AERO-SERVO-ELASTIC STABILITY ANALYSIS at flutter, the combined aero-servo-elastic stability analysis does not obtain physically meaningful frequencies and dampings at flight conditionsAuthors:

(7 7 Report No.

WPR (NASC) TR-79-1

AERO-SERVO-ELASTICSTABILITY ANALYSIS

byWilliam P. Rodden, Mildred R. Zeifman

andJohn M. Powers, Jr.

Prepared forDEPARTMENT OF THE NAVY

NAVAL AIR SYSTEMS COMMANDWashington, D.C. 20361

UnderContract N0001 9-76-C-0346

April 1979

DISTRIBUTION UNLIMITED:APPROVED FOR PUBLIC RELEASE

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Repo"t NO.

WPR (NASC) TR- 79-1

AERO-SERVO- ELASTICSTABILITY ANALYSIS

byWilliam P. Rodden, Mildred R. Zeifman

andJohn M. Powers, Jr.

Prepared forDEPARTMENT OF THE NAVY

NAVAL AIR SYSTEMS COMMANDWashington. D.C. 20361

UnderContract NOOO1 9-76-C-0346

April 1979

DISTRIBUTION UNLIMITED:IAPPROVED FOR PUBLIC RELEASE

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10. SUPPLEMENTARY NOTES

It. KEY WORDS (Continue onl towers* aide 1 iffl*0*0d& aid Ideifyj~ &V block number)

Aer-servo-laticityAsroelasticityStructural Dynamics

Control System Stability5~STRACT (fihnlnuo en reveriee side of ne~tcssar end Identify or block number)

A georal matrix formulation of the aero-servo-elastic stability problemis presented based on the British method of flutter analysis. The analysisapplies to the closed-loop system, and the digital computer program (HPASES)developed to perform the analysis is described. An application is madeto a typical air-to-air missile with three feedback loops.

DO I JH17 1473 EDITION oil I Nov 6 s as otSOrTE I 1Q *>

SECURITY CLASSIFICATION OF THIS PAGE (Whlen Doe Entered)

SLM44AY

A general matrix formulation of the aero-servo-elastic stability

problem for a closed-loop system is presented. It is based on a classi-

cal method for determining servomechanism stability (using linear

differential equations) and the British method of flutter analysis which

represents the aerodynamic forces as frequency dependent springs and

dampers. The combination of the aerodynamic forces with the mechanical

springs and dampers in the equations of motion and then with their

electro-mechanical equivalents in the servo system leads to a consistent

formulation of the linear equations of motion for the closed-loop aero-

servo-elastic system. An iterative real eigenvalue solution accounts

for the frequency dependence of the aerodynamic forces which is only

secondary.

The digital computer program MPASES Cmodified program for aero-

servo-elastic stability) developed to perform the analysis is described.

Dynamic storage allocation is utilized throughout MPASES to provide the

most efficient use of core storage through the variable dimensioning of

all arrays. Program limitations are minimal allowing stability results

in one computer run to be obtained for up to ten velocities at each of

five altitudes. In addition, as many as twenty-five servo element gain

va~1ation, are peatitted.

41p-,,

. x , ,. L. . .. . . . . ¢ , i, - - , t -- .. .. .. i. ._ .2 . . .. . .. .. .. .. . .... i'lI

II [I II I I III I I"0i

CONTENTS

Sect ion Pg

I INTRODUCTION .. . . . . . . . . . .. . . . . . . . . . 1

II CLOSED-LOOP AERO-SERVO-ELASTIC STABILITY ANALYSIS ....... 4

New Equations of Motion ............................ 4

The Eigenvalue Problem ............................ 10

Lining Up the Reduced Frequency .................... 13

Linear Spline Interpolation of Aerodynamic Terms ... 15

III AERO-SERVO-ELASTIC STABILITY ANALYSIS PROGRAM .......... 20

Program Description ............................... 20Subroutine Description .......................... 22Programming Symbols............................. 25Processing and Programming Considerations .........31

Input Instructions ................................ 33Units ........................................... 33Data Deck Setup ................................. 33Input Data Description ...........................34

Program Output Description .........................44

IV SAMPLE PROBLEMS ...................................... 46

Case I Description ................................ 48

Case 2 Description ................................ 49

Case 3 Description ................................ SO

Keypunch Forms .................................... 52

output ............................................ 58Case 1 ..........................................58eCase2 .......................................... 65Case 3 .......................................... 75

V PROGRAM FLOW CHART ................................... 89

VI PROGRAM LISTING ...................................... 92

MIAIN............................................. 92

MIASS............................................. 95

SERVO............................................ 96

FORI4AB ............................................ 99

ection baze

W~AS * *.*..*.* * ****......,. 103

SY1~ESUL ****.*****.**e****.*..*ooooo.*.e*a****o...* 19

INTERP 10

SPLINE 10

INVERS 11MAHP Ill-- - --- ~ n - --

ZGN=~~~~~~~ ooo4oooo~oooeoo~ooeo*o 1

I LLUSTRATIONS

F inurePage

1 IUniform Free-free Missile .............. 46

Servo Block Diagram . . .... *S****........ .... 47

3 Idealization of Typical Air-to-Air Missile .... 129

4 Idealization of Flipper Mass Distribution ...... 129

S Servo System Block Diagram ............ 138

iv

SECTION I

INTRODUCTION

This report documents a modification to Ref. 1 which presented a digital

computer program for the analysis of aero-servo-elastic system stability.

Reference 1 formulated the problem by combining one of the classical methods

for determining servomechanism system stability with the American method of

flutter analysis. Since the American method of flutter analysis represents

the aerodynamic forces as frequency dependent masses and utilizes the mathe-

matical concept of an artificial structural damping to detetmine the reduced

frequency at flutter, the combined aero-servo-elastic stability analysis does

not obtain physically meaningful frequencies and dampings at flight conditions

other than neutrally stable ones. On the other hand, the British method of

flutter analysis represents the aerodynamic forces more realistically as

frequency dependent springs and dampers. This is only an approximation for

transient motion since the aerodynamic forces are generally known only for

harmonic motion and it is not completely correct to identify the forces in

phase with the displacements as aerodynamic springs and the forces in phase

with the velocities as aerodynamic dampers. However, comparlsons of results

from the various methods for flutter analysis have been made by Jocelyn

Lawrence and Jackson in Ref. 2, and it has been found that the British

method of flutter analysis leads to reasonable predictions of transient aero-

elastic behavior when the damping levels are low.

The British method of treating the aerodynamic loads as springs and

dampers permits combining the aerodynamic forces with the mechanical springs

and dampers in the equations of motion and then with their electro-mechanical

equivalents in the servo system. This results in a consistent formulation of

the equations of motion for the closed-loop aero-servo-elastic system because

the frequency dependence of the aerodynamic forces is only secondary.

Although the frequency dependence is secondary, however, it is important and

an iteration is necessary to "line-up" the reduced frequency for which the

aerodynamics are determined and the frequency determined by the equtions

of motion; one iteration, however, is sufficient to achieve sufficient

accuracy between the aerodynamic forces and the equations of motion.

The iteration begins at a particular velocity by assuming the reduced

frequency k=O. (The aerodynamic damping can only be found from a low fre-

quency kO.O but an interpolation scheme will estimate the damping at k-O.)

Solving the eigenvalue problem from the equations of motion yields either -

separate and real roots or complex conjugate pairs. Any real roots found

for kuO are correct, e.g., the rigid body roll-damping root or a structural

divergence root. However, the complex conjugate roots must be lined up and

the lowest frequency roots permit the aerodynamic forces to be revised (by

interpolation) to the corresponding reduced frequency. With the revised

aerodynamics, the eigenvalue problem is solved again and the lowest oscilla-

tory root is obtained along with an estimate of the next higher frequency.

The aerodynamic forces are then revised again to correspond to the estimate

of the reduced frequency of the second oscillatory mode and the new eigen-

value problem is solved. This results in the second oscillatory root and an

estimate of the third frequency. The process of revising the aerodynamics

and solving new eigenvalue problems continues until the desired number of

roots have been obtained for the particular velocity. The whole procedure

is repeated for the next higher velocity and is continued until the velocity

range of interest has been covered. From the solutions for frequency and

damping of each mode, root locus charts can be drawn for use in redesign of

servo components, e.g., amplifier gains, or mechanical components, e.g.,

mss balancing.

The modified program for aero-servo-elastic stability (MPASES) requires

four revisions to the program PASES of Ref. 1. The first is the inclusion

of the aerodyna . forces as spring and damping terms in the equations of

motion rather than as complex inertial terms. The second is the iterative

eigenvalue solution required to line up the frequencies between the aero-

dynamic forces and the equations of motion as tiscussed above. The third

problem is the interpolation of the aerodynamic forces necessary in the

iteration to minimize aerodynamic computational expense. The last problem

is the result of the new formulation dealing only with real matrices; a

specialized eigenvalue extraction method that analyzes real matrices which

have either real or complex conjugate roots may be utilized for computational

efficiency. Each of these modifications is discussed in the following sections.

3-

3 '

"7 SECTION II

CLOSED-LOOP AERO-SERVO-ELASTIC STABILITY ANALYSIS

New Equations of Motion

The theoretical derivation of the equations of motion for the PASES

computer program is taken from Ref. 1 and is reproduced in this report in

the Appendix for ease of reference. It is only necessary here to rederive

the aeroelastic equations of notion since the servo equations are not

affected by the change in representation of the aerodynamic forces from

complex masses to real springs and dampers.

Our new definitions of aerodynamic influence coefficients (AIC's) are

taken from the survey of unsteady AIC's in Ref. 3. The unsteady force is

given in terms of the deflections and their velocities by

The AIC's [Chs ] and [ChDh] may be regarded as static and dynamic stability

derivatives but are, in general, fLnctions of the reduced frequency

k wb /V (2)r

where

br /2 (3)

Assuming the AIC's to be constants independent of k simplifies the analysis

and is adequate for low frequencies, but it is inaccurate at higher fre-

quencies and the reduced frequencies should be lined up with the frequencies

obtained from the equations of motion as discussed in the Introduction.

However, the choice of lining up the frequencies or not is left to the

user in his choice of frequency dependence of the AICI.

41Igg "2- j.ai

Adefinition ofcmlxoclaoyACswas also given in Ref. 3 as

WF - PW 2b 2SC](h}(4

for use in the American method of flutter analysis. Equations (1) and (4)

must be identical for harmonic motion and we have

(qS/z)([Ch5J + i(wc/V)[CSh]){hl . pW2 b 2S(ECR + .9- ]C (5)

where (ChR] and (CMI] are the real and imaginary parts of (Ch] respectively.

Identifying the real and imaginary parts of Eq. (S) leads to

(Chs] 2k 2 (Es/S) [ChR] (6)

and

[ChDh1 kCEs/S)(C hI] (7)

Equations (6) and (7) determine the aerodynamic stiffness and damping AIC's

when the oscillatory AIC's are given in the format of Eq. (4), as is the

case with a number of available computer programs, e.g., Refs. 4 and S.

In terms of the stiffness and damping AIC's, the equations of motion of

the aeroelastic system including the rotations of control surfaces appear as

(cf. Appendix.A, Eq. (2-15))

(M](h} + (C](fi} + (K]({hl - (ha](6}) u(}(8a)

-(qS/E)([Chsj(h} + 1(SDh){E/V}) (8b)

or

Q (14(h} + (CJ(A} + (K](hl (K](h 6]l - 0 (9)

where

[C C] - PVS(Chh] (0

[R] C K] - CqS/E)(Ch] (1

The modal solution proceeds as before (see Appendix L). The series for

the deflections is

Wh - (hh1f a ) + (hp]{aR} + [h61{61 (12)

Substituting Eq. (12) into Eq. (9) anid pTOMUltiplying by (hF]T leads toF3.

the modal equations for the flexible degrees of freedom.

4- It (13)(F](aFl [tR.R + btFdJ (13

+*J (]ad} IJFR1 N) + 1( a)tS 0

where

IMF] - [hF] T([M]hF] (14)

(MFR] - hF]T(M](hR] (15)

IMF,] - (h,]T(M](hs] (16)

[h (]T C](hR (lea)

UaR 1 R

a hpSh 1T ~~h h I(1b

[FS - [hF]T[C] [ha] (19a)

A -IPVS[hF] TChvh] (h6] (19b)

C1F] [ [hF]T[C]hF] (20)

(jtFRJ [hF]T(R hR] (21a)

- -(qS/E)EhF] T (Chs] ChR] (21b)

(tFa - thF3T JR11h6l (22a)

= -(qS/E) h (F]T[Chs] [hs] (22b)

Equation (13) may be compared to Eq. (2-20) in Appendix A. We have utilized

the fact above that the rigid body displacements cause no internal damping or

structural forces. We have assumed the vibration modes are either free-free

modes for the entire system or restrained modes for individual components,

but may not be arbitrarily chosen modes. The limited orthogonality of the

free-free or restrained modes (i.e., [tMF] is not a diagonal matrix unless

all modes are free-free modes) leads to a diagonal form for the generalized

stiffness matrix rKFj. If we denote the diagonal elements of [V.] by fMFJ,

then the generalized stiffness matrix is

rxF- [hF]T[K](hF] (23a)

a [(23b)

and

( •~ * - (qS/E)[h] [hF] (24)

7

The generalized structural damping matrix does not have a diagonal form, as

does the generalized stiffness, but is assumed so as an approximation that is

Justified by low levels of structural damping. The approximate form of the

equivalent viscous structural damping is

rcFJ - L'F/'WP'XFI (25a)

fCFI rgF/ yF J (2Sb)

so that

Ea,) fC J - VpVS[hF][Ch~h](hp] (26)

Next, substituting Eq. (12) into Eq. (9) and premultiplying by (hR]T

leads to the modal equations for the rigid body degrees of freedom.

(MRF]I(!F} + (MR] {MR + (MR6] {tS

" (IaIFI( +~(~ [ CR]'R IRd(S) (27)

" (.a) + (kit] {N) (R 8](6) 0

where

,J " ,(28)

SR,]. [hR] T t4] [hR1] (29)

(MMRJ - [hRIT[t] [h 6 1 (30)

(;] . ChR]T[] Ch,] (31a)

I IOVS(hR ]TChDh hF] (31b)

8~ *

h- j-Tt [h(j_ (32a)

S- 4pVS~hRJ T [ChDh ]hR (32b)

[;6] ChR]T[C] [h6 ] (33a)

= - 4pVS[hR]TChDh ](h6] (33b)

tNFI - hIT (] [hF] (34a)

C(qS/E)[hR T [ ]hF I (34b)

*kI [hR)T (R] ChR] (35a)

= -(qS/c) [hR] T (Chs (hR] (5b)

[kRu] [ [hR]T(Chs] [h6] (36a)

-(qS/c)hRI Chs ] [h6 ] (36b)

and we have again noted that rigid body displacements produce no internal

damping or structural forces. Equation (27) should be compared to Eq. (2-29)

in Appendix A.

The new matrix partitions in the aero-servo-elastic equations of motion

now become

[MFI~~~~ -NR

IF ) I

a 11 [[ (MR] (MR6] 'RCK3j](Wi - aR (37)

0 0 Ccj CS2]

PLFSAJ LRSAJ (SC] [SS2], i2

9

• -. . . . . . .* .. * . ... .... ........ . .-

*XlItl (YF p"R (R61 0R (38)0 c 0 (CC [CS2] AI

LFSGI LRSGJ [SCI (SS2. ;2

~FJ (ku c;FO 0 af

(X X)- [I Ril Cl (Z 0 SR (39)0 0 rCCj [CS2] 8

0 0 (SC] [SS2J .2

These may be compared with Eqs. (2-43), (2-44) and (2-45) -in Appendi A.

The coefficient matrices (X;] and (Xx] each have five new nonzero partitions

as expected from moving the aerodynamic terms from the mass matrix to the

stiffness and damping matrices.

The Eigenvalue Problem

The new representation of the aerodynamic forces changes the eigenvalue

problem only to the extent that the matrices are real now rather than complex.

The equation to be solved is still

(y(A] + [B])(V} % 0 (40)

where the amplitudes of motion, (V), are defined by

(vI - (V}eCxpyt) (41)

and instability occurs when the airspeed and/or control system gains are

such that the real part of y is positive.

10

Although (A] and [B] are now real matrices, they still may be singular

and obtaining the canonical form of the eigenvalue problem by a shift in

eigenvalues is still appropriate. We let

Y U YO - l/A (42)

where y is an arbitrarily chosen real number, and then the new eigenvalue is

a 1/Cy o - y) (43)

and the new eigenvalue problem is

-- A N CYo[A] (B])'[A](V} (44)

The shift value yo is arbitrary to the extent that it must be chosen so the

linear combination ya[A] + CB] is nonsingular. A value which scales [A) to

be the sam order of magnitude as [B] and of the same sign is recommended.

The eigenvalues of Eq. (44) are either real or complex conjugates. A V

subroutine ALLMAT (Ref. 6) for complex matrices was used in Ref. 1.

A more recent development for the real case of Eq. (44) is the sub-

routine LIGRP given in the International Mathematical and Statistical Library

(UL, Ref. 7). Subroutine EIGRJ calls I14SL routine EBALAF to balance the

matrix. Then INSL routine EISS reduces the balanced matrix to an upper

Hessenber8 form and routine EQIM3 computes all of the real and/or complex

conjugate pairs of eigervalues of the essenberg matrix.

The IfML Packap is universally used and is usually incorporated into

the scientific libraries of major computer systems. When requested, the

eigeuvectors are found in two iterations by the Inverse Power Method with

Shifts (Ref. 8, pp. 323, 626-628) in subroutine E QVCT (Ref. 9). Subroutine

11

908VC finds the sigeuvector (u) from a complex matrix [u] and its complex

eigevalue A by solving the equation

[U - AIJ(u) - 0 (4S)

The subroutine is used in the present development by setting

A -o (46)

and

[U] y y[A] [B] (47)

in Eq. (45). If eigenvectors are requested, all real eigenvalues are used

in Eq. (47), but only the complex conjugate eigenvalues with positive

imaginary parts (positive frequencies) are used, since the eigenvectors are

also complex conjugate pairs.

Writing the complex eigenvalue as

Y a * + iW (48)

where 0 is the decay rate and w is the damping frequency, we find the cyclic

frequency to be

f a w/2w (49)

and the fraction of critical damping C to be

C - * (" (SO)

For comparison to structural damping levels, twice the damping ratio C is

a preferable output quantity since

C- g/2 (s)

12

for a structurally-damped single degree of freedom oscillator. For a non-

oscillatory root a different definition of damping ratio is necessary and

we chose the time to half amplitude

T (S2)

If the motion is unstable, i.e., V > 0, Eq. (52) gives the (negative) time

to double amplitude. The stability of the oscillatory roots can also be

compared using Eq. (52) and this will be an additional output quantity.

Lining Up the Reduced Frequency

The need for lining up the reduced frequency for a specified mode of

motion with the frequency determined by the eigenvalue problem for that mode

was discussed briefly in the Introduction and at some length in Ref. 2. The

necessary equations for the iteration are given in this section.

It is possible to begin the iteration with any value of k. However, a

finite value of k may be representative of an oscillatory mode but we are

equally interested in static modes. Therefore, we begin with knO and any

real roots will be determined, e.g., the roll-damping root, a static struc-

tural divergence root, and any over-damped roots from control system compon-

ents. The first oscillatory root for a free vehicle may be either the short

period mode or the Dutch roll mode and the choice of k-O provides a good

estimate of that.

Let the complex conjugate pairs of roots be denoted by

Yrs a urs ± iWrs (53)

where r denotes the oscillatory mode number ordered by frequency,

(w15 <2s < "" ) and s denotes the number of the mode under investigation.

13

The reduced frequency is

k a wc/SV (54)

and the aerodynamic forces are determined for k s which should be

ks - wssE/2V (s)

However, w is not known at the outset- From the initial solution with k0 no,

we estimate the nonzero frequencies (w10, W20 1 W30 1 -"). Refining the aero-

dynamics by interpolation for k1 - W10E/2V and repeating the eigenvalue solu-

tion, we find a new sat of frequencies (w1 1, W2 1 W311 "") The value of

W 11is taken as the correct value for the first mode and its damping all

determines its stability. The eigenvectors for the first root w may then

be calculated if the mode shapes are desired.

The aerodynamics are next determined by interpolation for k2 a W2 1E/2V

and the next eigenvalue solution yields the frequencies (w12 1 w2 2 ' '32 ' "" )"

The frequency w2 2 is now taken to be correct and its damping a22 measures

the second oscillatory mode stability. We continue with k3 M W3 2 br/V,

finding the aerodynamics for k3 , and then the eigenvalues; w3 2 and a are

assumed to be correct. The process continues until all roots of interest,

and all eigenvectors, if requested, have been found.

Care must be exercised in tracking the oscillatory roots because, as

the reduced frequency is increased the roots which were real at k-O may

become complex at higher values of k or roots which were complex at k-O

may become real at higher k's. A suitable algorithm is to choose wss as

the closest value to w s 1,s- and then to choose w s+ls as the closest hither

value to W

14

Linear Spline Interpolation of Aerodynamic Terms

In the modal matrix equations of motion, twelve of the partitions

depend on the aerodynamic influence coefficients (AIC's) which are functions

of the reduced frequency k. As the reduced frequency changes when the roots

are tracked in the frequency lining up process, the AIC's will change and

interpolation on k is necessary. It is computationally simpler to inter-

polate the partitions rather than the AIC's. Two of the partitions depend

on structural parameters also but these are independent of k and can be

carried along in the interpolation. A linear spline is chosen for the

interpolation because it offers cubic accuracy and continuity throughout

the regions of interpolation and extrapolation.

A linear spline is a mathematical device for interpolating for a

function y(x) for all points x, when y is known for a discrete set of

points, Yi " y(xi). The spline passes through all of the known points.

The "thematical spline takes its name from the plastic spline used by

draftsmen for drawing curves through specified points. If the plastic

spline may be regarded as a uniform beam, the linear spline representing

it mathematically is a solution to the uniform beam differential equation.

The following derivation for the linear spline is taken from Ref. 10, App. E,

by R. L. Harder.

We wish to determine the deflection curve of a continuous beam over

aultiple supports. Consider the fundamental solution to the deflection

equation

El dx4- w (S6a)

a 0 (S6b)

is

that is symetrical about a support at the origin x O. The solution is

y An ak *j Cx2 + Dlxi' (S7)

Continuity of slope requires B 0, and equilibrium with the support load P

requires

c/2 c/2li" f/ dx- Iii EI AIX dx )

c -0 - .£ 0 - I d x(

a lim El !y +c/2 (S8b)_-0 dx3 I-/2

a P (S8c)

From Eq. (S7) the third derivative is found to be

!9X 6D sgn x (59)dx3

and Eq. (S8b) becomes

(12 1) D p (60)

Combining Eqs. (57) and (60) and generalizing for all supports at x a x

leads to

y Ix) a + C i(x -x) QiIx -xi') (61)

16

where Qi Pi/12EI. At a large distance from the supports, the deflection

curve should be linear. For large x " -, Eq. (61) behaves like

y(x) - x 13&n x (Ci " 3xisXgn x) o(x) (62)

and the required linear behavior necessitates

Q - 0 (63)

C - 0 (64)

qixi o. 6(6S)

Equations (63) and (65) are equilibrium equations, and Eq. (64) permits

writing

*(Aj + C (X - xi)) ao + ax (66)

in Eq. (61), where a0 and aI are new constants, so that Eq. (61) becomes

y(x) - ao + alx *+ Qi x - xiV (67)

In the case where the spline is symmetrical about the origin, as is

the case for the AICs (Cs(k)] and [hh(k)] which are symmetrical functions of

the reduced frequency k, we may use the method of images and Eq. (67) becomes

y(X) - &0 + I q(X - I!' + Ix * xil') (68)

and only Eq. (63) needs to be satisfied in addition to Eq. (68). The matrix

form for Eqs. (63) and (68) in terms of the specified values of y(xj) is

17

mV

0 1 ... I a

I K, K12 K 13 K1N qi

2 K12 K22 K23 ... K2N q2 (69)

YS1 K13 K2 3 K 33 K K3N (Q3

1 K1N K2N K 3N NN N

where Kji Ix - xil * Ixi + xil 3 (70)

The left hand side of Eq. (69) may be written

o 0 o ... o ,

YI1 0 0 ... 0 Y

2Y 0 1 0 . . 0 Y3(71)

0 0 1 ... 0

0 0 0 ... 1

17 NThe matrix form for an interpolated value of y(xk) is

ao0

Yk Ll Kk1 Kk2 ... KN & Q (72)

where

r *k j " x k x ' + I . x 1 3 (7 3 )

18

.,. . . . . .-m

Combining Eqs. (69), (71) and (72) leads to the desired interpolation coeffi-

cients I defined by

(74)~~Yk "Lkjl{Yj} C4

where {yj} denotes

Yl

' Y2

and

L'kjl =-l

0 1 1 1 ... 1 0 0 0 ... 0

1 1 K11 KI2 K13 ... KIN 1 0 0 ... 0

Ll Kkl Kk2 ... KNJ 1K 12 K22 K23 ... K2N 01 0... 0 (75)

1 KI3 K23 K K . . K 0 0 1 ... 0

1 KNK 2N K3N .. 0 0 0 ... 1

Equation (74) provides the linear spline interpolation scheme needed to

obtain the AICs in the frequency lining up process. The inverse required

in Eq. (7S) is ill-conditioned for a large number N of interpolated points,

say N=SO; however, a large number are not needed in the flutter analysis and

N will be limited to N £ 10 in the present.applications.*

A lower limit, say N - 5 is adequate for most applications unless anextremely wide frequency spectrum must be covered.

19

SECTION III

AERO-SERVO-ELASTIC STABILITY ANALYSIS PROGRAM: MPASES

Program Description

MPASES is a general purpose digital computer program for the analysis of

the closed-loop stability problem. This program, which is a modification of

PASES (Ref. 1), formulates the problem by combining a classical method for

determining servomechanism system stability with the more realistic British

method of flutter analysis in lieu of the American method used in PASES. With

the input of an arbitrary number of elastic degrees of freedom, aerodynamic

influence coefficients and the control system description, the stability

of a missile or aircraft configuration as an aero-servo-elastic system can

be investigated. When no aerodynamics are involved, the servo-elastic system

stability can be determined.

Dynamic storage allocation utilized throughout MPASES provides the

most efficient U3e of core storage through the variable dimensioning of

all arrays. Because the program is based on the theory that permits the

inclusion of the aerodynamic forces as spring and damper terms in the

equations of motion rather than as complex inertial terms, only real

matrices are used in the eigenvalue solution. This makes for additional

economy in computer usage.

MPASES uses the method of revising the aerodynamics by lining up the

frequencies and performing the spline interpolation of the aerodynamic

terms necessary in the iterative eigenvalue solution. This process is

repeated until the required number of modes and eigenvectors are obtained

for a particular velocity. Moreover, if the analysis Mach number (velocity

determining factor) differs from the AIC Mach numbers by more than a speci-

fied 'deviation' value (input by the submitter), spline interpolation of the

20

C77_

aerodynamic forces is performed also. In this case, the program interpolates

for Mach number before the reduced frequency interpolation for each mode is

carried out.

The variables in the stability analysis of an aero-servo-elastic system

are servo-gains, velocities (through Mach number and speed of sound input)

and altitude (air density input). An option is provided to vary the gain

of a single servo component or control surface with the density and Mach

number held constant. When this option is executed, the coefficients of the

output of the desired servo element or control surface are divided by the

gain factor, K. This method facilitates programming, since there may be

more than one input.

Since dynamic dimensioning is used, the program limitations are minimal.

Stability results can be obtained for up to five (S) altitudes and for each

density, as many as ten (10) different velocities can be analyzed. In

addition, twenty-five (25) gain variations are permitted. The number of AIC

matrices that can be input is limited to ten (10) reduced frequencies for each

of five (5) Mach numbers. There are no size restrictions for any of the other

parmters existing in the progpam.

The total number of memory units (words or bytes) required to execute

MPASES is completely dependent upon input. The size of the program is

reflected in the length of the blank common block found in the 'MAIN'

section of the program. The length can readily be altered to accomodate

different analyses. Likewise, dimensions of the arrays limiting the pro-

gram as stipulated above can be changed to the desired size.

21

Subroutine Description

MAIN Main program for MPASES.

Reads and prints basic input data.

Allocates dynamic storage for all arrays.

Calling program for subroutines MASS, SERVO, FORMAB, AIC, MPSEand MPAS.

AIC Reads and prints complex AIC matrices.

Separates AIC's into real and imaginary parts.

Formulates generalized aerodynamic forces and stores on temporaryFile I.

Calls subroutine GENMAT.

ZGIVCT Finds the eigenvoctor from a real matrix for which the complexeigenvalue is known.

Uses the inverse power method with shifts.

The eigenvoctor obtained is complex with the largest valuenormalized to unity.

A singular matrix returns an esigenvector of zeros.

ZIGRF Computes eigenvalues of a real, triangular matrix.

Calls routines from the IMSL Package (see Ref. 7).

Called by subroutine SOLV.

FPIKA3 Generates the A and B matrices which constitute the sigenvalueproblem.Stores A and (B] on temporary File 2.

GMT= Formulates the pueralized mass and aerodynamic force matricesutilizing rigid body, control surface and vibration modes.Calls subroutine MKZAML.

Called by subroutines AIC and MASS. 4

22j

-ITERP Generates the constant portion of the linear spline interpolation

coefficients matrix.

Prints message if spline interpolation matrix is singular.

Calls subroutine INVERS.

INVERS Finds the inverse of a real, square matrix.

Called by subroutines INTERP and S0LV.

KASS Reads and prints the weight matrix in pounds. Converts weight

to mass units (slugs).Reads and prints rigid body control surface modes, free-freevibration modes and restrained (rigid body) modes.

Stores modal data on temporary File 4.

Reads and prints damping coefficients and vibration frequencies(Hertz). Converts frequencies to rps units.

Formulates generalized mass matrix.

Generates generalized stiffness and damping matrices if flexiblemodes are input.

Calls subroutine GENMAT.

MATMPL Multiplies real, two-dimensional matrices.

If requested, transposes post-multiplier matrix to perform as apre-multiplier.

Called by subroutines INTERP, SERVO, GENMAT and SOLV.

MPAS Obtains stability results from aero-servo-elastic analysis for

altitudes and velocities requested.

Reads Mach numbers at each altitude for which analysis is to be made.

Calculates velocity (from Mach number and speed of sound) andconstants for the aerodynamic forces.

Performs Mach number interpolation if necessary.

Reads File 2 - brings (A] and [B] into core.

Executes gain option if requested; stores A and B matrices re-flecting gain factor on tempoerary File 3.

13

PAS Carries out spline interpolation of the aerodynamic forces for(Cont.) reduced frequency for each mode.

Generates new A and B matrices with changed aerodynamics for theparticular mode.Calls subroutine S0LV for eigenvalue solution of each mode.

Prints stability results.

Prints eigenvectors, if requested, for the analysis.

Reads File 4.

Computes and prints structural deflections at system mass pointsfor each mode.

Calls subroutines INTERP, SPLINE and SOLV.

4PSE Obtains stability results from servo-elastic analysis for number

of modes requested.

Reads File 2 - places [A] and [B] into core.

Executes gain option if requested.

Calls subroutine SOLV for eigenvalue solution of required modes.

Prints stabil~y results.

Prints eigenvectors for each mode.

Note that the eigenvectors are always computed and printed fora servo-elastic analysis.

Reads File 4.

Computes and prints structural deflections at system mass pointsfor each mode.

Calls subroutine S0LV.

RESULT Forms the array where stability results for each mode are stored.

Calls subroutine EGNYCT to compute the eigenvectors (only uponrequest for aero-servo-elastic analysis, always for servo-elasticanalysis).

Called by subroutine S0LV.

SERVO Reads coefficients from servo differential equations.

Prints control system description.

Computes input to rate gyros and accelerometers from differentia-tion or interpolation matrices.

Determines maximum order of servo elements.

Prints pertinent information concerning all servo elements.

24

SERVO Generates (X2], [Xl], [Xo1, [Yl], [Yo, :[Zo].(Cant.) Calls subroutine MAThPL.

SOLV Prints the A and B matrices.

Generates the dynamic matrix (eigenvalue problem).

Calls subroutine EIGF for sigenvalue solution.

"LLnes-up the frequencies" in aero-servo-elastic analysis.

Determines real and oscillatory roots for each mode.

Estimates new reduced frequencies.

Calls subroutine RESULT to compute stability results and sigen-vector for each mode.

Calls subroutines MATMPL, INVERS, EIGRF (from IMSL Package),and RESULT.

Called by subroutines 11PAS and 11PSE.

SPLINE Generates the linear spline interpolation matrix.

Determines the Lagrangian coefficients.

Used in the frequency lining-up process.

( Called by subroutine MPAS.

Programming Symbols - A partial list of FORTRAN symbols used in HPASES is

presented. Additional symbols are defined in the Input Instructions.

1. Integer Variables

ID Density index

IE Mode counter

IG Gain index

IK Reduced frequency index

IM Mach number index

IMODE Actual number of modes calculated for stability

analysis

2SI

L0B First real root (eigenvalue with positive real

part and zero imaginary part)

IOE Last real root

ISING Singular matrix control. If ISING = 1, matrix isnon-singular; return from subroutine INVERS.

Ji BLANK COMMON length required to call subroutine MASS

J2 BLANK COMMON length required to call subroutine SERVO

J3 BLANK COMMON length required to call subroutine FORMAB

J4 BLANK COMMON length required to call subroutine MPSEor subroutine AIC

J5 BLANK COMMON length required to call subroutine MPAS

JT0T Largest values of Jl, J2, J3 and J4 for servo-elastic analysis; or largest value of Jl, J2, J3,J4 and JS for aero-servo-elastic analysis. JT0Tis the minimum BLANK COMMON length required toexecute MPASES.

KOP Column assignment in A and/or B matrices for zeroand first order servos.

KOI Column assignment in A and/or B matrices for secondorder servo-velocity.

K02 Column assignment in A and/or B matrices for secondorder servo-displacement.

KX Control for Mach aumber interpolation.

K1 Order of servo element for which the gain varies.

K2 Column assignment in the X, Y and/or Z arrays forservo whose gain varies.

L Counter for coefficients from servo differentialequations including inputs to control surfaces,outputs from servos, and inputs to servos.

MDF Number of mass points; used as a variable dimensioninvolving flexible modes. (MDF a 1 when NFM a 0)

MDR Number of mass points; used as a variable dimensioninvolving rigid body modes. (MDR - I when NRM a 0)

MFM Number of flexible modes; used as a variable dimen-sion. (MFM a 1 when NFM 0)

26 4-

MRM Number of rigid body modes; used as a variabledimension. (MRM a 1 when NRM a 0)

Ml Number of flexible modes or rigid body modes, which-ever is larger.

M2 Ml or the number of control surfaces, whichever islarr.

M3 Number of reduced frequencies or Mach numbers, which-ever is larger. (AIC input)

M31 M3 + 1.

NFR Sum of the number of flexible modes and rigid bodymodes.

NIO Counter for real roots.

NK1 NK + 1

NMODE Sum of the number of flexible modes, rigid bodymodes and control surfaces.

NI NM. I+

NPART Number of partitions in AIC matrix. (NPART a NAif NA > 1)

NPR Counter for number of actual modes in the servo-elastic analysis.

NRO Control for determining the number of real roots.

NSERV Total number of coefficients from servo differentialequations including inputs to control surfaces,outputs from servos, and inputs to servos; used asa variable dimension.

NT Sum of the number of flexible modes, rigid bodymodes, control surfaces and all servo elements.

NTC Sum of the number of flexible modes, rigid bodymodes, control surfaces and second order servoelements.

NTT Order of the eigenvalue problem, 2*NTC+NO+Nl.

NVEC Control for calculation and printing of eigenvectors.

NO Number of zero order servos

N1 Number of first order servos

N2 Number of second order servos

27

2. Real Variables

ANOPM Test for normalizing structural deflections.

COMAI Constant for generali:ed aerodynamic forces-imaginary part, -pVS.

Cow Constant for generalized aerodynamic forces-real part, - 0Va/S/Z.

CONI Constant for imaginary part of AIC matrix, k(Zs/S). VCONR Constant for real part of AIC matrix, 2k2 (Es/S).

DET Value of determinant returned from subroutine INVERSif matrix is non-singular.

EK Reduced frequency, br4/V, used in the iterative pro-cedure; determined by the estimated frequency, w,

EW Estimated frequency for successive iterations of theeigenvalue solution; determined by the 'lining-upthe frequencies' process.

VEL Velocity for the stability analysis, ft/sec.

VELK Velocity for the stability analysis, knots.

.MNR4 Normalizing factor for structural deflections.

3. Integer Arrays (Variable dimensions indicated in parentheses)

IANA(NTT) Used in subroutine SOLV as an argument when subroutineRESULT is called. This vector is eventually used inZQVT as Li described below.

INDEX(WTF,3) Used in subroutine INVERS for working storage.

KO(NSE,2) Stores information concerning the maximum order ofall servo elements and their column assignments inthe X, Y and Z arrays.

LI(NTT) Used In subroutine EGNVCT to restore order of theelements in the eigenvector.

L2(TT) Used in subroutine ZGCVCT as working storage.

NL(NSUV,2) Stores information concerning the nuber of theservo element from which there is input and output.

NOR Stores information concerning the order of the(NSK,NSnV) servo element coefficients.

28

- a .A_. --

4. Real Arrays (Variable dimensions indicated in parentheses)

A(N r,,Tr) Matrix in the eigenvalue problem formulation.

AFI Generalized aerodynamics used for reduced frequency(NFR,NMODE,NK) spline interpolation - imaginary part.

APR Generalized aerodynamics used for reduced frequency(NFR,tU40DE,NK) spline interpolation - real part.

A I Generalized aerodynamics used for Mach number spline(NPR,NODE,NM) interpolation - imaginary part.

A4R Generalized aerodynamics used for Mach number spline(NFR,NWMDEoNM) interpolation - real part.

B(NTT,NTT) Matrix in the eigenvalue problem formulation.

BINV(,NT,l) Column of constants used in subroutine INVERS.

BC(NT,,YT) YO [A] + [B], where y is the eigenvalue shift.

C(NTT,NTT) Dynamic matrix used for the eigenvalue solution.

CF(FM) Structural damping matrix in its equivalent viscous

form (diagonal)

CHDH(NDF,NDF) k(Es/S)(ChI]. where [C] is the imaginary part of

the AIC matrix [Ch].

CHS(NDF,NDF) 2Y.2(Zs/S) [ChR ] , where [ChR ] is the real part of the

AIC matrix [Chl.

DFM(1,MFM) -K[D][h.]. where K is the rate gyro or accelerometer

gain, D is the differention or interpolation vector,

and hF are the flexible modes.

DRM(I,MR) -K(D][ha], where K is the rate gyro or accelerometer

gain, D is the differentiation or interpolation

vector, and hR are the rigid body modes.

FRC(2,M2) Partition of generalized mass or aerodynamic matrixused in subroutine GENHAT.

GAI(NFR,NM0DE) Matrix of generalized aerodynamics as output fromsubroutine GENMAT - imaginary part.

( GAR(NFRNMHDE) Matrix of generalized aerodynamics as output fromsubroutine GEAT - real part.

29

'

GENM Generalized Mass matrix(NFR.MPOE)

P(M31) Row vector for spline interpolation used in sub-routine SPLINE.

PSX(NK) Vector of LaGrangian coefficients from reducedfrequency spline interpolation

PSM(N) Vector of LaGrangian coefficients from Mach num-ber spline interpolation

RI(NrT) Imaginary part of eigenvalue

RR(NrT) Real part of eigenvalue

SAPI Matrix of generalized aerodynamic forces for esti-(NFR,NDE) mated reduced frequency obtained from spline inter-

polation - imaginary part.

SAFR Matrix of generalized aerodynamic forces for esti-(NFRMPDE) mated reduced frequency obtained from spline inter-

polation - real part.

SI(M31,M3) Matrix of 'ones' for spline interpolation - usedin subroutine INTERP.

SM(NM1 ,4) Constant spline interpolation matrix for Mach num-bers.

SNK431,M31) Matrix of Mach numbers or reduced frequencies inthe constant portion of the spline interpolationmatrix - used in subroutine INTERP.

SK(MFM) Generalized stiffness matrix (diagonal).

SPK(NKlNK) Constant spline interpolation matrix for reducedfrequencies.

STAB(MODE,6) Stores stability results for each mode.

Th(41 ,NDF) Intermediate array used in subroutine GENMAT.

XMO)E Matrix of flexible, rigid body and control surface(NDF,NMPDE) modes.

XO(fN,NT) Coefficient matrix of second order variables, dis-placement.

XI(NTNT) Coefficient matrix of second order variables, velo-city.

X2(NT,NT) Coefficient matrix of second order variables, ac-celeration.

30

YO(NTNSE) Coefficient matrix of first order variables, dis-placement.

YI(NT,NSE) Coefficient matrix of first order variables, velo-city.

ZO(NT,NSE) Coefficient matrix of zero order variables, dis-placement.

z Dynamic dimensioning array in BLANK COMMON.

5. Complex Arrays (Variable dimensions indicated in parentheses)

C2(='T) Complex working storage for subroutine EQ(VCT.

C3(bT) Complex eigenvector for a particular mode,computed by subroutine EGHVCT.

DUL (NDF) Structural deflections for the system mass points(detatrined by input modal data and eigenvectors).

(NTT) Couplex eigenvalues resulting from the aigenvaluesolution in subroutine EIGRI.

U(4TT,NTT) y[A] + [B], where y is the complex eigenvalue fora particular mode; used in subroutine EGNVCT todetermine oiSenvector.

V(ll) Psuedo complex $igenvector storage in subroutine11GZIG (not used).

VC(MODRONTT) Stores elganvectors for each mode (complex).

VV (KTT) Work area in subroutine EIGUP.

Procossing and Prognaing Considerations

1. Operation

Standard FORTRAN IV processor system. Operable on the CDC com-puter; model 6600, Cyber 11S-Scope 3.4.3 system.

Fote: MPASES can be made operable on all computer systems withminor modifications. Probable necessary changes arelisted below:

Deletion of PROGRA4 statement at beginning of 'MAIN'section of program.

END-iF-FILE (EO ) statement alterations.Alphanumeric modifications dependent upon the number of

characters per word in the operating system used.Single precision to double precision accuracy. (IPASES,

as presented in this report, has single precisionaccuracy.)

2. Core Storage

Number of memory units (words, bytes, etc.) required to execute

is dependent upon input data reflected in the length of BLANK

COMMON. Estimate of BLANK COMMON length can be accnmplished as

follows:

Servo-Elastic Analysis:

LENGTH = 1 + NTT(4NTT + 2MDE + 9) + NDFCNMODE + 2) + 6M0DE

OR

LENGTH. 1 + NDF(NDF+NMODE + Ml) + NFR(NI40DE) + 4NF1 + (M2)2t

Use the larger of the above two estimates.

Aero-Servo-Elastic Analysis:

LENGTH - 11 + NTTC4NTT + 2MDE * 9) I" 2CNFR)(NMODE)(NK + NM + 1) * 2(M43 1)2" NH(NM + 2) + NK(NK + 2) + NDF(NMODE + 2)" DE

OR

LENGTH a 1 + NDF(4NDF + NIMDE + Ml) + 2(NFR)(NM0DE) + (,[2)2

Use the larger of the two. If a number greater than NTT was

assigned to the parameter 'MODE,' use MODE-NTT in the above

calculations. Note that the first estimate for each analysis

is eigenvalue problem size oriented, while the second is

dependent upon the number of system mass points.

Refer to this section under Progrening Symbols for definitions.

Actual BLANK COMDN length required for the analysis follows the

stability results in the program's printed output. This information

will enable the user to make more efficient use of core storage in

subsequent analyses by recompiling the 'MAIN' section with a more

realistic BLANK COHMN length.

Note. MPASES, as presented in this report, contains a BLANK COONlength of 10,000. This length was more than adequate to per-form the analyses in the sample problems used as examples.(see Section 4.0.)

3. Auxiliary Files

Standard input read file (5).

Standard output print file (6).

Four temporary utility files (1, 2, 3, and 4).

32i S

Note: To avoid the cumbersome handling of data cards, it is sug-gested that data sets be generated containing the AIC's,modal data, etc. Consequently, MPASES can be modified toaccept this input from auxiliary files in lieu of cards.

Input Instructions

Units - All units are taken in the pound-feet-second system with the ex-

ception of density, which must be in slugs/cubic feet. The weight, which

is input in pounds, is converted to slugs internally in the program. Like-

wise, MPASES converts the frequency from Hertz to rps units within the

program. The AIC's must be non-dimensional when input with no altitude

consideration (i.e., p-1).

Data Deck Setup

1. Title cards (2).

2. Control card describing data input.

3. Eigenvalue shift, y0

4. Geometric properties of air vehicle (s, , S) and Mach number

deviation (DM).

S. Mach numbers and reduced frequencies, i.e., for AIC input.

6. Density and speed of sound for each altitude.

7. Number of velocities to be analyzed for each altitude.

8. Gain factors, if gain option is to be exercised.

9. Weight matrix.

10. Rigid body control surface modes.

I1. Structural damping coefficients if NFM > 0.

12. Frequencies of vibration (flexible) modes if NFM > 0.

13. Flexible modes, if any.

33

14. Restrained (rigid body) modes, if any.

IS. Coefficients of differential equations describing the servo

elements in the control system.

16. AIC matrices.

17. Mach numbers to determine velocities to be analyzed at eachaltitude.

Items 4, 5, 6, 7, 16 and 17 are included for aero-servo-elastic

analysis only.

Detailed description of input data follows.

Input Data Description

FORTRANNO. CARD FORMAT COLUMNS NAWE DESCRIPTION

1. TITLE 18A4 1-72 Any alphanumeric statement.

2. TITLE 15A4 1-72 Any alphanumeric statement.

NOTE: Two cards must be input. May be blank.

3. CONTROL 1415 1-5 MCODE = 1, diagonal weight matrixinput.a 2, coupled weight matrix

; input.

6-10 NDF Number of mass points in thecomplete system - degrees offreedom CNDF > 0).

11-1S NFM Number of flexible modes(NFM > 0). If NRM a 0, NFM> 0).-

16-20 NRI Number of rigid body modes(NRI42 0). if NFM a 0, NRM> 0).

21-25 NC Number of rigid body controlsurface modes; i.e., numberof control surfaces (NC > 0).

26-30 NSE Total number of servo elements

in the control system, includ-ing rate gyros and accelero-meters (NSE > 0).

34

FORTRANNO. CARD FORMAT COLUMNS 1AM DESCRIPTION

31-3S HA 0 0, no aerodynamics, i.e.,servo-elastic analysis only.> 0, AIC input, i.e., aero-servo-elastic analysis. Inputas follows: NA w number ofpartitions contained in eachAIC matrix.

NOTE: NA a 1, AICs derived from Doublet -Lattice,Mach Box theories, etc.

NA > 1, AICs derived from Strip, PistonTheories, etc. (NA a number ofstrips).

It is assumed that all AICs are obtainedfrom the same methoTor any one aero-servo-ela stc analysis.

36-40 NM Number of Mach numbers for

which there is AIC input

(NM<5). IfNA- , NM- .

41-45 NK Number of reduced frequencies(k) for each Mach number

_<(NKl10). If NA-0, NK-0.

NOTE: The reduced frequencies must be the samefor each Mach number.

46-40 ND Number of altitudes (ND < 5).if NA - 0, ND - 0.

51-55 MODE Number of modes requested for thestability analysis. Use a largenumber if all modes inherent to theanalysis are to be1 consi.dered.

NOTE: Due to the nature of the 'lining up thefrequencies' method of analysis, the eiger-value solution may result in less modesthan requested.

56-60 NG - 1, go lain variation.> 1, gain option executed.Input as follow: NG a numberof gain variations (K).(1 5 NG _5 25)

61-65 NGB - 0, no gain variation.> 0, NGB - number of the par-ticular servo element forwhich the gain varies.< 0, NGS - number of the con-trol surface for which thegain varies.

35

FORTRANNO. CARD FORMAT COLUMNS NAME DESCRIPTION

NOTE: The gain factor (K) can vary for only oneservo element or control surface in anyone stability a7nalysis.

66-70 NAB 0 0, A and B matrices notprinted.aI, A and B matrices printedby rows.

NOTE: If NA - 0, (A] and [B] as generated forthe analysis are printed.If NA > 0, (A] and (B] as generated fork-0 only are printed.

4. SHIFT E12.0 1-12 GA4A Shift eigenvalue (yo).

Choice of Y is left to theuser.

NOTE: See Section 2 (Theoretical Development)and Section 4 (Sample Problems) fordetails.

If NA * 0, OMIT the following cards: 5, 6, 7, 8 and 9.

S. AERO 4E12.0 1-12 SS Semi-span (s), feet.CONSTANTS 13-24 CBAR Mean aerodynamic chord (E),

feet.25-36 S Surface area (S), sq.ft.37-48 DM Mach number deviation. Choice

of DM is left to the user.

NOTE: If the difference between the Mach numberfor which the analysis is to be performedand the Mach number for which there is AICinput is greater than DN, Mach number inter-polation of the AICs will be made. IfNM - 1, no interpolation.

If NA = 0, OMIT Card S.

6. AIC DATA 5E12.0 XM(I) Mach number, I - 1, NM.

If NA a 0, OMIT CArd 6.

7. AIC DATA 6E12.0 XK(I) Reduced frequency, k; Il,NK. Continue on next cardif necessary.

If NA a 0, OMIT Card 7.

36


S. AER0 DATA 6E12.0 1-12 DENS(l) Air density, P , associatedwith the first altitude atwhich the analysis is to bemade; slugs/cu.ft.

13-24 S0S(l) Speed of sound associatedwith the first altitude;ft/sec.

2S-36 DENS(2) Density for second altitude.37-48 S0S(2) Speed of sound for second

altitude.

DENS(I), SOS(I); 1-1, ND.Continue on next card ifnecessary.


9. AERO DATA SI5 MD(I) Number of velocities (deter-mined by Mach number input -

see Card 26) for each alti-tude; I- 1, ND. (MD 1C 10)


10. GAIN 6E12.0 GAIN(I) Gain factor, K. I - 1, NG.

Continue on next card ifnecessary. (NG < 25)

If NGS a 0, OMIT Card 10.

11. WEIGHT 6812.0 WT(I,I) If 4CODE a 1, vector of di-agonal elements of weightmatrix, pounds. I * 1, NDF.Continue on next card ifnecessary.

WT(I,J) If MCODE - 2, elements ofupper triangle of coupledweight matrix, lbs. Inputby row; each row starts ona now card. I a 1, NDF,J - I, NDF.Continue on successive cardsto complete each row.

37


12. MODE 6E12.0 C4DE(J,I) Rigid body control surfacemodes. Each mode starts ona new card. I - 1, NC, J a 1,NDF.Continue on successive cardsto complete each mode.

13. DAMPING 6E12.0 GF(I) Structural damping coefficientcorresponding to vibrationmode (I). I a 1, NW. Con-tinue on next card if necessary.

If NFM a 0, OMIT Card 13.

14. FREQUENCY 6E12.0 FREQ(I) Frequency of vibration mode(I), Hz. I = 1, NFM. Con-tinue on next card if necessary.

If NFM a 0, OMIT Card 14.

15. MODE 6E12.0 FM0DE(J,I) Free-free vibration (flexible)modes. Each mode starts on anew card. I - 1, NFM, J a 1,NOF. Continue on successivecards to complete each mode.

If NFM O, OMIT Card IS.

16. MODE 6E12.0 RODE(J,I) Restrained (rigid body) modes.Each mode starts on a new card.I a 1, NRM, J 1, NDF. Con-tinue on successive cards to pcomplete each mode.

IfNRM a O, OMIT Card 16. V

Input description of the servo differential equation coefficients follows:

IMPORTANT: All servo differential equations must be equated to zero,i.e., all terms should be on one side of the equation, be-fore attempting to input the -efficients.

NOTE: Reference to servo elements and control surfaces is by num-ber; therefore, numbers should be assigned to each servoelement (1 to NSE) and control surface (I to NC). Servoelements also include the rate gyros and accelerometers inthe control system.

38


REPEAT the following cards 17, 18 and 19 for each control surface, i.e.,RE =tnes.

17. Output from Control Surface. FORMAT (3E12.0)

Columns 1-12 XO(I) Zero order coefficient for Control Sur-face (I).

13-24 Xl(I) First order coefficient for Control Sur-face (I).

25-36 X2(I) Second order coefficient for ControlSurface (I).Note: I-I, NC.

18. Input Control Card. FORMAT (IS)

Columns 1-5 INC Number of inputs from servo elements tocontrol surface (W-T.

REPEAT card 19 for input from each servo element to Control Surface (1),., "INC times.

19. Input from Servo Element. FORMAT (21S,2X,3E12.0)

Columns 1-S K Servo element from which there is input. [6-10 No Order of servo element coefficients.13-24 CO Zero order coefficient.25-36 Cl First order coefficient.37-48 C2 Second order coefficient.

If NO a 0, zero order; only CO input.u 1, first order; Cl must be input (CO may be

zero.a 2, second order; C2 must be input (CO and

C1 may be zero).

REPEAT the above card 19 INC times.

REPEAT the above cards 17, 18 and 19 NC times.

REPEAT the following cards 20, 21 and 22 for each servo element, i.e.,FTJmes (not necessarily in sequential order).

39

a7, h- !I "


20. Output from Servo Element. FORMAT (215,2X,3E12.O)

Columns 1-S I Servo element number from which there isoutput.

6-10 NO Order of servo element coefficients.13-24 CO Zero order coefficient25-36 Cl First order coefficient.37-48 C2 Second order coefficient.

If NO - 0, only CO is input.- 1, Cl must be input (CO may be zero).- 2, C2 must be input (CO and Cl may be zero).

21. Input Control Card. FORMAT (IS)

Columns 1-S INS Number of inputs from servo elements (orcontrol surfaces) to servo element (I).

If there are no inputs to Servo Element(I), INS- 0.-

REPEAT card 22 for input from each servo element or control surface toServ rvo Element I), i.e., INS times.

If INS - 0, OMIT Card 22.

22. Input from Servo Element or Control Surface. FORMAT (21S,2X,3E12.0)

Columns 1-S K Servo element number from which there isinput.

-K Control surface number from which thereis input.

6-10 NO If K > 0, NO = order of servo element co-efficients.If K < 0, NO a order of control surfacecoefficients.

NO - -1, input from body angular rate to rate gyro.- -2, input from body acceleration to accelero-

meter.

NOTE: If N < 0, K O.

13-24 CO Zero order coefficient.25-36 Cl First order coefficient.37-48 C2 Second order coefficient.

40m


If NO n 0, only CO input.a 1, Cl must be input (CO may be zero)a 2, C2 must be input (CO and Cl may be zero).- -1, only CO input, where

CO a -K& (rate gyro gain)a -2, only CO input, where

CO a -Ka (acceleremeter gain)

NOTE: When NO < 0, Card 22 must be followed by Card 23.

23. If NO -1,Differentiation Row Vector used to determine angular velocity forthe rate gyro.

if NO a -2,Interpolation Row Vector used to describe body acceleration for theaccelerometer.

FORMAT 6E12.0 D(I) Element of differentiation or interpolation rowmatrix. I a 1, NDF.Continue on next card if necessary.

NOTE: See Ref. 1 for description.

REPEAT the above Card 22 (and Card 23, if applicable) INS times.

REPEAT the above Cards 20, 21 and 22 HSE times.

Input instructions for the AIC matrices [Ch] follow:

NOTE: It is assumed that the AIC's are dimensionless and do notreflect density (i.e., p a 1).

If MA a 0, OMIT Cards 24 and 25.

If NA a 1, input only Card 25.

If NA > 1, input Cards 24 and 25.

The AICs are input for all the reduced frequencies, i.e., for XK(I),where I a 1, NK for each Mach number .W(J), where J a 1, INN.

NOTE: The reduced frequencies, k, must be the same for each Machnumber.

If NA - 1, REPEAT Card 25 for each AIC matrix, i.e., CNK*NM4) times.

41


If NA > 1, REPEAT Cards 24 and 25 NA times for each AIC matrix.See Card 3. NA a Number of partitions (NPART) in the AIC matrix.All the input AIC's must have the same number of partitions.

REPEAT until all AIC's are input, i.e., (NK*NM) times.

24. Partition Control Card. FORMAT (215)

Columns 1-S MS Size of partition.

NOTE: The sum of the sizes (MS) of all thepartitions in each AIC matrix must equal NDF.

6-10 NZERO = 1, all elements of partitions are equalto zero. Do not input. OMIT Card25.

- 0, non-zero partition. Input Card 25.

2S. AIC Partition or Matrix. FORMAT (6E12.0)

CH(I,J) If NA = 1, complete AIC matrix (complex).I - 1, NDF, J a 1, NDF.

If NA > 1, AIC partition (complex).

I * 1, MS, J - I, MS.

Input by rows. Each row starts on a new card. Continueon successive cards to complete each row.The imaginary part of each element follows the real part,e.g.:

Columns 1-12 CH(I,1) Real

13-24 CH(, l) Imaginary2S-36 CH(1,2) Real37-48 a ( , 2) Imaginary

OMIT the following Card 26, if NA a 0.

REPEAT Card 26 for each altitude, i.e., ND times.

42

. . . . . .

FORTRAN

NO. CARD FORMAT COLUMNS NAME DESCRIPTION

26. Mach Numbers for Analysis. FORMAT (6El2.O)

X14CH(I) Mach numbers selected for stability analysis at aparticular altitude. I - 1, 1#4ACH, where N?4ACH isthe number of velocities (determined by the Machnumber, XMKCH(I) and the speed of sound, SOS(J) foraltitude J). i.e.. !t4ACH4ID(J). Refer to Card 9.(NMA&CH C 10). Continue on next card if necessary.No eigenvectors are obtained or printed with thestability analysis when XMACH(I) > 0.

-XM4ACH(I) Same as above.Complex eigenvectors are calculated and printed foreach mode when XMACHTFC 0.

NOTES: XMACH(I) may be the same or different for eachaltitude. Do not confuse X?4ACH with XM whichare the Mach numbers for the input AIC's.

REPEAT above Card 26 ND times.

4 3

Program Output Description

Input Data

1. Upper triangle of weight matrix (lbs.).

2. Damping coefficients and frequencies (if free-free vibrationmodes are present).

3. Mode shapes - flexible, rigid body and control surface.

4. Control system description - coefficients from servo differ-ential equations, rate gyro gains and/or accelerometer gains.

S. Maximum order of each servo element (determined internallyin the program).

6. Rows and columns (assigned by the program) in the A and/orB matrices for the coefficients of each servo element.

7. Eigenvector element associated with each servo element (velo-city and/or displacement).

8. AIC matrix (by rows) for each reduced frequency and Machnumber, if aero-servo-elastic analysis.

A and B Matrices (if requested)

1. For an aero-servo-elastic analysis, only [A] and [B] fork-O are printed.

2. For a servo-elastic analysis, [A] and [B] as generated forthe analysis are printed.

Stability Analysis Results

The results are identified and tabulated for each gain factorvariation (if any). In addition, when aerodynamics are included,the results are printed for the velocities analyzed at eachaltitude.

1. For each mode calculated, the following is printed in tabulatedform:

449

Complex eigenvalue, y - real part, I., I/sec.

Complex eigenvalue, y - imaginary part, w, rad/sec.

Duped frequency, H:.

Udeped frequency, Hz.

Reduced frequency, k, used to detemine elgenvalueproblem for particular mode.

Traction of critical damping, .

Tim to half amplitude, sec.

2. Complex eigenvectors corresponding to each eigenvalue, ifrequested for the saro-eervo-elastic analysis of a particularvelocity. The eiSenvectors are always printed with the sta-bility results of a servo-elastic analysis.

3. Structural deflections (complex) representing the system masspoints for each mode are printed whenever eigenvectors arecalculated.

BAKCftW0 Length

The minaimim BLANK C*M(N length required to execute the program.This length is dependent upon the input data.

(i

45

ISECTION IV

SAMPLE PROBLEMS

As examples of the program capabilities we consider a simplified

missile and servo configuration first, Case 1, as a rigid body in vacuo,

second, Case 2, as a flexible body in vacuo, and third, Case 3, as a flex-

ible body in an incompressible airstream at sea level. The missile is

idealized as a uniform beam and control surface as shown in Fig. 1.

GYRO0 0

Figure 1. Uniform free-free Missile.

The modal characteristics and differentiation matrix for this configuration

are given in Ref. 11. For the numeri~l work we assume the missile to weigh

10m - 1000 lbs., to be 1S0 inches long, and to have a fundamental frequency

of 45 Hz. From the data in Ref. 11. this results in higher frequencies of

f2 a 125.4 Hz and f3 = 248.2 Hz; only three modes will be considered. The

structural dampings in the three modes are assumed to be g, 0.03, g2 -0.0S,

and g3 a 0.08.

A simple servo system is considered whose block diagram is shown in

Fig. 2.

46

ZL - .

Figure 2. Servo block diagram.

NO assUMe the transfer functions appear as follows.

If For the command error,

01+ *0 a (76)1 4. C

for the servoposition error,

a 01- 2 ~03 *0 (77)

for the actuator,

for the position potentiometer, KS(~ )(8

0 * (79)

and for the rate gyro,

s4/hi Y c(s 2 2 +2 /.+1) (80)

The differential equations corresponding to Eqs. (78) and (80) are

(11w82 * (2C 2/w8 ).9 e4 Kh 0 (82)

47

r

The actuator is seen to be a first order servo element and the rate gyro is

a second order element. Therefore the second order variables are

(Z) a- (83)

the first order variable is

(Y) e 3) (84)

and the zero order variables are

- e(8S)(Z) 2

For the numerical work we assume Ka 1 /6 deg. per deg./sec., T a 0.01 sec.,

a 1 deg./deg., wg a 376.991 rad/sec. (60 Hz), C. a 0.70, and K. N 0.3 deg.

per deg./sec. The differentiation matrix for the rate gyro from Ref. 11 is

(0] - (1/241 ]O 0 +1 -27 +27 -1 0 0 0 0] (86)

where Z- 1S inches.

The case descriptions below present the variables in the eigenvectors

and discuss the solutions. The data input code sheets and the program

printed output for the three sample problems appear following the case de-

scriptions.

Case 1 Description

The first case is the rigid body in vacuo. There are two rigid body

(modes of plunging and pitching. The eigenvalue calculation is made choosing

48

r

tt

YO -1.0. The 11 h order eigenvector that appears is in the following

order:

{V) a LaRa A;4 aRla'R2 4* 3* 2

There are four zero eigenvalues from the two rigid body modes and four

infinite eigenvalues from the command error, the servoposition error, the

position potentiometer, and the actuator. The three non-trivial eigen-

values consist of the real actuator damping and the complex conjugate rate

gyro frequency and damping. However, since only the eigenvalues with

positive frequencies in the conjugate pairs are printed in the stability

results, only nine solutions are presented in the printed output. The

structural deflections at the ten system mass points for each mode are also

printed.

Case 2 Description

The second case is the flexible body in vacuo. The addition of three

flexible modes to Case 1 results in a 17th order eigenvector which has the

variables printed out in the following order:

~~ Lala 2 aFS ;R1 ;R2 1 *4 a. 1 a. 2 aF3 aft 1 aft2 d 4 3 e1 e21 (88)

The eigenvalues are obtained by again choosing yo a -1.0. Nine non-trivial

eigenvalues are obtained in this case. Three are those obtained in Case 1

and the additional three complex conjugate pairs correspond to the three

flexible modes. The stability results present the modes with the frequencies

in ascending order, the negative values being omitted. Therefore, only

thirteen solutions are printed; the structural deflections for the thirteen

modes are also printed.

49

An unsatisfactory design is seen to exist from the negative damping

in the 133.0 Hz mode; the missile, will "buzz" in this mode.

Case 3 Description

The third case is the flexible body in an incompressible flow at sea

level with a density p a 0.00237692 slugs/cu. ft. and velocity V a 500 fps.

It is the same as Case 2 with the addition of aerodynamic loads. The

aerodynamic loads are assumed to act only on the control surface and are

derived from the incompressible strip theory presented in Ref. 15. The

control surface has an exposed span of 20 inches and a semichord of 15 inches.

Five reduced frequencies are chosen for the aerodynamic interpolation,

k - O.0S, 0.10, 0.20, 0.50, and 1.00; the minim value of k a 0.0S is

chosen because of the singularity in the aerodynamic damping at zero fre-

quency when it is computed from Theodorsen's function.

Although there are 17 degrees of freedom in the eigenvalue problems

solved, only nine modes can be obtained. This results from the frequency

"lining-up" process. The reduced frequencies determined by the estimated

frequency in each eigenvalue solution are shown along with the stability

results for each mode. In addition, the time to half the amplitude as well

as the damping ratio is indicated to facilitate stability evaluation.

The solutions for Case 3 are seen not to be significantly different

from those of Case 2. However, we note that the short period mode shows

up in place of the rigid body pitching mode. The reduced frequency of

the short period mode is k a 0.03346 and its damping ratio is C - 0.1401. hThe aerodynamic loads do not cause a large disturbance for the (impractical)

so

r *

configuration chosen as the example. The servoelastic "buzz" observed

in Case 2 is still unstable here. More practical configurations should

be studied that have more critical hinge line locations and mass balancing

on the control surface.

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113

APPENDIX A

THEORETICAL DERIVATION FROM REFERENCE 1

Z.0 THE CLOSED-LOOP AERO-SERVO-ELASTIC

STABILITY PROBLEM

2.1 NOMENCLATURE

A Element of coefficient matrix of Jv] in Equati (2-4)

AIC Aerodynamic influence coefficient

a Amplitude of generalized displacement coordinate

B Element of coefficient matrix of 1v I in Equation (2-4)

b Reference dimension

C Element of discrete damping matrix

C Element of generalized damping matrix

D Element of differe ntiation matrix

e Servo signal

F Element of forcing function matrix

gF Modal structural damping coefficient

H Element of interpolation matrix

h Deflection of aeroelastic system

h6 Deflection due to unit shaft rotation

I Element of unit matrix

i Imaginary unit

K Servo system gain constant; element of stiffness matrix

k Element of generalized stiffness matrix

k Reduced frequency (Strouhal number)

M Element of discrete mass matrix

m Element of generalized mass matrix

Q Element of generalized force matrix

- a Element of AIC matrix

114

.... : " '-- .... . ... .... . ..... -...... ' ' '" **, ,' ." --'-- :. .""-%, " .. ,,,. " -'.* "" ................................".......,--

Re( ) Denotes real part of ( )

s Laplace transform parameter

T Servo system time constant

U U =

V Amplitude of v; velocity of flight

V IvI= Lu x y zi T

X Element of coefficient matrix of second order variables

x Second order variable; forward Cartesian coordinate

Y Element of coefficient matrix of first order variables

y First order variable

Z Element of coefficient matrix of zero order variables

z Zero order variable

y Eigenvalue, i. e., coefficient in Equation (2-5) todescribe the time dependence of transient motion.Note Re(y) = 0 indicates neutral stability.

YO Shift value of y in eigenvalue problem

6 Shaft rotation of control surface

Ratio of viscous damping coefficient to the criticalviscous damping coefficient

X Eigenvalue of Equation (2-8)

X 0 Shift value of k in eigenvalue problem

WAngular frequency

(-) Denotes complex amplitude

Subscripts

a Actuator; aerodynamic

F Flexible body motion

g Gyro

h Hinge Line

115

p Potentiometer

R Rigid body motion

x Corresponds to magnitude of second order variable

it Corresponds to velocity of second order variable

x Corresponds to acceleration of second order variable

y Corresponds to magnitude of first order variable

y Corresponds to velocity of first order variable

z Corresponds to magnitude of zero order variable

6 Control surface rotation

Matrix Notation

[ I Square or rectangular

[IT Transpose

( ]-I Inverse

Column

L J Row

r i Diagonal

11

2.2 THEORETICAL DERIVATION

All components of an aero-servo-elastic system can be regarded

as being composed of elements whose characteristics are described by

second order (or lower) differential equations. Let lxj denote the set of

second order variablesfyi the set of first order variables, and faz the

set of zero order variables. The aero-servo-elastic equations of motion

can be written in the general form

- I [X~ROM + [X)l + PCX:IIxi

+ (YPIJ} + (Yyl]lI + zJ = P (2-)

where JFJ denotes a forcing function. The mathematical formulation

of the stability problem requires first order differential equations. We

introduce the variable

lul 1*1 (=-2)

and combine Equations (2-1) and (Z-2) into the matrix form

(2-3)

+L0 x Yy Z (F)10 0 0] x 0

Y t' Iwhich we may abbreviate as

=-1(A) Ivi +(B lvB I I (2-4)

117

U. 7

% ., ' :,=. =u . .. . . ........ ... .... .... * ",' ,,, -. L - '. * . , : ; . , __ , - . .. . . . J

By setting the forcing function to zero, and letting

lvi = VI exp (NtO (2-5)

we obtain the eigenvalue formulation of the stability problem

(-y[Aj + [B]) iV[ = 0 (2-6)

Instability occurs when the velocity and/or system gains are such that

the real part of y is positive.

Since [A] is singular and [] may be singular, we let

YO . i/(X-X 0 ) (2-7)

where y and X are arbitrarily chosen complex numbers. Then0 0

Equation (2-6) can be rewritten in the canonical form of the eigenvalue

problem as

Xlvi = (YoCA) +[B])' [A] + 'o (LAI +(B3 v (Z-8)

where the new eigenvalue is

X= X l/(,,, - yo) (Z-9)

The eigenvalue extraction leads to convergence to the eigenvahues X

corresponding to the value of y closest to y . The non-Hermitian

matrices yield roots that appear separately, as close pairs, or as

complex conjugates.

The frequency and damping for each mode are found from the real

and imaginary parts of each eigenvalue. Letting a denote the decay

rate coefficient, P the damped frequency, ; t l-e viscous damping

coefficient, and w the undamped frequency of each mode, we may write

CL + t3 (2-l Oa)

= + (2-lOb)

from which we find

W 2IT+ = (2-11)

118

and

,a/l (2-12)

The equations of motion of the aeroelastic system including Lhe

shaft rotation of the control surface (flipper) appear as

[M)t1; + [C) l a + [K)(hI - [h6j 161) = IFI (2-13)

where (hi denotes the displacement of the control point masses in the

lumped-parameter aeroelastic system, (M] is the mass matrix, CC]

is the viscous damping matrix, [K] is the stiffness matrix, [h 6) is the

flipper displacement matrix (the elements of [h,] are zero for points

off the flipper and equal (xh - x,) for the i- h control point where x h isthe hinge line coordinate), and (Fl is the external force matrix. For the

purposes of closed-loop stability analysis, the external force of interest

is the aerodynamic force induced by the motion. A survey of unsteady

aerodynamic influence coefficients (AICs) has been given in Reference 3.

For present purposes it is sufficient to write the aerodynamic force as

jF}-= -[al] {i~ (2-14)

where [aa) is a complex matrix of oscillatory AICs valid only for harmonic

motion. The AICs [ a] are dependent on the planform, the altitude, the

flight Mach number, and the reduced frequency (Strouhal number) of the

motion k = wb r/V where wo is the frequency, br is a reference dimension,

and V is the velocity of flight. This limitation to harmonic motion re-

flects the state-of-the-art of unsteady aerodynamic theory, viz. , consider-

ably more solutions have been found for harmonic motion than for arbitrary

transient motion. For this reason flutter analysis has traditionally re-

quired a trial-and-error solution to find the velocity and frequency for

neutrally stable oscillations. To the same extent the aero-servo-elastic

stability analysis must be carried out by trial and error. Equations (2-13)

and (2-14) may be combined to appear as

[lil + c) fI I + (<K)hl - ChG) 161) 0 2-15)

where the mass matrix now Includes the aerodynamics.

[X)~ •CM]> [n (+-16)

11.9

The inputs to the hydraulic actuators come from rate gyro and

accelerometer feedback loops. The angular velocity of a rate gyro

may be found by numerical differentiation of the displacement velocities.

L DJ {} (2-17)

where IDJ is a differentiation row matrix. Methods of numerical

differentiation have been discussed thoroughly, e. g., by Milne. If the

differentiation Is carried out locally, i. e. , "in-the-small", then the

elements of LDJ will only be nonzero for control points .urrounding the

gyro location, e. g., parabolic differentiation would involve only the

three control points closest to the gyro. The acceleration at an

accelerometer is found by numerical interpolation of the displacement

accelerations.

LHJ L 1J (2-18)

where IHJ is an interpolation row matrix. Methods of numerical inter-

polation have also been discussed by Milne4 . If the interpolation is also

carried out locally, the elements of lHJ will also only be nonzero for

control points surrounding the accelerometer location.

Since the aeroelastic system has a large number of degrees

of freedom, it is desirable to reduce them by a series or modal method.

We write the deflections in terms of a series of vibration modes, rigid

-body modes, and the shaft rotation(s) of the flipper(s).

ihi = [hF]a.1 + [hR)IaR} + (h)11 (Z-19)

where (hF. is a matrix of restrained and/or free-free vibration modes,

laFlis the corresponding set of generalized coordinates of the vibration

modes, [hR] is the matrix of rigid body deflection modes, and faRl is the

set of amplitudes of the% rigid body motions. Substituting Equation (2-19)

into Equation (2-15) leads to a truncation error in the solution since the

series, Equation (2-19), is finite. The error matrix may be interpreted

physically as a distributed force on the system. If we impose the con-

dition that this error does no work in any of the flexible or rigid modes

(N. B. P this is the method of Galerkin) then the generalized equationsT

of motion are found by premultlplying Equation (Z-15) by [hF). and setting

120

T7to zero, and also by premultiplying by ChR]T and setting to zero. Sub-

stituting Equation (Z-19) into Equation (2-15) and premultiplying by (hFJ

leads to

(IF{FI1 +CMFR]IIR +L7F6]l'

(2-20)+ (CiJ{&F4 +rFJIaFI- 0

where we have noted that rigid body displacements produce no internal

damping or structural forces, and

[ ( hF)T [M][hF] (2-21)

[ MR E ChF) T ]a) (2-22)

[rF6)- [hFEjT R][h6 ] (2-23)

(C F = (hF)T[C) hF] (2-24)

and

[kFJ P [hF] [K][h ] (2-25)

We also have noted that the generalized stiffness matrix has a diagonal

form by virtue of the orthogonality of vibration modes, whether they be

restrained modes or free-free modes, and can be written in terms of

the generalized mass matrix and frequency matrix as

= F mFJ (2-26b)where w Fj z [h F ]T (M ] [hFl , diagonal elements (2-27)

This is also the case for arbitrarily chosen modes, e. g., modes given

by polynomial expressions, if their associated frequencies can be defined.However, inclusion of arbitrary modes leads to a convergence requirement

( of a large number of terms in Equation (2-19) and therefore willnot be per-mitted in this formulation.

121

- -- . , . . ;- - - *.-.: ;. - .- .

and w F is the frequency of the vibration mode 1hFI. The generalized

damping matrix does not have a diagonal form, as does the generalized

stiffness, but may be assumed so as an approximate means of including

structural damping. If we represent the structural damping in its

equivalent viscous form we may write

(CFJ - fhgF/W A(-2a

-g e F Im~J (2-28b)

where gis the structural damping coefficient (N. B. , 2 ' 2 'Fwhere T

Is the ratio of viscous damping coefficient to the critical viscous damp-

ing coefficient) corresponding to the vibration mode 1hF[.

Substituting Equation (2-19) into Equation (2-15) and premultiplying byT

(hR] leads to

[RFIFFI PTRIRI +10-- 6 14 = 0 (2-29)

where we again have noted that rigid body displacements produce no

internal damping or structural forces, and

tMR F] Ch RhIT [ MXhFA (2-30)

[6R] (hR] TC M] hRI 231

and [M R61 Ch R)IT[Fj] [h 6J (2-32)

If the generalized aerodynamic forces for specific modes are

considered, we may rewrite Equations (2-2l)-(23) and (30)-(32) as

IMF]J + £PF] (2-33)

("FRI)= (mFR) + [""FRI 2-4

(A F63 2 EmF6) + t[5F6) 235

122

*MW '.t~g

[RI = ("R] + [ZR] (2-36)

[7RF] =["FRIT +[ 5RF (2)-37) !

[R6 1 = (M61 + MR61 (2-38)

where (FRI [hF]T [M][hR) (Z-39)

(mF ChF3J CM3 [h] (2-40)

T('R ] [hRT [M]hRI (2-41)

and [mR6 C R [hs] (2-42)

Note t'atCmFR) vanishes if frec-frce modes are used throughout. Also,

(mR] is the rigid body mass matrix consisting of total mass, static un-

balances about the coordinate origin (not necessarily the centroid), and

moments and products of inertia about the coordinate origin. The generalized

aerodynamic forces are found (e. g. , from Volumes I or II of this report) for

the appropriate vibration, rigid body, and flipper modes, including their

coupling.

The matrix partitions in Equation (2-3) may now be defined as follows:

m n c s

"-' 'FR] ('FF6] 0 &F

Vx] fIR n (2-43)0 0 rccJ Cs?] g

AS LFsAJ IRSAJ (SC] [ss lz

123

t4 .....

mn n c s

Mc.FJ 0 0 0 a,

n 0 0 0 0 aR[x ] } -- ,(2-44)

c o o rccJ zsz]

n, ,LFSGJ RSGJ [sc2]52] .i2j

m n C s

mrJ o 0 0 0 a

n 0 0 0 0 aR

I] x, = I 6 (Z-45)c 0o rccj [cs2] 6

ns 0 [c] (ss2] e?.L

where eZ refers to the set of second order servo elements and

CC = Control surface coefficients.

CSZ = Coefficients for second order servo element from which

there is input to a control surface.

FSA = Elements of the matrix -KaIHJhF ] for the accelerometer

into which there is input from the body acceleration.

FSG Elements of the matrix -KgL DJhF] for the rate gyro into

which there is input from the body angular rate.

RSA Elements of the matrix -KaLHJ(hR] for the accelerometer

into which there is input from the body acceleration.

RSG = Elements of the matrix -KgLDJ IR for the rate gyro into

which there is input from the body angular rate.

SC z Coefficients for control surface from which there is

input to a servo element.

SS2 Coefficients for second order servo element from which

there is input to another servo element.

1114

The number of rows and columns in each partition is indicatedby

m = Number of flexible modes.

n = Number of rigid body modes.c = Number of control surfaces.us = Total number of servo elements (zero, first and second

order), ns = s + q + r.s = Number of second order servo elements.q = Number of first order servo elements.r = Number of zero order servo elements.

qm 0 'i

n 0

(2-46)

nns s

q

mm 0 1n 0(yr] {r (2-47)

nal reSSI

where el refers to the set of first order servo elements and

CSI = Coefficients for first order servo element from whichthere is input to a control surface.

SSI = Coefficients for first order servo element from whichthere is input to another servo element.

125

rm 0 JeO

n 0(Z] Izl (C s (2-48)

ns (SSO0J

where eO refers to the set of zero order servo elements and

CSO = Coefficients for zero order s',rvo element from which

there is input to a control surface.SSO = Coefficients for zero order servo element from which there is

input to another servo element.

A study of the format of Equations (2-43)-(48) shows that each of the

six matrices is composed of partitions that may be described as aero-

elastic, servoelastic, or'simply servo terms. PASES, a cornpu.er pro-

gram designed to calculate clcsed-loop aero-servo-elastic stability, formsthe various matrices from the servo partitions which are input directly and

the aeroelastic partitions which are generated from basic input (e. g. , modes,

frequencies, mass, and aerodynamic data).

126

iiiiiii iii

APPENDIX B

AN APPLICATION OF PROGRAM MPASES

TO A

TYPICAL AIR-TO-AIR MISSILE

Introduction

The computer program MPASES developed in this report used the same

example problem to demonstrate its usage as had been used in the original

version of the program. That example of an air-to-air missile was not a

very practical one to the extent that it had no wing, the weight of the

aft end of the fuselage was assumed to oscillate with the control surface

(flipper), and there was only a single (rate-gyro) feedback loop. In

addition, the example used an incorrect transfer function for the actuator

[Eq. (78) should have read 6/e 3 - Ka/s(Tas + 1)].

The new example studied here is still somewhat idealized but is more

practical to the extent that a wing, with its attendant aerodynamic loads,

is added, and a light-weight dynamically balanced flipper with bending and

torsion degrees of freedom is considered separately from the body. A more

realistic servo system is also considered with three feedback loops, a rate

feedback loop (with a rate-gyro as before), and attitude and acceleration

feedback loops.

In addition to being a more practical problem which demonstrates a

more general use of Program MPASES, the sample also illustrates certain

peripheral aspects of the problem. One is the calculation of a coupled

mass matrix for the flipper. Another is the calculation of the wing and

127

flipper aerodynamic influence coefficients from Piston Theory for a more

practical range of supersonic Mach nmbers and their assembly to obtain

the proper partitioned format for the complete vehicle.

The servoelastic stability problem is considered first, and then the

aerodynamic influence coefficients are added next to consider the aero-

servo-elastic stability analysis.

Typical Air-to-Air Missile Example

The typical air-to-air missile is idealized as shown in Fig. 3. The

fuselage is idealized as a uniform beam with ten equally spaced masses.

The wing is assumed to be rectangular and massless, rigid in the spanwise

direction and to bend with the fuselage in the chordvise direction. The

flipper is also rectangular and is assumed to be rigid in both spanwise

and chordvise directions but to have root flexibility in bending and tor- H

sion. The flipper mass distribution is simulated by eight masses connected

by rigid uassless bars to the four degrees of freedom of the flipper. The

coupled mass matrix for the flipper is obtained from the least squares

solution in Appendix B of Ref. 12. The various input matrices to NPASES

are derived below.

128

- "Mod

RATE GYRO & ACCEL 6"

S150"

Fig. 3 -Idealization of Typical Air-to-Air Missile

3" M13 13-14 14

3" Ml1. 13 1 M12.14

. .2I

3" M12, A1.12 1

II

3I II:,

L-51/3"-.

Fig. 4 -Idealization of Flipper Mass Distribution

129

The Mass Matrix

The missile fuselage is assumed to weigh 1000.0 lbs. and

each flipper weighs 5.0 lbs. The ten equal fuselage weights on

the half-fuselage are 50.0 lbs. apiece. The flipper idealization

is shown in Fig. 4. The flipper center of gravity in assumed to

be 0.50 in. forward of the hinge line and 3.86 in. outboard of the

fuselage side; the hinge line is located 5. 333 in. aft of the leading

edge. The pitching moment of inertia about the hinge line is 50.0

lb-inZ and the rolling moment of inertia about the fuselage side is

137 lb-in2 . The flipper is balanced so it has no product of inertia

with respect to the hinge line and fuselage side.

The input to Program MASS consists of the inertial data and

the coordinates of the eight mass points. The annotated input deck

and program output are shown below in Tables 1 and 2, respectively.

The coupled mass matrix for the four flipper degrees of

freedom is:

130

"p#i

m V.

4.4

4- it It 11 1-m

P- P

4.4

113

Table 1 Inpuit Cards for Program MASS

C TITLE ANO SUBTITLE CARDSLEAST SQUARES MASS MATRIX

--...MASS tLEMENTS FOR TYPICAL AIHT-AI-MISSLE............___C NUMBER OF STktPS

I_- NUMBER OL-MASS(L.......... ___________

C INERTIAL PROPERTIES

C MASS POINT COORDINATES1*333333333 3.0 -2.b66666667 3.0 -696666664 3.0lo.J33333333 b.O_____ 1.333*33333 9.0.. Tbbq66eb%6 *,4___

-60660066667 9.0 -2*66666.667 90

Table 2 - Ouxtpuxt from Program MASS

LkA1I is.U~mis "AS MiTOIAmA&S KL&AWNS FOR TYPICAL. AIN-TO-AIkM ISSILt

NtUI4Ut Of "Asbsby %?Ri

I~ it S 10 ' i ItIt t3 1 l IS 16 It t9 40

lQrAL "ASb A 2 *S0044409%.01A utNuALAhCt. 0A DAN N e"000OG'fI

___________I 1l UWALANCL %Y *am a .IvituoaouluV Muf~tm W IntwitA I %US I U .%0900090Lt.04JA NUPIiNT U1, £NC0ILA I Wuu A. R *.31 00009t.0

.MuCTOF IMCAfIA I bSUO AT a 0g

At 11 *&sjjjjjjP.01 It 11 .3UOGUOO(.10it__ _______ 21 iiU .'el~eftaOJc.G It .31090a0 .*3fl.- -ju I'

At 3) *-.bs6e',e..I0. I J0000000001 1'@

A( 61 *.A1JJjjjk'Ui It 41 a *[email protected] *-A( SI -. .JJJJJ~jf.OIj Y1 S) a VOGQOOGO&-Gt

At 61 a .&0eataikeGl It 61 a .64000009C.014t 71 *-eefel.I it ?I *.usoovU0164

______ __________ _At .1 .aaIaCQ t a) .400Ou00066I041

itabIAb A'.U *1LUUN4JARI NAISS G&UJPtl IN ArNICtb -.

S040001001______ *~J~O~6 .1400k4 *IQOOUWGi *Ig1uut' .IJ0alvuc.41 .loG00i)E.OL .10, A-O

*[email protected] *3iiJjit.@I *.a.ab&0k -. 0*006lh.4OI *1JJ3j~ito *Ijjj~jt*0I*e1~ ~ tQ6193400t.02 .3J080111.@1 *3VIGuuvuot. *Ju.&904004tLI*'.l~uPG *AOJU.0I

0. ~.Ou1jiefta *.O0000.41 -. 0ui('* *OG6O.h **0O'I .O00t.@l

.000@049*61 *1S0Ooavk'I -. 000101

**e~ta~*O~..%'G4Gk*G1A.eI00u~e0 461i00iast.0

*.*OtUI*E .l~0I~.~uZNA"b MAINIA

** I6*ftft*3k-G4

132

As discussed in Appendix B of Ref. 12, the negative values have no

particular significance. The numerical value of the coupled mass

matrix becomes

4.433333 0.0010417 -0.396181 0.0

0.970139 0.0 -0.726042

1. 133333 -0. 296181

symnmetrical 1. 297917

133

Modal Matrices and Frequencies

The modal matrices and frequencies for the fuselage are the

same as those used in the MPASES example problem. with the '

addition of the deflections of the four flipper mass points. These

are found from numerical interpolation and differentiation among

the last four fuselage control point deflections, i. e. , mass points

7, 8, 9, and 10, to find the deflection and slope at the flipper hinge

line from which the deflections on the flipper follow. The

Lagrangian interpolation formula is

h (x) = h (i- x 8 )(x- x 9)(x- x 10 )

h~x = 7 ( 7 x".)x 7 - x 9 )("' 7 - xI 0 )

+ h8 (X-x 7 )(x- x9 )(x- x1 0 )

x8 -x 7 )(x,-x 9 ( 8- 10)

+ h 9 (x 9 - X7 )(x 9 - x 8 )(x 9 - X1 0 )

(x- x 7 )(X- x 8 )(x- x9 )+ h 10 (x 0 x7 (x 0 x ) (x1 0 -79

The hinge line is at x = 139. 333 and the remaining coordinates are

X = 97. 5, x8 = 112. 5, X = 127.5S. and xl1 0 = 142. 5, so the interpolation

formula becomes

hLa0. 049654 h7 0. 232235 h8 + 0. 526618 h9 + 0. 65 5963h

By differentiating the interpolation formula, we obtain the fuselage

slope at the hinge line, and the result is

134

hHL = -0. 009633745 h7 + 0.048160494h 8

-0. 134086420 h9 + 0.095559671 h o

From the hinge line deflection and slope the flipper deflections are

found from

h 11= h1 3 = hHL- 1.333333 h H L

h = h h + 6. 666667 hH*12 14 = nL HL

This generates the four flipper deflections required to complete the

description of the three fuselage modes.

The fourth and fifth vibration modes are bending and torsion

of the flipper uncoupled with the fuselage. In bending, if h13 = hi 4 =

1.0, then hll =h 1 2 =0.333333 since the flipper is assumed rigid,

and in torsion, if h 1 2 =h 1 4 = 1.0, then h11 = h1 3 = -0.2, as can be

seen from the geometry in Fig. 4.

The control surface deflection mode is similar to the flipper

torsion mode except that it corresponds to a unit control surface

rotation (one radian, positive trailing edge up), so that h1 l = hi 3

1.333333 and h 1 2i=h 1 4 =-6.666667.

The previously assumed frequencies for the fuselage were

45.0, 125.4, and 248.2 Hz. The flipper frequencies are assumed

to be 100.0 Hz in bending and 115.0 Hz in torsion.

The damping coefficients in the three fuselage modes are

assumed to be g 2 0.01, 0.02, and 0.03, respectively, and g =

0.03 i also assumed for both flipper modes.

C1'135

The rigid body modes require the addition of the four flipper

mass points: four unit plunging displacements, and four x-coordinates

for pitching about the fuselage nose, x, 1 =X 1 3 138.0 and x 1 2 = 14

146.0.

136

Servo System

The servo system block diagram is shown in Fig. 5. The

subscripts on the outputs are the same as the servo element

numbers. The system consists of three loops with rate, attitude,

and acceleration feedbacks. The transfer functions and their

corresponding differential equations and numerical values are dis-

cussed in the following sections for each component in the three

loops.

Control Surface Actuator - The actuator servo is assumed to move

the control surface shaft with an angular rate proportional to the

difference between the desired and actual angular displacements.

Its transfer function is6 1

YFS e 5 S/K+ I

and the corresponding differential equation is

(l/Kvi6 + 6 - e5 - 0

The numerical value l/K v :0.016 sec. is used in the analysis.

Servo Element No. I - The rate gyro transfer function is

I,?'/w Z+ (2 CW)

and the differential equation is

137

.0

CCC

- U10 UA

z U

2 'U

I- be8

al

zIrI Z p

ic8

a J"

cc0

OR4

ca.

ID.

a

138

<'/'>',+ (ZC//wge, + elK< . 0

The numerical values are taken as l/w2 = 0.659 X 10"5 sec 2

21 f.i * 0. 359 x L0 2 sec, and K = 1. 0 deg/sec per deg/sec.g g gThe numerical differentiation matrix to obtain the rate e t

based on a cubic spline fit among the deflections of mass points

2, 3. 4, and S. The differentiation matrix is (see Ref. 11)

LDJ = (1/Z4L)L0 +1 -27 +27 -1 0 0 0 0 0 0 0 0 Oj

where 4 = 15.0 in., the distance between fuselage mass points.

The coefficient 1/Z44 = 0.002777778 is input as the rate gyro gain,

and the integer elements of the differentiation matrix are input

directly.

Servo Element No. 2 - The noise filter transfer function and

differential equation are

Y RC = ez/el rs+

andP

?+ e- 1 20

where r =0.001 sec.

Servo Element No. 3 - For the variable gain amplifier we use

e 3/eZ = K B

or

(3 - K Be e 0

139

where K B =0. 1 deg/deg/sec for a Mach nuamber of 3. 0 at sea level.

Although the aero- servo- elastic stability analysis covers a range

of supersonic Macb numbers, a constant value of K B in used.

Servo Element No. 4 - The compensating damping amplifier equations

are

Y D = 4/3 ZW ZTds + 1

d + (2 C /wd)s a

and

dl/ 4 + Zd/wd); 4 + 4 - Td 3 - *3 0

2 5-2zwhere 1/wd =0. 253 X 10- sec , 2cd/wd =0. 223 X 10 sec, and

TdO= .O0l6 sec.

Servo Element No. 5 - The equation for the rate loop feedback

junction is

e 5 =ea8 -ea4

or

a+e a ea8 = 0

Servo Element No. 6 -The attitude amplifier gain is

0 /z= K 0

or

- K 9ea2 =0

140

whereK 2. 0 dog/sac per deg/sec.

Servo Element No. 7 - The equation for the attitude loop feedback

junction is

a 7 = a1 1 -ea

or

a7+a6 11ll= 0

Servo Element No. 8 -The electronic integrator in the attitude loop

has the equation

a 1/8

or

08 -a7=

Servo Element No. 9 -The accelerometer has the transfer function

1M~ z s/w:z + (2Ca/wIa)s + 1

and equation of motion

21/a e9 + 2/Wi9 +09 a m

2 5 2 -2where l/wa = 1. 013 X 10~ sec and 2C a/w &= 0. 6366 xI10 sec.

The numerical interpolation matrix to obtain the acceleration

a is also based on a cubic splin. fit among the deflections ofm

mass points 2, 3, 4, and S. The interpolation matrix is (see Ref. 11)

141

LHJ (l/16)LO -1 +9 +9 -1 0 0 0 0 0 0 00 OJ

The coefficient 1/16 = 0.0625 is input as the accelerometer gain,

and the integer elements of the interpolation matrix are input

directly.

Servo Element No. 10 - The acceleration loop feedback equation is

el 0 = ac - e910 C 9

which reduces to

e + = 010 9

for the stability problem.

Servo Element No. 11 - The acceleration loop gain is

ell/e 1 0 : Ka

or

ell-Kaelo = 0

where K =4.0 deg/sec per g = 0. 002170 sec/ft.a

142

Servoelastic Stability Analysis I,

The input data cards are prepared according to the MPASES

input instructions in Sect. TII. The data are found in the previous

sections of this appendix with the exception of the fuselage vibration

modes which are found in the original MPASES example problem;

the vibration modes for the present example are found by combining

the previous fuselage modes with the additional flipper mass point

deflections derived above in this report. The shift eigenvalue

Y x +100. 0 is chosen to scale CA) to be of the same order of

magnitude as [B] and with the sam sign. The input data deck,

with comments added, is reproduced in Table 3.

The printed output is reproduced in Table 4 except that only

the modes and deflections corresponding to eigenvalues of practical

interest are shown. (The modes and deflections corresponding to

zero or infinite eigenvalues are not shown.)

The stability analysis results come from the solution of a

30th order eigenvalue problem. The order of the eigenvalue

problem i found from

N x Z(m+n+c) + ns + s

where N a order of the eigenvalue problem

m a number of flexible modes

n a number of rigid body modes

143

c a number of control surfaces

no a number of servo elements

a a number of second order servo elements.

In this example, m z 5, n x 2, c = 1, no = 11, and s a 3, so that

N a 30. The stability analysis results are printed for 22 modes

since 8 of the eigenvalues are complex conjugate pairs. Only

modes 4-7, and 16-22 are of practical interest. The seven

oscillatory and four non-oscillatory solutions are all seen to be

stable. The motions involved in the modes are summarized as

follows.

Mode No. Dominant Motion

4 Actuator

5 & 6 Accelerometer (critically damped)

7 Noise filter

16 Rate gyro

17 First body bending

18 Compensating damping amplifier

19 Flipper bending

20 Flipper torsion

21 Second body bending

22 Third body bending

Table 3 - Inpuat Cards for Program MPASES forServoelastic Stability Analysis

c r1TLF CAROSTY'tQ~LC, &tw-rtj-4tw tAUSLp,StWVOELASTIC AN~ALfSLSC C01tTWOL CA4L

2 14 b 2 1 11 ai 0 a so I 1C SMLFT I1'AkANVALUt.

.c MASS MAYR12C 1S? Raw

0I0 0I.00 '

C T6 ? NO RUN00 00Wi I ICIO0 CIO 00001 0,0 CIO

000

Sol 0,.0 0 0 0.0 CI 0.00.0 CI0 0.0 0.0 0,0 0.006 qrM RQW

Solo 0,0 0.0 0 .0 0I0 0.00.0 0.0 0.0 0.0 0I0CT" Ku"

50.0 0,0 0.0 0.0 0.0 0.00.0 4,0 0.0 0'.0

50.0 0.0 0.0 0.0 0"0 000

10.0 0.0 0.00,

50.0 0.0 060 0.0 0I0 CI0

s0l0 0.0 0.0 0.0 0I0 0.0

C loT'. ROWs.600.0 0.0 0,0

C I IT14 ROW4*4333333 0.0010417 -09396 180 6 0.0

c IS'TMROW

C 14TH ROW1 ,29791&6___________________

0.0 0I0 0.0 0.0 0.0 0.000.0 0.0 0.0 0.0 1,333333333-6,666666b7

c OAMP[NC COEFFICIENTS, G0.01 0.0P 0.03 0.03 00

45.0 125.0 24a8.z 100.0 11560c 1 StRTO "OD 31APE

1.0 o9atti~r) *o.122oa -0.532567 *01756204 *00.?520G*0.53Z4b? w0.12R424 COMMit 1.0 0962tbao 16138681T8v a216 6T -- 71186-81C 2ND HOOE1'.0 .00150871 -6*910nA4 0.o.Q90216 .0,419043 0,419f643

145

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136.0 1460C CONTROL SURFACE nUTPUT

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1 2 1.0 3So q -a 0.659 5C NUMBER OF INPUTS TO SERVO ELEMENT

t'CT PUTROCTrIr'TCH RTrE

0 at -2.7777778.3C DIFFERENTIATION MATRIX FOR RATE GYRO-0.0 logo 27,0 *1,o o0.0 0,0 090 0.0 00 0.00.0 0.0

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C %UMBER OF INPUTS TO SERVO ELEMENT

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Cb P U~I~U~1SERVO ELEMENT3 0 to

C NUMNER OF INPUTS Tn SERVO ELE NTi

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146

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INPUT FROM RUOY ACCELERATION0 -2 -0.0625

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~~162

Aerodynamic Influence Coefficients

The aerodynamic influence coefficients (AIC's) at supersonic

speeds are found from Piston Theory and are generated by Program

PISTON (Ref. 4). Three Mach numbers are considered, 2.0, 3.0,

and 4.0, and four reduced velocities are considered, 1000.0, 20.0,

5.0, and 2.0 based on the wing semichord of 1.5 ft. The AIC's for

both the wing and the tail (flipper) are found for the same reduced

velocities by using the wing semichord as the reference for both

surfaces.

The wing planform is shown in Fig. 3 and its motion is deter-

mined by the deflections of mass points Nos. 7, 8, and 9 which are

located at the fractional wing chord positions 0. 152778, 0.569444,

and 0.986111, respectively. One strip is sufficient to represent the

wing since it has been assumed rigid in the spanwise direction.

The basic Piston Theory is used without thickness (a small thickness

ratio T = 0.001 is input). The input data cards are listed below

with comments and follow the input format described in the Program

PISTON User's Manual (Ref. 4). The input data are also echoed in

the program output which follows the input data card listing below.

At the end of the printed output the punched card output is listed.

The punched AIC's are input appropriately according to the proper

partitioning format into Program MPASES.

The tail (flipper) planform is also shown in Fig. 4, and its

motion is determined by the deflections of its four mass points,

Nos. 11, 12, 13, and 14 which are located at the tail quarter- or

163

three-quarter chord positions. The tail has been assumed rigid in

both spanwise and chordwise directions but to bond and twist about

the actuator at the root. Two equal width strips are used to account

for the bending motion. Except for the chordwise and spanwise

locations of the aerodynamic control points, the tail is treated the

same as the wing in terms of input to Program PISTON. The input

data cards are listed below those for the wing in the following pages

and the printed output of AIC's for the tail follows that for the wing.

Again, at the end of the printed output the punched AIC's are listed.

164

Table 5 -Input Cards for Program PISTON

C TITLE ANO SUSTITLE CARDSPISTON TIHORY AEROOYNANICS FOR WING

_Nt. ANO IMPROVED MAC..MISSILEC CONTROL CARDS

0 1 1 1C PL.ANFORM 6IEOMETRY0.0 1.5 1.0 3.0 3.0C STA IGEOMETRY

1.O 1.S O.15377778 0.569444444 0.986111111 0.4C THICKNESS DATA

J,0010 0. 0C SEriESu MACH NUMBERSt.0 3.0 4.0

_._..TR IMN_ANOLLO ,_O.A.TACK FO E*a_(CN_MACN¢._ NUJMaE A_______

0.00.,h*

C SEWIES OF REOUCE VELOCITIES FOR EACH MACN NUMBER104O.0.20.0 __ .0 2.0

1000.0 .00 S.o 2.0

V C TITLE AND SUBTITLE CARDS-- PISTON TNEONY AEROO"NAM;S.F.R TAIL

NEW MAC MISSILEC CONTROL CARDS

k 3 0 4 4 4C PLANFORM GEOMETMY

0 o .. .... I _S.. ___ . _ so 3.0C STMIP GEOMETRY0.5 0.666666467 O.2S 00.75 004

-- _9.95 S...... O 66666 _9t . O S S 0*4

C TNICKNESS DATA0401 0.0 000

sC , IuS_OF_.A,_N ERS2.0 3o0 40C TRIM ANGLE OF ATTACK FOP EACN MACH NUMBER

OoO

0.0 --. 0- s StO1000.0 2160 see 206. .G2006 See Joe

165

Table 6 - Output from Program PISTON

M z :'%

Z. 0l :006 a 0 ZDO

4D*

40. 0000O~ g

no :. 41 .9¢ 0a .-

wxw

° 1 Z

W z

a. o

44

U1. .

I '?~

. . .SM :

Alp"gnMawIc NVLutmCE COEFFICIENTS By PISTON THEORY withe CAMSER

- - OSCILLATORY CASE

14ACH fi, a 2,000000

JimC1 (N a 10000L*44

1M1i SIZE a I BY 3I

£EROYEADStc INFLUEN4CE CnIFFICIENTS BY PISTON THEORY with. CAlobaf

OSCILLATORY CASE

I STRIPS

__________________CM( 1) SIZE a 3 By 3

_*sIlqoco$E,3 ?1109807419*02 __28125040604E~3 *132147~ 4941 0999SlJeE603 $:1Th3,.aE001*lIOS790C.03 a 1324u8e6104 ,149781809.03 *,a91salsotoda 0,3107199t.03 **l?9ooqa~.oot.

*,4~000~*,~*l1&3001 *211~S70031#6S *.1"70 E.101 *00402I4043 *.sl3Wfs~E.0

AERODYNAMIC INFLUENCE COEFFICIENTS BY PISTON THEORY WITH CAMDEN

-______ -. OSCILLAIORy CASE

_____________MAC" HO,. a 2,006000

Ii()a STRIPS

____________________________CM( 1) WIE a3By 3

AIEEOYAHIC INFLUENCE COEFFICIENTS IT P13TON THEORY WITH CAMbtR

OSCILLATORY CAUt

________MACNM, o a,040680

-~ ISTRIP$

____________________________CHC 1) SIZE a 3 IT I

*IS091?q6S.0-;IZ0Tglgcoe----.-9149761841#41 0.0118S4041 6,310739T91[o0i *,l700aT9eE.@0

167

1jai

ACRNO@HMIC INFLUEN4CE CO9FPICZIE%?3 SV PISTONS ?"COOT MIT" CA*§tft

____________________ - - - OSCILLATORY CASE

- ___________ MACH '40, $ ,*Go000

I STRIPS

________________________ Cht 1) Sin *3Sby3 -

hf300YmAwIC INF9LUENCE COEFFICIENTS my PISTON Thica? WIT14 CA*GUN

__________________________________ -OSCILLATOOT CASE

MACH '40, * 3,0440000

_____________ ____ ___ ___ __- - - I STRP$

ACROOYNAMiIC INFLUENCE CaIFftCtENT3 81 P13TON ThIG'? MITH CA"VtO

_______________________ * - OSCILLATORY CASE

-~ --- ---. .. ACH No, a 3,000000

_____________________ -- I STRIPS-.

£ENOOvNAmfC INPLUINCI COEFFICIENTS BY PISTON THEORY WITH CA"690

OSCILLATOY CASC

____________.MACY* No, 3,090000

I STRIPS

____________________ CM( 13 SIZaISSY,3 .~

,315lltatI1oo *0,11tlc4l 044yS41732 tEol. *SSO60184@l 1)33SS146001 4TT06OQ99tool

*:1071500310G Sesa.I *.e.0I@ .,£TS3bSIoo 2,07036631*41 .119111811*00

168

-7-

AEmoNA"Ic IMPLUINCEI COEFFICIET$? *w PISTON THEORY "It" CAMBER

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3 1 1 1PIN4 4002

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Aero-Se rvo- Elastic Stability Analysis

The input data for the aero-servo-elastic stability analysis are

obtained by adding the aerodynamic influence coefficients (AMC's) to

the servoelastic stability analysis data deck and by adding the

necessary aerodynamic control cards and related data. The complete

input data deck is listed in Table 7. The AIC's were based on the

three Mach numbers M = 2. 0, 3.0, and 4.0. The stability analysis

is carried out at sea level for which the density is 0.002378

slugs/cu. ft., and the speed of sound is 1100.0 fps, and at the

three Mach numbers 2.5, 3.0, and 3.5; the eigenvectors are

requested for M = 3.0.

The results of the analyses at the three Mach numbers are

shown in Table 8. Table 8 shows an abbreviated output: it shows

the first page which echoes the aerodynamic control data, a typical

printout of the AIC's for M = 2.0 and k = 0.001, the three sets of

stability results, and the eigenvectors for M = 3.0. There are some

differences from the servoelastic results in Table 4. Both the servo-

elastic analysis and the aero-servo-elastic analysis have 22 roots. How-

ever, the 1.5th root, which was a co0plex conjugate "zero" in the servo-

elastic analysis, now has become the short period root in the presence of

the airstream. In addition, an unstable root, No. 3, has also appeared.

A perusal of the 3rd eigenvector and structural deflections indicates it

to be some kind of divergence in altitude. This instability is a subject

for further research.

176

The motions involved in the eight oscillatory and five non-oscillatory

Ssolutions are sumarized below.

Mode No. Dominant Motion

3 Altitude divergence

4 Actuator

5 & 6 Accelerometer

7 Noise filter

15 Short period

16 First body bending

17 Rate gyro

18 Compensating damping amplifier

19 Flipper bending

20 Second body bending

21 Flipper torsion

22 Third body bending

177

Table 7 - Input Cards for Program MPASES forAero-Servo-Elastic Stability Analysis

C TITLE CAROSTYPICAL AIR-tQj;'AH ML.AENO-SERVO-CLASTIC ANALY!)I UbING P[srUN THEORY

C CUNTROL CARO2 -1 - - 1 II 5 3 1 30 1 0

C SHIFT EINENVALUE100.0C ACOCN1A4T

1.0 3.0 3.0 0.05C ARC DATAC tAC' NUUkda2.0 3.0 4.0

C REW.ICED FREQUENCIES ______________________

0.001 0.05 0.20 0.0SOC AERO DATA0*002378 .1100*0

C NUM4HOFV ELCTTI-0-3

C MASS MATWIXC 1ST 'Odii50,0 0.0 0.0 0.0 0,0 0,00,0 0.0 0.0 0.0 010 0'00-;0 0oo

C 2ND Rnw50.0 0.00 0.0 0.0 0.0 0,0

00 0 0, 0,0 0,0 0,0.0

C SO 0 3RD RaW ________08___0________0__0___0__0____

-0o0 4 i00 0900 0.0 0'0 0,0

0,0 0.0 0.0 0.0 0'0 0.0C 0 TH ROW0-00 0,0 0.0 0.0 0,0090 0'0 0.0 0.0 0.0

C STH RW-070000 0.0 0,0 0,0 OV

0,0 0,0 0.0 0.0C 0 TH Row 0_0_ _60

50,U 0,0 0.0 0.0 0'0 0'00.0 0,0 0.0

C 7TH ROW pOt 0, 0.0 0.0 0,0 0p0

0.0 0.0C 8TH ROW

0,0C 9TH ROw-So 0 10 0. 0,5 0,0C 10TH ROWS0,0 0.0 0.0 0.0 0.0

4,4333333 0,0010417 -0,396806 0.0C 12TH Row_

9 7-03TV-, 0 0,u164417C 13TH ROW181333332 -0.2qb1806

1*2979166C CONTROL SURFACE OEFLECTION mODE

u.-To yU0 0 0.o 0,0 0 00.0 0.0 0.0 0.0 133333333ah,6b&6&66e?1,333333333-..bbabeo67

178

AMoak

C DAMPING COEFFICIENTS, G0.01 0,02 0,03 0.03 0,03

C NODAL VIBRATION FREOUENCIES4S,0 12S,4 148,2 too 115,0

C VIBRATION MODE SHAPES. C 1St MODE

IO 0411195 -0,122424 0.S32567 -0.7s6204 -0,756204.0,532467 w0.122424 Og4111qS 1.0 0,821620 1,1388810,821620 1.13g88

T 2ND MODE1,0 -O.tSO871 -0,910842 -0.990216 -0.419043 0,4190430j990o1S 0,910842 0.150671 -1.O -0.63025b -1.2819 4

.0.60256 -1.2019s4C 3"0 MODEWO-03195 0.64S103 1.0 0.105916 -0 9 ?7894 -947893

-.IOslff 1.0 0.645103 -0.03195 -0,259298 -1,188204"00259298 o1,186204C 4TH MODE00 0.0 0,0 0.0 0,0 0,00.0 0.0 0,0 0.0 ,3333333333 ,3333333333

0_00,0 0*0 0,0 00

C RIGID BODY MUOEC PLUNGE MODEto 1.0 1.0 1.0 1.01,0 1SO ISO 1.0 1.o toS1.0 1.0

C PITCH MODE7.5 22's 37,5 12,5 67'5 82,597,5 1123 127,5 142.S 138.0 146.0

C CONTROL SURFACE OUTPUT1.0 0.016 0,0

G NUMBER O9 INPUTS T CONTR(L SURFACE

C CONTROL SURFACE INPUTS FROM SERVOS5 0 -100

C OUTPUT FROM SERVO ELEMENT1 2 1.O 0,359 -2 0,659 5

c UBER OF INUTSTOSRvO ELEMENT1

C INPUT FROM BODY PITCH RATE-0 . -,*77 778a3

C OIFFERENTIATION MATRIX FOR RATE GYRO0.0 10 o27.0 27.0 -1.0 0,0O00 000 000 0 0,00,0 0.0

C OUTPUT FROM SERVO ELEMENTa I IO 0,001

C NUMBER OF INPUTS TO SERVO ELEMENt

C INPUTS TO SERVO ELEMENT1 0 -1.0

C OUTPUT FROM SERVO ELEMENT

C NUMBER OF INPUTS TO SERVO ELEMENT'G INFUT3 TO 3ERVO ELEMENT

O OUTPUT FROM SERVO ELEMENTc NUMBER OF INPUTS TO SERVO ELEMENT

179

2 ,,. - ,* . , *

*C INPUTS TO SERVO ELEMENTI- 3 ,-i .. -0,O1b

C OUTPUT FROM SERVO ELEMENT5 0 IS0

C NUMBER OF INPUTS TO SfRvO ELEMENT

iikTPiFnSEVO FLEMENT4 0 1,08 0 a10lso__ _ _ _ _ _ __ _ _ _ _ __ _ _ _ _ _ __ _ _ _ _ _

O~OttPITFROM S9-ERVO ELEMENT6 0 1.0C NUMBER OF INPUTS TO SERVO ELEMENT

C INPUTS TO SERVO ELEMENT2 0 02.0 _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

-C ur~ur sOTUTFOW-s rwiti-ErWT7 0 1.0

C NUMRER OF INPUTS TO SFRVO ELEMENT

C INPUTS TO SERVO ELEMENT6 0 IS0

---I ~ F~C OUTPUT FRnM SERVO ELEMENT

8 1 0.0 1.0oC -- NU,4@EWP-F TP'UT3 T"ERVO ELEMENT

C INPUTS TO SERVO ELEMENT

C OUTPUT FROM SERVO ELEMENT9 2 1.0 .06366 -2 1.(013 -5

C-- -WER OF IMPUtS 1U -ZWVVtL1tNI

C INPUT FROM BOOY ACCtLERAtION~207ii45

C INTERPOLAT ION MATRLA FOX ACCELE'QQMETEQ0.0 -1.0 9.0 9.10 -1.0 0.000.0 0.0 0.0 0.0 -070.0.0 000

C OUTPUT. FkOM SERiVO ELEM4ENT1 0 I~U

C NUMBER OF INPUTS TO SERVO ELEMENT

9 0 1.0C OUTPUT FROM SEHiVO ELE.ENT

C NUMBER Of INP~UTS 70 SERVO ELEM4ENT

10 0 -0.002170C AICS

C AIC PARTITION CONTROL

3 0C AIC PARTITION

4,0239SL401 6,62394E#01 3,744SSE,05-9.S92b3E,0a-7,7&eMQEtOS.eq724oEo1PTN 1 13-1910424E#05 5.88196FiO0 &.28929E.05-.t.97240E.01-5,1850lF,0i?.3e62O&.o2PTN 1 14

C AIC PAPTtTInN CnNTROL

C AIC PARTITION5.@1&02F.05m2,b0043E,0Z.5,016Q2E,0S 3'.71087P.#01 PYN 1 12

180

'a 9a01EO51 ItAE0-.60.E0-.~~f0 PYN 1 12c AIC PAR'T~fITIONCiONTRO-L

2 0C AIC PARITTIONS'01E,02EO5in2.bO3E+02w ,01b02E,05 3.71087E+01 PTN 1 22.4,98401aoE05 3.71087E,01.aq84o1E+05.p.S8620E,02 PYN 1 22

T -NO *k.TST -14C AIC PARTITIOIN CONTROL

C-AjtC--PRTflT N CONTROL3 0

C AIC PARTITION9,-i-58$E;02:f.-iq8b8E+01.7.2S0atE#02 1.32479E+00 1,99983E+02 1.1760 0PT4 2 12

1,609SSE+02 1,32479E+00 1,49782E,02-1.91853E*01.3.107U0E,02-1.7Qaa8EOOPTN 2 13w4,41bqSE,01 1.17b'40E-01 2.51570E.02-1,7944$E,00-2,07400E,02.5,124ooEOOPTN 2 14


C ATC PARTITION I2.O0balr.02-5.20AhE+00-2,oobalr+02 7.G2t74E-01 PTN 212I '96E+2--,4-'--Z0 PIN 2 12


-C ATC PlaIfrra2.,0Ob41E,02-i.2008bE,00.?.o06a 1EO,02 7.42174E-01 PTN 2 221.993bOE*02 7.42174E-01-1,99360E.02-5.l72a0E,00 PIN 2 22

-C 3RD (3C AIC PARTITION CONTROL

-c re, PATIToN CUNTROL3 0

C AIC PARTITION-3;2s1b1r,0'1-3;TUb70E+pp-4,51151E#01 3.3119YE-01 1,24989E+Ot 2,94099YE-OYPTN 3 12-1.00599E+01 3,3lt97E-01 9.16t3bE.0oma6.79E3IF,00.1,94212E,01-0i1aa620E-01PTN 3 13-2.760btE,0O 2.94099E-02 1,%7?31EOlw4.48620E-01-1,29625E.01-1.28100E+OOPTN 3 14

-- AIC-PiilfT'ZOW ON ROL

c IT ATTO OTO

C AIC PARTITIONCOTL

1,2S4OIP.01-1 30022E+00.1.25401FO1 l,.B55a3E-01 PmN 3 1QT.2oo~f5~Sao@24z600E+1-1 .93tOE~oo PTN 3 12

C AIC PARTITION CONTROL2 0

* 1.25401t.01in1.30022E,00m1,25401E.01 I.S5541E-01 PTN 3 221,2400E+..0 t.65'S43E.01.1.24600E*01.1..2931,E*00 PTN 3 22

-C--4TW T7TIC AIC PARTITION CONTRO3L

b 1G AIG PARTITION CONTROL3 0

C ArC PARTITION-57Z5I5RCT00i.'5c0op.,p908zE+00 1.32479F-01 t*99983E,00 1.17640E-02PTN 4 12-1,609M+E00 1,32479f-al 1,98E0-.lSE0-4170+0174R-IT a 13

w4,01b9Stae0 1,17640E-02 2,%1S70E,00.1.79445E-01-2,0740OE,00.5,12a4o E.0PTN a 14


*C AIC PARTITION2.0064tE+00-5,300A6E-ot-2 00641E+00 7'42t74E-02 PIN a 12

* 1.h0..217uo-1q93oE~o.5.~2qo~o1PTN 4 12C AIC PARTITION CONTROIL

30

181

AZ Z* A.- I

c AlC PARTITION__________ ____________________

f707&-R1iE;77zes-O4 PTN 'I 221,9936*E#00 7,42174(-02.1*993boE,00-5.172ooE-01 PIN 4 22

C 13T go 2N140C AIC PARTITION CflNTROL

C AtC PARTITION _________ ___________ _____

-,"71S .1 -T-O2 21 -b- .470- E*01 S.3388s5,O', 3.8MIT-OWPN 1 12'2,67875E+OS 4,u3037E+01 ?.aq71bE,05 6.39268F02-S,17S92F,0O%-%.9b5S9E.O1PTN 1 13e7.33a73E*04 3.R5431EgO0 4.18408V,05.5.9b5SQb,01-3,L15020F,05-1.705 7EO2PTN 1 14

1AI' 1P' A H"7 '*ZNr l

C AlC PARTITION CONTROLf

C AtC PARTITION

3'.34q31E.05-1,73b2oEO2.3,3493?F.05 2.47633E+01 PYN 1 12T3135~op~7~1Eiii)3Eo 1220VI +02 PTN 1 12 b

C AIC PARTITIOIN CUNTRnL2 0_____________________ _

AM FrKARTIT N ~r3,3G93?E.05-1.73b24F.02-3,3493?F,05 ?.47b33E+01 PTN 1 223.31735E.05 P,47633E.01.3.31735Ei05.722O1E,02 PTN 1 2

-C NU Xv NO-C AIC PARTITION CONTROL

6 1AM VAKrL11u149 GIJNTOL

3 0C AIC PARTITION _______

-3,0'q+02TO-TI+I4M2'E0 8.96074t-Of 1,3QE*077T9 L-OPTN 2 12F1.07t501.02 8,fl60?4E-Ot 9,A6E0-.75LO-,007+21132+O 2 13

o2,9354qE,01 7,70863E-02 t,67363E,02-l.19312E,00-1310O8E,02.3.aalO53EOOPTN 2 14

C AIC PARrITION CONTROL

C AIC PARTITIONI.33Q751.QZ-3,47?4aEiO0-t.33975E.02 0*.95266E401 PTN 2 12


-C' AreP-ARTrr LUll1,3397Q.#02-3,47a4aE1.o-1.33975r,02 4*.95eb6E-01 PTN 2 221.3269GE402 4,9S2&6E-01-1,32694E,02-3.44402E,00 PTN 2 22


* t~t~PA RzITUM CU14TKUL3 0

C AIC PARTITION _____________________

6.b9h8aE#00 2.21%~19E-01 bP4291E400-3.19614E,00-1,2939Rk,01-2.9R2AOE-QlPTN 3 11-105306a1+04 1.927tbE.02 1.0U602F.,01.9.82R0E-01-d.b2551E,00.8,S2633E-G1PtN 3 L4


C AlC PARTITION8.373421,00.8,66120E-01-e.17342!,OnI 1'23816E.01 PTN 3 12

-- r29337E.Q4rr;--23lti6rZ0 P-8 3?37 12.7T~5 0C AlC PARTITInN CONTROL

6.373421.00-6.08120E.01.8.373027.00 1.238tbE-01 PIN 3 226.293371.00 1.23a16E-1.o..9337r+oo$e.bIOOSF-o1 PIN 3 22

182

C 47HK, ?NOM 7TTICPC rWR T TfON CONTROL

I PATTO OTO

C AIC PARTITIONCNTL

'TS7i9E;oo.1.-;op6fIT+-TbO-4,84273E+00.A7Q-21335s.0 7,70863E-03PTN 4 121,07150E+00 8,86074E-02 9.QRR8ebF-O1-t.?7$S4E100-2,07Oi7F,00.1.19312E.0IPTN 4 13

%2.935 9F-01 7.70863E-03 t h7363E.00-1.19312EZC-l,3008E00-3,QIO53EOI3 TN 4 14C A I CP A R fIflONhCU TRtlL

C AIC PARTITION C(lNTROL

C AIC PARTITION1.133q7SE00.3.47248E-01-le33979F.00 4*.952bbf-02 PIN 4 12

---.32 69 4EM0 0-T.-952 6 SE- 07Q * b 01 P fTVT-C AIC PARTITION CONTROL

1933975E.00-3,47208F-a1-1.139?5FO0 4.,95266E-02 PTN 4 22

C AtC PA4TITION CONTROL

C---AIr PARTITION tNTMOL

C AIC PARTITION ________________________________38~7ET8377T4~~o29,o7z7~o53.33141E+01 2.50ME+.05 2,85050F*00ptN 1 12

2.00614E+05 3.33141k+01 1 ,873iaE.05-a.?92sJr.OZ-3.87963E .Ob-o.AzA293EO1PtN 1 13-S.48687E*OG 2.85050E+00 3,I314qE.05"0.46293E.01-2.58281E,05-1 ,2lb8SE.O2PTN 1 i4-CkI-ICP RT TMW7iCONTROLC-

I IC AIC PARTITION CONTROL

9 0C AIC PARTITION2.516OSE+OS-1.304t5E#02-2.51605E*05 1'.659O71.01 PUN 1 12

--2,3-40E:;S-T7S9-6F.+l-,7, a~?E+50j,9992#zP T N 1 127C AIC PARTITION CONTROL

2 0

2.51Ei05E405-1.30415E*02-2.51&05E.05 1h8 907E+O1 PTN 1 222.48402E+05 1.8 907E+01-2,48402F,05-1.?8992E.02 PTN 1 22

A,3RC AIC PARTITION CONTROL

-C- IC' PARTITION CONTROL3 0

C AIC PARTITION~E331ETO7R~si>.o-3,1eo1~o26,66282E-01 1,00340E+022 5,7OO49E-02PTN 2772

8902458Eg01 6,66202E-01 7*aaq34%E.1. 158~bE.00-1,S5185E.2O2..258?E.01PTN 2 13-2.194751,01 S.70099E-OZ 1,2S2eoE.02.- .92587E-01-1 *0331?L.02-2 ,55375E+OOPTN 2 14


C AIC PARTITION1.0Oh42E.O2-2.E,0830E.00-1.0064?F.02 3.7?1814E.01 PYN 2 12

-991OFTr a PVN 1 12C AIC PARTITION CONTROL

a 0~~~~PATfI TION

1.00a42E*G02-2.boR30O.OO-1.OO642F,02 3.71814F-01 PTN 2 229,9360101 3.71814E-01-9,93*,OSE,01.pS?983E.00 PTH a 22

c AIC PARTITION CONTROL

183

Zia,~

C AIC PARTITION CONTRnL3 o

c AIC PARTITION1,41E0-,87E0%,73E0 1'.66571fo0I 6,2~7126F+00 1..?sZSE-02PTN 3 12S.OI536t.OO 1.6S'TlE.O1 49&A372E#00.m..39b??E7I,OO9b9q8&,OO2,23147E-OIPTN 3 13-1,37172E+00 1.aa2SZ5E-o2 7.2710-,34E0-,4?I+0b34$-IT 14CTIPWI 1IbFFCONTO0L-I I

c AtC PARTITION CONTROL

C ARC PARTITION*-.91E0-.~~~~_62o~~o2-q~f0 -PTN 3 12

6121OO, +,OG FTPN 3 12C ArC PARTITION CONTROL

t~rrARTTIONb.29012E.Oo-hS2075E-01-a.29o)2E*oo 9.295361-02 PTN 3 22b.21005E+00 9,2QS36E-02-6921005F+00-6 449SAE-01 PTN 3 22

-C 4TH -K ;3 RWWIC AIC PARTITION CONTRnL

-C lPXT VIT T ON F N TROL3 0

C ARC PARTITION_________________________________rl63SE40;?,Z2-F0-.39Eo b.bb2a E-02 1,00340E*00 S,700qgE-O3prN '4- IZ8,024I58v.01 6,66282E-02 7g9qF0-.80E0-.5IS*089S7-2T 4 13-2.1947SE-O1 5.70099E-03 lo56E0-.?aE0-*032*02535- T '4 14* C -A IC-PAR rTTMWtONMwdL

I IC AIC PARTITION CONTROL

C AIC PARTITION1,00b42E.O0-2.60830E-01-1.00ba2E,00 3.718l4E.02 PN4-2


*- -AC Cr r P( r I LOPE1.00642E.00-2,b0830E-01-t.OObGZF,00 3,7181iGt-02 PIN 4 2290936E-01 3.?1S14Eu02-Q,956o6E-01-.S?9783r-Qj PTN 4 22

c 4ACi -NUMa~ks- PL* VELOUCTLES ANO EWl.,NVECIUNREUEST

184

Table 8 - Output ftrou Progm NPASES for Aero-Servo-ElastLcStability Analysis

I I

Q r

a * A a

,! I * I

t:I 5

-I I 0 0 °a * , - . -X so

4 1

1* 4 o U,,,4*- -i I. *n- *i l l m

.t " -l II

; : i *g

L ' 4' 4'5 3bass !

4' 8e

• -a ...- . . -

WII

1C S p

ftN

A 0

* 4 41

ft5

0. *JJ

.*6* %4* a. a,.*0 4000 0 40 00 000 *000 000.; '00,4 '000 000:0 .

* 0 50

;ZI: 4z; :4;Z:: ft* -0O00 00

1866

*NOVO-4 44 ... aam. 4qf--

* - *A'aqu ~ .q v - kw

*~~ 0 Ij 4; *W

q?0 C PA 0* a-t a .1 It4 SA ,

4 j4ai

I ItI. W.. 0* 00 06 0 AI A40g' *

0000e.e..0004a.4-A ow A

I~~~ T ~4

a I , I000-

*~~~~~~' 04 a -0S4- ~ a j

0040 0000 00 2

I... 187

44047i4EqEEU4*4 i l-Ef

.0 4 Of% * 10

.. Alft ~ ~ of 5* .*Ansa

to~@6 0 i--

*~ ta Sof

A'A ~ -4 -4

*4*** 5.. .... 5oil

I Z Ift ft-e4

4Al0 I8 Al aIla60

. - SAgI ~ ~ ~ * .oi I* 11 .1 1 $ 1 ;1 1 1 0

~188

1,44I S ftS

I If I* on1,a*,.e~ Olt0 j 0400 00 0 0e 411

E~4i

Iq 1- 0 ia oI a:. 1 10 IV S % : r . "I I1s P,* s sI

ft 0 * * ft Ib ft0" 4 t

9 IdI

.SSt *!;Sm : T -4 - t.V

w AM ow ft

Z,... P' a.. * oft"*. . **

M~ ORMgN-g"

C11- *e0~I0~~Ae

-~ 189

1 4 1 114 ~ ~ ~ aA 0 6 1 * .

~ Ia WOO Po -V 4P.

ltI s I. $ifS. ftg S P.40,40,I 111111101S ;1" *0 "

08 -.1

Z 20* JI,0 R . %

ow lose S *d.2 ~

S. I. S t . .; . j t~ I,

I I & a TO I

*s--.-* *4 SaO-- *ftP. it "01H 5 lSSt

I ~ ~ ~ ~ Z 5 SSBS e sss*s a 0 18.:00 is PE~0~~ ~ d.4:H- 04A0 Oto

1VIAIs*. 0* of '0 up. ft * 4 #

A* 9* ;. ; P.7 i0

A 4*- 40 ft

T. -II IS oil'61me 1

I4 so~ v I; ft if** ' nil '-S m.t f4 a -.4 s

tsszftsFit a.fS~

at I 10 0 1 1 1 1 a I0

:0 00944 Go we J

AliAi~i 2#0P* Al Go-I.W*1 aw1K 0P.44P

SU M: 22;,TT"T00 a00

jgvvxp jjq;Vxa G

I I . pa

SM P." 1 .I a II, IS

1 000 *eo 0 0 104!0a SoI aggg*TT1 11 1 11 1 1 oil o f f

P.1 000 :0I -210000

I I fi

i In~KiGf -. I.Go, 114P @&m'4aK

np.-~ 4Fq6~ - k i..* *~~I n o * .* * * -* 4

I .dl : 1*U1- - 11Reafi~~~ao

1 11. ~ * *

.. . . -. ... . - = -, , +

-- . . . . . .

:~~ ~~ 01 0 l

t1 I II I I I It, I

AJ 04 Ift J

=ff -r .0 -O10O O'sO

0o0 =o : so""

-t 0 it' ' '' °, !" , ! i , ,

,,, 1,. 111 ,| 01 1 C, $10 71640

O4v O r

40AI *.o ! :- P1

dp 1

NO# 4ee0 V olee11 Jil IIl 01 t a S, 0-0 1| ' I ft I 1 11

NAA 610 fm1

Jil itH I : o4p

loldn I -a- ft Nis2? -ft ap-

0.... @.... .... o.0 . ... I oo.. 0o..I ...0I , II III , fI P' I' I I + ~ ' II,;I I

!( ----I *I 2 ,

0 0 0 0 0 1 00 - - ...

1:i 2:. 40 1 1 lo Is. vr- OI I; I I of1

!A 0

4~ A.,~ Id 1:9 1 . ., ~ 0 f.49.4*p ;. r-4. 400% v It.* ;

ft4 I454 'a25 pal 't* ~ l

I I II *. I z I 1 1 11

'a0 ,** ot 1 .i0i *5 tlq

00.1 MPi On III VA at 0 1 1 I IftS

A*t1 '*P PP4- Ai@ @40 0.-

Ala.. V.1i4 to:. 114

ft' 20feISI. t105 **.1 -1 T* T' * 9 2

if vt **49m I' a91~ .4ISIi 4 ..

I IF

-0,1 61.001.16 44 1.14

' ~ f -V? 0 4 b 7?0.0 . me 7?'?? *&*O* 6 011_*SO

41PA. ft ,. P. [email protected] on.5 rS i*A0 .Ul i

2

I a it

48W~~ ~ 845 20"l~ ,t E 4

4.004.4.4Eed0iISIA

v I

gx 0400 0 a... a 00 a

w pa 2* a* * a

op 1 1 8

saleg1g

0 0a

i ooe oee oo ee*0E qooo o

ItI3b -0 op. Md .WMW4 .E .4 .I ...

1 I "lllMH

3ww I

! 193

" a-

• I I ,E tO, l$

-|.. i. _. I ..-... a .. .. * a ~ * .. a... { I I+

~ I !~V .

SYMBOLS

In this modification to the original program PASES, the general

notation has been retained. The nomenclature used in the basic develop-

ment (Ref. 1) has been reproduced in Appendix A. The new symbols intro-

duced in the present modification are defined here. The reader is referrea

to Appendix A for the definitions of symbols not appearing below:

A Coefficient in spline fit, Eq. (57); also see Appendix A

a0 ,a1 Coefficients in spline fit, Eqs. (67) and (68)

B Coefficient in spline fit, Eq. (S7); also see Appendix A

br Reference semichord

C Element of discrete damping matrix; coefficient in splinefit, Eq. (57)

C Element of generalized damping matrix

if.- Ch Element of oscillatory AC matrix

ChDh Element of dauping AIC matrix

Imaginary part of

ChR Real part of Ch

Element of static AIC matrix

c Reference chord

D Coefficient in spline fit, Eqs. (57) and (60)

E Modulus of elasticity

f Cyclic frequency

g Aeroelastic damping coefficient

I Element of unit matrix; spline bending moment of inertia

Ikj Element of spline interpolation matrix, Eqs. (74) and (75)

Ct KjiKkj Spline parameters, Eqs. (70) and (73)

K Element of generalized stiffness matrix

194F

N Element of generalized mass matrix

• P Spline reaction force

Q Spline loading parameter, Q=P/12EI

q Dynamic pressure

S Reference area

s Reference span; Laplace transform variable

T1/2 Time to half amplitude (negative value denotes time todouble amplitude)

U Element of matrix in canonical form for eigenvectoranalysis, Eqs. (45) and (47)

w Spline loading, Eq. (56a)

x Coordinate along spline; also see Appendix A

y Spline deflection; also see Appendix A

A Eigenvalue (-0) in eigenvector analysis, Eqs. (45) and (46)

V ii Decay rate coefficient

P Atmospheric density

C) Denotes complex amplitude- denotes inclusion of aerodynamicdamping and stiffness in b and k, respectively.

Subscripts:

r Oscillatory mode number in frequency lining-up process,Eq. (53)

s Number of mode being lined-up, Eqs. (53) and ($5)

I

195

REFERENCES

I. Anon., "Modal Flutter Analysis Study; Vol. IV, Aero-Servo-Elastic

Stability Analysis Program: PASES," Rpt. No. SD 15204R, Hughes Air-

craft Co., April 1971.

2. Jocelyn Lawrence, A., and Jackson, P.. "Comparison of Different

Methods of Assessing the Free Oscillatory Characteristics of Aero-

elastic Systems," British A.R.C. C.P. No. 1084, 1970.

3. Rodden, W. P., and Revell, J. D., "The Status of Unsteady Aerodynamic

Influence Coefficients," Inst. Aerospace Sci. Paper FF-33, January 1962.

4. Anon., "Modal Flutter Analysis Study; Vol. I, Subsonic Strip Theory

Unsteady Aerodynamics Program and Supersonic Piston Theory Unsteady

Aerodynamics Program," Rpt. No. SDIS204R, Hughes Aircraft Co., April

1971.

S. Giesing, J. P., Kalman, T. P., and Rodden, W. P., "Subsonic Unsteady

Aerodynamics for General Configurations; Part I, Direct Application

of the Nonplanar Doublet-Lattice Method," AFFDL-TR-71-5, Part I,

November 1971.

6. Funderlic, R. E., and Rinzel, J., "ALLMAT, A Fortran IV Arbitrary

Matrix Eigensystem Solver," SHARE Program Library, SDA No. 3441,

2 March 1966.

7. Anon., "Computer Subroutine Libraries in Mathematics and Statistics,"

International Mathematical & Statistical Libraries, inc., Houstin, TX,

November 1977.

8. Wilkinson, J. H., The Alaebraic Eiaenvalue Problem, Oxford: Clarendon

Press, 1965.

9. Cunniugham, A. M., Jr., "EIGNVCT, a Subroutine to Obtain Elgenvector

from Real Non-Symmtric Matrices for which the Eigenvalue is Known,"

General Dynamics, Pt. Worth, 1973.

C 10. Harder, R. L., MacNeal, R. X., and Rodden, W. P., "A Design Study for the

Incorporation of Aeroelastic Capability into NASTR iN," NASA CR-111918,

Xay 1971. 196

11. Rodden, W. P., "A Method for Deriving Structural influence Coefficients

from Ground Vibration Tests," AIAA J, Vol. 5, May 1967, pp. 991-1000.

12. Rodden, W. P., Parkas, E. F., and Malcom, H. A., "Flutter and Vibration

Analysis by a Collocation Method: Analytical Development and Computa-

tional Procedure; Appendix B, The Transformation of Inertial Data into

a System of Lumped Masses," Aerospace Corp. Rpt. No. TDR-169 (3230-11)TN-14,

31 July 1963.

197

Documents

AERO-SERVO-ELASTIC STABILITY ANALYSIS at flutter, the combined aero-servo-elastic stability analysis does not obtain physically meaningful frequencies and dampings at flight conditionsAuthors: