Effects and Importance of Penetration and Growth of Lift on Spacevehicle Response

NASA CONTRACTOR .. .. - REPORT

e c/1 e z

EFFECTS AND IMPORTANCE OF PENETRATION AND GROWTH OF LIFT ON SPACEVEHICLE RESPONSE

I .. . TECH LIBRARY KAFB, NM

0099bb4

NASA CR-326

EFFECTS AND IMPORTANCE OF PENETRATION AND GROWTH OF LIFT

ON SPACE VEHICLE RESPONSE

By Robert R. Blackburn and A. D. St. John

Distribution of this report is provided in the interest of informationexchange. Responsibility for the contents resides in the author or organization that prepared it.

Prepared under Contract No. NAS 8-11012 by MIDWEST RESEARCH INSTITUTE Kansas City, Mo.

for

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION

For sale by the Clearinghouse for Federal Scientific and Technical Information Springfield, Virginia 22151 - Price $5.00

PREFACE

This report was prepared by Midwest Research Institute under Contract No. NAS8-11012. Thework was performedunder thetechnical supervision of Mr. M. H. Rheinfurth, staff member of -the Dynamics and Flight Mechanics Division, Marshall Space Flight Center.

This report covers workon the contract during the contract period 21 June 1963 to 20 August 1964.

Mr. Robert R. Blackburn was the principal investigator for the anal- ysisin this volume. Mr. A. D. St.John was projectleader. Several members ofthe staff contributed in the preparation of numerical results: Messrs. Donald R. Kobett,John E. Scheu, Duncan Sommerville,and Dr. William D. Glauz.

TABLE OFCONTENTS

Notation ...... ix Abstract ...... xv

Sumrtlary ...... 1

I. Introduction ...... 2

11. Basic Goal of the Investigation ...... 2

111. Discussion of the Investigation ...... 3

A. Description of Saturn C-5 Model Used ...... 3 B. Description of Indicial and Impulsive Aerodynamic ForcesUsed ...... 3 C. Method of Solution for Indicial and Impulsive Responses of Saturn C-5 ...... 10 D. Wind-Induced Responses ...... 32

IV. Method Used in Analysis of Effects and Importanceof Penetration and Growth of Lift on Missile ResFonse ...... 35

V. Results and Discussion ...... 49

VI. Conclusions and Recotmendations ...... 54

Bibliography ...... 59

Appendix I - Equations of Motion of Saturn C-5 ...... 60

Appendix I1 - Develogment of Transient, Quasi-Steady and Steady Generalized Aerodynamic Force Expressions Resulting From a Unit Step and Unit Impulse Wind Profile . . . 70

Appendix I11 - Calculation of Initial and Steady-State Conditions Resulting From a Unit Impulse and Unit Step Wind Profile ...... 98

Appendix IV - Presentation of Extreme Excursion and Average Response Plots ...... 1C8

-v- TABLE OF CONTENTS (Continued )

List of Tables

Table No. Title Page No.

I Tabulation of Increment Sizes Used in Indicial and Impulsive Response Calculations...... 12 I1 Altitude Bands Used in Calculating Wind-Induced Responses ...... 34 I11 Ratios of Average Responses, ObservedMaximi Responses and Expected Maximum Responsesfor Profile No. 1 ...... 50 IV Ratios of Average Responses, Observed Maximum Responses and Exgected Maximum Responses for Profile No. 2 ...... 51 V Definition of Saturn C-5 Body Geometry ...... 77 VI Definition of Coefficients in(II-la), (11-25), and (11-36)...... 80 VI1 Elements of Forcing Function Matrix ...... IC) .. 100 VI11 Elements of the Transformed Forcing Function Matrix {E}...... 102 IX Initial Conditions Imposedby a Unit Impulse Wind For the Cases of Penetration with Lift Growth and Pure Penetration ...... 103 X Initial Conditions Imposedby a Unit Step Wind for the Cases of Penetration with Lift Growth, Pure Penetration and Instantaneous Immersion...... 105 XI Steady-State Values forImplsive Responses ...... 107 XI1 Steady-State Values for Indicial Responses...... 107

- vi - TABU - OF CONTENTS (Concluded )

List of Figures

Figure No. Page No.

1 Indicial Normal Force Growth Functions vs...... Time 2 Indicial 1st Bending Moment Growth Functions vs.. . Time 3 Indicial 2nd Bending Moment Growth Functions vs.. .Time 4 Impulsive Normal Force Growth Functions vs. ....Time 5 Impulsive 1st Bending Moment Growth Functions vs. Time. 6 Impulsive 2nd Bending Moment Growth Functions vs. Time. 7-24 Impulsive and indicia1 responsesof several generalized coordinates considering three aerodynamic representations - 70 sec . F.T...... 14-31 25- 33 Wind-induced responsesof several generalized coordinates considering three aerodynamic representations - 70 sec. F.T. band ...... 36- 44 34 Extreme Excursionof 2nd Bending vs. Cumulative Probability . 60 sec . F.T...... 48 35 Extreme Excursionof 2nd Bendingvs. Reduced Cumulative Probability . 60 sec . F.T...... 48 36 Approximate 4th Bending Impulsive Response vs. Response Time - 60 sec . F.T...... 55 37 Approximate 2nd Bending Impulsive Responsevs. Response Time - 60 sec. F.T...... 56 38 Actual 2nd Bending Impulsive Response vs. Response Time - 60 sec . F.T...... 57 39 Saturn C-5 Coordinate System ...... 61 . 40 Mechanical Analogyof Sloshing Fluids ...... 62 41 Saturn C-5 Body Geometry ...... 72 42-89 Extreme excursion of various generalized coordinates vs. reduced cumulative probability for five flight time bands...... 109-135 90-99 Average response of various generalized coordinates vS. flight time ...... 136-141

- vii -

NOTATION - Ais = see (11-27), (11-28), (11-29)and (11-30) a0 = gain value of attitude control.channe1 = gain valueof control damping -al Bim = see (11-27), (11-28), (11-29)and (11-30) b0 = gain valueof flow direction channel - Cin, = see (11-27), (11-28), (11-29) and (11-30)

CE = damping coefficient of swivel compliance ( = 25,p~@~)

DO = base diameter of missile

F = thrust fo( T) = see (11-4)

G(y;u) = see (11-19) gm = structural damping coefficient of mth bending mode

= Qg acceleration of gravity gsl = damping coefficient associated with first sloshing mode gs2 = damping coefficient associated with second sloshing mode

H( y;a) = see (11-26) I1 = instantaneous. immersion = mass moment of inertiaof missile about its c.g.

I(y;a) = see (11-37)

I( 7) = Heaviside step function % = spring constant of swivel compliance

- ix - = length of missile

= aerodynamic moment growth function due to a unit impulse wind

= aerodynamic moment growth function due to a unit step wind

= Mach number

= total mass of missile

= mass of swiveledengines

= sloshing RBSS associated with first sloshing mode

= sloshing mass associated with second sloshing mode

= aerodynamicnormal force growth function due to a unit impulse wind

= aerodynamicnormal force growth function due to a unit step wind

= pure penetration

= penetration with lift growth

Q ( T)~= aerodynamicbending moment growth function associated with the nlth %, bending mode anddue to a unitimpulse wind

Qli ( i)s= aerodynamic bending moment growth function associated with the mth m bending mode and due to a unitstep wind

9 = pU2 , dynamic pressure 12 R = radius of nlissile

S(X) = missilecross-sectional area

?E = first moment of swivel engine about swivel point

Tm = generalized mass associated with mth bending mode t = time

-x- U = missile freestream velocity vy,v(x,t) = horizontal wind velocity

X = coordinate of missile c .g. g = coordinate of gimbal point xE

331 = coordinate of sloshing mass associated with first sloshing mode = coordinate of sloshing associated with second sloshing mode 532 mass 9 = coordinate of top of missile = mth bending mode deflection (normalized to one at the gimbal ym curve point )

= lateral translation of rigid mode YO

Aerodynamic Integrals

(m,i = 1,2,3,4)

(tu = 1,2,3,4)

(m = 1,2,3,4)

- xi - (m = 1,2,3,4)

(k = 0,1,2)

(m = 1,2,3,4)

(k = 0J1,2)

Greek Symbols

% = indicated angle of attack eC = controldeflection (sometimes reQerredto as c.1 , first control)

& = actualswivel engine deflection

I&, = damping ratio of swivelengine llm = amplitude of the mth bending mode at gimbalpoint

BE = mass moment of inertia of swivelengine about swivel point

- xii - p = airdensity

T = time

51 = amplitude of first sloshing Inass relative to tank wall = amplitude of second sloshing mass relative to tank wall 52 # = pitchangle

6, = indicatedattitude angle

w = naturalfrequency of xuth bending mode Bm wDE = naturalfrequency of swivelengine

cosl = natural frequency of first sloshing mode

us* = natural frequency of second sloshing mode

Subscripts

i,m = integers

(I) = denotes differentiation with respect to x

(') = denotes differentiation with respect to time

- xiii -

ABSTRACT

The wind induced responsesof the Saturn C-5 without fins arecal- cula'ted with three aerodynamic representations. The most accurate representation uses unsteady aerodynamics and accounts for penetration gusts.into the The second uses pseudo steady aerodynamics and accounts for penetration. The third uses pseudo steady aerodynamics and assumes equal wind cross flows over the missile length.

The responses, which include two sloshing and two bendingmodes, are affected by penetration to a detectablebut insignificant. degree. The useof unsteady aerodynamics causes very little change. The conventional, third aerodynamic representation provided in these cases slightly conservative (large) responses

-xv- SUMMARY

Missile reeponses to winds are usually calculated using pseudo steady, slender body aerodynamics and the assumption that wind induced crossflows are equal at all stations along the missile.. However, it has been recognizedthat the time delays aaadciated with missile penetration into a gust could significantly affect the actual forcing hctiohg, especially those for the bending modes.

The wind induced responses of the Saturn C-5 (without fins) are calculated and compared for three aerodynamic representations. The most accurate representation accounts for the time delays of penetration and uses unsteady aerodynamics so that growth of lift smoothingand delays are included. The second representation, called pure penetration, uses pseudo steady aerodynamics butincludes penetration delays. The thirdrepresentation is the conventional one,defined here as instantaneous immersion.

The analytical model includes the following degrees of freedom: translation, rotation, first andsecond bending, two sloshing modes, and control. Frozen coefficients are used with each set applicable to a 10-sec. flight time.

Responses are calculated for the unit step and unit impulse gusts. These responses are used in a Duhamel integration to obtain the responses to two wind profiles defined by values at 25-meter altitude increments.

The incorporation of penetration effects causes a detectable but insignificant change in the responsescalculated for the Saturn C-5. The ad- ditionof grGWth of lift causes a very small change. The conventional,instantaneous immersion, responsesare conservative (large) for the cases com- put ed .

The present calculations do not provide an example of significant responsechanges due to penetration and lift growth. However, it appearsthat the possibility and nature of significant effects could be detected in a comparison of the unit step andimpulse responses based on instantaneous immersion and pure penetration.

It is recommended that the response comparisons be extended to the third and fourth bending modes and to a Saturn C-5 with (simulated) fins.

-1- I. INTRODUCTION

The calculation of missile responses due to winds requires the incorporation of a satisfactory description of the aerodynamic forces into the responsecalculations. In a previous program [lJ, two aerodynamic effects were pointed out which have been neglected in previous investigations of missile response to horizontal winds or gusts. The first is the gust penetration effect and the second is the lift growth lag due to aerodynamic inertia. In the same reference these effects were incorporated in rigid body indicia1 and impulsive aerodynamic loadsbased on slender body theory. Some numerical results were presentedfor a multistage missile entering a unit step and impulsegust. The results, however, were not utilized in response calculations.

Some effects of penetration and growth of lift were postulated in [J. For rigid body motions, the effect will be to filter out the high wind fr.e- quencies. On the other hand, when missilebending modes are included,there is the possibility of augmenting some wind frequency responses.

The project reported here is an investigation of the effects and importance of penetration and grarth of lift on missile responses when both rigid body and bending modes are considered. The Saturn C-5 configuration, excluding fins, is used in the study.

11. EASIC GOAL OF T€E INVESTIGATION

The basic goal of the work presented in this report was to determine the effects and importance of penetration and growth of lift on missile re- ponses. To accomplish this goal,the investigation was dividedinto two phases :

1. The indicial* and impulsivwresponses of thevehicle were com- putedusing three aerodynamic considerations: (1)penetration with lift growth effects, (2) purepenetration, and (3) instantaneous immersion. Slender body theory was used throughcut the investigation.

2. The wind-inducedresponses of the vehicle for the three aerodynamic environmentswere computed from the impulsive responses using the Duhamel superposition integral.

* Response to a unitstep. ** Response to a unit impulse.

-2- r

In both phases, numerical comparisons were of maderesponses calculated both with and without the effect of penetration and liftA growth. mathematical model of the SaturnC-5 configuration, excluding fins, was used in the analysis.

111. DISCUSSION OF THE INVESTIGATION

A. Description of Saturn C-5 Model Used

The SaturnC-5 configuration was used in the investigation of the effects and irnportance of penetration and growth of lift on missile response. The equations of motionof the missile system are presented in AppendixI. Seven generalized coordinates were considered in the response calculations: lateral translation, yo ; rotation, (d ; the first two bending coordinates., Ill and q2 ; two sloshing coordinates, El and Q2* ; and control deflection, Bc**. The control system considered in this report utilizes both an attitude reference control anda flow direction indicator.

The equations are valid afor swivel engine controlled vehicle where the swiveled engines account for four-fifths of the total thrust force. The missile and atmospheric parameters appearing in the equations are considered constant in predetermined time or altitude intervals.

The actual SaturnC-5 configuration contains engine shrouds and fins located on the aft section of the vehicle. The effects of these empennages were neglected in the analysis.

B. Description of Indicia1 and Impulsive Aerodynamic Forces Used

Tfie development of the transient and quasi-steady, generalized aerodynamic forces resulting froma unit step and impulsive wind profile is presented in AppendixI1 of this report.The development is based on slender body theory. The forces corresponding to rigidbdy and bending coordinates are presented first fora general wind profile. The indicia1 and impulsive forces are then derived.

* The two sloshing coordinates considered are associated with the fundamental fluid motion in thelox and fuel tanks located in the boosterof stage the C-5. ** The actual engine deflection, & , and the control deflection, PC , are considered tobe equal.

- 3- / The wind-induced forcing functions which are compared in this report have botha geometric and an aerodynamic aspect.

In the geometric considerationtwo cases are used.In the simplest case, instantaneous immersion, all stations along the missile are assumedbe to immersed in the same wind-induced crossflow, namely the wind crossflow occurring at the nose. The more accurate geometric representation, called penetration, assigns to each station along the missile the wind crossflow which exists at the altitude occupiedby the station. With penetration, the missile nose enters a side gust first and in subsequent time successive stations along the missile length move into the crossflow.

Two representations of the aerodynamics are used; they are quasi- steady and transient. In the quasi-steady representation the airforcesa at missile station are those which would exist if the local crossflow persisted unchanged for an extended time.The transient representation is basedon the theory of unsteady motionof slender bodies and includes the grcwth of lift with time. 0 Three types of wind-induced forcing functions are assembled using combinations of the geometric and aerodynamic representations. The simplest type is called instantaneous imlersion and uses the instantaneousimersion geometric representation with quasi-steady aerodynamics.A more accurate type, called pure penetration, uses penetration geometrics and quasi-steady aerodynamics. The most accurate aerodynamic forcing functions are called penetration with lift grcwth. These latter functions use the penetration geometrics with transient aerodynamics.*

The simpler function types, instantaneous iEmersion and pure penetration, can be obtained from penetration with lift growth {see I1Appendix 1

The develosment of the indicial and impulsive transient and quasi- steady generalized aerodynamic forces for rigid body motion follows the work presented in [l] and [2].

The derivation of the indicial and impulsive transient and quasi- steady aerodynamic forces, , corresponding tothe bending coordinates, &7h

* The crossflows inducedby missile motions are in all cases treated with quasi-steady aerodynamics. These crossflows are small in comparison to the wind-induced crossflows and appear in the left hand side of the equations of motion (see AppendixI } .

-4- 12 , requires a description of the mode shapes of the missile.A considerable savings in computation can be obtained if the mode Ym(x)shapes, , are approximated by polynomials. The components of Qqm corresponding to the constant and linear terms of the polynomials can then be rewritten in terms of the wind- induced aerodynamic force expressions corresponding to the rigid body coordinates. In addition, the polynomial representations forY,(x) need only apply to specific regions of the missile length, since the kernel of the integrals describing Q take on values only over the conic sections of the missile. qm The use of polynomial approximations for Ym(x) does not detract a from general approach to numerical solution, since the important mode shapes of the Saturn C-5 configuration were efficiently describedby low order polynomials in the regions of interest.

Mode shape segments, corresponding to the conical ofregions the C-5, were fitted with quadratic polynomials of the form

where the coefficients x , 5 and F** are considered constant fora discrete flight time or altitude.

The subscripts m and i designate a specific mode and conical region, respectively. Four conical regions(i = 1,2,3,4) were considered in the analysis (see AppendixI1 } . For theC-5 configuration , the quadratic polynomial mode shapes yielded values which are 1 withinper cent of the actual mode deflections.

* Penetration and lift growth lag effects are potentially important when missile bending modes are consideredin an analysisof missile response. Since the wind-induced aerodynamic forces corresponding to the bending coordinates were not developed [l], in it was necessary to formulate these forces in this report. These forces were derived using slender body theory and include the geometric and aerodynamic aspects discussed above. ** Numerical values of these coefficients were computed for the first four bending modes (m= 1,2,3,4) between flight timesof 10 and 140 sec. at 10-sec. intervals. However, values for only the first two bending modes between 30 and 100 sec., inclusive, were used in the numerical computation of the indicia1 and impulsive responses.

-5- The fundamental expressions required to compute the wind-induced forcing functions for the SaturnC-5 are given in Appendix11. The equation numbers of the indicialand impulsive forcing functions are given below for each of the three aerodynamic considerations.

AerodynamicConsiderationNumbersEquation

1. Penetration with lift growth

a. Indicial a. (11-13)~(11-20) J (11-31)

b . Impulsive (11-38)) (11-39)J (11-40)

2. Fure penetration

a. Indicial a. (11-Sl), (11-53), (11-55)

b . Impulsive(11-57), (II-Sg), (11-61)

3. Instantaneous immersion

a. Indicial a. (11-66), (11-67), (11-68)

b . Impulsive (11-69), (11-70),(11-71)

For purposes of general interest, plots C-5of normalthe force, first bending moment and second bending momenta unit for step and impulse side wind are given in this section. The curves are presenteda Mach for number of 1.345 (70-sec. flight time). The consideration of penetration effects causes the generalized forces to be distributed over time.+,he- These dependent forcing functions are called growth functions. The indicial normal force, first bending moment and second bending moment growth functions are given in Figs. 1, 2 and 3, respectively. The impulsive normal force, first bending moment and second bending moment growth flrnctions are given in Figs. 4, 5 and 6, respectively. The effects of penetration with lift growth, pure penetration and instantaneous inmersionon the generalized forces can be easily seen from the figures. The impulsive forces resulting from the effects of instantaneous immersion are not presented since they can only be described from a limit consideration.

-6- -- -Instantaneous Immersion i//1 ------Pure Penetration Penetration with Uft Growth = I M 1.345

0 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28 0.30 T Time, I Fig. 1 - Indicial Normal Force Growth Functions vs. Time in Seconds

2.0

0 r(

0 0.02 0.04 0.060.08 0.10 0.12 0.14 0.16 0.180.20 0.22 0.24 0.26 0.28 0.30 Time, T I Fig. 2 - Indicial 1st Bending Moment Growth Functions vs. Time in Seconds

-7- ,2- - .o-

.D -

"-Instantaneous Immersion - - - - Pure Penetration G- Penetration with Lift Growth M = 1.345 4-

r" - 2- I I I 0 1 1 I1 I I 1 I I I 1 I 1 I 1 I 0.00 0.14 0.16 0.20 0.22 0.30 \ 0.10 0.12 0.18 0.24 0.26 0.28 Time, 7 2-

G- / Fig. 3 - Indicia1 2ndBending Moment Growth F'unctionsvs. Time in Seconds

- - - - - Pure Penetntion Penetration with Lift Growth M = 1.345

b Fig. 4 - Impulsive Normal Force Growth Functionsvs. Time in Seconds

-8- r

7 2 21b C I .rl r' * I1 11 I1 - - Pure Penetration 2 20- It - - II Penetration with Lift Growth 16- I I M = 1.345 I 3 I 2 12- I I I B I x Q- M I r,*l f I ;I

W 0.02 0.04 0.06 0.08 0.12 0.14 0.16 0.10 0.200.22 0.24 0.26 0.28 0.30 .A Time, T 10 -4L r( 2 H - I1 - Fig. 5 - Impulsive 1st Bending Moment Growth Functions vs. Time in Seconds

- - - - Pure Penetration Penetration with Lift Growth M = 1.345

.4 Fig. G - Impulsive 2nd Bending Moment Growth Functionsvs. Time in Seconds

-9- ... ..

The half cross section of the C-5 configuration is given at the bottom of each of the figures for convenience of interpretation of the growth functions. At T = 0 , the first region(escape tower) penetrates the gust front and the generalizedforce buildup begins. The times T = 0.0216, 0.0500, and0.0930 sec. correspond to the successive gust encounters of the second, third and fourthconic regions. The purepenetration and penetration with lift growth curvesreached steady-state conditions at T = 0.1065sec. and T = 0.2895sec., respectively. The time for total immersion ofthe vehicle was 0.2629 sec.

For reasons of brevity, a detailed description of the growth functions will not be given in this report. The reader is referred to @-]for some general coments concerning the rigid body growth functions. The general formof the bending moment growth functions are very similar to those of the normal forcefunctions. The bending moment growth functions, however, reflect the characterof the mode shapes. The negativegrowth functions for the fourth region in Fig. 5 and the first andsecond regions in Fig. 6 correspond to regions of negative displacements of the first andsecond bending modes, respectively.Figures 2 and 3 reflectan integrated effect of the growthfunctions in Figs. 5 and6, respectively.

C. Method ofSolution for Indicia1 andImpulsive Responses of Saturn C-5

Themethod ofRunge-Kutta was used to calculate the impulsive and indicialresponses of theSaturn C-5. A fourthorder* Runge-Kutta integration equation was used tonunerically integrate the equations ofmotion. Saturn C-5 missile response calculations were obtained for six different sets of aerodynamic environments.These six sets offorcing functions {see Appendix 111, mble VI1)contain the indicial and impulsive aerodynamic forces resulting from

1. Penetrationwith lift growth effects,

2. Purepenetration effects, and

3. Instantaneous-immersioneffects.

* The Runge-Kutta integrationexpression used is considered to be offourth- order accuracy { see c3] } .

- 10 - The Runge-Kutta integration process requires knowledge of the initial conditions (or starting values). Since the system of equations used in the investigation can be written a assecond order set, (see(11-11) through (11-19)) , only the initial conditions for the dependent variables and their first derivatives are needed. The initial conditions forof theeach six sets of forcing functions were found throughof useLaplace transform techniques {see Appendix I11 3 . The indicial and impulsive responses ofC-5 thewere computed at discrete flight times or altitudes a fromsystem of linear, differential equations with constant coefficients*{see (111-1)} . Each set of constant coefficients, and consequently the associated responses, is applicablea specified in time or altitude interval.* A computer program used to obtain the response calculations is discussed Vol.in 111.

In each flight timeor altitude band, the responses were computed for a real time interval 20of sec. Extremely fine increments of response time were used in computing the forcing functions and the indicial and impulsive responses of the deflections and their first derivatives. Coarser increments were used in the printing and plotting of the output. Only the deflections were plotted on theSC-4020.

Twenty-second response records were computed to insure that the responses had achieved satisfactory steady-state values.For the flight times considered, all of the irrrpulsive*** responses had converged 1 toper cent or less of their maximum values at the 20 end sec. of

The calculation of 20-see. response records does not violate the assumption of constant coefficients as might be expected.From a subsidiary response analysis, it was found that if the sloshing degrees of freedom were removed from the system, the responses of the remaining system (excluding translation) would achieve satisfactory steady-state values 7 withinor 8 sec.

* Previous work has shown that it is permissibleto use sets of constant coefficients. ** Response calculations were obtainedat every 10 see. of flight time between 30 and 100 see. The applicable intervalof flight time for each set of these responses is takenas 10 see. Thus, for example, the 30-sec. response calculations (obtained using the coefficients30 sec at . ) are considered valid between flight timesof 25 and 35 see. *The decay of the indicial responses will be discussed later in this section.

- 11 - of response time. The sloshing modes ofC-5 the are very slightly dampedin comparison to the other uodesof the system (except translation). Therefore, since we are consideringa coupled system, the response calculations for rotation, first and second bending and control after7 or 8 sec. reflect the influence of the sloshing modes.This influence is basically governedby that part of the equation of motion which contains time invariant coefficients.

The selectionof the increment size used in the numerical calculation of the responses was dictated by two requirements:

1. The increment size should be sufficiently small as to permit an accurate calculation of the aerodynamic forces.

2. The increment size should permit32 calculated response values per cycle for the highest frequency componentof the system.

Based on theabove requirements, the increment sizes,AT , used in the response calculations are presented in TableI.

TABLJ3 I

TABULATION OF INCREMENT SIZES USED IN INDICIAL AND IMFULSIVE RESPONSE CALCUIATIONS

Flight Time T3 (sec.) (sec.(sec.)(sec.) (sec.) ) (sec. )

30 0.0005 0.02 0.001 0.75 0.004

40 0.0005 0.02 0.001 0.60 0.004

50 0 .0005 0.@2 0.001 0.50 0.004

60 0.0005 0.02 0.001 0.45 0.003

70 0.0005 0.03 0.001 0.40 0.003

80 0.0005 0.03 0.001 0.40 0.004

90 0.0005 0.04 0.001 0.35 0.004

100 0.0005 0.05 0.001 0.35 0.004

- 12 - The responsetimes, T , at which theincrement size was changed* are alsogiven. A71 was used inthe calculations from 0 to T~ ; AT* was used from 71 to 72 ; and &r3 was usedfrom T~ to the end of theresponse calculations.

Extremely small increments ( A71 = 0.0005 sec. ) are used at the begin- ning of the response calculations to adequately describe the high-frequency components ofthe control system. The 71 valuescorrespond to the times at which thesefrequency components are negligible. The 72 valuescorrespond to conservative estimates of the times when the aerodynamic forcingf’unctions reach steady-state conditions.

Since a voluminous amount of indicial and impulsive response data were generated,only a representative quantity of these datais presented. Plots of the indicial andimpulsive responses of first bending, ql , secondbending, *1]2 , and controldeflection, Bc (noted by first control) are given in Figs.. 7 through 24 for a flighttime of 70 sec. The impulsiveresponses are presented inFigs. 7 through 15; the indicial responses are given in Figs. 16 through 24. For eachcoordinate, the first figure reflects the effect of instantaneous immersion; the second figure reflects the effect of pure penetration; and the third reflects the effect of penetration with lift growth.

The numerics presented alongside the plots in each figure pertain to an analysis** of the zero crossings, maximum and minimum values of the respec- tivecoordinate response. The formatof the informationpresented is as follows :

R esponse ResponseResponse Response Time Value Maximum,Minimum or Crossing

Themaximum, minimum and zero crossing values are the calculated points which precede .(in time) the event.

* It was found expedient, from thestandpoint of conserving computer running time, to increase the increment size, when permissible, during the response calculations. ** The write-up on the computerprogram used to analyze the indicial and impulsiveresponses is presented in Vol. 111.

- 13 - InPULSE-INSTANTAhEBUSIHHERSIEN 7C SEC F-T. EW..~ST BENDING -0. -0. 0. CRBSS 0.17950 5.518€@38E-C3 PAX 0.37550 1.446i733E-03 WIN

1 .04 25 a 6.3163321E-041.0425a PAX 1 .10 85 0 5.9477S81E-C41.10850 PIK ".J&Xb50" 1.176oe45E-03 PAX 1 .47 15 0 2.165d975E-051.47150 CRESS

2.17949 -3.9832917E-05 WAX

.. . . 2 LW9" =LzzeEL(U+-c4"Plti. ~ 2.74648 -tl.0317C931-07 CRBSS

. ~ "_IsZ"%L45aE-c4- - 3.70947 1.2221843E-07 CRESS 4.13246 - 162248E-04 PIN 4.59745 -1.1195937f-06 CRBSS

5.06843 1.1445733E-04 ~WAX ~" 5.59342 1.453C485E-07 CRESS ___"~6.06140- -8.24C!109E-05 MIK ~- I 6.59838 -5.2775082E-07 CRBSS P "03937 5.153C246E-05 PAX tP 7.54335 2.8985308E-07 CRBSS I p--8.02334. - . -4~091C984E-05 . 1IIN~.-- 8.58430 -2.3885C74E-07 CRESS

10.02419 - 4C115E-05 HIN 1 0.63 61 5 -7.5187C65E-OS10.63615 CRBSS

-- -ILOZ(LU --.6&9W85 E-116"- EAX ~ ~ 1 1.5 03 08 2.66a:357t-ca11.50308 CRESS

~ --11.9pBQ5 ~~ --5d2%35.kQk ~ "PLN 12.84099 -1.1025705E-06 PAX 13.47394 -2.409C301E-06 PIK 14.16689 -3.734t965E-OS CRESS

14,31988 "" 2.25Llt&8E-97. ~ PAX 1 4.4 66 87 1.085C119E-0914.46687 CRBSS

_L5L3.1S81__.-4~.462_.~35E~Q6 ~p~ "IN ~ 1 6.0 32 76 -1.001tC88E-C816.03276 CRESS 16.35073 1-2185482E-06 MAX 16.66871 8.2923363E-09 CRBSS 17.40666-4.493E273E-06- -M&.-.- 18.21060 -I.~o~s~~~E-osCRESS 1 8.4 80 58 7-252CS39E-0718.48058 PAX 1 8.7 56 56 2.5024163E-C918.75656 CRBSS 1 9.5 66 50 -3.718C747E-C619.56650 PIN

Fig. 7 - Impulsive Response of 1st Bending Considering Instantaneous Immersion - 70 sec. F.T. I P 3.092SC78E-07 7.54935 CRBSS

a.o2614~-.-~~li656~-05. MUI ~~ I 8.59329 -1.6232790~-07 CRBSS

9Ju526 1.92lifWLt!AX ~ 9.54423 1.931~2oo~-ae CRESS 10.01 219 -1.340:4551-c510.01219 MIN 10.63015 -6.88215901-08 CRBSS 10.98112 4.255C05!&06--L.~ - 11 .410 09 2.7931521~-0811.41009 CRBSS -6 -3.8925285k06 WN" - 12 .64600 -5.44150951-0112.64600 MAX 11.44094 - 26783E-06 MIN 14.0 7990 -1.518i175E-0814.07990 CRESS

18.30959 -6.9446182~-09 CRBSS

~18.564575.8PQ535lf"X" ~ 18.82555 1.2171443E-08 CRESS 19.65949 - 85846E-06 MIK

Fig. 8 - Impulsive Response of 1st Bending ConsideringPure Penetration - 70 sec. F.T. IMPULSE-PENETRATIEh k/ LIFT GREbiTt- 7C SEC F.T. EUls 1ST BEKCIhG -0- -C . c. CRESS 0.18350 5.57C'S09E-C3 PAX 0.40650 1.1389137E-C3 PlFi . 0.4185C 1.13S!949k.C3 MAX 0. 53850 1.7985563E-050.53850 CRESS 0.72750 -3,985.~C45.~-~.__P-lN 0.93750 -4.4471409E-C5 CRBSS 1 .045 5~ 7.2167C63E-C41.0455~ PAX 1.13550 6-443t514E-C4 PIN 1.3095C -__1.0469549E-O3 PAX 1.47750 1.079SC26E-06 CRESS

1.61250 ~ ~ ~ :6.9Uf3&5€-C4 -__Ab..-- 1 .E4049-2.285'915E-04 PAX 1.95149 - 67331 E-04 PIN 2. 18849 -3.6375487E-052.18849 PAX 2.42549 -ZnB92E533E-Q4 -. MIF: 2. 71649 -1.512i278t-Cb2.71649 CRBSS

3.07348 ~ 2.8951020L-_c4- PAX 3.69747 8.9112263f-C7 CRESS 4.13246 -1.151?365E-C4 MIh 4. 59745 -1.1653625t-064.59745 CRBSS . 5..06843----.- -la0584C93E-C4 __ -.WAX .- 5.59342 4.5991439E-07 CRESS

. -. . . 5.46441: - ~ ~-7~529_312.€-525.. .."IN 6.60738 -5.1405142E-C7 CRESS I 7.04537 4.5602548E-05 PAX P 7.55235 9.0385733E-C8 CRESS

-. ~~ BAL2434 . 3r54(LUb2U.~_._FIlK I 8.59629 -1.9707877E-07 CRBSS

. ~ 9.Q2524 ~ . l,&9E!LUE:45r_- . PAX 9.54423 8.0065459E-C8 CReSS

11.40409 3.451 ZS89 E-08 CRESS

- ~ .11r88106 ~ -3,866A6_4_f-X6-.- .-PIN 12.6400C -6.409C415E-07 MAX 13.43494 -3.692i762E-06 PIN 14.08590 -3.1305355E-10 CRBSS

~"17.49465-. - ---4.3604968E-06 ClK"" 18.32459 -1.3042376E-C8 mess _18.56_7S7__~_5,359Et49.~~-07"_MAX 18.81655 6.9240C37E-09 CRESS 19.66549 -3.463C740E-C6 FIN

Fig. 9 - Impulsive Response of 1st Bending ConsideringPenetration with Lift Growth - 70 sec. F.T. I

IWPULSE-INSTANTAhEEUS IMMERSIPN 7C SEC F.T. ETI2t 2NO BENCING -0. -0. 0. CRESS 0.1545@ 3.4OC126LE-030.1545@ PAX 0.24050 1-481?480€-05 CRESS 0.33150 -3.2784S40E-C3 PIk 0.43050 -1-358C451E-C4 CRESS 0.5355c 2.9217769E-03 PAX 0.64950 8.597EC62E-05 CRESS 0.75751-1 - 7- F-0 3 VIN 0.88650 -2.611iZlSf-05 CROSS 0.97050 9-2321383E-04 PAX 1.06350 1.850194CE-05 CRBSS 1.16550 -8.8081745E-041.16550 PIN 1.27050 -1.0457354E-05 CRESS 1.37850 8.081C466E-04 PAX 1.50150 4-8960832E-66 CRESS 1.60050 -4-884f773E-04 PIh 1.7235C -1.112t1133~-05 mess 1.79849 1.8355871E-C4 PAX 1.87949 1-0831702E-06 CRBSS 7.00749 - e5864 6-124 PIN 2.14049 -5.888?571€-06 CRESS 2.22449 1.274e119~14 PAX 2.32049 2.963!439€-06 CRESS 2.42849 -1.305;271E-C4 PIN 2.54849 -1-1642305t-C6 CReSS 2.64749 8.017C153E-05 PAX 2.77048 1-8371378E-06 CRESS 2.83348 -2.349?297€-C5 PIh 2.89648 -2.94781513~-07 CRESS 3.06448 1.0397S87E-04 PAX 3.28648 2.793i388FC5 MIN 3.44248 5.003S04EL-C53.44248 PAX 3.61341 6.009iS05F-073.61341 CRESS 3.72447 -1.967i556E-053.72447 PIN 3.87441 -9.9644t53E-Cb PAX 4.09647 -3.0571266E-05 PIh 4.53745 -9-550f525€-09 CRESS 5.06R41 7.6994aE-05 PAX 5.52742 1-588SS57t-CB CRESS 5.99840 -1.9636817E-055.99840 PIN 6.55039 -l.a17ea26~-07 CRESS 6.98537 1.1961C05E-056.98537 PAX 7.49535 5.2301SZ2t-C8 CRBSS 7.98434 - 2 1507f-Cb PIh 8.5723G -1-677e188E-C8 CReSS 9.01326 ~.199:71e~-ot .PAX 9.55023 1-424CCl3E-C8 CRBSS 10.04219 -3.613544CE-C610.04219 MIh 10.73514 -4-7717727E-09 CRESS 11.09811 9-1845755f-07 WAX 11.60508 4.5301221€-0$ CRESS 12.99098 -1.0706583E-C6 PIh 13.75292 -1-697CeOlE-09 CRESS 14.12189 4.365€5701-07 PAX 14.46387 3.9309‘16E-09 CRESS 15.19882 - 6Oi308E-C6 PIN 15.90076 -4-083t504E-09 CRBSS 16.27574 5.672’248E-0716.27574 PAX 16.65371 4.009i487€-0916.65371 CRESS

Fig. 10 - Impulsive Response of 2nd Bending Considering Instantaneous Immersion- 70 sec. F.T. IMPULSE-PUREPEKETRATIBN 7C SEC F.T. ETA2t 2ND BENCING -0. ._

Fig. 11 - Impulsive Response of 2nd Bending Considering Pure'penetration - 70 sec. F.T. IMPULSE-PENETRATIBN YI LIFT GREWTP 7C SEC F.T. ETAZ. 2M BENOLNG "(1. .~ -0. 0. CRESS 0.03350 - E-C5 HlN 0.0435C -2.852~504~-06 CRESS

. . 0.1655L- ~~ 3.35(1@122E-C3 - FAX 0.25250 2.0458693E-05 CRESS

~ .. .O.~~LUL-~-~-~934~170m3-- PIL 0.4425C -5.602i199t-05 CRESS 0.54750 2-941 F-03 MAX 0.66150 4.611;245~-06 CRESS __ - 0.26W . -2.QBBB226L-llF ._PIH 0 .8985 0 -3.9334636E-060.89850 CRESS

.- . ILQZ9vL ~ - 32141394E~04 PAX 1.075 50 2.7424960E-051.07550 CRESS 1.17450 - lw9~-04 PIK 1.27950 -L.~Z~C~ZOE-O~ CRESS 1.39050 7.36OCllPFL-!l~~~"X

1.81 04 9 1.6LE456f-041.81049 WAX 1.88849 5.8824443E-06 CRESS 2.01449 . . -2.9755274~44 MIN 2.1494 9 -6.675742OE-062.14949 CRESS

2.23649 . 1.195t5351-04. ~ MAX 2.3324 9 3.965619OE-062.33249 CRESS 2.44049 - 696f-04 FIN 2.5574 9 -1.5316531E-C72.55749 CRESS

. - 7.7253612E-05 HAX I . .. 2.659-49 2.791 48 7.2507600f-C72.79148 CRESS p ." 2.84548 ___-l.L9CU)baE=Q5.. ~ PIN co -1.321I121E-062.89648 CRESS 3.a7348 9.~516665f-05 WAX I 3.30148 2.6344008f-C5 MlN 3~45148 4.483C5bl€-05 MAX 3.61947 6.7824255E-07 CRESS

.- ~1m7~ .--~4'071~-m MIN 3.88647 -~.o~~~ozoE-o~MAX 4.10547 - 5i275f-05 WIN 4.5434 5 -2.2082184E-074.54345 CRBSS

" ~~ 5.07443. . 2.3753786-t-95 MAX 5.533 42 1.7567123E-075.53342 CRESS -1.7890317E-05 PIN - ~ 6.QO74Q. 6.571 38 -7.9996862€-086.57138 CRBSS 7.000 37 1.026- 7.00037 2802f-05 MAX - 7.51335 1.5306532f-08 CRESS

~~Lp99~.---~~,U6442.~-44_.-~ !IN 8.59 929 -8.5867455f-098.59929 CRESS

9.0222.6 . 3.8632021~-~6 ~ PAX 9.559 22 6.597e689E-109.55922 CRESS 10.04219 -2.5437776~-06 VIN 10.87 013 -2.8601182f-0910.87013 CRESS

~ 11.05.QLZ.-. 1.25425531-C7 .. PAX 11.27510 1.044CC52E-05 CRESS

" .. 11.116UP.9,&76?32&EzQ-.. MIN 12.17 503 -1.482ez83f-0712.17503 VAX 13.18896 - 7~700f-06 PIN 13.8 549 1 -7.433e169~-0913.85491 CRBSS -~"~am~~.et~1maf-a7 -. . PAX 14.607 86 3.1367442E-0914.60786 CRBSS IMPULSE-INSlANTAhEEUS IPPERSIPN 7C SEC F.T.

~-~ BEIAle 1ST CENTREL -0. -0. 0. CReSS 0.00050 0. CRESX 0.00050 0. PIN . .. . a.wx 2.5954368~-02 Pax 0.11150 4.3127782E-04 CRESS 0.16450 -1,970E192E-02 PIN . 0.2 325 0 -2.6075112f-C40.23250 CRESS 0.30 55 0 1.4607260E-02 0.30550 MAX 0.41550 1263E-057.723 CRESS - .- - ~aam-.-:b.wt591f-~ YLN-. .~ 0 .61 35 0 -9.988ea47~-~50.61350 CRBSS .. (1.70390 3.8842a4~~~0.3. rA.8 0.79050 1.0825266E-04 CRESS 0.91350 - 3i551kC3 PIN 1.05150 -1.1975808E-C4 CRESS

1.14450 2.524659OE-03 ~ PAX 1.25850 4.1371278f-C5 CRESS 1.3425C~-1,2341838t-03 - PIN 1 .429 50 -5.7231533E-051.42950 CRESS 1 .5345 C 1.3901921E-03 1.5345C MAX 1.6395C 4-443213Of-05 CRESS

~ 1.76550 -1.5852152E-C.3__~ .PIN ~ . 1.93649 -1.270C570f-CS CRESS

~~-~- .. .. 1.97849 . 1.234%29L!4- . ..MAX ~ 2.02349 2.607i323E-06 CRESS 2.1674 9 -7.0741691 E-04 WIN 2.30849 -8.198F162E-OC CRESS

I ~ ~ 2.39249 ~ 2L63-8.328-04~. . PAX 2.49149 8-111!216E-07 CRLSS Iu 2.58749 - 2,290 1 04 LE: 0-4. . P 1-h 0 2.68649 -2.4624095E-06 CRESS 2 .830 48 3.451:457c-c4 2.83048 PAX I 3 .001 48 1.9041e56f-043.00148 PIN 3.20848 - .. 3.949C803€~04_ ... PAX . 3.52347 9.0432810E-05 PIh 3.55347 - - . 9.084557BL-C5-~-~ PAX 3 .721 47 2.307ft8OE-063.72147 CRBSS 3 .892 47 -1.0264533E-043.89247 PIN 4 .0754 7 -7.171 1289E-05-7.1714.07547 PAX

4.21046 ~ ~-8.3045C91E-C-5- PIN 4 .438 45 -8.396e896~-074.43845 CRESS

4-93044 ~ ~ 1.2?/4t43E-04 PAX 5.48542 5.9566417E-08 CRESS 5.95340 - 5e764 E-C5 PIN 6.50239 -2-6204852E-07 CRESS

. 6-922.37. ._ ~ 4.4662217E-05~~~~~ ?AX 7.39036 1.66oe741f-07 CRESS 7.87034 -3C953C205E-0-5 . PIN 8 .422 31 -1.7437256E-078.42231 CRESS 8.85128 2.280?C02E-05 PAX 9.34324 1.711!761E-C7 CRESS 9 .817 21 -1.833157Of-059.81721 PIN

10 .3811 7 -1.18EC748E-0710.38117 cRess~ ~~. 10.78614 9.253i342E-06 PAX 11.24810 3.63OS429E-08 CRESS - 11.75507 -9-1575969E-C6 PIN 12.39702 -3.377I755E-Oq CRESS 1 2.712 00 2.5284407E-0612.71200 WAX 1 3.06 597 2.903C178f-0913.06597 CRESS

Fig. 13 - Impulsive Response of 1st Control Considering Instantaneous Immersion - 70 sec. F.T. I

ILPULSE-PUREPEtiETRATIEN 7C SEC F.T. _.BETAIS UTCENTReL -0, -0. c. CRESS 0.0005C 0. Cm 0.00050 a. PIh - 0.0635k . 2.644tC5OE-C2 WAX 0.112 50 3.0896129E-040.11250 CRESS

-0.166511 ~ :LII~L;~~~L-QZ ti 0.23750 -1.153~199~-04 mess a 1.3Ri5SlE€-O2 WAX 0.424 50 1.7385110E-040.42450 CRESS __~- 0.514W -- -L54oUlEE-[13_. YLH 0.62550 -1.262Z948E-04 CRBSS

-...... !LUZlo_~ 1.4044584.E-tl3 MAX 0.79950 6.8OOC552E-05 CRESS 50 - 5E-C3 PIN 1.060 50 -7.066651OE-051.06050 CRBSS

1.15350 ~ 2.400~597~-03 PAX 1.27050 4.471€165E-C5 CRESS 1.35150 -1.107116~~-~3- .PLN 1.44 15 0 -2.902t951f-051.44150 CRBSS 1.546 50 1.ZXL%lEZE-C31.54650 HA X 1.64850 1.8OOC533E-05 CRESS

1.77450 - "-1.469:e82~-03 PIN 1.945 49 -3.3051448E-061.94549 CRBSS 1.99049 1.3.173512E-04 WAX 2.03849 4.414t739E-C6 CRESS 2.17649 - - PIN 2.314 49 -1.051'721E-062.31449 CRESS 2.401 49 2.6706744E-042.40149 PAX I 7.1367866E-06 2.50649 CRESS

~~ 2.~9949 ~~~~ZL.~SLC.~~E-.C~ PI^ l-0 -1.7036265E-062.68949 CRESS P 2.83948 3.381?948E-C4 PAX I 3.01348 1.8904782E-04 WIN z - 3.21448 3,621S714PC4 PAX T 3.53547 7.2145537E-05 PIN . - 3.56247 7.243i995E-C5 PAX 3.72 141 2.2871842f-C63.72141 CRBSS 3.89847 - 73826E-05 HIti 4.09347 -6.484r541E-C5 MAX 4.21346 -7.2262908E-05 CIFi 4.43245 -1.1768577E-Cb CRESS I 4.93044 1.12a1594t-04 MAX I 5.47342 3.7572587~-07 mess O 5.94740 - 85446E-05 PIN L 6.496 39 -2.948;C94E-076.49639 CRESS

691337 . 4.081C973E-fi ~ !W 7.38136 2.845i74CE-07 CRBSS 7.86 134 -3.61LiS871-057.86134 PIN 8.41 631 -1.294t311E-C78.41631 CRBSS 8.842 28 2.0564075E-C5 8.84228 WAX 9.334 24 1.7644768E-C79.33424 CRESS

" 9.80821 -1.6473256E-05 PIN 10.37 517 -3.2554756E-0810.37517 CRBSS

~ " 10.77414 _. 8.08OC476E-C6 PAX 11.22 710 3.964I410E-CB11.22710 CRESS 11.73707 - .229iE94E-O6 PIh 12.37902 -7.642EC15E-04 CRESS

p1L68+M- ~ __.2.1595669E-C6 MAX 13.02 397 4.4860564E-0913.02397 CRESS

Fig. 14 - Impulsive Response of 1st ControlConsidering Pure Penetration - 70 sec. F.T. IMPULSE-PENETRATIEN Y/ LIFT GREWTH 7C SEC F.1.

"CBNTRRL -0. ~- ~ -0. 0. CRESS 0.00050 0. CRESS 0.00050 a. MlN o.owa 2.6G-W=- 02 PAX 0.11250 3.6707809~-a4 CRESS 0.16650 - 97496E-Q2 MIN 0.2375C -1.7929068E-04 CRBSS 0.31050 1.384~845~12 MAX 0 .4245 0 2.8451068E-040.42450 CRESS 0.51750 - 14403E-03 MIN 0.62550 -2.0241444E-04 CRESS -0.7125(3 3.3796085~-03 MU 0 .799 50 1.2526956E-040.79950 CRESS 0.92550 - ~C~OBE-O~ MIN 1.060 50 -1.1027411E-041.06050 CRESS 0 UE-0 1.27050 ?3~~~:848-0~:t"eSp 1.35450 - 92W%€43 -m. - 1.44150 -4.7012845E-05 CRESS 1.546 50 1.22305b9E-C3 1.54650 MAX 1.64850 3.4311120E-05 CRESS

"" - 1.77450 -1.4624959E-03 PI& ~ 1.94549 -8.198714OE-06 CRESS -1m9 1.3405m-C4 MAX _~~_ 2.03849 1.020e489~-05 CRESS 2 .17 649 0555412.17649 - E-04 PIN 2.31449 -6.1087382E-06 CRESS

&4-0449 A6561415E-C4 MAX ~ . I 2.50649 1.031e989~-05 CRESS nl ~~ ~ "229949%=L4434u-04 MLpl Iu -4.9355479E-062.68949 CRESS 2.842 48 3.3639428E-04 2.84248 MAX I 1.8802685E-043.01348 MIN

" 1.2U48" - 3599"~ Pa p. 3.53 547 7.1172699€-053.53547 PIN - 3156547 -LL&i&XQZL-E .- PAX 3.721 47 2.6171317E-063.72147 CRESS 3.89847 -9.8125691E-05 PIN 4.09347 -6.4760510E-05 MAX 4.11646 -7.20749.46E:C5- Mlh 4.43 54 5 -3.5385700~-074.43545 CRESS 4-93044_"1.121?761_€-~04 ~.MAX 5.47342 4.7374576~-07 CRESS 5.94 740 -7.6015723E-055.94740 HIN 6.496 39 -5.0367093E-076.49639 CRESS

~ ~p~ 6.91337 4.0545192E-05 MAX- 7.38136 3.0686155E-07 CRESS 7-86 43 4 -3.6025829E-057-86434 PIN 8.41 931 -4.62Ot139E-088.41931 CRESS" 8.84228 2.036i847E-05 MAX 9.33424 1.4475565E-C7 CRESS

.- - -~ 3.81121 ~ -p-1.646f4245705 ~ - ~ ~ ~~ PIN 1 0.37 817 -5.9233357E-0810.37817 CRESS

-~ " .10-77414 ~~~ 7.927'902E-0t PAX 11.22410 4.8774739E-08 CRESS 11.73707 -8.2610t18E-06 PIH 12.38502 -3.425'349E-08 CRESS I Iuw I

(-3 1CIP41C l.IN UIT WCP -ImlAWANW 1~~1(110 I(( P.1.

Fig. 16 - Indicia1 Response of 1st Bending ConsideringInstantaneous Immersion - 70 sec. F.T. I

(u P I

Fig. 17 - Indicia1 Response of 1st Bending Ccnsidering Pure Penetration - 70 sec. F.T. -STEP-PENETRATIBN bI/ LIFT GF0IdlN 7C SEC F-T.

EZbla- 1U BENCltlG ---a. . ~ -0. 0. CRESS O.ZiB50 1.197C714E-03 PA X 0.94050 2.688!101E-C4 PIN __ . ~'~z75a. __ 6,411m18~-04~~~ 2.71949 3.2981583E-C4 PIN

Lm4L 2A5ZC416E-(LLA~~ 0000 4.60045 4.232E296F-04 PIN 000 0.0

8.59929

~ -. ~- PA4123 10.64215 a 11.40709 14 e08890 14.66485

. ~ . 16.11075 16.75870 18.33359 4.36L53.59€-04 Plh 18.81355 4.3631812E-04 WAX

C 1 A 1

I s 1 n c U D I U C

Fig. 18 - Indicia1 Response of 1st Bending ConsideringPenetration with Lift Growth - 70 sec. F.T. ” ___~~~~ - .. STEP -1NSTANTAbEBUS IMMERSIPN 7C SEC F.T. ETA2.ZNDaENCLNG -0. . -0. 0. CRESS 0.74050 4.2006465E-04 MAX o ~3350 1.9034908E-05 HIN ow0 -0.652504.241339LE-04 MAX 000 O#? 0 .8865 0 9.03C5e716-050.88650 WIN 1.06350 1.9526847~c4 - MAX 1.27050 7.9540582€-05 MIN 1.50150 1.969i84EE - 04 MAX 1.72650 1.286C186E-04 PIN 1.87949.~69576a~-a4 nu 2.14349 9.351C990E-05 MIN 2.32049 1.W92E-Q4 A 2.54849 8.955C158E-05 MIN 2.77348 1.0009949E - 04 PAX 2.89648 9.878E294E-05 MIN 3.61647 1.3498096E-04 MAX 4.53745 1.2104362E-04 MIN 5.52742 1.3867525~-~~ 6.553 39 3.25722466-046.55339 WIN

8.57230 1.26351276-04 PIN 9.55023 1.295i803E-04 MAX 10.73514 1.2696037E-04 WIN 11.60808 3.2746617E-04 MAX I 13.75292 1.2592400E-04 PIN 10 041.2613676E 14.46387 - MAX In 1.2613676E-0414.46687 MAX I- 15.90076 1-24777656-04 HAX 15.90376 1.2477765E-04 PIN 16.65671 1.25056301-04 MAX 18.12960 1.237t7426-C4 MIN 18.78656 1.23912866 04 MAX 18.78956 1.2391286f-04 FAX

C-Y ICICOIIC TIW WIT Ill? -Il*TAMlAWOW IM~III~ 70 ICC 1.1.

Fig. 19 - Indicia1 Responseof 2nd Bending Considering Instantaneous Immersion- 70 sec. F.T. 1

.. . STEP -PURE PEhTRATI0N 70 SEC F.T.

ETAZr-ZND BENGLkG-~O* - " -0. 0. CRBSS 0.04550 - 88E-C7 PIN 0 .056 5C -9.8486670~-080.0565C CRBSS ILZl5L". 32mlU64€-ilW 0 .442 50 2.954189OE-050.44250 PIN

5-42 1.383d348E-04 ClAX 6.57138 1.264C342E-04 PIN 7.51335 _- 1.326C170E-04 PAX 8.59929 1.270'560E-04 MlN 9 55922 1.294C993E-04 PAX 10.86713 1.2741127E-04 PI# 11.29010 1.275t979E-C4 lnAX 13.85791 1.260!452~-04 MIFi 14.60L86 1.2631804E-04 MAX 16.00276 1.24946311-04 PAX

~ 16.00576 1.2494631E-04 MI& 16.74370 1.252C737E-04 PAX 18.25559 1-2395141E-04 MAX 18.25859 1.2395141E-04 PIN

-~ ~ ~. -18.85855 1.240520lE-C4 r4 x

Fig. 20 - Indicia1 Response of 2nd Bending ConsideringPure Penetration - 70 sec. F.T. I 1.8427769E-04.0785a MAX 1.28250 8.336t141E-05 PIN

~~ ~~ - 1.5105Q ~ _1.904%6L"O4 1.7 3850 1.259t456E-C41.73850 PIK --~~~_L&U!tL-- 1-419'493E-a4- __ WJ. - 2.15249 9.3855295E-05 PIh' 2.33549 1.079 1097E-C4 PA X 2.55749 9.162C648E-05 t4Ih 2.79148.- .~ L0271%9.U-C4__ PAXIx. 2.89940 1.016tC55E-C4 PIN 3,62243. ~. -1.1VLPYik-W.- !!AX. 4.5 4645 1.22li740f-044.54645 PIR 5. 53642 1.3833564E-045.53642 PAX 6. 57438 1.2642321E-046.57438 PIN 7,51335- ~..~L~~ZL!XYXL~CPL-!~AX- - 8.602291.270t664E-a4 Y Ih'

Iu 11.2691C 1.275 1424FC4 PAX 0, 1.2603e26E-0413.85791 PIN . I ~ .~.. 14.60706 ~ 1.263iC951-04 ~- -?AX 16.66576 1.2495193E-04 PIN . 16,74679 lLL!zLlW4- PAX 16.7497C 1.25214671-C4 MAX 18.26159 1.2395765E - C4 PIN 18.86155 1.2405846E-04 MAX

Fig. 21 - Indicia1 Response of 2nd Bending Considering Penetration with Lift Growth - 70 sec. F.T. STEP-1NSTANTANEEUS IMMERSIeN 7C SEC F.T. BETII.1STCBNTBPLL -a. ~- -0. 0. CRESS a 0. CRBSS 0.00050 0. MIN

o. ua ~ L4BPiEUkL-..PAX 0.221 50 3.044i425E-070.22150 CReSS

~a.zm~.--05 .m- ~~ 0.244 50 -2.5691210E-060.24450 CRESS PA X 0.613 50 8.333t432E-040.61350 MIFl

0 .7 -0 U76I!245E-QL~JM.0.7-0 ~ 1.05 450 3.2466637E-041.05450 PIh ."653Qm39+-C4 ."653Qm39+-C4 -PAX 1.43250 5.1621032E-04 MlN 1.64 25C 7.015E- 1.6425C 193 E-04 FAX 1.93949 4.240~112t-a4 MIH

- .- 2.02Yt9~-4.3Q%5941t-04 "MAX" 2.30849 3.092L222E-04 MIL 2.49.149 .3AQQU6BBCL FAX 2.6 864 9 3.1~2e705t-042.68649 Plti 3.774 47 5.3220t01E-P43.77447 PAX 4.441 45 4.828eC97E-044.44145 PI& 5.4-42 p5.5~~416~-~a4 PAX 6.50 239 5.0732055E-046.50239 PIN

7.39036 5.1263019E-04 ~ PAX 8.425 31 5.068S631E-048.42531 MIti I 5.203ca47f-049.34624 MAX ro 5.083dtllE-0410.38416 PIN a -11.24810_.. ~ ~ 5.1.347905E-04 FAX .ow0 I 12.39702 5.07CC638f-04 MIN

~~~ -3=Qb!i9L.~.QQLC.~LOEYQ~ . . ~.PAX

L 1 1 I

I 8 1

C 0 n I n 0 L

Fig. 22 - Indicia1 Response of 1st Control Considering Instantaneous Immersion- 70 see. F.T. _____~~ ~ . . ~~ ~ .. " ". STEP -PURE PENETRATIBN 7C SEC F.1. BETdl.1STT.mIL "(3..- ~. -0. 0. CRBSS 0.00050 0. CRBSS 0.00050 C. WIN 0.1U50 1.5221367E-03 PAX 0.21650 - 1.0437631F-06- CRBSS " - 23750 82910E 05 PIN - 0.2 5 95 0 -5.1729705E-060.25950 CRBSS 0.4 2 45 0 [email protected] - 03 PAX 0.6 2 85 0 1.9406686E-04 0.62850 MIN 0.7 99 50 1.17-16E-Q30.79950 MAX 1 .06 05 0 3.1004462E-041.06050 MIN 1.27350~90~03~-CLPAJ 1.44450 5.067iO37E-04 MIN 1.64850 6.698i831E-04 MA X 1.94549 4.120C918E-04 MIN -2.038494.2052436E 04- PAX - 2.3 14 4 9 3.17575052.31449E-04 MIN -2.5094U3.5115tS65E 04 - MAX .- 2.68949 3.2852259E-04 ClIN 3 .72 44 7 5.3387014.E-043.72447 MA X 4 .43 5 45 4.886I987E-044.43545 MIN 5 .47 64 2 5.5847012E-045.47642 MAX- 6.49939 5.097ma~-o4 MIN 5.3274815E - 04 UAX 7.38436 , 8.41931 5.0915702E-04 MIN w 9.33724 5.2107427E-04 MAX 0 5.102'288E-0410.38117 MIN I -11.22410- 5,1454515E -114 PAX 11.22710 5.1454515E-04 PAX 12.39102 5.086 L?84E-Q4 MIN- 0 13.01497 5.0943919f-04 MAX 1 1 I

I I 0 L

Fig. 23 - Indicia1 Response of 1st ControlConsidering Pure Penetration - 70 sec. F.T. ax PIK -PAX CIh ClAX PIN PAN VIh rAx PIN nAX PIN CIA X rIN .M4X

c-I RUr(1lI TIR -11-ITIC -wcmTtrn WJ UMM IO arc P.T.

Fig. 24 - Indicia1 Response of 1st ControlConsidering Penetratiorl with Lift Growth - 70 sec. F.T. The discussion of the effects and importance of penetration and growth of lift on the indicial and impulsive responses of the vehiclebe presented will in a later section.

There is, however, some information readily derivable front the indicial and impulsive responses which has a direct bearing on the calculation of the wind-induced responses. The evaluationof the wind-induced responses can be determined from either the indicialor irupulsive responses. An imgortant feature in this choice is the convergence characteristicof the response. From Figs. 1 through 18, it is seen that the indicial responsesdo not converge satisfactorily within the 20-sec. record. The path or drift root component has a predominant influence on theslow convergence of the indicial responses. The impulsive responses, however, are asnot greatly influencedby this component and do converge satisfactorily within the record. Thus, the inlpulsive responses were selected for use in the calculationof the wind-induced responses. The evaluation of the wind-induced responses is discussed in the next section.

D. Wind-Induced Responses

This section presents the method used and procedures followedcom- in puting the wind-induced responses of the SaturnC-5 configuration. A sample of the numerical calculationsis presented for illustrative purposes.

The responses of the vehicle to any side wind input are formulated as Duhamel integrals {see [4] } . The kernel functionof these integrals can be written internis of either the indicial or impulsive responsesof the system. From the discussion given in the previous section, it is advantageousto compute the wind-induced responses in terms of the impulsive responses.

Thus, from @I,the wind-induced responses are givenby

t R(t) = Ri(t-T)vy( T)dT

where Ri( T) is the ith coordinate resFonse to a unit impulse wind,T) vy(is an arbitrary side wind profile and R(t) is the ith coordinate responseto the wind, vy( t ) .

- 32 - r

The wind velocity used in the above integral was availablea as function of altitude. The profile was interpreted in terms byof using time a constant missile velocity associated with the midpoint of the altitude band. This, in essence, produceda slight shift in the effective wind frequencies at the extremesof the altitude interval.

The responses of the SaturnC-5 to two wind profiles* were computed. The wind profiles used were: the -West East components of8116 and 4768.** (In this report8116 is referredto as profile1; and 4768 is referred as to profile 2.) These profiles had been measuredby a modified spherical balloon (Jimsphere) -. radar technique and were tabulated at 25-meter increments. The 8116 profile representsa moderately severe wind environment (maximum wind velocity of 46.3 meters/sec at an altitude 13,200 of meters). The 4768 profile represents a mild wind environment (uaximum wind velocity17.8 ofmeters/sec at an altitudeof 19,000 meters).

The vehicle responses to the two profiles were computed at numerous altitude points during the flight trajectory. The responses of the following generalized coordinates were computed:

Translation,yo

Rotation, $

First bending, r),

Second bending, ?12

First sloshing, t1

Second sloshing, 5,

Control deflection, pc

* The wind data were supplied by C.George Marshall Space Flight Center. ** Wind values for profile8116 were given between altitudes2,350 of and 19,525 meters; values for profile 4768 were given between450 and 19,700 meters. The wind-induced responses were evaluated in five flight time bands. Each band was initially chosen to encompass10 sec. of flight time with missile parameters evaluated at the band midpoint. Subsequently, each of the actual calculations was extended ato length of15 sec. so that 5 sec. of overlap were available for comparison. The comparisons werea practical test of the ofuse frozen coefficients. The agreement of response values in the overlap regions was, in general, satisfactory.As expected, the high-frequency responses showed better agreement than low-frequency responses.

The correspondence between the altitudes and flight time bands used in the numerical calculationof the wind-induced responses is presented in Table 11. The increment used in each band to calculate the responses is also given. These increments were chosen to provide sufficient calculated points to adequately define the highest frequency component.

TABU I1

ALTITUDE BANDS USED IN CALCULATING WIND-INDUCED RESPONSES

Bands Flight Time FlightTimeInterval Altitude Interval- Responsea/ Increment (sec.) (sec.) (meters (sec.) (sec.) ) (meters )

50 45 to 60 3,594 to 7,1C4 8.33

60 55 to 70 5,787 to 10,200 12.5

70 65 to 80 8,573 to 13,940 12.5

80 75 to 90 11,987 to 18,361 12.5

90 85 to 92.5 16, C63 to 19,525 12.5

-a/ It was not possible to obtain an adequate descriptionof the slowly con- verging responses at the lowest of end some of the intervals because the wind profile did not extendto a sufficiently low altitude.

- 34 - To avoid burdeninG the report with an excessive amount of data, only a representative quantity of the wind-induced response data is presented at this time. Plots of the first bending,Ill second bending, 7)~, and control deflection, cc (noted by first control) responsesto profile 8116 (noted by profile 1) are given inFigs. 25 through 33 for the altitude band,8,573 to 13,940 meters. The 70-sec. impulsive responses were used in computing these responses. For each coordinate, the first figure reflects the effect of instantaneous immersion; the second figure reflects the ofeffect pure penetration;* and the third reflects the ofeffect penetration withUft growth.

The numerics presented alongside the plots in each figure pertain to an analysisof the response curves.The format of the information presented is as follows:

Response Response Response Response Response Altitude Value Maximum, Minimum orMinimumMaximum, ValueAltitude Crossing

The maximum and minimum values, indicatedby a check mark, define the approximate envelope of the extreme excursiansof the response. These values were used in the analysisof the wind-induced responses discussed in the next section.

IV. METHOD USED- IN ANALYSIS OF EFFECTS AND IMFORTANCE OF FENETRATION AND GROWTH OF LIFT ON MISSIIE RESPONSE

The analysisof the indicial and impulsive responses is largely qualitative.

The three aerodTynanic representations yield similar responses to the indicial wind. The responses to the unit impulse wind are also very much alike. Samples of these responsesare shown in the section, "Method of Solution for the Indicia1 and Impulsive Responses." The most noticeable differencesOCCUT during the highly oscillatory response shortly after immersion in the gust.

* An undesired translation of altitude was made in the calculationof the responses to profile1 considering pure penetration effects. The altitudes recordedin the plots and tabulationsshould be reducedby 100 meters.

- 35 - PR0FILE.l,_A~.~llUDE~.- VS. RESP0NSE-lNSTANTANE0US IMMER-SI0N "--_70_SEC F.T. ETAls 1ST BENUING

ALTITUDE^ "- .. - ~- ." . - ~-~ RE SP BN SE TYPERESPBNSE SEQ 8650. 1.4404573E-02 MIN 9013. 1.6436338E-02 MAX 1,

~"9150.1.56.37597E-02- MIN -.- 9338. 1.7740874E-02 MAX 1.4863462E-02 lr -~ -9530. -"IN DDD0 9800. 1.7691638€-02 MAX m0 011 10100. 1.4604546E-02 MlN L" 10300. 1.7447110E-02 MAX ...... I,., .... MIN J 10413. 1.66327_1_4E-02 ,I,. :.., 10588. 1.8907347E-02 MAX J )I,. ,... c" -J # lOe00. 1.4053984E-02 WIN .... *,I.. 10975. 1.6250993E-02 MAX v" I .... MlN :.I. .... 11213. 1.3777776E-02 ;&:I .... 11375. 1.4129932E-02 MAX w MIN r/ ...... 11488. ~ 1.3703871E-02"_" I I': 6. .. 11725. 1.5146002E-02 MAX :!I: Q .... II ...... 12925. 1.9371116E~02 M_!N. v -.. .- 13200. 2.0593875E-02 MAX V ..... - .. ___13575. 1.9190341Ey02 MI& f ~ ~ ~ ...... 13713. 2.0111562E-02 MAX r/ . . -. I -...... - w .... cn .... I ...... -...... -.... -...... I:: ...... 1 ...... _...... (...... ,... (...... OlJ I 8, - - ....I.:.. ;:.:..:,. :. ...,,,. ,; ...... -, ...... ,...... -...... " .I..:.;:' ...... cI_ I

""z-, i .. i ....j ...! ...... I...... - I...... ,...... ' Ill! !Hi ii/i.I

Fig. 25 - Wind-Induced Response of 1st Bending Considering Instantaneous Immersion - 70 sec. F.T. Bans ..PRBFILE. 12 ALTITUDE VS. RESPBNSE-PURE PENETRATIBN - 70 SEC E. ETAl. 1ST BENDING .. .~ - ALTITUDE - .RESPBNSETYPE SEO 8613. 1.4653077E-02 MAX J 8750. 1.4160251E-02 HIN V - 9113?. 1.6166161E-02 _.MAX r/ 9250. 1.5422924E-02 MlN r/

"". 9450. 1.7487065E702~ -_MAX v'" OOW 9650. 1.4774020E-02 MIN J OW 03. 9900. 1.7484932E-02 WAX J 10200. 1.4611643E-02 HlN I/

- "1040@. 1.7323195E-02 _WAX - J~ __ 10513. 1.6497581E-02 MIN J

" 10688. 1.8722725E-02_pMAX r/ 10900. 1.4144631E-02MlN W 11063. 1.6218176E-02 MAX r/ 11313. 1.3823870E-02 MIN MAX ~ 11475.~ 1.4163373E-02 11588. 1.3740399E-02 MIN fl

- - -- - ." 1181.3..- . .1_&084230E-02 .MAX .. 11863. 1.5053488E-02 MIN 12238.2.0980438E-02 MAX V 12463.1.9039713E-02 HIN J ..... -~12725,2.2518~15_2E_02 --.MAX r/ 13 025 . 1.9265627E-0213025. MIN V

"~~ 13300.-Z1O46375.4E-02 .MAX V I 1.9171460E-0213675. HlN Y 13825.2.0088750E-02 MAX Y Crl ". 4 I

Fig. 26 - Wind-Induced Response of 1st Bending Considering Pure Penetration - 70 -sec. F.T. Band PRBFILE 19 ALTITUDE VS. RESP0NSE-PENETRATIBN W/ LIFT GRBWTH 70 SEC F.T. ETA1.1ST BENDING

ALTITUDE RESPBNSE TYPE SEO 8650. 1.4236685E-02MlN y 9013. 1.6253431E-02 HAX Y 9150.1.5516338E-02p M.1) 9350. 1.7583695E-02 MAX I/

9550. 1.4879114E-OZ ~ MlN L/ 9800. 1.7586579E-02 MAX I/ 10100. 1.4719502E-02 MIN V 10300. 1.7422382E-02 MAX y 10413. 1.6598139E-02 MIN V 10588. 1.8822923E-02 MAX v 10800. 1.4252848E-02 MIN -V 10963. 1.6313982E-02 MAX 11213. 1.3918215E-02 MIN 11363. 1.4252233E-02 MAX 11488. 1.3826443€.-02.. MIN V __ 11713. 1.5162708E-02 MAX 11763. 1.513JS8~07€-02_~ Ml~N_

12138. 2.1063884E-02 MAX "~ 12363.1.9139434E-02 HlN r/ 12625. 2.2622252E-02 MAX Y

" 12925. 1.9387_4_16-€:02 "IN W .01. 13200. 2.0582911E-02 MAX Y - 13575. 1.929083E:0_2_~"Ml_N I/ 13725. 2.0205618E-02 MAX Y I .Dl. w 0) I .OI?

.ole

Fig. 27 - Wind-Induced Response of 1st Bending Considering Penetration With Lift Growth - 70 sec. F.T. Band 1

-______~

ALTITUUE RESPBNSE TYPE sEa ~ ~ 8613. 4.3744482E-03 MAX J 8725. 4.0023295E-03 MlN 8900.. 4.508_427_3Er03 MAX. 8 95 6. 4.4853886E-038956. MIN 9013.4.5133249E-03 llAX--- 9 06 3. 4.4713711E-039063.. . .~ MlN 9163. 4.7270079E-03 MAX 9225. 4.6558270E-03 MIN 9325. 4.8.68302E-0_3_- MAX J -~ 9425. 4.5769562E-03 MlN - 9463.. 4..6183988E-03 MAX 9575. 4.4374572E-03 MIN /- 9800. 4.9639040E-03 MAX /

10025.""~~ 4.3506055E-03 MIN L/ 10100. 4.4606026E-03 .. MAX 10125.4.4583065E-03 MlN 10263.4.6101188E-03 MAX 10300.4.6029136E-03 WIN 10450.5.1743199E-03 MAX / 10538.4.9490079E-03 MIN 10 57 5. 4.9866819E-03 10575. .

10700.~~4.6580511E-03 ~ MlN 10725. 4.6716678E-03 MAX I 10850. 4.1802588E-03 MIN 10938. 4.3903562E-03 MAX Y w 11000. 4.3079751E-03 MlN u) 11063. 4.3995984E-03 MAX I 11188. 3.9612482E-03 1123-8. 3.986790s-03 11338. 3.8722376E-03 11413. 3.9520928E-03 MAX

11450." ~ 3.9398412E-03 MIN 11550. 4.1447524E-03 MAX 11625. 4.1042637E-03MIN -___..12150. 5.989227OE-03 MAX r/ 12363. 5.6847295E-03 MIN L/ 12638. 6.2802830E-03 MAX 12950. 5.5422813E-03 MlN L/ 13213. 5.8223675E-03- _MAX y 13538. 5.5234889E-03 ' WIN V

PI~ILI I, btllllOC VI. RCI?WII-I~14MlAKWIMNRIICU TO KC r.1.

Fig. 28 - Wind-Induced Response of 2nd Bending Considering Instantaneous Immersion - 70 sec. F.T. Band PRBFILE 19 ALTITUOE VS. RESPBNSE-PURE PENETRATIBN 70 SEC F.T. ETA2. 2NO BENDING

ALTITUDE RESPBNSETYPE SEP 87 25. 4.2176469E-038725. MAX Y 8825.3.8937714E-03 MlN v 9013. 4.3667L2&E-p3_ MAX 90 50. 4.3585067E-039050. MIN

9113. 4.4024892E-0-3 ~ MAX 91 75. 4.3599912E-039175. MIN 92 63. 4.5815882E-039263. MAX 93 38. 4.5304561E-039338. MIN 9438.4.7557Q35E-0_?-- MAX v 95 25. 4.4887494E-039525. MIN 95 63 . 4.5174341E-O3MAX_....9563. 9675.4.3385074E-03 MIN v- 9900.4.8553365E-03 MAX V 10138.4.3016424E-03 MIN 10200.4,3873003E-03 MAX 10225.4.3827401E-03 MIN

10375. ~-4.565.1)>4_2+-0_3_ MAX ~~ 10413.4.5587956E-03 MlN 10550.5.0410824E-03 MAX v 10625.4.8679379E-03 MIN

10688. ~~~ 4.92885f~lEr03. .. .MAX 10813.4.5982428E-03 MIN 108255L 4.600875_4€-03 -MAX- 10950.4.1527604E-03 MlN I 11038. 4.3669858E-03 MAX - 11100. 4.2859800E-03 WIN rp 11163. 4.3571311E-03 MAX 0- 11288. 3.9353110E-03MIN I 11350. 3.9587723E-03 MAX 11438. 3.8631847E-03MlN 11513. 3.9378367E-03 MAX 11550. 3.91576116-03MIN 11663. 4.0993641E-03 . MAX 11725. 4.0729089E-03MIN 12250. 5.8181860E-03 MAX 12463. 5.5339883E-03 MlN MAX V- 12738. 6.1153229E-03 13050. 5.4488079E-03 MIN Y 13313. 5.7280810E-03 MAX ~ .." - ~______v 13650. 5.4588858E-03 MlN V

13788. ~ 5.-6195624E--03-. MAX Y

Fig. 29 - Wind-Induced Response of 2nd Bending Considering Pure Penetration - 70 sec. F.T. 'Eand 1

PRBFILE 1. ALTITUDE VS. RESPBNSE-PENETRATIBN W/ LIFT GREWTH 70 SECF.T. ETA2r 2ND BENDING

~ -. ALTITUDE RESPBNSE TYPE SEP 8625. 4.2531948E-03 MAX ~0 8725. 3.9324676E-03 MIN J 8913. 4.4080828E-03 MAX 8950. 4.4009144E-03 HIN 9013. 4.4455560E-03 MAX 9075. 4.4032969E-03 MIN 9163. 4.6252174E-03 MAX 9238. 4.5766608E-03 MIN 9338. 4.803-9.E-03 MAX / 9425. 4.53631156-03 MIN - 4.5640465E-039463. MAX 9575.4.3866286E-03 MIN 9800.4.9047810E-03 MAX J 10038. 4.3509600E-03 MIN

10100. 4.4359268E-03 MAX . .. . ~ 10125. 4.4318246E-03 HIN

10275. 4.6159303E-03 MAX " 10313. 4.6083646E-03 MIN 10450. 5.0889627E-03 MAX 4 10525. 4.9193029E-03 MIN 10588. 4.9797933E-03 MAX 10713. 4.6461850E-03 MIN 10725. 4.6484539E-03 "nAX I 10850. 4.2023506E-03 MlN P 10938.4.4151754E-03 MAX 10 I-J 11000. 4.3336198E-03 MlN

I 11063.4.4024632E-03 . MAX ~ ~~ 11188. 3.98040186-03 MIN 11250. 4.0033435E-03 MAX 11338. 3.9077162E-03 MIN"7" ~- 11413.3.9808055E-03 MAX 11450.3.9583264E-03 MIN

1156~3.4.1.4R9405E-03- MAX- ~ ~. - 11625.4.1148040E-03 MIN 12150. 5.86434616-03 .~.MA>. 12363.5.5839833E-03 MIN L, 12638.6.1705226E-03 MAX V 12950.5.5080432E-03 MIN c/

13213. 5.7872030E-03-.MA.X /-~ 13563.5.5172216E-03 MIN V 13688.5.6780403E-.02.- MAX r/ .

Fig. 30 - Wind-InducedResponse of 2nd Bending Considering Penetration With Lift Growth - 70 sec. F.T. Band PRBFILE 11 ALTITUDE"~ VS. RESPBNSE-INSTANTANE0US- ~-~ lM.MER_SSIBN 70 SEC E;T. BETA19 1ST CBNTRBL ___~______... ALTITUDE RESPBNSETYPE SEQ 85 75 . 1.7410416E-028575. MAX y 87 63 . 1.6703387E-028763. MIN r/ 8963.1.8356686E-02- MAX 90 13 . 1.8063814E-029013. MIN -~ 9088. 1.866_5588E-02 MAX 9138. 1.8280481E-02 MIN 9288. 1.9121487E-02 MAX 9338. 1.8910399E-02MIN I

9400. 1.930_40_96E-O2--MAX / ~ - 9613. 1.7242196E-02MlN L/ 9825. 1.9931168E-02 MAX-. V- . . t 10088. 1.6927401E-02 MIN r/ 10138. 1.7548994E-02 MAX 10163. 1.7482192E-02 MIN

"" 10375.- " 1,26667.88E-02 MAX 10413.1.9600492E-02 MIN 10513.-2.0454207€-02_ MAX r/ "-~ "~ .I, . .L!...... 10913.1.6726357E-02 MIN r/ .... I ...... 11000. 1.7756241E-02 MAX r/ ;m:.,I1 ).a...... 11100. 1.6610196E-02 MIN (11, ...... " -- 11113. . 1.6629184E~02 MA-X- ..14', : ...... 11263. 1.5316690E-02 MIN 7- ......

11.350, 1,56-5-8986€-02 ~ MA!". .- ~~ 11400. 1.5393791t-02 MIN I 11463. 1.6173211f-02 MAX /...... _% 11500. 1.6048744E-02 MIN MAX ;:;;I;:::/::::!;:\; a 121751_~_2_.488~7~E-02 r/ 12338. 2.3452254E-02 MIN I' .... - . I I... ..,...... 12350. --223452491E-02 MAX - ...... I.. 12388. 2.3390457E-02 MlN I/ ...... 12638. 2.5474401E-02 MAX Y ...... 12938. 2.1837827E-02 MIN r/ ...... I ..... A: 13250. 2.3008297E-yO2 MA3 I/ ::::I::::/ ...... - .. .L. 13550. 2.1766546E-02 MIN 13638. 2.2199959E-92 MAX

Fig. 31 - Wind-Induced Response of 1st Control Considering Instantaneous Immersion - 70 sec. F.T. Band 1

PRBFILE 1s ALTITUDE VS. RESPBNSE-PURE PENETR~AJIBN - 70 SEC F.T. 8ETAl. 1ST CBNTRBL -~ ALTIrUDE RESPBNSETYPE St0 86 75 . 1.7073567E-028675. MAX V 87 88 . 1.6606990E-028788. MIN 8800. 1.d6Cb07365E-02 MAX 8850. 1.6406069E-02 MIN -v 9063. 1.7967731E-02 MAX 9113. 1.7684730E-02 WIN 9188. 1.8364903E-02 MAX 9238. 1.7996660E-02 MIN 9388. 1.-6798708E-02 MAX 9438. 1.8530783E-02 MIN

. 95OQ.L1.89744835-02 MAX w ~ - 9713. 1.7108991E-02 MIN 9925. 1.9684783E-02 MAX W 10188. 1.6953144E-02MIN v 10238.1.7546383E-02 MAX 10275.1.7463198E-02 MIN 10475. 1.9628280E-02 MAX __ ._ _ 10513. 1.9532677E-02 MIN 10613.2.02158lOE-02 MAX V 11013. 1.6757926E-02 MIN 11100. 1.7740327E-02 MAX W -. 11200. 1.6602312E-02 MIN

11225. 1.6630280E-02 MAX _ ~~ 11363. 1.5420731E-02 MIN 11450. 1.5769569E-02 MAX 11500. 1.5499354E-02MIN 11563. 1.6.27_qZ91_E-02 MAX 11613.1.6147811E-02 MIN 12288. 2.4312204E-02- MAX- 12438.2.3017929E-02 MIN 12463. 2.3032048E-02 MAX 12488. 2.3003346E-02 MIN 127.38.,- ~~2,49_~0061E-OZ-~ MAX V _. 13025. 2.1685508E-02 MIN L/

"" 13350. -_2..2864062E-02 MA& t/ . . 13650. 2.1769869E-02 MIN 13738. 2.2204046E-02 MAX

Fig. 32 - Wind-Induced Response of 1st Control Considering Pure Penetration - 70 sec. F.T. Band PRBFILE lr ALTITUOE_VS~~R€~SPBNSE-PEN_ETRAT_IBN~)1/LIFT GRBWTH 70 SXF2T. BETAl, 1ST CBNTRBL

~~~~ .- - ALTITUDERESPBNSE TYPE sta 8575. 1.7154690E-02 MAX v 8688. 1.6691650E-02 MlN L/ 8700. J.6694309E-02- MAX . 8750. 1.6500505E-02 MIN 8963. 1.8065161E-02.. MAX ... 90 13 . 1.7785577E-029013. MIN 90 88 . 1.8472157E-029088. MAX 91 38 . 1.8105457E-029138. MIN

9283!-1.8910677E-02-. "A!-- - - 93 50 . 1.8641195E-029350. MIN -, 9400." 1.9091090E-02 MAX - . _____ r/- .Ok?4- 9613. 1.7232920E-02 MIN r/ 9825. 1.9804101E-02 MAX 10088. 1.7083355E-02 MIN 101x8.- 1.,7671509E-02 MAX 10175. 1.7585936E-02 MlN

10375. ~ -1.9751828E-02 ?AX 10413. 1.9653051E-02 WIN _. .. 10513. 2.0329306E-02 MAX r/ .om- 10638._.~~~~ 1.9689135E-02 MlN 10650. 1.9689291E-02 MAL- 10913. 1.6877996E-02 MIN " 11000. 1.785612_4€~02... MAX- J ... 11100. 1.6716338E-02 MlN I 11125.1.6744404E-02 MAX t? 1.5531965E-0211263. MlN _. .. +__ 1.5878151E-0211350. MAX .om- I 11400. 1.5605651E-02 MIN 11463. 1.6382551E-02 MAX

11513.". -~ ~ 1.6248956E-02 MIN 12188. 2.4405416E-02 MAX r/ .019- 12338. 2.3127962E-02 MIN 12363. 2.3144265E-02 MAX 12388. 2.3117316E-02 Mi-N- - .. 12638. 2.5073410E-02 MAX Y .Dl.- 12925. 2.1831702E-02 MIN Y 13250. 2.3007213E-02 MAX / 13550. 2.1914103E-02 MIN -. , 13638. 2.2344534E-02 MAX

_. .. .ala- _. . -.

Fig. 33 - Wind-Induced Response of 1st Control Considering Penetration With Lift Growth - 70 sec. F.T. &nd The pure penetration and penetration with lift frowth representations yield slightly smaller oscillatory excursions than instantaneous immersion. These differences are small (not more 10 than per centof the excursion amplitudeor area) and itis difficult to predict the effect on calculated responses due to actual wind profiles.It is anticipated, however, that the aerodynamic repre- senations which include penetration will provide smaller oscillatory excursions .*

A second feature was expected and verified in the responses to impulse and step winds.. Those responses which exhibita long, slm approach to their steady-state values are, during this approach, nearly independentof the aerodynamic representation used. This similarity in responses indicates that the local average responses to wind profiles should be nearly independentof the aerodynamic representation used.

The indicia1 and impulse responses are seen to provide clues for the analysis of wind profile responses.In the wind-induced responses calculated with the three aerodynamic representations, the local averages are expected to be similar while the deviations from the local average will be largest using instantaneous immersion. These ideas about the wind profile induced responses are tested using formal analytical procedures.

Responses are calculatedfor just two wind profiles. The main problem is hcw to use this small amountof data fora comparison of the responses from the three aerodynamic representations. The comparisons are made using extremals. The postulated differences in excursions from the local average are tested by comparing the distributions of extreme excursions. The local averages are compared using the averageof the envelope of points which in pairs define the extreme excursions.

The following procedure is used for each responsea calculated in altitude interval.** The response is examined and those extreme points are selected which appear to lie on or near envelope curves which would enclose

* The reduction in excursions is important because maximum responses very often will be due to excursionsor deviations froma "local average." ** The altitude intervals varyfrom 3,510 to 6,375 meters. Normally, constant missile characteristics should probably not befor used intervals which are this large. However, in this case, the additional adjunctpro- wind file can be considered typical of the altitude region and thus provides additional typical response data.

- 45 - the response.* The values of the response at these points are read and then adjunct values are differenced. The magnitudes of these differences are extreme excursions and are analyzedas a sample of extremals from a single population. The analysis of the excursions include ordering, transforming and curve fitting. The transformation usedis

where y is called the reducedvariate, pm -m- - , the accumulative probabil- n+l ity in a sample of n maximums and m is the order from the smallest =(m 1).

The transformation isdesigned to providex-. as a linear function Ul of ym"" , where yta is the observedmth ordered maximum in the sampleof n maximums. The curve fit is made in the x,y coordinates using the methodof least squares.

A local average response is calculatedas the unbiased averageof the extreme point values initially selected from the response. Vhen an odd number of points is selected, the number of maximums and minimums are unequal. This bias is removed witha weight factor of one-half on the first and last points.

The analytical techniques described here constitute unconventional employments of conventional methods. Ordinarily, a sample of extremes is obtained by taking from eacha numberof of equal-sized samples the largest (or smallest) value. The resulting distribution is used to predict probability of occurrence of an extreme value in an even largerof similar-sizednumber samples taken from the same population. An associated interpretation provides the return period which is the expected number of equal-sized samples which will be required to locate an assigned maximum.

* Since the responses being compared are very much alike, the near envelope points selected are in all cases equivalent points. ** See "Statisticsof Extremes'' byE. J. Gumbel, Columbia University Press,or "Statistical Theoryof ExtreEe Values and Some Practical Applications"by E. 5. Gumbel, National Ijureauof Standards, Applied Mathematics Series33. The transformation is applicableto .maximums from populations with probability functions of the "exponential type." Many important distributions are of this type including the exponential itself, the normal, the chi- square, the logistic, and the log normal.

- 46 - In the present analysis, the basic sample size is the altitude increment in which responses are calculated with frozen missile propedies. However, not just one maximum excursion is taken from this sample of the calculated response. Thenuntber of excursions taken are all those whichappear to extend to envelopecurves. This selection takes the attitude that the response excursions are a superposition of responses involving the spectral content of the wind and admittance properties of the vehicle. With this viewpoint, the successive "maximum excursions" may be considered extremals in a superposition of independents,*thereby justifying the use of the analysis. It is clear that theconventionalconnotationof return period has been alteredhere. It might be possible to recover this predictive capability by reinterpreting sample size. However, this predictiveuse of thepresent analysis is not recommended. Instead, the analysis is recommended for the comparison of excursions obtained using the three aerodynamic representations. The analysispermits the corapari- son to be made in an integrated rather than point fashion and uses the majority of pertinentdata. The trend with y , the reduced variate, may beinterpreted simply as the nonlinear dimension of increasing numbers of samples or increasing numbers of missile transverses.

The calculation of an average from the average of the extremes is not conventional but is an accepted and often powerful technique.

Figure 34 shows an example of the excursion analysis and resulting curve fits plotted on paperdesigned for this purpose.In the adjacent Fig. 35, the same data are shownon rectilinear coordinates, the observed variate (ordinate) and the reduced variate (abscissa) which is the transformed cumulativeprobability. The similarcurve fits for theanalyzed responses are shown on rectangular coordinates in Appendix N.

* The problem of independence in the primary data arises in the analysis of river flows. There, it is aninferred assumption that the flow during each 24-hr. period is anindependent measure, although correlation between the flows of successivedays is easilydemonstrated. The theory of theextremes is still very successful in this case.

- 47 - CUUULbTlVE PROBABILITY

Pig. 54 - Extreme Excursion of 2nd Sending vs. Cumulative Probability - 60 sec. F.T. -

Profile No. 1 GO sec. and /

ReducedCumulative Probability Fig. 55 - Extreme Excursion of 2nd Bending vs. Reduced Cumulative Probability -- 60 sec. F.T. - 46 - V. RESULTS AND DISCUSSION

In this section the results are referenced and discussed not only for their immediate import but also as a guide to future prediction of penetration and lift growth effects.

Samples of the wind-in'duced responses are shown in Figs. 25 through 33 for the three. different aerodynamic 'representations. The most important result is the small difference between the responses and between the maximum responses. The same result is observed in the comparisons for otherresponses and flight times. Generally, when thedifference in maxim responsesexceeds l.per centthe instantaneous immersion resultsare conservative (large). The differences in maximum responses with the aerodynamic representations do not appear significantin an engineering sense. For instance, with profile No. 1, a moderately severe profile, instantaneous immersion provides first bending which is conservative by -0.7 percent to 1.3 percent. Second bendingwith instantaneous immersion is conservative by -1.Oper cent to 2.3 per.cent, while the first sloshingresponse is conservative by -0.8 per cent to 4.8 per cent.

The results and interpretations which follow indicate that the per cent difference may be larger in weak wind profiles but the magnitude of the differences will remain unimportant. % The results indicate that the responses are insignificantly affected by the inclusion of penetration effects and lift growth effects. However, the conclusion can be drawn only for the missile configuration used and the modes included. The possibilities for more general results are explored next.

Samples of the indicia1 andimpulsive responses are shown in Figs. 7 through 24. There is a small butnoticeable difference between the responses which do and do notinclude penetration. The oscillatoryexcursions with penetration (PP and PWLG) are smaller thanthose from instantaneous immersion. This is especially true for the bending and sloshing modes. It was thenanticipated that the same type of difference in oscillatory excursions mightappear in the responses to wind profiles. An analysis of the wind-inducedexcursions was de- signed and applied as described in Section IV. The results are shown in Figs. 42 through89 (pp. 109 through 135) where the anticipated differences are observed.In these figures the ordinate is the magnitude ofexcursions and the abscissa is a nonlinear dimension of increasing transverses or sample sizes.

Figures 42 through89 illustrate the close correspondence between responseswith pure penetration (PP) and penetration with lift growth (PI-m). The difference between responses calculated with and without penetration are seen to diminishwith altitude. This would be expectedsince with higher speeds the penetration cases are approachinginstantaneous immersion.

- 49 - .I The indicial and impulsiveresponses (Figs. 7 through 24) also indicate an important similarity in responses with all three aerodynamic representations. This similarity occurs in the long slow approach to steady-state values. The local average values of thewind-induced responses depend largely on this long tail and the history of wind inputs. Thus, local average responses to winds were expected to be about equal for all aerodynamic representations. This expectation is borne out by the averages presented in Figs. 90 through 99 (pp - 136 through 141).

The differences and similarities in the indicial and impulsiveresponses seem to have their logical counterparts in the wind-inducedresponses taken as a whole. However, since main interest attaches to the predictionof maximum responses we must test the extension of the same general logic to esti- mation of maximum responses calculated with the three different aerodynamic representations. The results of this test are shown in Tables I11 and IV. TABLE I11 RATIOSOF AVERAGE RESPONSES, OBSERVED MAXIMUM RESPONSES AND EXPECTED MI" RESPONSES FOR PROFIW NO. 1

Flight Avg . Response Observed Max. Expected Max. Time Rat ios Rat ios Rat ios Band -I1 a/ PP a/ -I1 -PP -I1 -PP ( sec .) Response PWIG PWLG PFJLG -PWLG -PWLG -PWLG

1st Bending 50 0.9967 1.0039 1.0092 1.0034 1.0254 1.0036 60 0.9972 1.0028 0.9973 1 .0028 1.0122 1.0025 70 1.0024 0.9941 1.0129 0.9954 1.0110 0.9952 80 0.9924 0.9959 1.0082 0.9972 1.0097 0.9989 90 0.9746 1.0020 0.9932 1.0032 1.0054 1.0028 2nd Bending 50 0.9970 1.0049 1.0045 1.0043 1.0362 1.0046 60 0.9917 1.0113 0.9907 1.0113 1.0039 1.0106 70 1.0091 0.9899 1.0178 0.9911 1.0182 0.9910 80 1.0113 0.9786 1.0229 0.9817 1.0236 0.9839 90 0.9737 0.9993 0.9884 0.9999 1.0004 1.0000 1st Sloshing 60 1.0028 1.0093 0.9921 1.0080 1.0480 1.0054 70 1.0117 0.9869 1.0472 0.9953 1.0376 0.9930 80 1.0204 0.9506 1.0397 0.9926 1.0480 0.9938 90 1.0103 1.0045 1.0458 1.0029 1.0363 1.0026

~~ ~ ~ ~~ -a/ I1 denotes instantaneous immersion; PP denotes pure penetration; PFJLG denotes penetration with lift growth. - 50 - r

TABLE IV

RATIOS OF AERAGE RESPONSES, OBSERVED RESPONSES

AND” EXPECTED . MAXl#LI” AESPONSES FOR PROFILE NO. 2

Flight Avg . ResponseObserved Max. Expected Max. Time RatUo s Ratios Rat io s Band I1 PP I1 PP “I1 -I1 -PP - - ( sec ) Response PWLG ‘PWLG PWLG . “PWLGPWLGPWLG - - - - 1st Bending 50 1.0065 1.0033 1.0669 1.0030 1.0578 1.0029 60 0.9777 1.0033 1.0178 1.0024 1.0286 1.0025 70 1.0035 0.9945 1.0329 0.9977 1.0283 0.9978 80 1.0015 0.9965 1.0140 0.9994 1.0264 1.0017 90 0.9928 1.0031 1.0048 1.0033 1.0114 1.0033 2nd Bending 50 1.0116 1.0045 1.0747 1.0038 1.0766 1.0041 60 0.9605 1.0134 1.0275 1.0104 1.0384 1.0107 70 1.0073 0.9900 1.0445 0.9935 1.0418 0.9944 80 1.0211 0.9811 1.0432 0.9846 1.0372 0.9889 90 0.9985 1.0004 1.0208 1.0006 1.0075 1.0006 1st Sloshing 60 1.0017 1.0075 1.0987 1.0002 1.0892 1.0008 70 1.0129 0.9886 1.0694 0.9987 1.0739 1.0004 80 1.0201 0.9577 1.0501 0.9972 1.0528 0.9990 90 0.9314 1.0001 1.9179 1.0018 1.0181 1.0020

Here, thepenetration with lift growth results are used as a referencesince this representation is the most accurate.

The expected values of maximum responseused in the ratios of Tables I11 and IV were formed as the local average plus one-half the expected maximum excursion. Where theexpected maximum excursion was thevalue indicated by the curve fits in Figs. 42 - 89. Comparison of thetabulated ratios for profile No. 1 indicates some correlation between observed and expected values es- peciallyfor the sloshing response. However, a significant part of thedevia- tions from 1.0 are due to the averages which constitute a sizeable part of the response.

The tabulated ratios for profile No. 2, a weak profile, show excel- lent agreement between expected and observed ratios.

,It appears that the differences and similarities observed in the indicia1 and impulsive responses can be used with partial success to predict the effects of penetration an4 lift growth. The differencein impulsive excursions appears in wind-induced responses as a change in extreme excursions. Vhere these excursions play the important role in the maximum response (a weak wind profile) the maximum responses are affected (by penetration primarily) and to about one-half the extent indicated by the comparison of indicial and impulsive responses.

In a moderately severe wind profile the differences in averaged responses are likely to be as important as the oscillatory excursions. The differences in averaged responses are difficult to predict from the comparisons ofindicial and impulsiveresponses. In addition, it appearslikely that some intermediate frequencies may play a role of equal significance and be difficult to detect in the comparison of indicial and impulsive responses.

For responses vhich would be significantly affected by penetration and lift growth it is likely that this fact would be apparent in the comparison of indicial and impulsiveresponses based on the three aerodynamic representations.

There is a simple and appealing idea which in the past has been em- ployed tospeculate about the effects of penetration. This idea is presented here and is shownby comparison with calculated results to be insufficient for predicting the effects of penetration.

The displacements in the bending modes change sign along the length of the missile. The generalized forcing function which drives one of these bending modes takes on different characteristics when penetration is neglected and included.

When penetration is neglected each station of the missile is imuersed in the same wind-inducedcrossflow.* The resulting generalized force for the bending mode is an algebraic sum of local contributions where the changes in mode shape sign lead to cancellations.

* The discussion here pertains only to wind-inducedcrossflows. The crossflows due to the local transverse body velocities will always provide damping in the analyses reported here. If growth of lift delays were added to the forces from these crossflows the damping might bereduced or eliminated. The crossflows due to the local body angle of attack are 90' out of phase with body velocity and in the long run neither add nor remove energy from the bending mode. Adding growth of lift delays to the forces from these latter crossflows could provide either damping or undaaping.

- 52 - When penetration is included the situation is best illustrated by a unit impulsegust. The gust crossflow is applied to successive stations along the missile with delays appropriate for the time required to penetrate the gust. It is apparent here that the energy first added to the mode at the initial penetration my be augmented or canceled during the penetration by sub- sequentstations. The critical factors are the phaserelationships between the modal responseand the succession of inputs. The character ofpreviously calculated responses indicated that the critical phase relationships might be estimated.

Each indicia1 and impulse response has alwaysbeen dominated by a frequency of the coupled system which lies fairly close to the uncoupled modal frequency.This led to the idea thatpenetration effects could be estimated by assuming that each bending mode responds primarily as an uncoupled mode and at its naturalfrequency. The comparisonof modal responseswith and without penetration would reduce to a comparisonof the following forms.

and

R (t) = The approximated mth bending mode inpulseresponse in- flm cluding penetration.

r (t) = The approximatedresporlse neglecting penetration. f\m = An unknown admittance amplitude coefficient. crlm

- 53 - Qqm(7) = The growthfunction for the mth bending mode. (Similar to those shown in Figs. 5 and 6.)

% = The uncoupled natural frequency. 6(0) = The unit impulse imposed at time zero.

= Theassumed modal response to a unit impulse imposed at sin %t time zero.

The approximations R,, (t) and r (t) havebeen evaluated for the m % fourth andsecond bending modes.They are shown inFigs. 36 and 37. The approximations for the second bending mode response may be compared with the actual, coupledsystem, impulsive responses shown inFig. 38.

For both the fourth andsecond bending modes the approximated responses indicate that incorporation of penetration results in increased response afterpenetration is completed.During penetration the expected sequence of augmenting and canceling effects are observed.

The approximatedsecond mode responses do not correspond to the coupledsystem responses. Further, the major implicationof the approximated responses (increased response with penetration) is refuted by the responses for the actual coupledsystem. The main reasonfor this disagreement appears to lie in strong coupling effects, probably with the control system and swivel engines. These couplingeffects raise the predoninant frequency of response and provide a significant response to the high frequencies generated during penetration. If thebasic idea of theapproximation is to be usedfor the prediction of penetration effects it will be necessary to include some important system coupling.

VI. CONCLUSIONS AND RECCl.IVIENIIRTIONS

Specific conclusions about the importance of penetration and lift growth must be restricted to the Saturn C-5 without fins and to the modes included inthe analysis. The specificconclusions are:

1. The calculated responses to winds are changed by a detectablebut insignificant amount when the aerodynamics are revised to include penetration and lift growth.

2. The change inresponses is due primarilyto penetration; the addition of lift growth has very little effect.

- 54 - "" I1 PP - / \ \ \ / \

I 0.2 Time, 7 (sec.) / \ \ / \ /

Fig. 36 - Approximate4th Bending Impulsive Response vs. Response Time - 60 sec . F .T.

- 55 - Fig. 37 - Approximate 2nd BendingImpulsive Response vs. Response Time - 60 sec . F .T.

- 56 - Fig. 38 - Actual 2nd Bending ImpulsiveResponse vs. Response Time - 60 sec. F .T.

- 57 - I

3. The maximum responsescalculated with instantaneous immersion aerodynamics are conservative ( large) .

General conclusions are:

1. The intuitive idea that penetration effects can be predicted from the unccupled bending mode periods and corresponding penetration delays is in- correct.

2. The calculationsperformed here provide no example of significant effects. However, it appearslikely that thecases in which penetration (or penetraticjn and lift growth)plays an important role can be detected by comparisons of the indicial and impulsive responses using aerodynamics with and without penetration.

It is recommended that:

1. The comparison of indicial and impulsiveresponses vith and without penetration be used as a measure of the adequacy of instantaneous immersion aerodynamics.

2. Comparisons (1 above)be carried out for the third and fourth bending modes and for a Saturn C-5 model including fins. (The fins canbe simulated by a conic section which provides equivalent cormal forces in the steady state. )

3. The conclusions of this report be checked with a larger amount of wind data by using one or two of the existing impulsive responses to calculate wind responsesfor a nmberof profiles in the high g altitude band.

4. In the event a simplepredictive technique for penetration and lift growth effects is sought, consideration should be given to refinement of the isolated mode idea. The refinement would include coupling the mode to the important control frequencies.

- 58 - BIBLIOGRAWY

1. Yates, J. E., "TransientAerodynamic Loadin@; on Nultistage Missiles, Midwest Research Institute Phase Report, Project No. 2544-P, March, 1962 (Confidential).

2. Miles, J. W., The PotentialTheory ofUnsteady Supersonic Flow, Cambridge University PES (1959).

3. Hildebrand, F. B., Introduction to NumericalAnalysis, McGraw-Hill Book Company, Inc . ( 1956 ) .

4. St. John, A. D., and €3.R. Blackburn,"Research on the Loading of Missiles Due to Turbulence and Wind Shear," Midwest Research Institute Final Report,Project No. 2544-P, October, 1962 (Confidential).

5. Rheinfurth, M. H., "Control-Feedback StabilityAnalysis," ABMA Report, DA-TR-2-60.

6. Ward , G. N., LinearizedTheory of Steady High-speed Flow,Cambridge University Press (1955).

7. Luke, Y. L., "A Procedurefor the Inversion ofa Class of Laplace Transforms," Midwest Research Institute Report, Project No. 2383-P, July, 1961.

- 59 - APPENDIX I

EQUATIONS OF MOTION OF SATURN C-5

The equations of motion (see [5]) of a flexible missile system in vertical flight are given in this section. Ten generalizedcoordinates are considered(see Vol. 111) : lateral translation, yo ; rotation, fl ; first bending, 71 ; secondbending, 12 ; third bending 113 ; fourth bending, Tl4 ; two sloshing, J1 and 22 ; actualending deflection, PE ; andcontrol deflection, Bc (see Fig. 39 ). The equationsare valid for a swivelengine controlled vehicle trhere the swiveled engines account for four-fifths of the total thrust force.Slender body theory is used todescribe the generalized aerodynamic forces.

The oscillating propellants are described by a mechanical analogy (see Fig, 40 ). Only the motion of theliquid in the booster tanks is investi- gated. The first sloshing mode is associated with the furthest aft tank (tank A), while the secondsloshing mode is associated with the adjacent tank (tank B) '

For simplicity, the following terms of theequations of motion (see [SI) are neglected on the basis of being small bycomparison:

1. The rotation of the missile crosssections during bending.

2. Icorr (the difference of mass moment of inertia 03 the frozen liquid and liquid propellant in the full tanks about the c.g. of the missile).

3. Generalizedforces due to the flowingpropellants. (This elimi- natesthe terms containing the time derivatives of the mass of the propellant.)

Assuming that the missile and atmospheric parameters are constant in predeterminedtime or altitude intervals, the equationsof motion become for translation ,*

* The numericalconstant, 4/5, appearingin (I-l)> (I-2), and (1-3) as a multiplier of &I canbe generalized to accountfor any percentage of the total number of engines which are swiveled.

- 60 -

ii . I

Deflection Curve

Fig. 39 - Saturn C-5 Coordinate System I cn Iu

I Tank B

ks2 = 4PS2

cs1 = wslmslgsl cs2 = Ws2ms2gs2

mSl,~2 = Sloshing Masses mol,m02 = Fixed Masses in Sloshing Analogy 101,102 = Mass Moments of Inertia of Fixed Masses in Sloshing Analogy about Their c.g.

Fig. 40 - Mechanical Analogy of Sloshing Fluids for rotation,

QJ1 ..

for mth bending,

- 63 - for first sloshing,

- 64 - for second sloshing,

4 + g )TI. + WS-2522 = g1Y!(x s2 1 0 , i=l for the swivel engine,

The last equation of motion describes the control and actuator system of the missile. The control system considered in this report utilizes both an attitudereference control and a flowdirection indicator (see r.51). The differential equation describing the relationship between the control deflection, f3c , theindicated attitude, Bi , andthe indicated angle of attack, &i , is given by

where

- 65 - and

V CYW = Y (1-10) U

The quantities a's in (1-7) are time-independentcoefficients while thegain values a. , a1 and bo are time-dependentvariables.

It was convenient for the investigation of the effects andimportance of penetration and growth of lift on missile response to simplify the above system of equations in anticipation of generating a voluminous amount of numerical results. Thus, for the study presented in this report the following additional conditions were imposed on (1-1) through (1-9):

1. The third andfourth bending mode contributions were neglected.

2. The complianceof theswivel engine (difference between the actualdeflection angle, @E , andthe control signal, 8~)was assumed to be zero.This, in essence, is similar to making the linkageconnection between the two variables infinitely rigid.

3. The mss of theswivel engine and the mass moment of inertia of the swivel engine about its swivel point were neglected.

Incorporating the above conditions into the above equationsyielded for translation,

- 66 - for rotation,

(1-12)

for first bending,

- 67 - (1-13)

for second bending,

for first sloshing,

(1-15)

- 68 - for second sloshing,

Substituting (I-a), (1-9) and (1-10) into (1-7) and reducingto a set of second order equations givesfor the control system,

(1-17)

where

(1-18) and .. P-N=O (1-19)

The preceding equations of motioni(1-11) through (1-19)) describe the missile system considered in the numerica investigationof the effects and importance of penetration and growthof lift on missile response (seeVol. 111).

- 69 - APPENDIX I1

DEVELOFMENT OF TRANSIENT, QUASI-STEADY AND STEADY GENERALIZED AERODYNAMIC FORCEEXPRFSSIONS RESULTING FROM A UNIT STEP AND UNITIMPUISE WIND PROFILE

The developmentof the transient, quasi-steady and steady generalized aerodynamic force expressions resulting from a unit step andimpulse wind profile is presented in this Appendix. The development is based on slender body theory.For simplicity, the details ofthe analysis are omittedand the reader is referred to the original work of Miles [23 and the extension to multi-staged vehicles by Yates [l]. for additional information.

Transient aerodynamic force expressions corresponding to rigid body and bending modes of vibration are presented first for a general wind profile. The indlcial* andimpulsive** transient, quasi-steady and steady force expressions are thenderived. Quadratic polynomials are used to curve fit segments of the mode shapes in evaluating the generalized forces associated with the bending modes.

The wind induced forcing functions which are compared in this report have both a geometric and an aerodynamic aspect.

Inthe geometric consideration two cases are used. Inthe simplest case, instantaneous immersion, all stations along the missile are assumed to be immersed in the same wind-induced crossflow, namely the wind crossflow occurring at thenose. Themore accurategeometric representation, called penetration, assigns to each station along the missile the wind crossflow which exists at the altitude occupied by the station. With penetration,the missile nose enters a side gust first and in subsequent time successive stations along the missile length move into the crossflow.

Two representations of the aerodynamics are used; they are quasi- steadyand transient. In the quasi-steady representation the air forces at a missile station are those which would exist if the local crossflow persisted unchanged foran extended time. The transientrepresentation is based on transient slender body theory and includes the growthof lift with time.

* Response to a unit step wind profile. +F-E Response to a unit impulse wind profile.

- 70 - Three types of wind-induced forcing functions are assembled using combinations of the Geometric and aerodynamic representations. The simplest type is called instantaneous-immersion and uses the instantaneous-immersion geometric representation with quasi-steady aerodynamics.A more accurate type, called pure-penetration, uses penetration geometrics and quasi-steady aerodynamics. The most accurate forcing functions are called penetration-with-lift- growth. These functions use the penetration geometries with transient aerodynamics .*

The simpler function types, instantaneous-immersion and pure-penetration, canbe obtained from penetration-with-lift-growth which is derived first.

I. TRANSIENT AERODYNAMIC FORCESWITH PENE!TRATION AND LLFT GROWTH FROM A GENERAL CROSSFLQW VELOCITY

Consider the multi-staged pointed body of revolutionshown asin Fig. 41 . The Cartesian orthogonal coordinate system(x-y-z) has its origin at the nose. The vehicle is considered to be traveling in the negativex direction with a constant velocity, U . At time zero, the nose encountersa side wind of magnitude v(x,t) directed along the positive z axis. Now, from [J,, and [d the transient aerodynamic forces corresponding to translational, rotational and bending coordinates are given,respectively, by

(11-1)

* The crossflows induced by missile motions are in all cases treated with quasi-steady aerodynamics. These crossflows are small in comparison to the wind-induced crossflows and appear in the left ofhand the side equations of motion. c

Fig. 41 - Saturn C-5 BodyGeometry (11-2)

and

(11-3) where thenotation (p’ means thatthe x integration is performedover (0, Ut) or (0,L) as the body has partially or totallypenetrated the side wind profilev(t- $) . Here xg is thedistance from the nose tothe center of gravity and the sign of the moment equation is chosen so that positive M(t) gives rise to a clocldse rotation of thevehicle as viewed in Fig. 41 . In (II-l)>(11-2) and (11-3), primesdenote total differentiation of thefunction with respect to its argument.

The exactexpression for fo(T) (see El]) is rather cumbersome to work with in obtaining numerical results. A numericalevaluation of this function is given by various authors as are their expansions for large and small arguments(see [2] and [6] ) . A convenientapproximation of this function, valid for all valuesof the argument, is givenby Luke [7] . The approximate mathematical form of fo(T) is

- 73 - -0.21005~ -1.30637 fo(T) = 1 - 0.0405e - 2.7077e-' + 0.0016e

-0.210057 0.3920Te-~ - 0.0001-re -

- Te -o*616827 (1.0204 cos 0.407317- 0.6574 sin 0.40737)

+e-o-61682T (2.2466 cos 0.40737 + 0.2066 sin 0.40737) , (11-4)

This representation is usedin the numerics and computer program presented in this report.

11. TRANSIENT AERODYNAMIC FORCES FOR SATURN C-5 CONFIGURATION* ENCOUNTERINGA UNIT STEP AND UNIT IMPULSE WIND PROFILE

A. Unit Step and Unit Impulse Input

For the special caseof a unit step and unit impulse wind profile, the generalized forces corresponding to translational, rotational and bending coordinates reduce toa relatively simpleform. 17e will first consider the indicial transient aerodynamic forces.

Let

v(7) = I(5) , (11-5) where I denotes the Heaviside step function. Substituting (11-5) into (11-l), (11-2) and (11-3), the indicial transient normal force, moment and bending moment become

* The SaturnC-5 configuration considered in this report does not include fins.

- 74 - (11-6)

(11-7)

and

(I1 -8)

where thesubscript s denotes that theseentities are due to a unit step wind profile.

If we consider the unit impulse wind profile

v(7) = a(?) , (11-9) where 5(7) is theDirac delta function,the impulsive transient normal force, moment and bending momentbecome

(11-10)

17he re

= 0 for UT > L

- 75 - (11-11) where

(uT-x~)s'(uT) = (x-xg)s'(x) for 0 S UT I L X= UT

= 0 for UT > L and

(11-12) where

= 0 for UT > L

andthe subscript i denotesthat these entities 8;-e due to a unitimpulse vindprofile. Equations (11-6) through(11-8) and (11-10) through(11-12) definethe indicial and impulsive transient aerodnamic forces. Thesefunc- tions will be used in the following section to describe the forces for the Saturn C-5 confiGurations.

Before proceeding, it should be pointed out that the impulsive €orces canbe obtained from the first timederivative of theindicial forces. Thus,

- 76 - r

in thefollowing sections the indicia1 forces will be developed first. The latter forces %rillthen be differential to obtain the desired expressions for theimpulsive forces.

B. Development ofTransient Indicialand Impulsive Aerodynamic Forcesfor -~ .. "" ." .- .. - . " ~ ~ ._. " . .. ~. The Saturn C-5 Configuration

The basic configuration of the Saturn C-5 missile is given in Fig. 41. The nose spike on thefront of themissile corresponds to the escape tower which is attached'to the vehicle throughout the boost flight. The geometry of the configuration is defined in Table V with conic character- isticsdefined by en = tanan for n = 0,1,2,3 .

TABLE V

DEFINITION OFSATURN C-5 BODY GECMETRY

Body Radius Area Derivative Region Nx) S'(X) Applicable

2npgx o

Substituting the appropriate definitions of the body geometry into (11-6) yields

- 77 - (11-13)

where the components , NL2)(7) , Ni3)(?) and N~*)(T) correspond to the normal forcegrowth on the first, second, third and fourth missile regions, respectively (see Fig. 41 ). Thesecomponents are defined as follows: for the first region(escape tower)

forthe second region (nose cone)

X1 -%T

-ST<-.x5 (11-17) U

The limits of integration in the above expressions follow from a consideration of the various penetration times of the individual regions.

Carrying out the above integrations and simplifying for ComPuation, the transient indicia1 normal force is given by

(11-13)

where the quantity Nin) ( T) (n = 1,2,3,4) depends on the value of the independentvariable, T , and is defined by the following two expressions:

for Xn I T < Q ,

n = 1,2,3,4 (11-18)

The coefficients in (11-18) are defined in Table VI for the four regions of the C-5.

- 79 - wm VI

DEFINITIONOF COEFFICIENTS IN (11-18) (11-25) AND (11-36)

-n hn- -an -Region

2 =oxo sox0 1 so - First 3 2

2 -(x1U1 - 2) %-So 28: I

0) O3 yX3U - s2-s1 2) 2e5

S R2-S R2 4 s3-s2 FouI-~~ The f'unction G(y;a) in (11-18) is defined as

(I1 -19)

vherefo(y) is given by (11-4) and (x, corresponds to theprduct Mf3n (see Table 11). Bn is thetangent of a particular region connectionangle, a, .

Next, we consider the moment growth on the C-5. Substitutingthe appropriate definitions of the body geometry into (11-7) yields

(11-20) where

(11-21)

,ST<",x1 (11-22) U

557

Now, carryingout the above integrationsand simplifying, the transient indicia1 moment is given by

where thequantity MLn)('r) (n = 1,2,3,4)depends on the independentvariable, 'T , and is defined by the following two expressions:

n = 1,2,3,4 . (11-25)

The functionH(y;a) in (11-25) is defined as

- 82 - (I1-26)

Now, we will consider the derivation of the transient indicia1 forces corresponding to the bendingcoordinates. From (11-8) it is seenthat a description of the mode shapes , Ym(x) , is needed in order to evaluate the Q~(T)~'s. Themode shape datafor the Saturn C-5 were availableonly in discretenumerical form. These data were inconvenient to use in this form since they prevented evaluation of (11-8) in a manner similar to that used for (11-6)and (11-7). However , by approximatingthe mode shapeswith polynomials, the preparation of (11-8) for computation could be readily accomplished.

From (11-8) it is easily seen that the evaluation of the integral requiresinformation about the mode shapeonly over those sections of the missile which have a changingradius. Thus the mode shapeapproximations need onlyapply over relatively short streamvise distances. From ananalysis of the first fourbending modes of the C-5 configuration, it was found that quadratic polynomialscould be used efficiently to approximate Y,(x) withan acceptabledegree ofaccuracy. In thisinstance, the quadratic polynomial mode shapes yielded values which were within 1 per cent of the actual mode deflections.

Now, let the mode shapes (m = 1,2,3,4)in the desired region (see

First Region (11-27)

Second Region (11-28)

Third Region (I1-29) and

Fourth Region (I1-30)

- 83 .. - where the A's , B's and E's* are consideredconstant for a discreteflight time or altitude.

Substituting (11-27)through (11-30) and the appropriate definitions of the body geometry into (11-8) yields

(11-31)

where

(11-32)

I * The TIS , B's and E's for the first fourbending modes were computed between flight times of 10 and 140 sec. at 10-sec.intervals. However, values for only the first two bending modes between 30 and 100 sec., inclusive, were used in the numerical computation of the indicia1 and impulsive responses.

- 84 - x3 -ST<<, (11-34) U

x5 -ST

Performingthe above integrations and simplifying, the transient indicia1 bending moments are given by

where

- 85 - (11-36)

The functionI(y;a) in (11-36) is defined as

(11-37)

The advantagesof using polynomial curve fits (especially quadratic polynomials)for the mode shapes, Ym(x) I can be easilyseen from (11-36).

The evaluation of theQ(n) (7) 's can be mde from the normal force, moment %I and anadditional integral, I(y;ct) . If quadraticpolynomials had not pro- videdsufficient accuracy in descrlbing the Ym's , the number of integrals to evaluate would be equal to one less than the order of polynomial approxi- mation. However, for the C-5 configuration the transientindicia1 normal force, moment and bendinc moments can be expressed in terms of integrals similar to (11-19),(11-26)> and (11-37).

- 86 .. As discussed at the first of this Appendix, the norm1 force, moment and bending moments due to a unit impulse wind profile are given by the first timederivative of theindicia1 force and moments. Therefore,differentiating (11-13),(11-20) and (11-31) with respect to time yields

and

where

for dn s T < cr~ , n = 1,2,3,4 , (I1-41) - a7 - - 2fnanU( 7-bn) 3 E' ($ ; fnj) for d, I; T 5 xn en n = 1,2,3,4

- 6f,a,U(~-b,)~ c" (-7-dn ,. f,) - H (5- ; \ en en

for dn 5 T < 43 ,

n = 1,2,3,4 , (11-42) and

- 88 - 2fnanU2( 7-bn) 4 - I' (3; fnD for dn 5 7 S -dn , en n = 1,2,3,4 +-cc,,- U

- for dn 5 7 < m

n = 1,2,3,4 .

(11-43)

(I1-44)

(11-45)

and

- 89 -

I (I1-46)

When n = 1 , the terms 1G' (-T-% ; fn) , 1R' (-T -dn ; fn) and en en en en 1I' (- -4 ; fn) in (11-41) , (11-42) and (11-43), respectively, must be en en omitted.

The equations (11-13), (11-18), (11-20),(11-25), (11-31), (11-36), (11-38), (11-39) and (11-40)are the fundamentalexpressions required to compute the indicial and impulsivenormal force, moment and bending moment growth functions for theSaturn C-5 configuration. These aerodynamic forces and moments were used in calculating the indicial and impulsive responses of the C-5 missile system for the case "penetration with lift growth."

C. Development of Qcasi-SteadyIndicia1 and Impulsive Aerodynamic Forces for the Saturn C-5 Configuration

The pure-penetration forcing functions are presented in this section. These functions are based on penetration geometrics and quasi-steady aerodynamics .

In penetration Geometricseach station along the missile eqeriences the wind crossflow associated with the station altitude.

In the quasi-steady aerodynamics the crossflow velocity is assumed to beeverywhere much smaller thanthe local speed of sound.Thus, the problem is one ofsolving Laplace's equation in each crossflow plane. This approach is justified only in the case of low frequency oscillation of the vehicle at moderate Xach numbers.

We vi11 consider first the penetration normal force, moment and bending moment growth functions due to a unit step wind profile. Assuming that the local speedof sound is essentially infinite with respect to the crossflow velocity,then

(I1-47)

- 90 - Substituting (11-47) into (11-6), (11-7) and (11-8) gives

(I1-49)

MR-0 and

(11-50)

MR-0

Substituting the appropriate definitions of the body geometry into (11-48),(11-49) and (11-50), recalling (11-27)through (11-30) andintegrating the results gives: for the normal force due topenetration only,

where

n = 1,2,3,4 for % s T <- , MR-0 1 n = 1,2,3,4 ; (11-52)

- 91 - for the moment due to penetration only,

(11-53)

where

2Ucnen (T-bn)3 - for d, S T 5 3% n = 1,2,3,4

n = 1,2,3,4 ; (11-54) and for thebending moment due to penetration only,

(~r-55)

where

- 92 - MR+O . MR+o

I n = 1,2,3,4 (11-56)

The penetration normal force, moment andbending moment due to a unit impulse wind profile are obtained from (11-52),(11-53) and (11-55), respectively, upon differentiating with respect to time. Thus, forthe normal force

(11-57) MR-0 MR+O MR-90 MR40 MFi-0 where

I n = 1,2,3,4 ; (11-58)

for t.he moment,

(11-59)

MR+o m-0 MR-0 MR+o MR40

- 93 - where

MR-0 MR-0

p(-r-bn)2 for d, S 7 5 d, , n = 1,2,3,4 -"Iuo I n = 1,2,3,4 ; (I1 -60) and for the bending moments,

(11-61)

m-+o MR40 MR-30 MR -io MR-0 where

2anU2(7-bn)3 for dn 7 Zn - n = 1,2,3,4 + 2q c, U 0 for d, s T < ~1 , n = 1,2,3,4 . (11-62)

- 94 - The equations (11-Sl), (11-53), (IS-55), (11-57), (11-59)and (11-61) are the fundamental expressions required to compute the penetration indicial and impulsiveforce and moments forthe C-5 configuration. The coefficients given in these equations are defined in Table 11. These forces and moments were used in calculating the indicial and impulsive responses of the C-5 missile system for the case of "pure penetration."

D. Development ofthe Steady Indicia1 and Impulsive Aerodynamic Forces for the Saturn C-5 Configuration

This section presents the instantaneous-immersion forcing functions. These functions are based on instantaneous-immersion geometrics and quasi- steadyaerodynamics. In this geometric consideration every station along the missile experiences the same wind-inducedcrossflow, namely that crossflow occurring at the nose.

We will consider first the steady normal force, moment and bending moments due to a unitstep wind profile.Referrin& to (11-6), (11-7),and (11-8), the indicial forces resulting from instantaneous immersion are

L (11-63)

(11-64)

and

L (11-65)

It should be noted that the aerodynamic coefficients in the above expressions are identical to the coefficients multiplying the translational velocity terms in the rigid body and bending equations of motion (see Appendix I

- 95 - ......

(1-1), (1-2) and (1-3)). Thus, (11-63) , (11-64) and (11-65) can be rewritten* as

(11-67)

and

(11-68)

respectively, where I(7) is the Heaviside step function.

The instantaneous-immersion normal force, moment and bending moments due to a unit impulse wind profileare obtained from (11-66),(11-67) and (11-68) upon differentiating with respect to time:

(11-69)

(11-70)

and

(11-71)

* The evaluation of gs(7) , Ks(7) and %m( T)~can also be obtained from

(11-52),(11-54) and (11-56) by setting T equal to infinity.

- 96 - where 6(7) is theDirac delta function. - For T > 0 , Ni( T) , &(T) and T%(T)~are identically equal to zero(see (11-58), (11-60) and (11-62) for T = a ) .

The equations (11-66) through (11-71) are thefundamental expressions required to compute the instantaneous-immersion indicial and impulsive force and moments. These forces were used in calculating the indicial andimpulsive responses of the Saturn C-5 missile system for the case of "instantaneous irrmersion. "

- 97 - APPENDIX III

CALCUIATION OF INITIAL AhTD STEADY-STATE C0NDITI;ONS RESULTING FROM A UNIT IMPULSE AND UNIT STEP WIND PROF'ILE

The numerical solution for the indicial and impulsive responses of the vehicle requires a knarledge of the initial conditions imposed by the various forcing functions (see Appendix I1 ) . Since the Runge Kutta method of solution is used, the initial conditions for the generalized coordinates and their first derivatives are needed.

The steady-state values for the indicial and impulsiveresponses of the vehicle are required for computer logic which terminates the integration when response is sufficiently close to its steady value (see Vol. 111).

A procedure is given in this Appendix for computing the initial and steady-state conditions for the impulsive and indicial responses of the vehicle. For simplicity, the method is described for a system defined by the following generalizedcoordinates (see Appendix I): translation,rotation, first and second bending, first and secondsloshing and controldeflection. The procedure is general in nature, however, and can be applied to a system with mre degrees of freedom.

The initial and steady-state conditions are obtained through usage ofLaplace transform techniques. The initial and steady-stateconditions are fouhd in the limit of the transform as the Laplace variable approaches infinity and zero,respectively. The above conditionsare derived for six cases .of forcingfunctions: unit impulse and stepconsidering penetration with lift growth,pure penetration and instantaneous-immersioneffects.

I. CALCULATION OF INITIAL CONDITIONS

Writingthe equations of motion [see (1-11) through (1-19)) as a set of first order equations, we find

(111-1 )

- 98 - where the elements aij and bi. (i,j = 1,2, ....,18) of the square matrices [All and [B], respectively, corre$pond to the coefficientson the left hand side of (I-=) through (I-ig),

1 (111-2)

and {C] is a column matrix of forcing functions. Six different sets of forcing functions were consideredin the numerical investigation presented in this report.

For conveniencesof comparison the elements,ci , (i = 1,2,. .. . ,lS> Of {C} are given in TableVII for a unit impulse wind profile{vy(t) = 6(t)} and a unit step wind profile{v (t ) = I(t)) considering penetration with lift growth (FVE), pure penetrationY (PP) instantaneous-immersion (11) effects.

Expressions for the aerodynamic quantities in VI1 Table are givenin Appendix 11.

- 99 - 'IIABLE VI1

ELEMDIS OF FORCING FUNC!EON YATRIX {C)

ci Unit Impulse Wind Profile UnitStep Wind Profile i FWLG 'PP I1 mLG PP i1 ------1

0 0 0 @ 0 0

14 0 0 0 0 0 0 15 0 0 0 0 0 0 16 0 0 0 0 0 0 17 0 0 0 0 0 0

- 100 - Now, the initial conditions, (q(O)} , canbe found readily by using Laplacetransform techniques. Taking theLaplace transform of (111-1) yields

a {G} = {E) (111-3) where

{<}= .{q} (111-4) and

[G] = s[A] + [B] (111-5)

- The elements ci (i = 1,2,. ..,l8) of the transformed forcing function matrix (C] are given in Table VI11 for the six different sets of forcing;'func- bCI tions. The symbol @(') denotes"the order of -1 .$1 S S

The initial conditions," (q(o)), are now obtainedby solving (111-3) for each dependent variable by Cramer's rule, multiplying the ratio of two determinantsby s and takingthe limit as s+ cb .

The initial conditions imposed by a unit impulse wind are given in TableIXfor the cases where penetration with lift growth and pure penetration effects are considered.

* If T(S ) is givenas the Laplace transform of f(t) , then the initial condition,f(+o) , is found from

where s is theLaplace transform variable.

- 101 - -1

TABU VIII

ELEMENTS OF THE TRANSEURMED 3bRCING FUNCTCON MATRIX{E} - ci Unit Impulse Unit Step Wind Profile Wind Profile -i FGJLG and PP -I1 PWLG and PP -I1

1 0 0 0 0 2 0 0 0 0 3 0 0 0 0 4 0 0 0 0 5 0 0 0 0

6 0 0 0 0 7 0 0 0 0 8 0 0 0 0

9 0 0 0 0 10

11 e/($)

13 2m2 1

14 0 0 0 0 15 0 0 0 0 16 0 0 0 0 17 0 0 0 0

18 bo 1 bo 1 - " " U U us us

- 102 - INITIAL CONDITIONj IMPOSED BY A UNIT IMPULSE WIND FUR THE CASES OF “ITON WITH LXFT GROWTH AND PURE PENETRATION

P(0) = 0 P( 0) = -bo

The initial conditions imposedby a unit impulse wind for the case of instantaneous immersion are:

Yo@) = 0 N(0) = 0

P(0) = 0

&(o) = 0

i(0) = 0

(111-6)

- 103 - 111 II I I1111 111 II I . I II 111111.1 111 I II I II I . In addition, solutions for yo(.O) , d( 0) , T1( 0) , q2(0) , SI( 0) and &(O) areobtained from the matrixequation

(111-7) where

and e>= 2

(111-8) (111-9 )

The elements, di , of thesquare matrix [D] correspond tothe inertia coefficients in (1-11) through (1-16) (see Appendix I).

Finally, the initial conditions imposed by a unit step wind are given in Table X for thecases where penetration with lift growth, purepenetration and instantaneous-irrsion effects are considered.

- 104 - TABU x

N(0) = 0 i(0) = 0

P(0) = 0 +(O) = 0

11. CALCULATION OF STEADY- STATE VALUES

The procedure for computing the steady-state values of theimpulsive and indicial responses follows the analysis given in Section I except for the following changes :

1. The first ordersystem of equations of motion Eee (111-l)] are writtenin terms of Go and yo instead of yo and Jj, to avoid an inde- terminant form .

2. The steadystate values of theimpdsive responses are the same for thethree aerodynamic environmentsconsidered.* Thesame is true of the steadystate values of the indicialresponses. Thus, the steady state values for the impulsive and indicial responses will be found from the case of instantaneous immersion.

- 105 - 3. The steadystate conditions are obtained when the Laplace transform variable goes to zero in the limit .** -

Omitting the details, the steady state values of the impulsive and indicial responses are given in Tables VI1 and VIII, respectively.

TABLE XI TABU XI1

STEADY STATE VALUES FOR STEADY STATE VALUES FOR IMPULSIVE RESPONSES INDICIAL FGSPONSES .. . Yo(-) = 0 Yo(4 = 1

$(m) = 0 #(m) = 0

* The steadystate values of impulsiveforcing functions, given in Table 111, are the same forthe three aerodynamic environments. The same is truefor the steadystate values of theindicial forcing functions. Since the forcing function vector is the only quantity whichchanges in the set of equationsEee (111-l)] for the different aerodynamic environments, it is easily seen that the above statement is correct. ** If T( s ) is the Laplacetransform of f (t) , then the steady state value, f('m) f('m) , is foundfrom

- 106 - APPENDIX IV

PRESENTATION OF EX"REMf3 EXCURSION AND AVERAGE RESPONSE PLOTS

This Appendix contains the extreme excursion and local average response plots discussed in Section V. The method of analysis used to obtain these plots is discussed in Section IV.

- 107 -

Profile No. 1 50 GeC. F.T. Dand

1ns.tantaneous / hersion ( II)~, //

pure Pcnctration (PP) ant1 Pcnctrat.ion with Lift Growth (PWLG)

_I I I I -3 .O 1.0 2.0 3.0 4 .o Rcduccd CumulaLlvc l'robablllty Flc. 42 - Extreme Excursion of 1st Dcnding vs. Reduced Cwulntivc Probability - 50 sec. P.T.

I I 3.0 -2.0 -1.0 0 1.0 2.0 3.0 8.0 Rcduccd Cumulative ProbahLlity Fig. 43 - Extrcme Excursion of 1st Bending vs. Reduced Cumulative Probability - GO SCC. P.T.

- 109 - No 1.0 rl t Profile No. 1 x 70 sec. F.T. Band

2 rl I

1 I I 1 I 1 -3.0 -2.0 -1.0 0 1.0 2.0 3.0 4.0 Reduced Cumulative Prohability Fig. 44 - Extreme Excursion of 1st Bending vs. Reduced Cumulztive Probability - 70 sec. F.T.

No. 1 F.T. Band

I I I 1 -3.0 -2.0 -1.0 1.0 2.0 3.0 8.0 Reduced Cumulative Probability Fig. 45 - Extreme Excursion of 1st Bending vs. Reduced Cumulative Probability - 80 sec. F.T.

- 110 - -3.0 -2.o -1.0 0 1.0 2.0 3.0 4.0 Reduced Cusulative Probability

Fig. 46 - Extreme Excursion of 1st Bcnding vs. Reduced Cumulative Probability - 90 sec. F.T.

- 111 - X M 4.0- d *rl a Profile No. 1. 2 50 sec. F.T. Band 3.0- (u3 I d 0

I 1 -3.0 -2.0 -1.0 0 1.0 2.0 3.0 4.0 ReducedCumulative Probability

Fig. 47 - Zxtreme Excursion of 2nd Bending vs. Reduced Cumulative Probability - 50 sec. F.T.

Profile No. 1 60 sec. F.T. Band

-3.0 -2.0 -1.0 0 1.0 2.0 3.0 4.0 ReducedCumulative Probability

Fig. 48 - Extreme Excursion of 2nd Bending vs. Reduced Cumulative Probability - GO sec. F.T.

- 112 - K) 0 rl I

Profile No. 1 70 sec. F.T. Band

I I

I I -3.0 -2.0 -1.0 0 1.0 2.0 3.0 4.0 Reduced Cumulative Probability Fig. 49 - Extreme Bcursion of 2ndBending vs. Reduced Cumulative Probability - 70 sec. F.T.

Profile No. 1 (5.0 -

M 0 rl x 5.0 - w .rl a W 4.0 - a cu I C '$ 3.0 - k

-3.0 -2.0 -1.0 0 1.0 2.0 3.0 4.0 Reduced Cumulative Probability Fig. 50 - Extreme Excursion of 2nd Bending x. Reduced Cumulative Probability - 80 sec. F.T.

- 113 - Profile No. 1 90 sec. F.T. Band 5 .O pr) 0 rl x

I I I -3 .O -2 .o -1.0 1.0 2 .o 3 .O 4.0 Reduced Cumulative Probability

Fig. 51 - E,xtreme Excursion of 2ndBending vs . Reduced Cutdative Probability - 90 sec. F.T.

- 114 - 1.0 rl Profile No. 1 sll 60 sec. F.T. Band M

-3.0 -2.0 -1.0 0 1.0 2 .o 3.0 4.0 Reduced Cumulative Probability

Fig. 52 - Ektreme Excursion of 1st Sloshing vs. Reduced Cumulative Probability - 60 sec. F.T.

Profile No. 1 70 sec. F.T. Band x I

Reduced Cumulative Probability

Fig. 53 - Extreme Excursion of 1st Sloshing vs. Reduced Cumulative Probability - 70 sec. F.T.

- 115 - / 1.4}

o'2 t -3 .O -2.0 -1.0 0 1.0 2.0 3.0 4.0 Reduced Cumulative Probability

Fig. 54 - Ektreme Excursion of 1st Sloshing vs. Reduced Cumulative Probability - 80 sec. F.T.

1.2 L

I I I I 1 I 1 I -3.0 -2.0 -1.0 0 1.0 2 .o 9.0 4 .O Reduced Curmlativc Probability Fig. 55 - Extreme Excursion of 1st S!.oshing vs. Reduced Cumulative Probability - 90 sec. F.T.

- 11s - Profile No. 1 60 sec. F.T. Band

X I z 4 0.8 . m4 a / I //

-3 .o -2 .o -1.0 0 1.0 2.0 3 .O 4 .O Reduced Cumulative Probability

Fig. 56 - Extreme Excursion of 2nd Sloshing vs. Reduced Cumulative Probability - GO sec. F.T.

rl 4 Profile No. 1 x 70 sec. F.T. Band

.I I -3.0 -2 .o -1.0 0 1.0 2.0 3.0 4.0 Reduced Cumulative Probability

Fig. 57 - Extreme Excursion of 2nd Sloshing vs. Reduced Cumulative Probability - 70 sec. F.T. - 117 - / 1.4 1

Profile No. 1 BC scc. F.T. Band

-3 .O -2.0 -1.0 0 1.0 2 .o 3.0 4.0 ReducedCumulative Probability

Fig. 5G - Extreme Excursion of 2nd Sloshing vs. Reduced Cumulative Probability - 80 sec. P.T.

Profilc No. 1

d 90 sec. F.T. Band 4 x 1.0 - Note: There is an insufficient E: amount of data to adcquately .A 2 definethese curves d co a 0.0 - cu -.”-” (1113 - - ”- - - - 7” OE 4 (PP and I - s 0.G p1.m ) wE 8 E 0.4 -

0.2 -

I I I I I I I -3.0 -2.0 0 0 1.0 2.0 3.0 4.0 Reduced Cumulative Probability

Fig. 59 - Extreme Excursion of 2nd Sloshing vs. Reduced Cumulative Probability - 90 sec. F.T.

- 11c - 1.2 t Profile No. 1 cu 50 sec. F. T. Band

x 1.0

--3 .O -2 .o Reduced Cumulative Probability Fig. SO - Mreme FXcursion of 1st Control vs. Reduced Cumulative Probability - 50 sec. P.T.

Profile No. 1 (u 0 1.0 60 sec. F.T. Band rl / x rl 0 +J g 0.8 U

rl::

0.2

I I I I -3.0 -2.0 -1.0 1.0 2.0 3.0 4.0 Reduced Cumulative Probability

Fig. 61 - Mreme Excursion of Ist Control vs. Reduced Cumulative Probability - GO sec. F.T.

- 119 - 2.0

1.8 No. 1 F.T. Band

1.6

Profile No. 1 70 5ec. F.T. Band

1.2

I rz,P 0 I

I I I 1 1.0 2.0 3.0 4.0 Cunulative ProbabilityReduccd Cunulative ProbabilityRcduccd Cumulative

FiG. 52 - Ektreme Excursion of 1st Controlvs. Reduced Fig. 63 - Extreme Excursion of 1st Control vs. Reduced Cumulative Probability - 70 sec. F.T. CbulrtiveProbability - 80 sec. F.T. -3 .O -2 .o -1.0 0 1.0 2.0 3.0 4 .O Reduced Cumulative Probability

- 121 - 4.0 Profile No. 2 GO see. F.T. Band

9.5

Frofile IJo. 2 50 scc. F.T. Band

1.(

1 I I I I -2.0 -1.0 0 1.0 2.0 3.0 4.0 Rcduccd Cumulative Probability Rcduced Cunulativc Probability

Fig. 65 - Extreme Excursion of Rotationvs. Reduced Fig. 66 - Extreme Excursion of Rotation vs. Reduced Cumulative Probability - 50 sec. F.T. Cumulative Probability - GO sec. F.T. “c$; 0.8 Profile No. 2 50 sec. F.T. Band /

3 0.4 (PP and t’ PWLGI

-3.0 -2.0 -1.0 0 2.0 1.0 -.o 5.0 6.0 Reduced Cumulative Probability Fig. 67 - Extreme Excursion of 1st Bending vs. Reduced Cumulative Probability - 50 sec. F.T.

Profile No. 2 60 sec. F.T. Band ,/

I C 0 rl “/3Y

-3.0 -2.0 -1.0 0 1.0 2.0 3.0 4.0 5.0 Reduced CumulativeProbability Fig. GO - Extreme Excursion of 1st Bending vs. Reduced Cumulative Probability - 60 sec. F.T.

- 123 - % rl 1.0- X w Profile No. 2 E: .,-I 70 sec. F.T. Band 2 2 0.0- z 74 I

I I I I I I I -3.0 -2 .o -1.0 0 1.0 2.0 3.0 4.0 5.0 6.0 Reduced Cumulative Probability Fig. 69 - Extreme Excursion of 1st Bending vs. Reduced Cumulative Probability - 70 sec. F.T.

- Profile No. 2 00 sec. F.T. €!and

(W and PWIG)

1 I I I I I 1.0 2.0 3.0 4.0 5.0 6.0 Reduced Cumulative Probability Fig. 70 - Extreme Excursion of 1st Bending vs. Reduced Cumulative Probability - 80 sec. F.T. - 124 - Profile No. 2 1. 90 sec. F.T. Band

I I I I 1 f 1.0 2.0 3.0 4.0 5.0 6.0 Reduced Cumulative Probability

Fig. 71 - Extreme Excursion of 1st Bending vs. Reduced Cumulative Probability - 90 sec. F.T.

- 125 - 5.0- M 0 rl x Ml C 2 4.0-

p1E a C cu Profile No. 2 I 3.0- .c 50 sec. F.T. Band 0 .A v1 k 5 2.0-

-3.0 -2.0 -1.0 0 1.0 2.0 3.0 4.0 5.0 6.0 Reduced Cumulative Probability Fig. 72 - Extreme Excursion of 2nd Bending vs. Reduced Cumulative Probability - 50 sec. F.T.

M 0 rl x Profile No. 2 M 60 sec. F.T. Band 4.0 a E p1 a cu 3.0

Reduced Cumulative Probability Fig. 73 - Extreme Excursion of 2nd Bending vs. Reduced Cumulative Probability - 60 sec. F.T.

- 126 - Profile No. 2 70 sec. F.T. Band

-3.0 -2.0 -1.0 0 1.0 2.0 3.0 4.0 5.0 6.0 Reduced Cumulative Probability

Fig. 74 - Extreme Excursion of 2nd Bending vs. Reduced Cumulative Probability - 70 sec. F.T.

Profile No. 2 80 sec. F.T. Band

Reduced Cumulative Probability Fig. 75 - Extreme Excursion of 2nd Bending vs. Reduced Cumulative Probability - 80 sec. F.T.

- 127 - Profile No. 2 90 sec. F.T. Band

-3 -2 -1 0 1 2 3 4 5 Reduced Cumulative Probability

Fig. 76 - Extreme Excursion of 2nd Bending vs. Reduced Cumulative Probability - 90 sec. F.T.

- 128 - Profile No. 2 60 sec. F.T. mnd

1 -3 -2 -1 0 1 2 3 4 5 6 ReducedCumulative Probability

Fig. 77 - Extreme Excursion of 1st Sloshing vs. Reduced Cumulative Probability - GO scc. F.T.

1.0 4 0 rl x bD C Profile No. 2 70 sec. F.T. mna Q-O 30 rl v1 * 0.6 C 0 -4 5 2 0.4 w

-3 -2 -1 0 1 2 3 4 5 6 Reduced Cumulative Probability Pig. 78 - Extreme Excursion of 1st Sloshing vs. Reduced Cumulative Probability - 70 sec. F.T.

- 129 - Profile No. 2 4 80 sec. F.T. Band

L-IVI I I I I I 1 -3 -2 -1 0 1 2 3 4 5 6 Reduced (;‘umulatlve ProbablUty Fig. 79 - Extreme Excursion of 1st Sloshing vs. Reduced (=umulative Probability - 80 sec. F.T.

Profile No. 2 90 sec. F.T. ISnd 0

I I L -3 -2 -1 0 1 2 3 4 5 6 ReducedCumulative Propability Fig. 80 - Extreme Excursion of 1st Sloshing vs. Reduced Cumulative Probability - 90 sec. F.T.

- 130 - -3 -2 -1 0 1 2 3 4 5 6 Reduced Cumulative Probability Fig. 81 - Extreme Excursion of 2nd Sloshing vs. Reduced Cumulative Probability - GO sec. F.T.

Profile No. 2 70 sec. F.T. Band

Reduced Cumulative Probability Fig. 82 - Extreme Excursion of 2nd Sloshing vs. Reduced Cumulative Probability - 70 sec. F.T.

- 131 - -3 -2 -1 0 1 2 3 4 5 6 ReducedCumulative Probability Fig. 03 - Fxtreme Excursion of 2nd Sloshing vs. Reduced Cumulative Probability - 80 sec. F.T.

4 X t 48 0.61 Profile No. 2 0 d 90 sec. F.T. Band cn

Reduced Cumulative Probability Fig. 84 - Extreme Excursion of 2nd Sloshing vs. Reduced Cumulative Probability - 90 sec. F.T. - 132 - cu 4 Profile No. 2 x 1.0 I- 50 sec. F.T. Esnd

u c, 0.8 lo I-I

E 0.4 w / / 0.2

L ___I I 1 1 -3 0 1 2 3 4 5 Reduced Cmlatlve Probability

Fig. 85 - Mreme Excursion of 1st Control vs. Reduced Cumulative iProbability - 50 sec. F.T. cu F: Profile No. 2 60 sec. F.T. Band

-3 -2 -I. 0 1 2 3 -4 5 Reduced Cumulative Probability Fig. 86 - Extreme Excursion of 1st Controlvs. Reduced Cumulative Probability - GO sec. F.T.

- 133 - 1.2t Profile No. 2

N 70 sec. F.T. Band

Reduced Cumulative Probability Fig. 87 - Extreme Excursion of 1st Control vs. Reduced Cumulative Probability - 70 sec. F.T.

-3 -2 -1 0 1 2 3 4 5 Reduced Cumulative Probability Fig. 88 - Extreme Excursion of 1st Control vs. Reduced Cumulative Probability - 80 sec. F.T.

- 134 - Profile No. 2 90 6ec. F.T. Band

-3 -2 -1 0 1 2 3 4 5 Reduced Cumulative Probability

Fig. e9 - Extreme Excursion of 1st Controlvs. Reduced Cumulative Probability - 90 sec. F.T.

- 135 - - - - - Instantaneous Immersion (11) - - -he Penetration (PP) -Penetration With Lift Growth (PWLG)

Profile No. 1

Fig. 90 - Average Response of 1st Bending vs. Flight Time

50 60 70 80 90 Flight Time (sec.) Fig. 91 - Average Response of 2nd Bending vs. Flight Time - 136 - Flight Time (sec.) 60 70 80 90 I 1 I 1

4 -1.0 - x .rl2 4 -2.0 - 0 r4 rJl c, ""

rl "-(PPI t -3.0 - 0 I Profile No. 1 0 0 a -4.0 - B

4P -5 .o -

Fig. 92 - AverageResponse of 1st Sloshingvs. Flight Time

Flight Time (sec .) 50 60 70 80 90 0 I 1 I 1

-1.0 0"a rl x

? -2.0 5 r4 rn a PJ -3.0 I

0) C P 0) -4.0 a 9 F 4 -5.0

Fig. 93 - Average Response of 2nd Sloshing vs. Flight Time

- 137 - I I I I 50 60 70 80 90 Flight Time ( sec .)

Fig. 94 - Average Response of 1st Control vs. Flight Time

- 138 - Flight Time (sec.) 50 60 IO 80 90 0 I I I t

*) -1 ,o x

Fig. 95 - AverageResponse of 1st Bending vs. Flight Time

Flight Time (sec.) 60 70 80 90 0 I I I I

-0.2

Profile No. 2 rr) 4 -0.4

Fig. 96 - AverageResponse of 2nd Bending vs. Flight Time - 139 - Fig. 97 - AverageResponse of 1st Sloshing vs. Flight Time

Flight Time (sec.) Fig. 98 - AverageResponse of 2nd Sloshing vs. Flight Time - 140 - Fllght Time ( sec .) 60 70 80 90 05* I I I I

"- -( 11) "4pp 1 -1 - (pm> K rl Profile No. 2 R c, " -2- 2 I4

Q) ul IY 0 - 3 -3 a" 0 i?

NASA-Langley, 1965 CR-326 - 141 -