuator.

AACTN(J,I) and Jth coefficient of numerator and denominator of polynomial, AACTD( J, I respectively, of transfer function used to model Ith actuator.

IAN(1) and Number of coefficients in numerator and denominator of Ith actuator IAD(I) transfer function.

ACLN(K, I,J) and Kth coefficient of numerator and denominator polynomial, respec- ACLD(K, I, J) tively, of transfer function which relates Ith actuator input to Jth sensor output excludinq qain, scheduling, and phase error contributions.

ILN(I,J) and Number of numerator and denominator coefficients in transfer func- ILD( I, J) tion relating Ith actuator input to Jth sensor output.

IFILTERij(K) Indicates which types of filters are to be combined to define overall filter transfer function relatinq actuator i input to sensor j output. Maximum of 12 filters for each control- sensor pair. IFILTER(K) is Kth filter type (sequence must end with 1).

(= 1 NO filtering.

= 2 Notch filter.

= 3 Integral filter. IFILTER(K) = 4 Proportional plus derivative filter.

= 5 Lead-lag (or lag-lead) filter.

= 6 General filter comprised of numerator and denominator PO lynomia Is.

- ___ 'Internally qenerated parameter.

54 I APPENDIX F

WNI(K) and Natural frequency and damping ratio of notch filter if SNI (K) IFILTER(K) = 2.

KIBDO ( K) Gain employed in integral filter if IFILTER(K) = 3.

KIBD1 (K) and Gains employed for proportional plus derivative filters, respec- KIBD2( K) tively, if IFILTER(K1 = 4.

ANI(K) and Time constants of zero and pole in lead-lag (lag-lead) compensator AD1 (K) if IFILTER(K) = 5.

AFN(1,K) and Numerator and denominator coefficients (constant term first) of AFD(1,K) general-type transfer function if IFILTER( K) = 6.

ACTLN(K, I,J) and Kth computed coefficient of numerator and denominator polynomial, ACTLD (K, I ,J) t respectively, of overall transfer function (T~TLT~)~, relating actuator I output to sensor J input, excluding gain, scheduling, and phase error contributions.

ICLN( I, J) and Number of coefficients in numerator and denominator of overall ICLD( I ,J) transfer function (T TIT ) ALsij. DCT+ Array of coefficients in common denominator d of all transfer functions (T T'T ) i.e., common multiple of all ACTLD arrays. A L s ij' NCT Number of coefficients in DCT.

CNTLN(K, I, J) Kth coefficient of polynomial corresponding to product of ACTLN(1,J) and common denominator DCT = (1' TIT ) d. ALsiJ Subroutine CONTEXT. - In subroutine CONTEXT the overall matrix product of the sensor, control logic, and actuator matrices of transfer functions is formed and a c~?mrnondenominator is found. Primary subroutines called are SENDYN, CLOGIC, and ACTDYN. Key parameters used are aii except IFILTER, W?!I, SXI, KIROO; KIBOI, KIBO2, AN1, AD1, AFN, and AFD.

Subroutine SENDYN.- Subroutine SENDYN determines the numerator and denominator polynomial coefficients and order of the transfer functions relating sensor output to sensor input. The numerator coefficients are adjusted for the appropriate power of s to define a position, velocity, or acceleration sensor. Subroutine called is NCOF. Key parameters used are NS, ISDYN, IORD, XKS, ASN, ASD, ISN, and ISD.

Subroutine CLOG1C.- Subroutine CLOGIC sets up the input for calls to FILDYN in order to obtain the numerator and denominator polynomial coefficients which make up the transfer functions relating sensor output to actuator input. Primary subroutine called is FILDYN. Key parameters used are ACLN, ACLD, ILN, and ILD.

Subroutine F1LDYN.- Subroutine FILDYN contains the filter types that may be combined to construct control logic dynamics. The mathematical expressions that represent the various filter types are defined in the section "Control-System Repre- sentation." Any combination of the filter types may be utilized to generate an overall filter. For example, if IFILTER(1) = 3, IFILTER(2) = 6, IFILTER(3) = 6,

~ tInternally generated parameter.

55 APPENDIX F and IFILTER(4) = 1, the overall filter is the product of one integral and two general polynomial filters. When IFILTER(K) = 1 is encountered, no further filter dynamics are computed for the particular control sensor pair under consideration. Therefore, the IFILTER array must end with 1. Subroutines called are NCOF and PMULT. Key parametcrs used are IFILTER, WNL, SNI, KIBDO, KIBD1, KIBD2, ANI, ADI, AFN, and AE'D.

Subroutine ACTDYN.- Subroutine ACTDYN determines the order of the polynomials in the transfer function representing the actuator dynamics relating actuator output to actuator input. Subroutine called is NCOF. Key parameters used are IACT, AACTN, AACTD, IAN, and IAD.

Subroutine NC0F.- NCOF is a function subroutine which returns the number of coe€ficients in a polynomial; that is, it determines the order +1 of a polynomial.

Subroutine PMULT.- The utility subroutine PMULT multiplies two polynomials together. All polynomial arrays are assumed to have constant terms first.

Subroutine EQUATEP.- The utility subroutine EQUATEP sets one polynomial equal to anothe r.

Overlay PKF'LUT

The matrix in equation (1) or (2) is formed by overlay PKFLUT. A user-specified number of characteristic roots are determined as a function of variation in velocity, altitude, density, or feedback gains and the resulting output is stored for possible subsequent plotting (if IPKPLT # 0). (See section "Characteristic Root Determina- tion.") By operatinq in a multicase mode, the effect of variation in other parameters such as actuator or compensator dynamics or sensor location can also be studied. Primary subroutines called are PKEXT, MATIT, MNLOOP, DETIT, FINSCN, AT62, PREDICT, FUN, DEFINE, SCHEDUL, AERO, AEROSP, CXQR, MATINV, CXDEV, IUNI, and CSIUNI. Key parameters are defined as follows:

= 1 Use matrix iteration to obtain initial estimates of roots. IOPT2 = 0 Do not calculate initial roots using matrix iteration.

ITRACE ( I ) Array of integers indicating whether to trace Ith root either from input or using IOPT1 or IOPT2 options. These parameters also indicate type of curve used in extrapolating for Gil and 6i2-

(See section "Determinant Iteration. I' )

= 1 Linear.

= 2 pladratic.

= 3 Cubic.

56 APPENDIX F

KVAR Integer indicating type of variation to be performed. If negative, any corresponding plot output will be with respect to dynamic pressure. Can be reset interactively at plot time of IPKPLT > 0.

= -tl Velocity variation.

= +-2 Altitude variation.

= +3 Density variation.

IPKPLT = 0 No plots.

= +-1 Root loci.

= f2 ClW plots. n

= +_3 Both root loci and Clo plots. n = + Allows.alteration of titles, scales, curve deletion, etc., at plot time.

NV Number of increments in parameter being varied.

= 0 No gain variation. IGAIN = 1 Gain variation.

DV Increment in parameter variation

VO, HO, Initial values of velocity, altitude, and density and RHO0

DVEL+, _DRHO~, Incremental change in velocity, density, and altitude. Only one is set to DV, depending upon KVAR. Others are zero.

Current value of gain on transfer function relating actuator I output to sensor J input.

DELGAIN (I, J) Increment in gain GN(I,J) for gain variation.

VVi H, Current values of velocity, altitude, or density during parameter and DENS variation.

XVC and XV Current value of independent parameter being varied.

A A S1I and S21 Array of initial estimates, sil and si2, of roots to be traced. These need to correspond to initial value of parameter (velocity, altitude, density, or gain) variation. S21, only, is used initially in MATIT if IOPT2 = 1 as an estimate (not required) of roots to be traced. MATIT will return values for both SI1 and S21 in this case. S21 is always assumed to be the better estimate in determinant iteration.

tInternally generated parameter.

57 APPENDIX F

N F+ Number of characteristic roots for which to solve.

ITEST~ Integer indicating whether there is a convergence problem in tracing a specified root.

= i iteration procedure converged.

= 2 Iteration procedure failed to converge.

CONFAC1 Factor for converting input velocity to feet per second for use in subroutine AT6 2.

CONFAC2 Factor for converting input altitude to feet for use in subroutine AT62.

CONFAC3 Factor for converting density output by subroutine AT62 in slugs per cubic foot into units corresponding to input model.

IPH and JPH Control (IPH) and sensor (JPH) pair for which phase error is to be introduced.

PHASE Phase error, in degrees, to be introduced in IPH control, JPH sensor transfer function.

IPRINT Option indicating how often to print characteristic root data to screen (or to file OUTPUT) durincj interactive session; IPRINT = N, prints every Nth iteration. All data are written to file TAPE6, regardless of IPRINT.

XMACH Mach number at which aerodynamic data were generated.

QBAR~ Dynamic pressure corresponding to current velocity and density.

KFIT Type of interpolation to be performed to obtain aerodynamic force at particular frequency when k aerodynamic tabular data are beinq used rather than p-plane aerodynamic approximations.

If KFIT is negative, the interpolation will be for Re(Q(k)) and Q' (k) (see unsteady aerodynamic force subsection)

= fl Linear.

= f2 pladratic.

= +3 Cubic spline.

Plt and P2t Nondimensionalized estimates, Gi and $i2 of characteristic root prior to determinant iteration at each value of independent variable. Upon returninq from determinant iteration, P2 is current converged root, if found, and P1 is preceding converged root.

tInternally generated parameter.

58 APPENDIX F

Pt Value of estimate for nondimensionalized root in MATIT corresponding to root in determinant iteration procedure for which convergence difficulty was incurred. Upon return from MATIT, P is replaced by actual root, if found.

Pll(1)t P22(I) is value of converged Ith root hi for current value of and P22(IIt independent variable, and Pll(1) is value of converged Ith root for preceding value of independent variable. P22( I) is employed to prevent iterations for Ith root from converging upon previously determined root by defining determinant

i=l

PP(I,IN)~ Array of up to four preceding values of converged INth root. (1=1,4) and P11 (IN) is equivalent to PP(4,IN). Array is used for predicting PPF(1) new root estimates P1 and P2.

NORD+ Order of extrapolating curve currently being used to predict estimates P1 and P2. s TEPMX Maximum step increase allowed in amplitude of characteristic root in determinant iteration.

< 0 Indicates change in stability has occurred in current root SIGN^ be i n q trace d. i> 0 No change in stability has occurred. NCHECK~ NF x 1 vector recording successive number of times noncoverqence has occurred in roots being traced. If NCHECK(1N) 2 5, the INth root is no longer traced.

ITRACK+ NF X 1 vector initially iiaV%rig 1 ir, all elements. If NCHECK(1N) > 5, ITRACK(1N) is set to 0, and computations for INth root are bypassed

NFINE Number of subintervals into which increment in independent variable is divided in performing finer scan for particular root. Number of values at which INth root is determined in finer scan. N = NFINE + 1.

= 1 Make precise determination of point where first change in IDEF+ stability occurs.

= 0 Bypass precise determination.

Interna1 ly generated parameter .

59 APPENDIX F

Yt Vector array giving f requency, flutter speed, altitude, density, and damping obtained by interpolating for value of independent variable at which damping in INth mode is zero; i.e., at which change in stability occurred. - -+ 1.1'' and NIT Number af iterations employed in seeking characteristic root, and maximum number allowed.

ICUTt and NCUT If predicted step increases magnitude of determinant, step size is halved up to NCUT times in attempt to improve estimate of characteristic root. ICUT is number of times halving has occurred.

EPS I If either relative or absolute change in estimate for characteristic root is less than EPSI, convergence is assumed.

PROOT~ Estimate of characteristic root at which determinant is being e va luate d.

SCHEDUL Complex number, for each control sensor pair, returned from SCHEDUL function subprogram giving contributions of scheduling component and any phase error.

I DAMP

L.= 3 91 = "1". gs, ii

= 0 Aerodynamic forces are computed as function of k = Im(p).

= 1 Ferodynamic forces are obtained from p-plane approximation by using coefficients previously generated. ISPLAN E = 2 p-plane approximations to unselected set of modes is

determined.

= 3 p-plane approximations to selected subset of modes is determine d.

= 1 Control system is included. ICONSYS = 0 No control system.

= 1 Control system denominator is cleared. IDMULT = 0 Control system denominator polynomial is not cleared.

Inte r na 11 y ge ne rate d pa r ame te r .

60 APPENDIX F

= 1 Aerodynamic and inertial hinge moments are neglected in control rows of equations of motion ICSACT = 0 Control surface is included as full deqree of freedom, and effects of inertial and aerodynamic hinge moments are considered.

Subroutine PKEXT. - In subroutine PKEXT, modal, sensor, aerodynamic, and control system data are read-in from data storage file TAPE1; the parameter variation loop is controlled; and characteristic root data are stored on data storage file TAPE1 for plotting later if IPKPLT f 0. Primary subroutines called are MATIT and MNLOOP. Key parameters used are IOPT2, ITRACE, IPKPLT, ISPLANE, ICONSYS, KVAR, IGAIN, NV, DV, DVEL, DH, DRHO, DELGAIN, S11, S21, NF, and IPRINT.

Subroutines MATIT and MATITX. - Subroutines MATIT and MATITX, together, perform a matrix iteration to determine the characteristic roots. They are used to obtain initial estimates slil and gi of all the roots at the beginning value of the parameter (velocity, altitude, iensity, or gain) being varied if IOPT2 = 1 and/or to aid in obtaining a specific root when the determinant iteration process has convergence difficulties. (See appendix D. ) Primary subroutines called are AERO, AEROSP, SCHEDUL, CXQR, and MATINV. Key parameters used are VV, H, DENS, S21, P, and ITEST.

Subroutine MNLOOP. - In subroutine MNLOOP, the independent variable is incre- mented, initial estimates, of characteristic roots being traced are obtained, and (for each increment) , the characteristic roots are determined using the determinant iteration method, if possible, or the matrix iteration method if the determinant iteration process fails to converge. Primary subroutines called are AT62, FREDICT, DETIT, MATIT, and FINSCN. Key parameters used are ITRACE, KVAR, IPKPLT, IGAIN, DVEL, DH, DRHO, DELGAIN, W, H, DENS, NF, ITEST, P1, P2, P11, P22, PP, NORD, STEPMX, SIGN, NCHECK, ITRACK, NFINE, IDEF, IT, NIT, ICUT, NCUT, XMACH, CONFACl, CONFAC2, CONFAC3, XVC, and IPRINT.

Subroutine DET1T.- Subroutine DETIT performs the actual determinant iteration to obtain hIN, the INth characteristic root corresponding to P2* See section "Determinant iteration. Subroutines called are FUN1 AND FUN. Key parameters used are ITEST, P1, P2, STEPMX, IT, NIT, ICUT, NCUT, EPSI, and PROOT.

Subroutine F1NSCN.- Subroutine FINSCN traces the IN characteristic root for smaller increments in the parameter being varied (velocity, altitude, density, or gain) in the same mnner as MNLOOP in case of a change in stability or convergence difficulties. A new value is returned for P2 = h The first time a change in IN: stability occurs, the precise point where it occurs is obtained. Subroutines called are AT62, DETIT, MATIT, DEFINE, and PREDICT. Key parameters used are KVAR, IGAIN, DEVEL, DH, DRHO, DELGAIN, W, H, DENS, ITEST, PI, P2, PP, PPF, NORD, NFINE, IDEF, XMACH, CONFACl, CONFAC2, CONFAC3, XV, and IPRINT.

Subroutine AT62.- Subroutine AT62 determines the density and speed of sound at a specified altitude. Key parameters used are H*CONFAC2, DENS/CONFAC3, and V"CONFAC1 /XMACH.

Subroutine PREDICT.- Subroutine PREDICT fits a polynomial of order n < 3 to n + 1 points and evaluates the function at a specified value. It is used to extrapolate for Gil and at the current value of the independent parameter (velocity, altitude, densityf20r gain) based upon the values of the corresponding n + 1

61 APPENDIX F converged roots. It is also used to interpolate for the precise point at which a change in stability occurs. Key parameters used are PI, P2, XVC, XV, PP, PPF, and NORD.

Subroutines FUN1 and FUN.- In subroutine FUN1 and FIJN; the fizz1 matrix modei is ferze:! 2nd the vdiue of its determinant is obtained corresponding to the current e s t imate ($INn+l) of the root Primary subroutines called are AERO, AEROSP, 'IN .

CXDW, and SCHEDUL. Key parameters used are VV, DENS, PROOT, SCHEDUL, IDAMP, ISPLANE, ICONSYS, IDMULT, and ICSACT.

Subroutine DEFINE.- Subroutine DEFINE uses PREDICT to interpolate for the precise value of the independent variable (velocity, altitude, density, or gain) at which a change in stability occurs. (See section "Precise determination of point at which change in stability occurs.") Subroutine called is PREDICT. Key parameter used is V.

Subroutine SCHEDUL.- Subroutine SCHEDUL provides for scheduled compensation of the signals from the sensors prior to sending them to the actuators. It also provides for the inclusion of a phase error into the compensation for one control-sensor pair. See section "Compensation Cptions" and appendix B. Key parameters used are PROOT, PHASE, IPH, JPH, XMACH, and QBAR.

Subroutine AERO.- For ISPLANE = 0, aerodynamic forces as a function of k = Im(p) are determined in subroutine AERO. Subroutines called are IUNI and CSIUNI. Key parameters used are IM(PRO0T) and KFIT.

Subroutine AER0SP.- For ISPLANE f 0, the aerodynamic forces are evaluated at p in subroutine AEROSP by using the p-plane approximations generated in overlay PPLANE. Key parameter used is PROOT.

Subroutine CXQR.- The eigenvalues of a complex matrix are determined by subroutine CXQR along with selected eigenvectors.

Subroutine MAT1NV.- Subroutine MATINV determines the inverse of a real matrix.

Subroutine CXDEV.- Subroutine CXDEV computes the determinant of a complex matrix.

Subroutine IUN1.- Subroutine IUNI performs linear or quadratic interpolation needed to obtain the aerodynamic forces at a particular value of k, depending upon KFIT = *1 or 22, respectively.

Subroutine CSIUN1.- Subroutine CSIUNI performs cubic spline interpolation needed to obtain the aerodynamic forces at a particular value of p if WIT = 23.

Over lay AEROPLT

Plots are constructed which show how the oscillatory aerodynamic forces vary with reduced frequency and depict how the p-plane approximation fits the data.

62 APPENDIX F

Primary subroutines called are APLOT, MULTPT, IUNI, and CSIUNI. Key parameters are defined as follows:

0 Plot of particular element of aerodynamic force matrix shows input data and curve defined by interpolating for points between these data.

0 Additional curve showing the p-plane approximation is displayed.

1 Plot of (1,J) aerodynamic force element is made. MODEA ( I ,J) (1 0 (default) no plot is made for (1,J)th element.

KMINt, mAXt Minimum and maximum reduced frequencies for which aerodynamic forces are available. nese set the range of reduced frequencies over which the data are plotted.

I APLT Indicates not only that aerodynamic data are to be plotted but also indicates which type of curves used to fit data are to be plotted.

= 1 Plot interpolated curves, depending upon KFIT, and p-plane approximation if ISPLANE f 0.

= -1 Plot only interpolated curves regardless of ISPLANE.

LWIT Indicates type of interpolation to be performed in obtaining aerodynamic forces at a particular frequency. When KF'IT is negative, kQ'(k) will be plotted for the imaginary part of the aerodynamic force (see the unsteady aerodynamic force subsection).

= k1 Linear.

= *2 madratic.

= k3 Cubic spline.

Subroutine APL0T.- KMIN and KMAX are determined in subroutine APLOT. Aero- dynamic forces at 50 intermediate values of k between WIN and KMAX are also determined by using interpolation and PPLANE approximating functions (if ISPLANE f 0 and IAPLT > 0). Plot routine MULTPT is called. Subroutines called are IUNI, CSIUNI, and MULTPT. Key parameters used are all those defined for overlay AEROPLT.

Subroutine MULTPT.- Subroutine MULTPT actually calls the plot labeling routines, line plot routines, axes routines, etc., and creates the plot vector file which defines the actual plots. mese plot vector files can either be sent to the screen immediately for interactive plotting or simply saved and plotted after execution via any plotting device available. Subroutines called are library plotting routines for generating axes, drawing lines, labeling, etc.

tInternally generated parameter.

63 APPENDIX F

Subroutine 1UNI.- Subroutine IUNI performs linear or quadratic interpolation for the aerodynamic forces.

Subroutine CSIUN1.- Subroutine CSIUNI performs cubic spline interpolation for the aerodynamic forces if KFIT = 3.

Over lay PKPLOT

Plots are constructed which portray the stability characteristics of the aircraft by using data generated in overlay PKFLUT if IPKPLT f 0 during its execution. All available sets of data are listed if IPKPLT > 0. Also in overlay PKPLOT, the option of removinq unwanted sets of characteristic root data is available. Pri- mary subroutines called are PKPLT, PLOT, RLPLT, SCALE, and DELPLOT. Key parameters are defined as follows:

I NTERAC+ Indicates whether PKPLOT is beinq executed in interative mode. INTERAC is true if ICASE > 0.

.T. All input is from file INPUT and output is to file OUTPUT. These are usually assigned to the keyboard and screen, respectively.

.F. All input is from file TAPE2, and output except plots is to file TAPE6.

I PKPLT Inteqer indicatinq which type of characteristic root plots to qeneratc? and whether modifications to labels, scalinq, and other plot parameters are to be allowed on a plot-per-plot basis.

>O Modifications are allowed.

= +I Generate plots of root loci versus velocity, altitude, or density as determined by KVAR = 1, 2, or 3, respectively, or versus correspondinq dynamic pressure if KVAR < 0, or versus qain if IGAIN = 1 durinq root determination.

= +_2 Generate plots of c and w versus velocity, altitude, or n density dependinq upon KVAR during root determination (or corresponding to dynamic pressure if KVAR < 0) or versus qain if IGAIN = 1 durinq root determination.

= 23 Generate both (c,w ) and root loci plots. n

= 1 Delete unwanted characteristic root data from data storaqe IDELPLT file TAPE1.

= 0 Do not delete any data.

'Internally qenerated parameter.

64 APPENDIX F

ICOUNT Number of sets of characteristic root data to be plotted concurrently out of total number of sets of data generated during different cases.

IPLT and Array of numbers indicating which sets of characteristic root data IPLT2 are to be plotted or saved, respectively.

I COMB1 N Integer indicating whether to combine plots of different sets of characteristic root data into one plot (all on same page) or to generate plots for separate pages.

= 1 Combine plots.

= 0 Do not combine plots.

PTLTLE 80-character description of data being plotted. When plots are being combined, this will appear on plot being generated in lieu of the original plot titles.

IDASH Indicates type of line used to connect points in data currently being plotted. This may be changed for each set of data being plotted.

=o (solid line)

I SAME Integer indicating how many of next sets of characteristic root data are to be plotted by using same PLOTP namelist data as currently being read-in, including current set. (Default = 1). Program will not request another PLOTP namelist to be input until n = ISAME sets of data have been plotted.

KVAR Sign of KVAR indicates whether data are to be plotted versus velocity, altitude, or density (KVAR > 0) or versus dynamic pressure (KVAR < 0).

65 APPENDIX F

~ MODEPK( I) Integer indicating whether to generate a plot showing Ith characteristic root variation.

= 1 Generate plot of data (Default)

= U Bypass Ith root information.

XRLORG, XRLEND, Lower limit, upper limit, increment between major tic marks, and XRLDEL, and number of minor divisions between major tic marks for abscissa NDXRL in root-locus plots. (Optional)

YRLORG, YRLEND, Lower limit, upper limit, increment between major tic marks, and Y RLD EL, and number of minor divisions between major tic marks for ordinate NDYRL in root-locus plots. (Optional)

VORG, VEND, Lower limit, upper limit, increment between major tic marks, and VDEL, and NDV number of minor divisions between major tic marks for abscissa in and w plots. (Optional) n FORG, FEND, Lower limit, upper limit, increment between major tic marks, and FDEL, and NDF number of minor divisions between major tic marks for ordinate in w plots. (Optional) n GORG, GEND, Lower limit, upper limit, increment between major tic marks, and I GDEL, and NDG number of minor divisions between major tic marks for ordinate in plots.

I IOP Integer indicating whether to continue plotting. This option allows user to generate additional plots with characteristic root data that either were not included before or for which changes in plots are desired.

= 1 Continue plotting.

= 0 Return to main program and continue with new case or terminate execution.

NFR+ Total number of roots traced in current data set being plotted.

NVRt Total number of increments in independent variable for current data set

IDYNAM~ Indicates whether to convert to dynamic pressure rather than plot original independent variable.

= 1 Convert to dynamic pressure.

= 0 Do not convert.

___~

'In terna1 ly gene rated parame te r.

I 66 APPENDIX F

Subroutine PKPLT.- Subroutine PKPLT actually reads in the plot data, converts the independent variable to dynamic pressure, if desired, and calls plott ing rou ti ne s to generate desired plots. Subroutines called are PLOT and RLPLT. Key parameters used are IPKPLT, NFR, NVR, MODEPK, and IDYNAM.

Subroutine PLOT - Entry points PLOT1 and PLOT2.- Subroutine PLOT generates the plots of C and on versus the corresponding independent variable desired (velocity, altitude, density, dynamic pressure, or gain). Subroutines called are SCALE and library plotting routines for generating axes, drawing lines , labeling, etc. Key parameters used are ICOMBIN, IDASH, MODEPK, NFR, NVR, VORD, VEND, VDEL, NDV, FORG, FEND, FDEL, NDF, GORG, GEND, GDEL, and NDG.

Subroutine RLPLT. - Subroutine RLPLT generates the root locus plot. Initial points are identified with the mode number, Every 10th point is identified with a t for nongain variation plots. Gain points are identified on the gain variation plots, which are 0, .5, l., and 2. times the nominal gain incorporated by input into . Subroutines called are IUNI, SCALE, and library plotting routines for generating axes, drawing lines, labeling, etc. Key parameters used are ICOMBIN, IDASH, MODEPK, NFR, NVR, XRLORG, XRLEND, XRLDEL, NDXRL, YRLDRG, YRLEND, YRLDEL, and NDYRL.

Subroutine SCALE.- Subroutine SCALE determines an appropriate power of 10 to be used for tic marks on axes which are 2., 4., 5., or 10. in. in length and returns the corresponding scale factor required to scale the data.

Subroutine DELPL0T.- Subroutine DELPLOT deletes unwanted sets of characteristic root data, generated by using overlay PKFLUT, from data storage file, TAPEl. Key parameter used is IPLT2.

67 APPENDIX G

INPUTS FOR SAMPLE CASES

Tne 'TWE2 inputs required to generate the plots shown in fiqures 8 to 16 are presented in this appendix. These inputs exercise the major options of STABCAR. They are presented to serve as a guide for users to follow in preparing input data. Reference should be made to table I1 for the definitions of the input variables.

Cases 1 to 4 (Figs. 8 to 11)

?he inputs for sample cases 1 to 4 (fiqs. 8 to 11) are presented as a unit in table GI. These cases are run in a batch mode (ICASE < 0). Case 1, which qenerates the data for fiqure 8, has ICASE = -2. This means that TAPE5 (table 111) mst be local in order to make the airplane generalized aerodynamic forces available. In cases 2 to 4, ICASE = -1 means that a batch mode of operation is in effect and all data must be obtained from TAPE1 (table V), which was generated in the previous case and from TAPE2. Note that in case 3 the array S21 ) has been set to zero. This (G12 was done because, with IPS = 0 (the default), S21 contains the roots found at the end of the last case havinq IFLUT = 1. That set of roots is for h = 0 (high dynamic pressure), whereas case 3 beqins at a low dynamic pressure. Setting S21 to zero restarts the estimation process; thereby , converqence difficulties are avoided that would have occurred since S21 is used in startinq the matrix iteration (IOPT2 = 1 search for initial estimates. (See appendix D.) These cases could have been run interdctively with the plots being returned durinq job execution by making ICASE > 0 and loading LIBFTEK and Techtronix interface plotting routines, which support interactive qraphics, or their equivalent.

Case 5 (Fig. 12)

Tables GII, GIII, and GIV contain the inputs required to generate f iqure 12.

The modal deflections at two accelerometer locations are needed to implement the flutter control law. TAPE10 (table IV) which contains the mode shape data, TAPE5, and TAPE2 are made local. For clarity a new TAPE2 is generated; it is shown in table GII. Alternatively, one could have modified the initial TAPE2 correspondinq to case 1.

The modal deflections determined in this run are differenced and this difference is read onto TAPES. None of these operations are shown. Next a run is made which defines the control law. The inputs are shown in table GIII.

At this point the user saves TAPE1 which now contains the control-law dynamics and selected sensor data. This TAPE1 will be employed not only to complete case 5 but also as part of the input for cases 6, 8, and 9. Table GIV presents TAPE2 input required to complete case 5. 'Ihese runs could have been done interactively by setting ICASE = 1. Note that ISCHEDUL (3,l) = 1 which means that the scheduled control law factor defined in appendix B and figure 7 will be employed.

68 APPENDIX G

Case 6 (Fig. 13)

The input required to qenerate fiqure 13 is now shown. The TAPE1 employed is as it existed prior to making the last two runs in case 5. Thus, the aircraft is modeled including two rigid-body, six elastic, and one control mode, the corresponding sensor deflections, and the definition of the flutter suppression system control law. ICASE is set to +1 in any convenient TAPE2. For ICASE = +1 none of the rest of TAPE2 is employed. The values of all parameters are contained on the TAPE1 which must be local. The interactive session then proceeds as shown in table GV.

Case 7 (Fig. 14)

The interative session during which figure 14 was generated is shown in table GVI. This figure shows the effect of phase errors in the control system upon the elastic mode gain root loci. The TAPE1 created in case 6 is employed as the primary source of input. Three runs are made. The first two of these runs determine stability characteristics by using the p-p method after phase errors have been introduced. 'Ihe third run combines the root loci plots from these two runs with phase errors with the correspondinq plot from case 6 which had no phase errors.

Case 8 (Fig. 15)

Fiqure 15 presents gain root loci for the control-law poles (see fiq. 7) and one of the open-loop actuator poles (s = -401 rad/sec). The loci are shown at the open- loop flutter point for NMa = 0.86. mble GVII shows the interactive session which created the figure. The TAPE1 saved prior to running the last two segments of case 5 is the primary source of input. Modifications are made with the TAPE2 shown at the top of the table.

The initial segment of this session traces one elastic root, mode 8, one actuator root, mode 13, and three control-law roots from zero gain to a qain of 0.07. !&e scheduled pole is not traced because this denominator is not cleared and, for a gain of zero, would result in division by zero in equation (2). Xate that 1QMrJL.T = 1 in TAPE2 which means that the nonscheduled control-law denominator, actuator denominator, and sensor denominators are cleared in equation (2). The card image containing the three zeros tells the program, when IPKPLT = 1, that no plots are to be generated in this initial batch (ICASE = -1) run. This first segment is generated knming in advance that a series of runs will be required with the resulting plots combined into one composite plot. The mode number identifiers (fig. 15) are only placed on the first plot set. Consequently, one plot set - they can be plotted in any order - should contain all the modes starting from the initial value of the independent variable. The +1 ending TAPE2 makes ICASE > 0 so that the interactive mode of operation is invoked.

A series of interactive runs are made to generate the qain loci by using the p-k method. Note the first interactive run where IPS = 1. This value for IPS means that the initial estimates at the start of the last execution with IFLUT = 1 will be employed as initial estimates in the current run. Note also the runs where root 12 is being determined as a function of gain. This is an illustration of the case where two real roots combine to form a complex pair. The next to last run is a batch run where roots 11 and 14 are found with the p-p method. Finally, the last case combines the plot sets to form figure 15.

69 APPENDIX G

Case 9 (Fig. 16)

The inputs required to generate figure 16 are shown in table GVIII. Figure 16 is an open-loop locus of the short period root with altitude. Three interactive runs are made. The TAPE1 generated in case 2 is made Inca1 2nd ICASE is set to +i in a I'WLL. 'lhe first two runs obtain the root locus €or the p-k and the p-p method. The third run combines the two plot sets to produce figure 16.

70 APPENDIX G

TABLE GI.- INPUT FOR SAMPLE CASES 1 TO 4

-2 CASE 1. P-PLANE FTT AND AERODYNAMIC FORCES PLOTS. (FIG 8) $INPUT W=16 ,NR=2 ,NC=4 ,ICSACT=l ,NK=12 ,KFTT=2, TSPLANE=3,NCOEF=7 ,NKK( 1)=2*5,TCOF( l,l)=O,O, 1 ,O,l ,JCOF(1,2)=1,1,

NC0L(1)=0,BN(1.1)=0.0939.0.188.0.2816.0.376. ' CSURFEF( 3j= 1. ,TSELECP 1, f APLP1 ,C= .S96138, MODEA( 3,3)=1,MODEA( 5,3)=1 ,MODEA( 8,3)=1, MODEA( 3 ,!I)= 1,MODEA( 5,5)=1,?ZODEA( 8,5)= 1, MODEA(3,8)=1,MODEA(5,8)=l,MODEA(8,8)=1, MODEA(3,15)=1 ,MODEA( S,15)=1 ,MODEA(8,15)=1$ $SELECT NMODES=9,NRNEW=2,NCNEW=l,NOC(l)=1,2,3,4,5,6,7,8,15$ -1 CASE 2. OPEN LOOP STABJLITY CHARACTERISTICS VERSUS ALTITUDE $INPUT TAPLT=O,ISPLANE=O,IFLUT=1,NV=60,ICSACT=1, TSELECP-1, DV=O. 5 ,HO=30. ,KVAR=2 ,EPSI= .E-b, XMACH=O .86 ,CONFAC1=3.28084 ,CONFAC2=3280.84 ,CONFAC3 515.3787, NIP26 ,NCUP2 5 ,NFINE=4 ,I DAMP= 2 TOPTZ=l ,IPRINT=S ,IPKPLT=-3 ,MODEPK( )=8*1 , VORG=O . ,VEND=30. ,VDEb 10 ,NDV=10 FORGO ,FEND=60 ,FDEL=20 ,NDF=4 GORG=-1. ,GEND=l .O ,GDELiO. 5 ,NDG5 XRLORG=-300. ,XRLEND=100. ,XRLDEL= 100. ,N DXRL= 5 YRLORGO. ,YRLENP400. ,YRLDEP100. ,NDYRL=4 ITRACE( 1)=8*1 $ $SELECT NMODES-8 ,NRNEW=2 ,NCNEW=O ,NOC( 1)=1,2,3,4,5 ,b ,7,8$ -1 CASE 3. OPEN LOOP CHAMCTERISTIC ROOTS VERSUS DENSITY,DV=0.025 (FIG 10) $INPUT NV=28 ,DV=0.025 ,RH00=0.05 ,V0=269.64 , S21=25*(0,0), KVAR= 3 ,VORGO.O ,VEND=0.80 ,VDEb0.20 , NDV= 10 ,GORG-0.2 ,GEND=O.6 ,GDEL=0.2 , NDG10 ,XRLORG=-120,XRLENB20,XRLDEI;-20 , NDXRL=4 $ -1 CASE 4. OPEN LOOP STABILITY CHARACTERISTICS VERSUS DYNAMIC PRESSLIRE. (FTC, 11) SINPUT JFiiiT=O,IC'iAR=-? ,Vf2RC=n, ,VFND=3000P.. VDFL=5000., NDV=10 $ 0

TABLE GI1.- MODAL DEFLECTIONS AT ACCELEROMETER LOCATIONS

-2 CASE 5A. MODAL DEFLECTIONS AT ACCELEROMETER LOCATIONS $INPUT NM=16 ,NR=2 ,NC=4 ,NS=Z,IFLUT=O ,IPKPLT=O, ISENSB1, INTEW- 1,Ism= 1 ,NPMX=bO, TPLATE( 1)= 1, NSECTNS=3 ,NNODES=100 ,ISS( 1,1)=1,60,6 1,92,93,100, XCG=6.71576 ,XO ( 1)=6.4470 ,O. 0,9.1328,YO (3)=.25 15, R0(1)=1. 7799,l. 570796,. 5175 $ SSENLOC XS(l)=7.1438,7.3363,YS(1)=2.083,2.134, ITYPE(l)=l,l,NSS(l)=l,l $ 0

71 APPENDIX G

TABLE GII1.- SELECTION OF SENSOR DATA AND DEFINITION OF CONTROL LAW

-2 CASE 5K. SELECTTON OF SENSOR DATA, AND DEFINTTTON OF CONTKOL LAW. $INPUT N?I=16,NR=2,NC=4,TCSACT=l,NK=12,KFTT=2,NS=I, ~SPLANE=3,NCOEF=7,NKK(1)=2*5,TCOF(1,1)=O,0,1,0,1,TCOF(1,2)=1,I, KCOL(1)=0,BN(1,1)=0.0939,0.188,~.28l6,0.376, CSURFEF(3)=l.,TSELECTSl,TAPLFO,C=.596138, bIODEh( 3,3)=1 ,MODE!\( 5,3)= 1 ,MODEA(X ,3)=l, YODEA( 3,5)= 1 ,MODEA( 5,5)= 1 ,MODEA( 8,5)=1, MOI)EA( 3 ,a)=1 ,MODEA( 5,8)= 1 ,MOL)EA( 8,8)= 1, MODEA( 3,15)= 1 ,MOI)EA( 5,15)= 1 ,NODELI( 8,15)= 1, DV=0.5 ,NV=1 ,H0=4.572 ,KVAR= 2 ,EP s T= 1 .E-6 , XMACH=O .86 ,CONFAC1=3.28084 ,CONFAC2=3280.84 ,CONFAC3=515.3787, NTT=26 ,NCUT=2S ,NFTNE=4 ,T DA?IP=2 TOPT2= 1 ,I PRI NP5 ,T PKPLP-3 ,MOI)EPK( 1)=8* 1 , VOKGO. ,VEND=30. ,V DEL= 10 ,~~v=in FOKGO ,F ENB6 0 ,F DEL= 2 0 ,NDF=4 GOKC=- 1. ,GEND= 1 .0 ,GI)EL= (I. 5 ,NDG=5 XRLORG-300. ,XRLENB 100. ,XKLDElr 100. ,NDXKL= 5 YKLORG-0. ,Y KLEND=4UO. ,Y KLDEL= 1 00. ,NDYRL=4 , TTKACE( 1 )=8* 1 , ICONSYS=I ,GN(3,1)=0. , TSCHDUL(3,1)=1, ISENSE=O ,TFLUTSO ,IPKPLT=O ,TNTEKP=U , ISYM=1 ,NPMX=hO ,TCSKEABl, NSECTNS=3 ,NNODES= 100 , I SS( 1 ,I )= 1 ,hO, 6 1,92,93,100, TPLATE( 1 )=I , XCG=6.7157h,XO(l)=h.4470,0.0,9.1328, YO (3)=0.2515 ,KO( I)=1.7799,1.570796,. 5 175 S SSENLOC XS( 1)=7.1438,7.3363,YS( 1)=2.083,2.134, TTYPE(I)=I,I ,NSS(I)=I,I $ SSENINP TSI)YN=O ,IOKI)(I)=Z s 31 $ F T LTT N 1F I LTEK= 6 ,h ,6 ,h , I , AVN( 1,1)=25O. ,AE'I)( 1,1)=25O. ,I., RFN( 1,2)=2.07965,220. ,I. ,AFl)( 1,2)=2.704E5,400.,1., AFN(1,3)=7.841hE4,40.,1. ,AFU(1,3)=7.8612E4,100.,1., AFN( 2,4)=-1.1554 ,AFI)( 1,4)=2., I. $ n o SACTINP TACT(3)=1,AACTN(1,7)=6.583E14, MCTD( 1,3)=6.583E14 ,2.34HE12 ,2.9939E9,3.607Eh, 1.7373B3, I. $ $SELECT NMODES=9,NRNEW=2 ,NCNEW=l ,NOC( 1)=1,2,3,4,5,6,7,8,15$ 0

TABLE G1V.- INPUT FOR SAMPLE CASE 5

-1 CASE 5C. OPEN LOOP STARTLITY VERSUS VELOCTTY ( REDUCED ORDER MODEL, FTGUKE 12A) $INPUT TSELECF-1 ,TCSKERI)=O,KVAR=l ,IFLUT=I ,ISPLANE=O, KHOO=U. 6 10 1, VO= 50. ,NV=7 5, DV=4, TAPLF0 ,TPS=O ,SLT=2 5" (0,O) , TPKPLTz-3 ,VOKGO. ,VENU=350,VI)EL=50., NDV=10 ,COKCF-I . ,GENE= 1 .O ,GI)EI.=O. 5, NDG5 $ $SELECT NMOL)ES=4 ,NKNEW=O ,NCNKW=l ,NOC( 1)=3,5,8,15 $ -1 CLOSEI) LOOP STAHILTTY VS VELOCITY (KEI)UCEI) OKDEK MODEL, FIGURE 128) $INPUT GN( 3,l )=I. ,V0=50., I PS=O, S21=25* (0,O) $ 0 APPENDIX G

0 r.1 0 li m o) 0 z H Q L3 ++++++

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73 APPENDIX G

r?- +0 LL m m 0. ri 0. n P - ci h M M ri 4 r w 0 H 0 4 H F LL LL t- 0 .e .e ij in + m 0 H w E H I UJ ii) c I I- % -0 I I- w _I 0 c I: .€ 4 t- z cr900.n ri Q u .€ 5-0.0-MM7 4 I M P rlio n P m u if ri ri M 5- ci P il w ....* H - t- I Iil I t- ill t- b3 m ci 3 0 m H 0 1 + 3 if LL ijw 0 W zo. -J t- % L QO -I o iTI I r\ 0 % Q ii -4 LL ! io if b7ti om-is 3 Q z 0005- u L7 I 0 + + + r.1 W H www * w ri 0 t- t- t- 0 b7 0 GI > t- Q 0 0 n ci o m 0 t- ic 0 0 mso 4 t- -4 O w if x odri~ i - t- 4hr.O P Y a H w v sr H i Q MX io H c c2 0- Q U in i nic t- 0 ri t- m 1 ii t- OQ z u 0 .-+x 3 t- 0 0 10 ic mi ** a k HlSl La if 0 z H i H hfi- z ii Q ow- WH 0 -z ic w PHE t-0 t- w YEW Q z k iT La LL 0 Q w 3 r LL x _It- w z icwt- i Lu wij0 t- 13 >z b- om t- 0 IC * + IOt- I-a0 I 0x0 m zic H iUJ z 3 JQO LJ* H HX x-0 2 .=at-w u W I i H r\i 3w u wz U10 i 0 HH- .€ t- 9- I.- U M L-

74 APPENDIX G

0 M I3 M m 0

.* W E H I-

m 4 \ m 0 \ .. ri M c. m r 4 W W x XI l3 H e- H W H b- W I- J LL I- I- m C m H Gl H Gl H I- 0 I I I I- I I- I- m I- - I 0 + W I- c.1 h I- i 0 E Q n IT rl C li ww -i J b7 CI\ Eij . /I U u m 0 !A7 0) 00 E -I I 0 P a. 0 xE w ic ic ! u H 11 QP H I X w I- I I- I1 I- uu I- ci I- - m Lox ln HH z~ I- b-J H .0 - H I-I- w 3 r.1 ic m 1.0 ic mm z 0 i w 0 W im W CQ H t7 LL I- I1 I- JJ m I/ O XI m It 0 ww XI -1 L > 4 v UT Q 0 w u m ic Hv iL C G t- C h7 I > mci I 0 H> 0 u C.1 i- iL L >- C LUI- .> QQI-ic u L! 0 I- +Q I- m uuI-w J I- H H u H w ox u .. J Ll QH J G JI- L 0 H 0 icu H 0 UW G 9- v iL J QO El x IU W Q IJ 4 w3 z h-l ci H I- I- 0 !- # H 00 0 m O I- m Gla -+E 0 0 >o oz E 4 J LL ic I-0 LL I-C 0 0- 0 HZ 0 O 41 I- z J m *+ o Ln H Hz cn mC LnO I- C ZH I- z HE HC 0 D *QQ 0 0 H J -It- D J H - WZ u W I1 Ln ti I- 0- JH a I-w C -ic CQ w 0 ALL G W H L!-J Q Ha I- x uoo f- ic -> c 4 LZ Q C >L W ic Ll iLz iL > Q1 WGl r. W H WOO W J JO W z z LL =\.a HZ J W W 0 wu4 L I- a 0 > I- I/ U H W 001 I-H + 0 0 u Jm OI- I- I- ?- XiLC JLn 3 I- I-\ci LQ W I I an J z Ln m zz I-W H H t- WHL W z z 3 B W AI- 3 w* H Ln wo z iT.0 W 3 3 GlliJ H ic WE 0 0 I- J>-u I- 3w u4 2- > O w+.4 z urn >-Ln 0 --~i H 00 Q I- 0 ou .. . * L. LL. u 0 C-. ci r-. E r-. r r- P r f- C- c-

75 APPENDIX G

n ri

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APPENDIX G

3 11 Lo z H Q a ic 0 LL Lo if 0 I- Q z H x P. 0 w z x w H ci t- a Lo z c.1 rn H HLo z I _I H I- 3 c4 I- iL W Q I i a a LI 0) a 0 m =I l 0 - c a H u3 I- r.1 Lo 0 H ic z W >- H I- sr Q o a Q L!3 K 0 Q c;i I- I U 0 (3 I- I- I- z >- 0 0 H I- O 0 0 m H v if ic z _I H 113 w W P sr P _I J LO Q I 14 sr w I- I/ QI Q ic 0 Lu c I- iL Lo ffl Lo 0 LL Q z 0 0 5.-I 3 2 Lo Lo I- I- H 0 0 u 0 H 4 0 ic wo U _I 10 W +_I w I- z-1 H I- 0 HLo +I- Q 0 XI O K if wo wif w _I= z I- H W w LLz a m_I a Q I- 0 OH OW I- I-+ _I= H LO I 2-z XI Lo XH I-w H I- z 3 w. I- KP 3 O Ed 0 z 3 >- J Urn 0 0 !L H 0 lL- fi LL p 0

77 APPENDIX G

b3 b7 L3 ri I r.1 ri r.4 0 u 000 * - ..* > a i>>a 2 aJjm b3 t- b3 L7 b3 - Q f f *Ti ri r.i r.1 H 0 0 000b 0 i+I- i 0 000- i vvv 0 b3 0 HHb7P K UtiP I 0 0 I/ !I z P. JJWW H W lnm Q I: ++Qk H w 00x1 i OOLL + icK Lo H HH U t- zzou I HUOO t- I- Q-€-IJ Lo Lo Lo u 0 I- + + E: + O 0 0 0 Q -I J J L Li u u Lo ti K ic x w iu w H i i t- + Lo + + t- n 3 3 3 K -J 1 J W LL LL LL w I- I2 > o 'L z 'L m a Q u a u 4 ic i3 b3 Q I U u I- > w n Lo 0 + \ H w J a z d U 0 u - a E w Q b7 a li) c t- * 00 OU Lo dX -+z it 4 dH W LL 0- o+ 0 41 dl- Lo 0 Q Lo LoQ LoJ -- I Lo HU I- HG mu z H H 0 0 -w H J H - wv u + 0- 03 JO Q -x -z b7 Q b7 H I-A w H -I- H i ic -> t-t- Q Q >z >z -€I do O K > JU wo c W x -IK W z LL wo H J w 0 >I- LLZ I- o l2 H w O + I- 0 u I ow + >- IOY JG 3 i iH uo W I g3 x z Lo W I-ld U t- w3 z z 3 w w JO 3 H Lo >. z w- wic H icP w 3 uic a ic+ WX 0 I- ACIMP>O JO t- 3 UQ >- 0 I-o wa z 00 >-ln dJ U uo 00 LL- LL. u H t oli .. LL. c. LL iir c. ~ ~~

APPENDIX G

.-.-?.,-._,-, mmmrimm OrlOrlOC~ +I+! I+ wwwwww m~.omb-!o 0.0.+P)P0 9mmm0.0 n0.oqII-Jo hmOoP0 bim~-oom ri-rrt+mw...... t- I 0 0 03 P ...... P ri iri m m m 000000 W 0 ++++++ II) U? wwwwww A 0 riooo.oo u ri m -iOOOPO 4 z --_-,-.A mooooo U mmm m mooo--ro Q 000 0 0.000--10 -. (3 +I+ + -+ooLi)oo w www w P r.1 b? ri P ri x: ic 000 0 ...... H 0 000 0 IIIIII I- LL 000 0 000 0 co moo o vvv-vv a 4 E no9 0 0 ..- 0 0 moo. m -- \ t- ri .-, A P o r.1 0. Q ...... mm 0 z 00 PP \ H z r.l o ri ci ri r.1 -.. ri II Q++++++l000000 00 m 0 00 z awwwwww r.1 c.i W ...... w h0.m ~mn iI .. is1 ri rl r.1 m m m x: o. o. m I\ n II-J vv W 000000 o~omrnno I1 II t- m L? ++++++ ’~mnrnma -- H z tl uwwwww b-~m ri p\ m I\ rl PP 2 ffl Hffl 000000 .+ A rt o. m r.1 ,I irl >. J 000000 ~mmmam t- VV I J 000000 ...... 3 HH I 4 2 000000 J r.i A z W 0.D.0000 LL fflffl ic Q I 0.0‘0000 Y .... Q U rlD.OL?OO -- U K ffl P rl h-J r.1 P r.1 0 9b-J - YI -w ...... I1 0.0.- Q t- l!lIl/ 040 ,++o w drim t- t- h? 000 000 z ..-. rn 3 0 Q +++ +++ Q O0.h J + t- www www b? P b-J I LL PW z W mom00- c I I r? I z.c H N rlom 0 0 Li) ffl vv .. L s r! 4 i. 13 ii .=2-+ H I1 It 0. m I 0. L3 P. 0 =c OC-0 0 --0. 0 ii -PI 00- 00- rl I1 -4rl - H om44 0 ,-..-.-*-- 0 9 0 b-! oom 11 m 4.4- t- 0 0 or1 I- Plmm m rl 4 r.1 ,+4m I- t- --P ffl +..... & ir, 000 0 ...... J u HH I H www. fi +I+ 4- m 3 0 r.1 A - 17 ‘OM0 z www w H H ffl ffl I1 W PrlO H 000 0 .. -- t- 0 P Li) cd 000 0 zH --a u z 000 0 rl . .v QL c 000 0 oou KL L r.1 o o o w did.+rid - - r.i Q ffl ~i)o hi r.1 E 000 000 omm Id rl w LDOO. m 3 +++ +++ .+PI -0 00. - 00 m ic r.ir.irl P xi- WWW WWW 0 *dl1 Pm- m K ...... OH 000 000 ‘0 -- I10 >> b-J rl 0 00 mt- 000 000 >- II ,+d vv - t-m 0 rl 0 z-i b-J b-J b3 b-J II-J bl ulo .I /I 11 r.1 H 4 b-J WQPPP PPP - - b-J JPI bl m ffl 000 000 m M r.1 -0 ffl 0 i- nl\PI PInn 44 I a. H ,+ 0 ...... ‘P 999 999 Q 0 r.1 rl ri M m m z 1.1 I.. t-0 - 0 HuFr ic 000000 Q -.rn .c n- I- innus LLO P 0 v I >- ZU Q Offl J H ic I-i W 40 2 zico 0 HI-9 ~zrn I xu f t- W (3 t-+ z

79 APPENDIX G

...... 9 ci 3 ri M P) M r.i 000000 ++++++ ci UWWUUW b3 0 mb-l.oO.oI\ , ci -r o ri ri L> ri m riu. qLiiw D. rl z hO.MOP0 H MM mu.MoIl30 Q 00 QD.rlr\MO I. W I+ P.+Q3PCI W uw ...... XI ic 00 IIIIII H 0 00 I- LL 00 00 10 00 i ic 00 0 0 om a \ + -4- al 0. Q ... 3 0 z 0 c \ n -4 ri r m 0 [L c, z W C w -..- XI 0 .. G 3MM H U Li: 000 t- t- 0 +++ -5 z ri wwu Lo u nm 000 H ! i 000 I H I 3 000 I- ii 000 w 0-00 W + iL I 0.00 J Q 5 c 0 0.00 I- 0 L5 q r.1 H L1 w si ... I- i) l-2 0.+0--0 Q Ill H m ri 00 o o o o I- 3 t- a: <3 + + + + + + W io E.l wwwwww z H ill Li7 0 M 0 0 9 z ic oMoQOO3 H w c o.o~\oom Q I- c~-ohOom o o . rio .o o o L Q - I 0 iL , hoooom 34m3-M U .. , , I- -€ ...... - mil I :r OH 13 u - Ill oic z r 1 3 w b-l H- t. 0 11 I- ri u t- > 00 z H z e- J rl443.43 >- z H 000000 si cn m ++++++ W 6 Y w w w w w w b3 iT I- L7000000 ... in000000 rd 0 0 Lil -- z Lil bl Li3 b3 lJ3 hl 0 .L LL WPPqqPw 0 ri o 0 u?000000 I- Lo I\I\I\bI\h I- m ;i.O9.099.0 0 0 I- z...... 0 -imM 0 6 I ic 000 0 - 0 +++ u 2 ffl W UWW..I XI z- ooolnu Q W ri -Lo 000clm Ln t- 4 XI 000 Q ic ooowri -. Y i w OOOJ4 W t- mmoI- ? z W 0.0.0H4 >- W iiJ +mrit-~ I- 13 c ... 0- IIII-~ cc 0 I-I- 00I- H At- +z UU z I EH WE m t- ---+u 3 .-+ zx I- 3 t- ww W z 0 icic 3 H z W0.MPicW 0 -z ii .-ti30 I- > 0 0 0 UQ 0 E S icx- 0 0 I- L- i; (L

80 APPENDIX G

0. iil n it7 0 I/ c.1 m vi z H MM Q 00 .. U ++ w wu x: ic 00 H 0 00 I- LL 00 mo m h0 + ic Ill 9 a 0 0 b-J 0. Q) \ I- r.1 -r 3 0. Q f. 0 z d c.1\ H .r( x: JJ m 0 c z 0 -- U .* E c.1 r.1 w 00 m I I- U ++ m 4 H z r.1 ww H cl m HCn 00 m H > 00 > I -I 00 i ..I Q 00 Q 2 z D.0 z 4 0.0 Q 4 10 w 0 iz Q bl iz 4 it w .. w m 4 I- I/ I- 4 I- I- I- 3 P co ?d .. i L z L I H vv LL 4 i I 0 u li n --0 t- MM 00 ++m WWH 00 00 o oo-r 4 00 K r.1 o 4 lif Q i Q Li) 0. 0 P r.1 4 iz M .o > Ill - m c 0 z 6 W (0 m m I- H n .. -ij 0 r.1 c.i z A K 004 0 ++ W WW P z-l oom v Ha 00 > x: 00 4 ic 00 -I W b0 K I- iil 0 i L w w 640 200 > mi P b-J ic ZI-0 0 c - -I- HQ I OH I IZ rom I +I- 0 c U U >z icH K I- --a LL I- 0 z s 00 mx 0

81 APPENDIX G

w w U-J iL Ill bl Lil b3 b-J ill bl 0 0 300 3 0+ 0 0 m++ Ln + + + + mww m w w w w w w99 w -0 -0 9 9 9 r.i r.i Lc ri r.1 ri Ti r.i u 0. 0. 0. 0. 0. ii0.0. n n r. r. r. rl rl rl 4 ri r-i ri c: r.i ri r.i I.

- h A m m M m I9 0 0 0 0 0 + + + + w W w w w 0 0. 9 m m 0 L!l M n M 0. m 9 i L!l P ri 0 m -0 bl bl Lil Ill P P P P M c.i rr ci ri ri

I- L- 0 0 b? iL -. 0 ri ci + ri 00 w 0 ++ -0 + 0 ww ci w ,: -0. 0. L!l 0 m 9 Lil p\ h- 9P + I\ z rl H mm 0.M ri Q 00 90. ri bl w + + ci ~ii -4 ww "y r. ic 0 bl I 0 mi I I LL Pr\ II ri x Y Lo -or\ -- iL K ri q Q a 0 Ill -4 io a t- ri ri CI =1 Q .. z ri ri c H z ri ci 0 MZri ri ri ci .rl II IO0 0 01 0 000 0 0 0 Q++ + +Q+ +++ + + c, wuW ww w d z oww w m c W wmw a r.w -.. P 0. r. 0 mx,..*, 0. m m -4 Li ri ri r m m .r rl u 04.0 0 b7 00 Ll7 i 0 ++ bU7 i bl . P ci U r, ri ri rl P 0 m z 1-1 o rd 0 H 4 m ri H L!! 000 0 ! POP P m P I H i 9 ... t- H 3 c ut- > W JQ c7 I I- u uin u? ci W m 0 co .. 0 H II 00 z + 000000 i0-40.+0-40+0+0 0 3 I- L2 i 00 w in U + w A 0Q++/I E ~o+a+o+o+o+o+ H I t-wui r-lwlwl w z E-c ww~wtiw I\ X z WWP4X iu H --ON-4iQ i NOPSW~I m z 0. H t- Q 0 OMXI 0 bl P ir-l 0 9 u 0 rim + 9lll.oP-0 n 0 -ri 0 W Q w -m - iL 0 0 ri 7 ci P ri r. ii 0 ri ri ci >- c i I- I Li) V 0 P m z 0 - H W II 10 P t-- n H t- Y 9 In H -4 3 z K i 0 m u H w-4 4 w 0 I io GO Q z 3+ + c w 0 I- II 0 I-XW in H -00 0 I-mo 0 ** I t- ti3 L L LL C _I z L3 0 iL QWP PI 0 M w 100 ou m t- h h LL 0 9 b3 H t- 0 + H ii-0 . 1-1 0 0 w 2. 0 K M X Q i H 0 w 2 P X .. . r w 0) Q w z-! 9 I- M mmmmm t- 0. Q H> cn H 1'1 io d 'ZI 9 E ri m +w .. K ic -4-2 w w W >- 7 z io _I t- iL c H >- iu - 0 I- u ill t-0 w w i t- I- ic 0 z z 0 I- I> H 0 o ri uir I> 3 XI V 1- I wz ill u z m ic ill ill iL XLi PdK XI-4-40 w K c-4 3 H w rl 00000 --art u I- 0 t- 3 3 W LL LLLLLLLLLL W z L-2 ir 3 r( zzzzz 0 t- LL1 r- t- % z Y 0 t- 0 04 0 0 I 0 L1 XI ill I- ;i APPENDIX G

-0 .- -0 M ri ci m 00 M ++ -0 P 0 ww P /I P in ll3 n P d z r.1 0. n P)M 113 ll3 Q: 00 -4-0 .. (3 ++ nm W ww 4 b3 W r r.8 r.1 .* E H MA It n I- P)P t- llY M

in H in >- i Q z Q K W I- I- 3 i LL z I u

83 APPENDIX G

.. t-0 0 0 0 - P MM ri 00 ++ 0. 0 ww P I’ 00. (0 ci o z - m.0 H o ri Q om U r.P 44 W iL .. x1 0 I1 U LL I-

10 YY iz d 0 0 t- \ 0. 0 \ ri c m C w 0 x1 Y -. u t- w t- Ln In Q H ! +I ii Ln I > H t- i H I 4 > w t- z u J Q x Q I- Q H Ln 3 iL w t- 0 i4 w 4 H Q t- a! I- t- lI1 ’ c: o ri o 3 0 !=t 0Q+O+ w Ln 3 + E I rwI W z H i z wwowo iL LL w -ON000 z w Q 0 000 H c L (3 0 00.0 0 - 0 -0 W Q u MA- 0 -I\ Ob0 u ic 00 ci >- c ++ ri I- I U ww f MO w -0 Ln t- P Li7 H t- 9PLL 3 H 0.00 ic J * 9 i\ W r( 9 ri.0 w I i4 . ‘E t- c 9 3 0 t- r. in M Ln Lh in -- L LL 0 LLL W 4 00 iL L 0 0 u u t-W r.r. LL Ln in I-= 0 0 .!i 9m-a * H I- - H3H 0 O mz -z- o h Q i ob- x iL: 06 0 0 OH Q + W iL At- z w wic *. w2>- z4 OMi4 M in* Q W P Hin 0 2 Ln - >Q!i 3m t- Q t- *. Q id i 5 I-ri ciwm iz 6Q6 w 0 04 Air. w i HH I- !-I10 t-0 z iT c1 i w w kt- + W box 0 3 HH H iii *x iz --k- l3 Q 0 zz z .!- I- 0 I HH H 0- IZ z +I- iz 0 t-+ 0 00 0 I- I zz z H uiz u ir. QQ6 >-z w u0 z I xH xt-wxr. - iL in t- -0- 0 t- 3 13 LI L zci F 3 t- * w* W - 0 z iL 3 U zic i z wciu H izw 0 k w P-iQ 6 3u t- H H 0 2:” iz I 0 LL 3 I- - L- APPENDIX G

W ic L7 ill Ill iil iil L!Y iil ii? L!l 2-J iil iil hi iil Lii ui iil iil ii? iii iil "7 30 0 0 0 0 0 oooooo~ooooooooo Lo+ + + + + + ++++++++++++++++ mw W W W W W wwwwwwwwwwwwwwww w9 9 9 9 9 9 9 9 9 .o 9 9 9 9 9 -4 9 9 9 9 9 9 ic r.i ri r.1 r.1 r.i r.i r.1 PI ri ri ri r.i r.1 r.1 ri ri ri r.i ri r.i r.i r.1 u0. 0. 0. 0. 0. 0. o-a.0.0.0.*0.0.o.0.0.*0.0.0.0. n n I\ n r. n nr.r.r, bnr. nnnnr\nnnb ri * .+ -4 i .i rl-+3-4.+rl.iHrlrlii.+rl*.+ ri ri r.i r.i r.i ri ri ri ri ri r.1 r;I r.1 r.i r.1 r.i r.i r.i ri r.i r.i ri ri r.i r.i r.i r.i r.1 r.1...... ri r.1 ri r.1 r.i r.1 r.i r.i ri r.i ri ri ri r.i r.i

h h c h - ,. - ,.. h h rl ri - - r;l r.1 r.1 r.i r.1 ri ri r.i r.i r.1 r.i r.i r.1 r: 0 0 0 0 0 0 0 ri C1 ri 0 0 0 0 0 0 0 0 0 0 0 0 + + + + + + + + 1 W W + + + + + + + + + W W W W W W W w W W W 0. ri rl .I w W W W m P b m M h r.i r.1 iil 0 m 4 c 9 Y m 3 h 0 0 U-J .o ill 4 9 .+ ri 9 m I\ iil i n c 0 3 n 0 iil r.i r.i M 9 0 iil -i R 0. rl n ill n 0 0 iil -4 io 0. iil iil r.1 -i iil r;i m 0 ii) 9 n 0 h m M M 0 9 P rl -i iil P 0. F 0 P r.i M P n M M n 0 rl r.1 m M U 9 r.i M P P L!l iil P P iil iil 9 9 9 9 9 d 9 9 I- O 0 ic- M M M M m M m m M M M r.i r.i r.i ri r.i ri r.i r.i 0 0 0 0 0 0 0 0 0 0 0 + + + + + + + + + +0 + + +0 0 0 0 0 0 0 W W W W W W + + + + + + W w W W w W lll W W W W m rl 0 0 M 0. P W W n 0. 0. 0. 9 Ill 0. P .i 0. 0. rl rl m r.1 i P Li) M 0 6. M iil 0 m 9 9 M r.1 rl 0. 0 9 r.i 0. r.i 4 9 m ri 4 iii PI 0 n 0 m m r.i P h R PI q 9 rl I\ P M P 9 m m r.i 0 ri m 0 9 M 0. 9 M m ri M il 9 m 0 1, ri 0 4 0 rl iil m m m P + P m m r.i r.1 rl rl rt rl 3 0 0 m W rl -I rl ri rl 9 P 0 m ii7 P -i .-I rl i .i .+ ii rl r. 0 0. 0. 0. 0. m m m m 0 I I I I I I I I I 0 I I I I I I 0 0 v v v v v v 9 r.i h

z ri ri r.1 r.1 r.i ri r.i r.i ri r.i r.i r.1 r.i r.i r.i rt r.1 ri ri ri r.i 10 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Q+ + + + + + 0 0 0 0 3 0 + + + + + + + + + + t 4. aw W W W W W W + + + W W W W W w W w W W W W WP m 9 m 0 t!l 4 r.i 3 .i M !4l .o 0 I\ 0 M ill 0 ri xo bl m r.1 0 d m m m 9 0. m r.1 P rl 0 Oh bl m m 0 M ri r.i bl n iil r.i 0. 9 M P 9 9 q 0 P Lil a + m a R P rl 0. m m m 0 r.i m z r.1 W m 9 r. ri r.i m iil 1, 0 n m 0 9 m 0 I\ ii7 0. I\ P m 9 P 0 ! Hm 0 ri ri 4 rl rl 0 r.i 0. 0 0 0. 0. m m m R n I\ R 9 0 r.i r.i r;I r.1 r.i ri r.i r.1 rl i rl rl n 0 ri rl 4 .i ri d rl 0 9 C.l P W rl 4 I 0000000 000-000000000000 rn r.i o r.io + o - o rl o ii o ,+ o -1o+orlo.+- -0+odorlorlorl0-oriorl0rl0riorlo 4 oQ+o+o+o+o+o+o+ 0+0+0+0+0+0+0+0+0+0+0+0+0+0+0+0+ @ t+w IW I w I w fW I WI w IWIWIWI- IwIw1wI~I~IwIwIwIwIwl~lw z L'W plus WPWmu9 wr3.w r\ M PW iii M ~3 H ?~'mwri w .iw w w o w'o w 'ow o. wo w w w n w m w N rl 9 0. m o G- 0 0 r. M P iil 1-4 P LO M ~7 4 4 Q - o o o o o o o o o o o i,0 a -3 om o 0.0 o o o o '0 o ~omo~omo~or~io~om o M o bi o 1 w 0 b~ iii 0 U? U: 0 0 D.Obl PC.19 o o o o c mc? nu o no ri o o o onrl mmP bl ri .o ii3 h m m I o. iil o 0. rl n ri 0. M lil P lif iil maiil I\ 7 m .o UI o o r. - m -P -0 -I\ -P -r.i .o -0. -m -m -3 -0. -0 +ri -P -n -0 -m .R -i.~i) -0 -P 0 01 rl .+ P rl r.1 - m o. m + n bl + m rl o .+ 0. m rl n rl ri -a ri ~i)ri P ri M rim ri ri ri rl ri il ri o ri o M 0. I- -4 0.0.0.0.o.D.o. a.mm~mmmmmmmmmmmn 0 ...... *I..*..---- W +m mmmmmm z WH HHHHHH H 0 m 0 z 04 rlrlrlrlrlrl 0 z 0 wrl rl -4 rl rl rl 4 -l4rl.iArl.+rlrlii.irlrlrlrlrl m omooooo 0000000000000000 W 0 3+ + + r + 2 2 ++++++++++++++++ if PicFwawawxwnwawicw icw~wicwicwicwicwifwifwifwicwifwifwicwicwifwicw if 'OH0000000000000 00000000000000000000000000000000 0 mcomomomomomomo momomom0momomo~omo~omomomo~omom0 0 z l Lil z b-J z bl z li) z b-J z Ill z Ill z b-J z bl z M z bl z ill z Ill z iil z Ill z Ill z LI-J z Li) z bl z L!l z Lil z bl z L-J WQPWPWPWPWPWPWP WPWPW~WPW~WPWPWPW~WPWPW*WPWPWPWP 0) (0 omomomomomomo momomomomomomomomomomomom0momom0 I- Rnnnnhn r.nnnRhnRRnnnnnnn O -m 9m9m9m9mam9a9 m.om9P9m9m9P9~9m9P9m9a9m.om9~9m9mQ 0 mz -z -z*z -z -2.z I z 'Z 'Z -2 *z*z *z +z*z *z -z *z -z *z*z -z * if CQQQQQQQQQQQCQQQ W zrl Hm E if W I- W mJ Q OH +I- H >z CCH icL3 x bl ic iil if hl ic Ill ic ill ic P if P if P ic P ic Pic P ic c ic P ic P ic P I- 0000000000000000 LLLLLLLLLLLLLLLLLLLLLLLLLLLLL C

85 APPENDIX G

lil li3 lil li3 +0 0+ +0 0+ W W W W 9 9 .c1 9 c1 ti r.1 r.1 0- (E (E (E I\ I\ I\ 4 .- c.1 r.1 r.1 Cl

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.A 4.J C u0 ! H H> u APPENDIX G

Li3 Li3 ti) Lil ++++0000 wwww 9999 r.i r.i r.1 r.1 0.0.0.0. nnhh

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87 APPENDIX G

b2 117 Li) 117 117 +++++00000 wwwww 9aaaa r.i ci ci r.i ci hr-.br-.r-.@.*0.0.0. 3113i r.i c i c.i r.i ci ri:_-.. ci r.i c.i ri

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3c ci ci ci rr .A 0 0 0 0 CI + + + + c W W W W W c M 0 ci P r 0 r.ia .+ a P H u * m 4 t- W P t- in b> 117 m H r\ 4 ri m ffl ! I\ r-. a b7 H 4 5 in.,- 4 3 4 4 I I H t- t- I 1 H 3 I c W t- 1 z > J Q LL iL U u t- H c H 0) o if w c o 4 ili W 4 4 , l-l I Q +- ci o ri 0 c.i o ci o rn o 3 I- W t- t- 0+0+0+0+0+ W Lo z Ln 3 IWlWlW lWlW z H c -1 wow0wowowo iL I LL 0000000000 z W d 0 I 0000000000 H t- ffl I1 L [email protected] o H r.i Lo I -0 .o .o -0 .o W c t- U u qoMoc.io rl00.0 11 K rl u l-l 444rl.4 F- Q I/ 0 t- ..... t- I t- H in fflmmfflcn 0 -I H WHHHH W 3 0 K ffl F- 5 0 W H t- u 4 t- 4443rl 3 H H II ic 1 > 44437. W H __lil z _.. 00000 I sr q-. +++++ t- c W o ci o o r.1 4 iLwiCwifwifwicw 0 t- r r. -11 mm ri 0000000000 In H -0- PP 0 fflofflofflocnofflo .- t- -.. M 117 z bl z Lil z 113 z b7 z Lif L LL 00 0 L7 WWWPWPWPWP 0 0 z 00 4 -a fflocnocnofflocno 0 ci ci rl LL ffl H II Ln 0 3 H t- t- b-4 0 3 -I U - 0 W u W 0 t- -- 0. E: X MMrlrlMcnO Q W W 7 a - W HO cn t- h ffl c Q Q ff K ooooow -1 W 4 rl 4 rl 310 W W WH if K J-IJJJt-t- u z W U 00000H >- W - 0 t-0 I2 0 et-0 > z K if~ K K I-0. t- 0 il: WE 70 0 H t-I-t-t-t- n I zzzzzt-0 il: 0 if u 70s I 000000- 0 t- W m r t- -iL 27 t- t- uuuuu1 z mQ w- - 0 o -IIa z I 7 z iclJ7il:PicPKPifP w il: cn 3 -u t- H w W DOOOOI-I 3 J -1 LL LL LL LL LL zu t- 3 WW W - I2 xzzzzic ic 3 u HHHHHKW 0 J -I > W W QQQQQ3Ut- W H (3 0 (3 (3 (3 uc H 0 K I 0 LL LL t- cl @.

88 APPENDIX G

h- -e MM MM ++00 ++00 ww wu '0 o m r.i 40 MI\ MO nh 00 70. 00 ma 0.m hli, -4P Mli) .I If I- O X-.0 .. e r.i m r.i m r.i 00 00 ++ ++ M 0 ww wu P I/ 00 0.- Lo 0 0 M PI m z 0 0 CI 0 0 H MM 00 [\P Q 00 00 04 (3 ++ 00 coo .* WW !A3 r.1 Li) 4 W iL: 00 .I .. II 0 00 If II H E L 00 t- 00 Li) rn 00 -v vv rl if '00 r.1 0 0 mm 0 -c. + 4P \ li Q .* 0. > L z 0 -m I H z ri ri ri ri \ 0 iL r 100 00 C.1 b3 0 Q++ ++ m r >- .+ z (3ww ww w a ri LLL oii 0 -1 w~i)hnm II liiia .ti E r.i M ~mno.n .. ++ w h0 u 00 OMO ori W 1- J ot- L Li) L-J 4 ii) (3 ++ h.0 PO I- I- L r.1 r.1 o > ri z r.1 WW r.1 n o. 4 4 0 0 0 L-J ill iFI 0 HLo 00 + ri o o m 40 . -0r.1 * J 00 ~m '00. 0' 000. 00 .. .* I * >>- -o> 3 00 I w 00 I 00 U 00 4 Lo L-J r.i .I Ll I1 0 ic II Ll 0 0 Q+ + I-W $2 W z wn z.0 u vu N ri Q r.1 Q Lil 0. (3 P -h *P om44 0 A- 0 Lil 0 0 0 r.l I- MM r.1 + + + r.1 00 www * (3 ++ Lo 'OM0 Z ww H PA0 H 00 0 P L') E 00 4'0PL z 00 -t 00.00 00 +'on e 00 W+ 9 r.1 .Q w Lo bl r.1 GO . * ." w 0.m 3+ iL: 4P XI-w K ..OH0 0 Lo?-0 u z i bl WQP Lo Loo I- h 0 ..L c29 0 r.1 M z- iL: ++00 Q W ww iL z- 00 nm HLO 00 z II 00 K 00 +A W 00 04 I- 00 io W 0.0 OK =_I P r.1 ic ZZZZZZZZ '2 I. b- HHHWHHHH OH IIz QQQcQQQQ I-+ 0 13L3c0wowo H UiC >z w KH K+'0 I- YY 0- LII I- .B z Lo :,- w4-T H H a44 c I 0 0 (3 I- E r -4 0

89 APPENDIX G

ffl ffl cn ffl ffl ffl ffl ffl ffl m I- I- I- I- I- I- I- I- t- I- O O O O O 0 0 O 0 O 1 J J J J J J -1 J _I L u ti L L L L L LO L 4 ic ic ic K w w w W4b I- I- I- t- I- C I- I-I- 3 3 3W J J Jm LL LL LL I- x 'L ZO L L u-1 u a K 9) 0 a LL 5 rl 0 0 i- d W 0 U 0 0 M ri w Ill ri U P 0 E M .o m r.1 --. H ! r\ 0 h 0 PI 0 I- h 0 I\ 0 h 0 H 0 .-i 0 * 0 * z H 0 m Ln Lo m m Lo H H H H H H H I- 3 z /. U w 0 0 0 0 0 0 W I4 X m bl b7 bl lil bl b-J w Y v v v v 4 > >- * > B Q Q Q < -1 J J J z ic ic iL: 5 0 W W W w 0 > > > > w 0 0 0 O m I I I I I L I- I- t- I- I- 0 L3 a -$ -(=? z -(=? z OD w w w w w WP J -I J J J z.0

ii I2 ii ij ij Ha -1 -1 J -1 -1 bl w W W W w H H H H k-4 i00 LL LL LL LL LL 1-. \

90 APPENDIX G

TABLE GVII1.- INTERACTIVE EXECUTION OF SAMPLE CASE 9

71 TYPE IN TITLE 2 CASE 9 LOCUS OF SHORT PERIOD ROOT WITH ALTITUDE (P-K, FIGURE 16) TYPE IN /INPUT/ ' SINPUT ISPLANE=O,IFLUT=l,IPKPLT=-l,KVAR=2,HO=l6,DV=l,NV=lO,IPRINT=lO, 9 ITRACE=O,1,6*O,IOPT2=1S 1 DETERMINATION OF STAB1 LITY CHARACTER1 STI CS -- )STABCAR<-- DATE 83 / 09 I1 2 TIME 11 33 04 CASE 9 LOCUS OF SHORT PERIOD ROOT WITH ALTITUDE (P-K, FIGURE 16) MACH X= 8600

FIELD LENGTH-OVERLAY(4, 0) IS 104264 P-K FLUTTER ANALYSIS INITIAL ROOTS DETERMINED BY MATRIX ITERATION

FIELD LENGTH INCREASED TO 104320 lTHIS CASE IS FOR AN INITIAL DENSITY OF 1664707E+00 AN INITIAL VELOCITY OF 2537597E+03 AN INITIAL ALTITUDE OF 1600000E+02, AND AN INITIAL DYNAMIC PRESSURE OF 5359857E+04

DO NOT TRY TO DETERMINE ROOTS CORRESPONDING TO GAIN SCHEDULING DENOMINATORS FOR GAINS=O 0 INITIAL s1 52 MODE 2( - 33806443E+OO, 29525551E+01) ( - 34147922E+OO, 2982 3789E3+01) X- ITER ROOT NBR ALTITUDE ZETA OMEGA-N ROOT PRESSURE 2 2 1600000E+02 1137569E+OO 4777617E+00 ( - 3414830E+00, 2982379E+01) 5359857E+04 4 2 6000000E+01 2078029E+00 9194466Et00 ( - 1200489E+Ol, 5650944E+Ol) 2444549E+05 CURRENT PLOT TITLE IS CASE 9 LOCUS OF SHORT PERIOD ROOT WITH ALTITUDE (P-K, FIGURE 16)

HIT RETURN OR TYPE SAME, IF OK, OTHERWISE TYPE IN NEW TITLE ' SAME

DO YOU WISH TO GENERATE PLOTS OF STABILITY CHARACTERISTICS AT THIS TIME' '0

C ONT I NUE '1 TYPE IN TITLE ' CASE 9 LOCUS OF SHORT PERIOD ROOT WITH ALTITUDE (P-P,FIGURE 16) TYPE IN /INPUT/ ' SINPUT ISPLANE=l,IPS=O,SlI=25*~O,O~,S2I~Z~*~O,O~,IOPT2=1S 1 DETERMINATION OF STABILITY CHARACTERISTICS --)STABCAR(-- DATE 83/09/12 TIME 11 44 53 CASE 9 LOCUS OF SHORT PERIOD ROOT WITH ALTITUDE (P-P,FIGURE 16) MACH X= 8600

FIELD LENGTH-OVERLAY(4,O) IS 101750 P-K FLUTTER ANALYSIS INITIAL ROOTS DETERMINED BY MATRIX ITERATION UNSTABLE ROOT

FIELD LENGTH INCREASED TO 102004 lTHIS CASE IS FOR AN INITIAL DENSITY OF 1664707E+OO AN INITIAL VELOCITY OF 2537597E+03 AN INITIAL ALTITUDE OF 1600000E+02, AND AN INITIAL DYNAMIC PRESSURE OF 5359857E+04

91 APPENDIX G

TABLE GVIII a- Concluded

nn NCT T!?V TZ "UTERflINE ROOTS CORRESPONDING TO GAIN SCHEDULING DENOMINATORS FOR CAINS=O 0 INITIAL si S2 MODE Z( - 41972234E+00. 27741750E+01) ( - 42396196E+00, 2802 1970E+01) #-ITER ROOT NBR ALTITUDE ZETA OMEGA-N ROOT PRESSURE 1 2 1600000E+02 1495938E+00 4510590E+00 ( - 4239619E+@O, 2802197E+01) 5359857E+04 3 2 6000000E+01 2985435E+00 8603155E+00 ( - 1613783E+01, 51590@8E+01) 2444549E+05 CURRENT PLOT TITLE IS CASE 9 LOCUS OF SHORT PERIOD ROOT WITH ALTITUDE (P-P,FIGURE 16)

HIT RETURN OR TYPE SAME, IF OK. OTHERWISE TYPE IN NEW TITLE ' SAME

DO YOU WISH TO GENERATE PLOTS OF STABILITY CHARACTERISTICS AT THIS TIME' '0

CONT I NUE '1 TYPE IN TITLE ' CASE? LOCUS OF SHORT PERIOD ROOT WITH ALTITUDE (FIGURE 16) TYPE IN /INPUT/ 3 SINPUT IFLUT-0,IPKPLT-15 1 DETERMINATION OF STAB I L ITY CHARACTER1 STI CS -- )STABCAR (-- DATE 83 I09 I12 TIME 11 50 35 CASE? LOCUS OF SHORT PERIOD ROOT WITH ALTITUDE (FIGURE 16) MACH x= 8600

DO YOU WISH TO GENERATE PLOTS OF STABILITY CHARACTERISTICS AT THIS TIME' '1 PLOT SET TYPE OF VARIATION x MODES PLOT TITLE

1 ALTITUDE 8 OPEN LOOP STABILITY CHARACTERISTICS VER SUS ALTITUDE,DV=O 5 (FIG 9) 2 ALTITUDE 1 CASE 9 LOCUS OF SHORT PERIOD ROOT WITH ALTITUDE (P-K, FIGURE 16) 3 ALTITUDE 1 CASE 9 LOCUS OF SHORT PERIOD ROOT WITH ALTITUDE (P-P,FIGURE 16) " TYPE IN NUMBER OF PLOT SETS TO BE PLOTTED," 81 CORRESPONDING PLOT SET nos, AND WHETHER TO COMBINE THEM" '2 231 CURRENT PLOT TITLE IS CASE9 LOCUS OF SHORT PERIOD ROOT WITH ALTITUDE (FIGURE 16) HIT RETURN OR TYPE SAME, IF OK, OTHERWISE TYPE IN NEW TITLE ' SAME TYPE IN /PLOTPI -KVAR,IDASH,MODEPK, ETC FOR PLOT SET # 2 ' SPLOTP IDASH=@,XRLORG=-2,XRLEND=O,XRLDEL=l,NDXRL=2, ' YRLORG=O,YRLEND=6,YRLDEL=2,NDYRL=4S

FIELD LENGTH OVERLAY(5,O) IS 077256 PK FLUTTER PLOTS TYPE IN /PLOTP/ -KVAR,IDASH,MODEPK, ETC FOR PLOT SET # 3 ' SPLOTP IDASH=iS

FIELD LENGTH OVERLAY(5,O) IS 077256 PK FLUTTER PLOTS " DO YOU WISH TO CONTINUE PLOTTING' (110)'' '0

CONT I NUE '0 f

REFERENCES

1. Severt, Francis D. : Development of Active Flutter Suppression Wind Tunne 1 Testing Technology. AFFDL-TR-74-126, U.S. Air Force, Jan. 1975.

2. Roger, Kenneth L.: Airplane Math bbdeling Wthods for Active Control Design. Structural Aspects of Active Controls, AGARD-CP-228, Aug. 1977, pp. 4-1 - 4-11.

3. Hassig, trermann J.: An Approximate True Damping Solution of the Flutter muation by Determinant Iteration. J. Aircr., vol. 8, no. 11, Nov. 1971, pp. 885-889.

4. Woodcock, D. L.; and Lawrence, A. Jocelyn: Further Comparisons of Different Methods of Assessing the Free Oscillatory Characteristics of Aeroelastic Systems. Tech. Rep. 72188, British R.A.E., Dec. 1972.

Rodden William P.; Harder, Robert L.; and Bellinqer, E. Dean: Peroelastic Addition to NASTRAN. NASA CR-3094, 1979.

Field, E. I.; Herting, D. N.; and Morgan, M. J.: NASTRAN@ User's Guide (Level 17.5). NASA CR-3146, 1979.

Peele, Ellwood L.; and Adam, William M., Jr.: A Digital Program for Calculating the Interaction Between Flexible Structures , Unsteady Perodynamics and Active Controls. NASA TM-80040, 1979.

Bisplinghoff, Raymond L.; Ashley, Holt; and Halfman, Robert L. : Wroelasticity. Mdison-Wesley Pub. Co. , Inc. , c.1955.

Giesing, J. P.; Kalman, T. P.; and Rodden, W. P. : Subsonic Unsteady Aerodynamics for General Configurations. Part I, Vol I - Direct Application of the Nonplanar Doublet-Lattice Method. AFFDL-1R-71-5, Pt. I, Vol I, U.S. Air Force, Nov. 1971. (Available from DTIC as AD 891 403L. )

Giesing, J. p.; Kalman, T. E.; arid Ecdden, W, P.: Subsonic gnsteady Aerodynamics for General Configurations. Part I, Vol I1 - Computer Program H7WC. AFFDGTR-71-5, Pt. I, Vol 11, U.S. Air Force, Nov. 1971. (Available from DTIC as AD 892 535L.)

Cunningham, Atlee M., Jr. : A Steady and Oscillatory Kernel Function Method for Interfering Surfaces in Subsonic, Transonic and Supersonic Flow. NASA CR- 1 44895, 1976.

Rowe, W. S.; Redman, M. C.; Ehlers, F. E.; and Sebastian, J. D.: Prediction of Unsteady Perodynamic Loadings Caused by kading Edge and Trailing Edge Control Surface Motions in Subsonic Compressible Flow - Analysis and Results. NASA CR-2543, 1975.

Vepa, Ranjan: Finite State Modeling of Aeroelastic Systems. NASA CR-2779, 1977.

Edwards , John William: Unsteady Terodynamic Modeling and Active Teroelastic Control. SUDAAR 504 (NASA Grant NGG05-020-007) , Stanford Univ. , Feb. 1977. (Available as NASA CR-148019. )

93 15. Dunn, H. J.: An Analytical Technique for Approximating Unsteady Aerodynamics in the Time Domain. NASA TP-1738, 1980.

17. Merino? Luiqi: Steady, Oscillatory, and Unsteady Subsonic and Supersonic Aerodynamics - Production Version (SOUSSA-P 1 .1 ), Volume I - Theoretical Manual. NASA (33-1 591 30, 1980.

18. Harder, Robert L.; and Desmarais, Robert N.: Interpolation Usinq Surface Splines. J. Aircr., vol. 9, no. 2, Feb. 1972, pp. 189-191.

19. Anon.: U.S. Standard Atmosphere, 1962. NASA, U.S. Air Force? and U.S. Weather Bur.? Dec. 1962.

20. Tiffany, Sherwood H.; and Newsom, Jerry R.: Some Programminq Techniques for Increasinq Program Versatility and Efficiency on CDC: Equipment. NASA CR-3033, 1978.

21. Boeing Wichita Co.: Integrated Desiqn of a High Aspect Ratio Research Winq With an Active Control System for Fliqht Tests on a BQM-34E/F Drone Vehicle, Vol- ume I. DOC. No. D3-11535-1 Volume I (Contract NAS1-14665), Boeinq CO., 1979. (Available as NASA CR-166108.

22. Boeinq Wichita Cn.: Inteqrated Desiqn of a Hiqh Aspect Ratio Research Winq With an Active Control System for Fliqht Tests on a BQM-34E/F Drone vehicle, Vol- ume I Appendices. Doc. No. D3-11535-1 Vol. I Appen. (Contract NAS1-146651, Boeinq Co., 1979. (Available as NASA CR-166109. )

23. Boeinq Commercial Airplane Co.: Integrated Application of Active Controls (IAAc) Technoloqy to an Advanced Subsonic Transport Project - Winq Planform Study and Final Confiquration Selection. NASA CR-165630, 1981.

94 L TABLE I.- OUTLINE OF DATA ON TAPE2 AND/OR TAPE3 (INPUT)

When required Variable or Set Data type name li s t name

1 (a) ICASE Free-f ield integer

2 (a1 HEADER Eight-word (A101 alphanumeric array

(a) $INPUT...$ Namelist 31 I 1 4 ISENSE = 1 SSENLOC.. .$ Namelist

5 ICONSYS = 1 & ICASE = f2 $SENINP...$ Namelist or ICHSEN = 1 & ICASE = f1

6 ICONSYS = 1 & ICASE = f2 I&J Free-f ield integers or ICHFIL = 1 & ICASE = fl

7 ICONSYS = 1 & ICASE = +2 $FILTIN...$ Namelist or ICHFIL = 1 & ICASE = fl

8 ICONSYS = 1 & ICASE = f2 $ACTINP...$ Namelist or ICHACT = 1 & ICASE = f1

9 ISELECT f 0 $SELECT.. .$ Namelist

~ 10 IDELPLT f 0 IPLT2 ' Free-field interJer array

11 IPKPLT > 0 ICOUNT Free-f ield integer

12 IPKPLT > 0 IPLT Free-field integer array (Size = ICOUNT)

13 IPKPLT > 0 ICOMBIN Free-f ield integer

14 IPKPLT > 0 & ICOMBIN = 1 PTITLE Eight-word (A10) alphanumeric array

15 IPKPLT < 0 I $PLOTP...$ Name list 1

16 IPKPLT > 0 I OP Free-f ield integer

Repeat sets 10 to 16 as desired, terminate with IOP = 0

Repeat sets 1 to 16 as desired for each case, end with ICASE = 0

aRequired in all cases.

95 TABLE 11.- DESCRIPTION OF TAPE2 INPUT DATA

Alqehraic Namr 11s t When required FORTRAN variable orscription variable

xlways ICASE Inteqer indicatinq type of case to be run (Initialization or continuation) and which version of proqram iinter,>ctivr or nonlnter- active) tc Invoke.

= f_2 Inltlallzatlon case. All data are (rellnitial- ized hy namelist and option parameters from TAPE% and all array data (such as aerodynamic-, qeneralized masse-, and mode shapes) from TAPES and TAPElO.

= tl Continuation (-.%SF.. C~sehrinq run will use d

=o EXC(.,I t 10" s topp(1,i.

Always I4ERL)EH Alphanuwrlc description of case heinq run.

+INPIIT :CASE = _+2 NM Total number of modes: riqld, rlastlc, dnii rontru1.

NH Nitmher of riqld-hody modrs (st%? sei-tlun 011 "snsor input definitions" 1.

N(' Numhrr of cnrtr,>! modes.

NK Rumher of reduccd trrqurncirs ,it which d~rody- namic data haw hct,n computx-d.

Nl: hbmbrr of qust forces LnclndPd as column.; ~n arrodynarmc array Input. Provision has been made to read-in aerodyndmic mist force.; which may be included as additional (NGl columns in aerodynamic data. 'Ihese data are not used by the proqram. icefault = 01

I Number of sensors. (kfaillt = 0) TABLE 11.- Continued

Alqe brai c lamelist When required FORTRAN variable Description variable

INPUT lode1 definition arrays are read-in from TAPEl (a data storage file created and used by proqram durinq execution). These data represent modpl as it was at end of precedinq cas?, incorporatinq any modal and/or frequency selections up to that point. New modal and/ or frequency selections are from this input.

lode1 deflnition arrays dill not br altered for this case. (Default) If ICASE = tl, the arrays will be precisely those in effect at the end of the precedlnq case.

CASE = +2 CSllRfEF I I) Iultiplier which modifies column of aerodynamic force matrices correspondinq to ith control mode. he multiplication is only performed when ICASE = +_2 or ISELECT > 0. (Default = 1.0)

b or c C leference chord lenqth

mACH lach number at which aerodynamlc forces Were '4a qenerdted.

KFIT rype of tit to br employed when interpoldtinq input aerodynamic form 4ata to obtain aero-

fcrc-s at d Partlcrrl.ir frequPnry. ~ llsrd when ISPLANF = 0. Whpn KFIT < 0, the interpolation 1- for Re(?(kl) and p' !k\ keP SPCtl"" "unsteady aeruaynamir forreS" 1.

= +1 :Linear flt.

= t2 aadratic fit. (Default)

= k3 Xhic fit.

ISTIFF Inteqer indicatinq how stiffnes~matrix is ohtalilea.

=o romputr diaqonal stiffness matrix: KI1 = Mll(2T[rN, 1'. (?eflUlt)

Va<-l? fhll ctlffneqs matrix.

ISPLANF IntrqPr in,iic.itinq whptrirr to cmplux- F ;?a?- approximdtians to a~rodvnamlcs. furictlons nsrd for this fit have thr form:

=o DO not compute or use p-planp fit. (%fault)

=1 mploy p-plan- curve fit, usinq coefficients whlch have heen ;'reviously computed and stored on TAPEl.

=2 Compute and employ p-plane approximations for full set of aerodynamlcs from TAPES, rewindinq TAPE5 autonatically.

-3 Compute anti employ p-plane approximatlons for selected set of aerodynarmcs from 'rAPE1.

INTFRP

=1 Cornpnte and store geometric CorffIclents. used when compiitlnq senior ,ief lections for plate- type srct1ans.

=fl Do not compute.

'Individual parameters are optional if drfault values arc acceptable.

9 7 Alqehralc Name 1LS t 'ORTRAN variable variable

+INP~JI.

98 i I

TABLE 11.- Continued

Alqehralc iamelist when required ORTRAN variahii Description v6riahle

= t2 enerate plots of 5 and o versus velocity, altitude, or density de&nding upon KVAR during root determination (or corresponding dynamic pressure if KVAR < 0) or versus gain if ISAIN = 1 durinq root determination.

= i3 enerate both (S,w I and root loci plots.

=o o not qenerate any characteristic root plots. (Default)

DELPLT

=1 elete unwanted characterlstlc rwt data used for plotting from data storaqe file TAPEl.

=o b not delete anv data.

SPLANE f 0 lC0EF umber of coefficients desired currently for p-plane fit, includinq ratlonal term p/(p + baj). Maximum = 10.

4POLYC ( J I hlmher of polynomial coefflclents In p-plane fit of jth column of aerodynamic forces. (Default = 3)

3N (L,Jl 'onstant drnominator coefficient ~n Ith p/(p + b ) correspondlnq to jth column mode. If same 5Jlues of h are used for more than one mode, they stil?jnaed only he input oncc. See definition of NCOL(J).

NKK(J) hhlmher of reduced frequencies to be used for lth pressure mode fit. Program use5 first nkk. of total number nk. If 0, thc proqram use2 all nk frequencies for the jth mode. (Default = 01

NCOL( J) Inteqer indicatinq whether to use a precedinq set of hl for jth mode.

=N jet each h (Default = 1) a, = bm* =o Inpnt nrw bp, coeftrcients for jth voiir.

ICOF(N, ,l) In~ry~rir.,dicati~.r whether to lnciuiir ::kt constraint (as defined below) when fittlng the Jth column of drrodynaaic fnrres. [See appcndix &.)

=o Nth constraint not employed. (Default)

>o mnploy Nth constraint.

1COFLl.J) = 1 Force aqreement with tabular value for kl = 0. Q,, (01 = Qll (0).

ICOF(2,J) = 1 For k, = 0, fi I~[Q~~(EI] E

E =k

ICOF(3.J) = 1 For k, = 0, Force slope = -FRe[Qi2(0)I/ben.

ICOF(4,J) = 1 FO~kl = 0, Force slope = 0.

ICOF(5,J) = Pi Force A = 0. 'j ICOF(6,J) = 1

usinq a piecewise quadratic fit to Q.(k.) (i = 1, nk). 31 ...,

tIndividual parameters are optional if default values are acceptable.

99 TAB1.F 11.- Continued

IFRPRT

SPKI)

100 TARLE 11.- Continued

Alqehralc game 1is t When required 'ORTRAN variable Description variable

'INPIIT CONSYS = 1 [CSACT

=1 wtudtor transfer function relates actual control surface position to commanded position. (DefaultI

=o Zontrol stirfaces are treated as full degrees of freedom and actuator transfer functions can he considered to relate actuator-produced hinqe moments to actuator Inputs.

:N( I, J) ;ai" for Ith control and Jth sensor pair. me user will find it convenient (virtually essential in plottino qain KOOt loci) to non- ~3irensionalizeG by includinq desired dimensional qainl!n (TL)~.. hen gain variations will be in terns o? a multiple of a nominal or desired value. (kfaiilt = 0)

'HASE Phase err\>r (deqrees) in transfer function. (Eefat1lt = 0)

[PH & .JPH Zontrol (IPH) and sensor (JPHI palr for which phasr error 13 introiluced.

[SCHDIlL( I, ,T)

=c NO scherlulinq in control ioqic for cointrol I, sensor J. (Default)

=1 Scheduling is included ifi control l~qlcfor control I, scnqor J. -__ ------ICHSEN

=1 Indicates a chanqe 1s to he made in a denominator corfficient for sensor dynamics. (Sre namellst SFNINP.

Sensor dynamics R~Punchanqed from prrcedinq case. (Default)

ICHPIL

10 Indicates a chanqe 19 to he made in filwr 4,, ._ "I ^= (*e- ndrn~'~;:? CILTIN!.

No chanqe 17 to he made in filtrr dynamics. (DP f ilu 1t 1

=1 Chanqe only occurs In numerator coefflcl~nts.

=2 Chanqe cnly occur- in drnoninator ropfficirnts.

=3 Chanqe occurs in both numerator and denominator.

ICHACT

f0 Indirates a change 1s tr, he made in actuatur riyndmlcs (see namellst ACTINP).

No chanqe is to be made in actuator dynamics.

Chanqr only occurs in numerator coeffici-nts.

manue only occurs ~n denominator coefficients.

Clanye occurs in both numerator and denominator.

IFLUT = 1 KMR Intrqer indicatinq type of variation to he performed in fluttrr analysis. If negative, any correspondins plot output will he with respect to rdynmic pressure. Can h? chanserl at plot time ~n intera-tive rxrc'ition.

'Individual parameters are optional if default values arr dcce~tahle.

101 TABLE 11.- Continurd

4lq~hraic lame list FORTRAN variahl? Description va r i a hle ~- INPUT 'LUT = 1 KVAR

z ti

= t2

~ i3

IGAIN

I 1)MIIL'I'

- I1

EPS I

NIT

NVIIT

NFINE

I IlAM1'

-I TABLE 11.- Continued

Alqehrai< amelist When required FORTRAN variable Description variable

INPUT FLUT = 1 IDAMP

=3 Dampinq is (~jO,,~lq,,(if wn, f 0) 1 and 0 (if un, = 0:. corresponds to viscous darnpinq 25.

ITRACE ( I) Inteqer indicating whether to trace Ith root, pronded Ith initial que53 in nonzero.

= 1, 2, Or Trace root. lJse linear (11. quadratic (21, or cubic (31 curve fit to extrapolate for estimate of Ith root at new value of independent variable. (Default = 11

=o DO not trace root.

IPRINT

=N Print characteristic root ,data to screen (file OUTPIIT) every N times they are computed. (Default = 0) ------_ ------vn Initial velocity for velocity variation; COnstant velocity for density variation.

RHO0 Initial density for density variation; constant density for velocity variation. -__ ------. ~ FLUT = 1 6 HO lni~i~lnititu'ie for ill.titiirl~ variation. KVAR = +_2 CONFACl Factor for convertinq input velocity to feet per secnnA for use in subroutine AT62. (Default = 0.0833 to convert inches per second)

CONFACZ Factor for convertrnq input altitude to fept for use in subroutine AT62. (Default = 11

CONFAC3 Factor for convertinq density. output by subroutine AT62 in sluqs per foot3, into type of units correspondinq to input model. (Default = 0.482258-04 for convertinq to inch unit51 ___ . ------FLUT = 1 6 DV Step size for each increment of independent IGAIN = 0 variable: velocity, altitudr, cor density, ciirres~ondinqto KVAR = fl, +2, or i3, rrsnect1 ve ly . ------_- - FLUT = 1 6 Increment in GN(1.J) for proper root estlma- IGAIN = 1 tlon. Fach nonzero DELGAIN should he equal. (Default = 0) ------. SlI(1I 6 S2I(Il Two initial estimates (complex) for Ith characteristic root. S2I(I) should he best guess. In Same cases, these can he determined automatically by proqram. See definition of IOPTl and IOPT2.

IOPTl

=1 Compute initial estimates for elastic modes usinq natural frequencies and FS and GS (defined subsequently). (See eq. (171.)

DO not compute initial estimates in this manner.

IOPTZ

=1 Use matrix iteration to obtain initial estimatr (appendix D, 0 = 2 and 3). This method attempts to cof:verqe on elastic and rigid- body characteristic roots (0 = 21 unless some S2I(II f 0. (0 = 3), if: which case method will attempt'to converge on nonzero roots nearest those input in array S2IlI). See also definition of OMR.

Individual parameters are optional if default values are acceptable.

103 TABLE 11.- Continued

,&I'll 1

I'KI'L'I' I 0 TABLE 11.- Continued

Alqebraic game 1 is t When required FORTRAN va riahli Description I variable 'INPUT IPKPLT = +2, +3 FURG, FEND, Lower limit, upper limit, increment hetween I FQEL. 6 NDF major tic marks, and number of minor divisions between major tic marks for ordinate in w plots. (Optional)

GOQD, GENII, Lower limit, upper limit, increment between GDEL. & NDG major tic marks, and number of minor divisions between major tic marks for ordinate in 5 plots.

'SENLOC ISENSE = 1 XS(1) 6 Arrays defininq sensor locations. Tnese should he unrotated coordinates I = 1,NS.

ITYPE( I) Inteqer indicating whether sensor- measure linear or an~pllarmotion.

=l Linear.

=2 Anqular.

NSS(1) Number of the structural Section on which sensor is located.

'SENINP ICASE = +2 & Sd ASD(1,J) Ith coefficient of denominator polynomial of ICONSYS = 1 or transfer function which models sensor ICASE = +1 & dynamics for Jth sensor (cunstant term ICHSEN = 1 first). I = 1, ..., 11. (manges only) "Sn XKS ( I, J) Ith coefficient of nomerator polynomial of transfer function which models sensor Synamcs for Jth sensor (constant term first). I = 1. ..., 10.

IflRD(J) Inteqer indicatinq type of sensor that sensor J is.

=0 Position Sensor. (Default)

=1 Rate sensor.

=2 Accrleration sensor.

i T4nYNI 1) Inttiqer indicatinq whether ta inciude sen5ur lyn*s>L 113.

=1 Iislnde sensor dynamics.

=o Perfect senljur; do not include dynamics. (Default)

ICASE = t2 & I&J Inteqers indicating control-sensor pair for ICONSYS = 1 or which sensor data are to be filtered. ICASE = +1 6 I and J refer to identification prior to any ICHFIL = 1 selection. Only control-sensor pairs for which data are to he filtered need he Input. End all filter input witi, O,O.

' FILTIN ICASE = +2 & hW1 & SN1 Natural frequency and dampinq ratio in second ICONSYS = 1 or order notch. See section "Compensation

ICASE = +1 & options. " ICHFIL = 1 K 1 BDO Gain for inteqral feedhack.

KlRDl Gain for proportional feedback.

KIRD2 Gain for derivative feedhack.

AN1 & AN2 Rme constants'of zero and pole in lead-lag filter.

'Individual parameters are optional if default values are acceptahle.

105 TAR1.F: 11.- Contlnurd

Alcirhral When requ1re-l c "ORTRAN va rlahlp variahle

' F I ;.TI N CASE = f_2 & 4FN(K,I,l 6 ICONSYS = 1 or AFD(K.I.1 ICASE = f_l 6 IC!IFI!. = 1

IF I!.TFR

'ACT !N1' 'An aAd

1 A<.I(( .I I

NMiIIIY5

N HN FW

NCNW

NtX'

NKNW

= !I

106 TABLE 11.- Continued

Rlqehraic lamelis t When required 'OUTRAN variable Description variable

DELPLT Z 0 IPLT2 Wray of numbers indicating which sets of characteristic root data are to he saved from file. If ICASE > 0 (interactive execution) then these can he modified before actual deletions occur.

PKPLT > 0 ICOUNT iumber of sets of characteristic root data to he plotted, currently, out of total number of gets of data qenerated during different cases. Complete list availahle is printed out before this input is required.

PKPLT > 0 IPLT irray of numbers indicatinq which Sets of characteristic root data are to be plotted.

PKPLT > 0 ICOMRIN [nteqer indicatinq whether to combine plots of different sets of characteristic root data into one plot (all on same frame) or generate plots on separate frames.

=1 bmbine plots.

=o )o not combine plots.

PKPLT > 0 5 PTITLE 3Osharacter description of data being plotted. ICOMBIN = 1 %is will appear on plot being generated in- lien of oriqinal plot titles when plots are being comhined.

'PLOTP PKPLT > 0 IDASH Indicates type of line used to connect points in data currently bring plotted. n-iis may be chanqed for each set of data plotted.

=o (solid line)

ISAME Inteqer indicating ha, many of next sets of Characteristic root data can be plottee with qame PLOTP namelist data currently heinq read-in, includinq current set. me proqram will not request another PLOTP namelist to he input until n = ISAME sets of data have been plotted. (Default = 1)

KVAR Input negative of value of KVAR that was used when determininq roots if you desire to change the independent variable in the graphics output (see definition of KVAR on p. 101).

MODEPK( I) Integer indicatinq whether to qenerate plot showinq Ith characteristic root variation.

=1 Generate plot of data. (Default)

=o Bypass Ith root information.

IPKPLT = tl or +3 XRLORG, XRLEND, Lower limit, upper limit, increment between XRLDEL, L major tic marks, and number of minor divi- NDXRL sions between major tic marks for abscissa in root-locus plots. (@tionall

'Individual parawters are optional if default values are acceptable.

107 TAULE I I. - Concluded

Alqehrai c Name 1is t FORTRAN variable Drscription va r i a h le

'PLOTP IPKPLT = +1 OK +3 YRLORG, YRLEND, Lower limit, upper limit, increment hptwern YRLDBI., 6 major tic marks, and numher of minor clivi- NIIYRI. sions between malor tic marks for ordinate in root-locus plots. (Optional)

IPKPI.'r = +J or +3 vow), VENI), [nwer limit, upper limit, inrrrmerit between VDFL, & NllV ma~ortic marks, and number of minor division? between malor tic marks for abscissa in i and w plots. (Optional)

Lower limit, upper limit, incrrment hrtwpen malor tic marks, and number of minor dlvi- sions between malor tlc marks for ord1nat.e in w plots. (Optional)

tionti, GENU, Lower limit, upper limit, incrrment hrtwet,n GDFT,, 6 NDG major tic marks, and numher of minor divisions hetwren major tic mdrks for ordindtr in 5 plots.

IOP Inteqer indicatinq whether to continue plottinq. 'This option allows user to qenrratrx additiondl plots of characteristic rcmt d.itn thdt either were not included heforr <>rfor which rh.inqes in thc plots arc de-.;] rpd.

=l Con t i niir p1 ot t 1 n ii .

=n Ketiirn to miin proqr,im and rontlnrle with n new c,1sr ,>r tt.rmin.3tr *-x,.<-rltion.

tIndividual paramrters arc optiondl if default ValIIPs arc acceptable. TABLE 111.- DESCRIPTION OF TAPE5 INPUT DATA

[All data are read-in using free-field format]

When Algebraic FORTRAN Description READ statements required variable array name

ICASE = +-2 KK and/or aerodynamic data were READ*,KK(K) ISELECT > C computed where KK(K)=kK

ICASE = +-2 RRDR & VARDI Real and imaginary parts of FOR K=l,NK and/or or aerodynamic motion forces; FOR J=I,NM(+NG)+ ISELECT > AR & AI pairs are read-in b FOR 1=1 ,NM (per pressure mode) T per READ*,(VARDR(K,J,I),VARDI(K,J,I) frequency and transposed where VARDR(K,J,I)=Re{Q(kK)I,J} i nterna 1ly and VARDI ( K, J,I )=Im{ Q(kK )I ,J} MAT1 &neralized masses; read-in FOR I=l,NM and/or by rms FOR J= 1 ,NM ISFLECT > 0 READ*,MATl (I,J) where MAT1 (I,J)=MI,

ICASE = +2 {fn) FREQ Vatural frequencies (Hz) FOR 1=1 ,NM and/or READ*,FREQ(I) ISELECT > 0 where FREQ(I)=fnT

ICASE&= +-2 {9:} MOMSQ Zeneralized stiffness matrix; FOR 1=1 ,NM and/or read-in by rows FOR J= 1 ,NM ISELECT > 0 1 READ*,MOMSQ(I,J) where MOMSQ( I,J) =KI, J ISTIFF = 1

ICASE = +_2 G Structural damping coefficients FOR 1=1 ,NM and/or READ*,G ( I ) iSKLECT > 01 where G(I)=gs I ICASE = f2 H DS Sensor deflections; read-in by FOR J=l ,NS and/or columns per sensor FOR I=l,NM ISELECT > 0 READ*,DS(I,J) T & where DS ( I,J) =H IrJ ICSREAD = 1

ICASE = 22 HE Actuator elongation per unit FOR I=l,NC and/or generalized coordinate; FOR J=l ,NM ISELECT > 0 read-in by rows per control READ*,HE(I,J) & ICSACT = 0

= 52 Generalized force per unit ICASE FD FD FOR J=l ,NC and/or actuator hinge moment; read-in FOR 1=1 ,NM ISELECT > 0 by columns per control READ*,FD ( I,J) & ICSACT = 0

'If gust forces are included as last NG columns of aerodynamic data, provision has been made to read these in (NGfO), although they are not stored in these arrays.

109 TABLE IV. - DESCRIPTION OF TAPE1 0 INPUT DATA

[All data are read-in using free-field format]

- ~~ ~ ~~ When gebrai c FORTRAN Set A1 Description READ statements - required variable irray name 1 INTERP = 1 or x or y' TAB( I,1 ) x-coordinates if IPLATE = 1 FOR I=l,NNODES ISENSE = 1 or points along y'-axis READ*,TAB(I, 1) if IPLATE = 0 at which where TAB(1,l )=xI or y; - modal data are available 2 CNTERP = 1 or y Qr z!y') TAB ( I,2) y-coordinates at which FOR 1=1 ,NNODES ISENSE = 1 modal data are available READ*,TAB(I,2) if IPLATE = 1 or bending where TAB(I,2)=yI or z(y;) deflections along the y'-axis if IPLATE = 0

~~~ 3 INTERP = 1 or z or R'(y') Modal deflections at (x,y) FOR 1=1 ,NNODES Y ISENSE = 1 if IPLATE = 1 or rotatioi READ*,TAB(I, 3) about y' at points along where TAB(I,3)=ZI or R'(Y;) Y - y'-axis if IPLATE = 0 4 INTERP = 1 or 0 or %Cy' ) Zeros if IPLATE = 1, or FOR 1=1 ,NNODES ISENSE = 1 rotation about x' at READ*,TAB(I, 4) and points along y'-axis if where TAB(I,4)=O or Ri(Y' ) NDOF = 4 IPLATE = 0

Repeat sets 1 to 4 for each mode (1 to 3 if NDOF = 3)

110 TABLE V.- DATA STORED ON DATA STORAGE FILE TAPE1 BY PROGRAM

Record Algebraic variable ~ 3RTRAN array name Size Description

1 M MAT1 NMODES**2 Selected generalized masses NMODES ISelected Mun2 array 2 MOMSQ

3 fn FREQ NMDDES I Selected frequency (Hz) array

4 SCOF & BN NCOEF*NMODES**2+250 Selected p-plane coefficients and bnj array

5 IAlSUB I 102 I Subindex array for PKPLOT data

6 kIT & Re[Q(ki),i=l, ...,nkl RFRQ & AR NK+NK*NMODES* *2 Selected reduced frequencies and I real part of aerodynamic data 7 AI NK*NMODES* *2 Selected imaginary part of I aerodynamic data 8 HE NCN LW *NMODES Selected actuator elongation per I I unit generalized coordinate 9 FD NMODES*NCNEW Selected qeneralized force per FD unit actuator hinge moment

10 H CS NS* (NMODES-NCNFX) Selected sensor deflections

11 G I NMODES I Selected structural dampings

12 DCT, CNTLN, & ICLN NCTM+(MPM+l)*NS*NCNEW Control logic common denominator (d(s)1, numerator elements, and number of coefficients in each numerator element

SCOT & E?? NCnRF*NM**2+250 I puii sec of p-piane coefficients I and bnj array

-~~ INDEX4 for SPLINE 51 Subindex array for geometric coefficients used in surface spline interpolation

H cs NS* (NM-NC) Full set of sensor deflections

INDEX for FILTIN 51 Subindex used in storing filter data

20 Computed All intermediate control data aSn I a R7, .. . ASN,ACLN, .. . used in computing d and N. Ij All labeled commons All labeled common data from previous case for restart , capability 111 Inputs outputs

Fiqure 1 .- STABCAR dynamic elements.

Fiqure 2 .- Characteristic root determination with STABCAR.

112 U t

Figures 3 .- Geometrical relationship between (x,y) and

I I

Figure 4.- STARCAR overlay structure.

113 'I 1 I 7

Figure 5.- Flow diaqram of program STABCAR. If !IPLA'IE:?. call SFLA!:I 5 P L P.N E overlay to determnine p- -(-) plane fit to aerodyndmic, r----+o for selected rncdes. .c Prosram %PLANE

iets ur array Selected aerc:,naric data e.~ployeC storage areas. 7 1 ir fit are read-in from :APE! A constrained least-s~juares fit is perfcrtoed tn ''etermine coeffic1en:s

d-)

traisfer function

eacn sens?' iqw

Determine fi:ter dynamics.

Cverall matrix of transfer functions lS i)"idi?,el by performing natr,x product of sensor filter and actdator transfer frr nc: 1 DR ~d ?rI ces . These data are Stole6 on randon access file TfiPEI for (use in subse,uent overlay

Figure 5 .- Continued.

115 1

-1

Fiqure 5 .- Continued. If IAPLT=!l, call AEROPL: +-) AE ROP L I 4-) overlay to plot (selected , -QMULIPI aerodynamic motion forces f t Subroutine APLOI Subroutine MULTPT

Selected (MOOEA(I,J)=l] aerodynamic force Plot5 ,,?"?rated storage areas. elements are generatej. cbrves. If IAPi'=I. interpolated curve fits !hJsed upon Cnoicf df KrIT) dre qenerdted. Ii ISPLAhEtO. p-plane fits are also Generated If IAPLT-.! (ISPLGt~Cmust be nonzero!. onl the p-plane fits BW qenerated.

If IPKPLI10. call PKPLOT Enter OverlJy to plot data -(-) -(-) generated by overlay PKFLUT. i. i f Sdbroh:ire PiPLT SUbFOutine PLOT Read-in plot data from TrlPEl. 'lot flutter data Plot , n YS rer mode for which 'velocity, altitude lODCPI:=O. density, dynarnlc pressure or gain depending upon YYAR. IGAIN. and 7N o c '1~t:a:a from several previously j?nr,'dt@d Clot sei,. :?ps+ ,#,,mber of jets to be plattea. ~~n.e2p?nAlnaSet ?os. and whether to combine them. In- ut nanelist PLOT?. ;See description if &t,j fcr !PvPLT 0 In description if input.] I

Return to begiinlnq to input data to execilte another case or terminate execution.

Figure 5 .- Concluded.

117 Fiqure 6.- Paneling of winq and horizontal tail €or aerodynamic force computation.

if N < 0.61, set N = 0.61 bls Ma

Figure 7.- Control law and actuator dynamics.

118 2 p-plane fit 1 Interpolated

-.lo x

-.30 0 1 2 0 1 2 0 1 2 Red. freq. Red. freq. Red. freq. t

-50

- 7n .LY 0 I 5 z t

-1.00--50 - .40 0L 1 2 0tLLu 1 2 0 1 2 Red. freq. Red. freq. Red. freq.

(a) Column 3.

Figure 8.- Selected generalized force matrix elements.

119 2 p-plane fit 1 Interpolated

(3,5) -4r r t 0 I x h id id c 4t- c .d -d D 0 2 2 H H -1.0

1 1 -2.0 1 0 1 2 0 1 U 1 2 Ked. frcq. Red. freq. Red. frcq. r (3,5)

.8 -.2

4 id 2

-.6 -4 t- t -1.0 -2 0 1 2 0 1 2 0 1 -3 Ked. frcq. Red. freq. Ked. frcq.

(b) Column 5.

Fiqure 8.- Continued. 2 p-plane fit 1 Interpolated

120

x 0 x 8o LI k ld ld c d .4 4 m m ld E 2 H H -2 40

-4 L 0 1 2 0 1 2 0 1 2 Red. freq. Red. freq. Red. freq.

i I

15 -40

ri ld 2

11 -80

7 -120 0 1 2 0 1 2 0 1 2 Red. freq. Red. freq. Red. freq.

Figure 8.- Continued.

121 2 p-plane fit 1 interpolated

0 2o i1°-3

-.01 0 t x h &4 t h m m c 2 G -4 .r( rn rn 2 H H2 -.02 V' lob+0 -.lo -.03 -10 0 1.0 2.0 0 1.0 2.0 0 1.0 2.0 Red. freq. Red. freq. Red. freq. r (5,15) (8,15)

.18 .10

4 m m -I a, m P; 2 ? 2 .14 .09 F.. t P .10 r,llJ .08 L 0 1.0 2.0 0 1.0 2.0 0 1.0 2.0 Red. freq. Red. freq. Red. freq.

(d) Column 15.

Figure 8.- Concluded. Mode number

I+ /+ 0 h=O

fh = 5000 km

7 B

@ 6 e

+ 31 - - 2 0, 1 I I I 1 I I I I I 1 1 1 - 1 I 1 I 1

(a) Altitude root loci.

Figure 9.- Open-loop stability characteristic variation with altitude at NMa = 0.86.

123 (b) 5 and wn versus altitude.

Fiqure 9 .- Concluded. i124 8

0 p = 0.80 kg/m3

300 Ap = 0.25 kg/m3

+ I 7 Q

6 0 5 /+-+ @------

L 0 111 Ill Ill IIlII1I 111 d -120 -100 -80 -60 -40 -20 20 ~e(“i), raii/sec

(a) Density root loci.

Figure 10.- Open-loop stability characteristic variation with density. Nm = 0.86; U = 269.6 m/sec.

125 6 5

(b) < and wn versus density.

Figure 10.- Concluded.

126 400

Mode number 2 0 q = 30 kN/m l8 2 Ay = 9.08 kN/m

300

7 8 U a, m \ P

L4 6 e S

- + \3, - -

0 IlII1Il Ill Ill Ill 1 I rq d 20

Re( ?,i) , rad/sec

(a) Dynamic-pressure root loci.

Figure 11 .- Open-loop stability characteristic variation with dynamic pressure. = 0.86; U = 269.6 m/sec. NMa

127 200

~1 1,2 -

LL L J AJ

2 Dynamic pressure, kN/m

(b) 5 and an versus dynamic pressure.

Figure 11 .- Concluded.

128 6o r

IIIIIIIII 11111111I 111I11111 Ill11 1111 1 111 111 11

Mode number

3 0

-.5

-1.0

Figure 12 .- Characteristic root variation with velocity ( reduced-order model 1. p = 0.6101 3 *Ma = 0.86; kg/m .

129 Mode number 60 r

3: 5 c 3

3 :1 / F 0 t

Velocity, m/sec

(h) Closed loop.

Fiqure 12.- .Concluded.

130 G p-k analysis 31 ----- p-p analysis XO 0 0.5 0 1.0 0 2.0

rModsnurrJJer

0 II-~ L._- -200 -150 -100 -50 0 50

Re( ), rad/sec

Figure 13 Elastic mode qain root loci at = = 4.572 .- U,,. NMa 0.86; h km. 500

400

u 300 ar Lo \ a L-i -i E H 2LiO

100

0 -200 -150 -100 -10 0

RC(, ii.), rad/scc

Figure 14.- Effect of phase errors in control signal elastic mode gain root loci at UD (p-p analysis). NMa = 0.86; h = 4.572 km.

I132 600 G31 XO 0 0.5 0 1.0 0 2.0

400

p-k analysis

-700 -600 -500 -400 -300 -200 -100 0 Re( hi ), rad/sec

Figure 15.- Control root loci at UF. NMa = 0.86; h = 6.705 km.

133 / / /

f i f f / f

/ f I 2 x f ii / r: f f / / / / f t

I I I I 1 I I I I

134 1 Report No 2 Government Accession No 3 Recipient's Catalog NO NASA TP-2165 4 Title and Subtitle 5 feport Date STABCAR - A PROGRAM FOR FTNEING CHARACTERISTIC ROOTS June 1984 Sr SYSTEblS HAVING TRANSCENDENTAL STABILITY MATRICES 6 Performing Organiratlon Code 505-34-33-05

7 Author(sJ 8 Performing Organizatlon Report No William M. Adams, Jr., Sherwood H. Tiffany, L- 1 4861

Jerry R. Newsom, and Ellwood L. Peele 10 Work Unit No 9 Performing Organization Name and Address

NASA Langley Research Center 11 Contract or Grant No Hampton, VA 23665

13 Type of Report and Period Covered 12 Swnsoring Agency Name and Address Technical Paper

14 Sponsoring Agency Code

~~ 15 Supplementary Notes

16. Abstract

STABCAR can be used to determine the characteristic roots of flexible, actively controlled aircraft, including the effects of unsteady aerodynamics. A modal formulation and a transfer-matrix representation of the control system are employed. Oper- able in either a batch or an interactive mode, STABCAR can provide graphical or tabular output of the variation of the roots with velocity, density, altitude, dynamic pressure or feedback gains. Herein the mathematical mode 1, program structure, input requirements, output capabilities, and a series of sample cases are detailed. STABCAR was written for use on CDC@ CYBER 175 equipment; modification would be required for operation on other machines.

17 Key Words (Suggested by Author(sJJ 18 Distribution Statement ISAC Stability Unclassified - Unlimited Flutter Root loci p-k method p-p method Active controls Subject Category 61

19 Security Classif (of this report) 20 Security Classif (of this page) 21. No of Pages 22 Price

Unclassified Unc lass i f i e d 143 A0 7