TECH LIBRARY KAFB, NM

NASA CR-1540 ODb084b ~~ 4. Title and Subtitle ADVANCEMENTS lN STRUCTURAL DYNAMIC 5. Report Date June 1970 TECHNOLOGY RESULTING FROM SATURN V PROGRAMS - 6.' Performing Organization &de VOLUME II 7. Author(s) 8. Performing Organization Report No. P. J. Grimes, L. D. McTigua, G. F. Riley, and D. I. Tilden D5-17015 10. Work Unit No. 9. Performing Organization Name adAddress 124-08-13-04-23

The Boeing Company 11. Contract or Grant No. Huntsville, Alabama NAS1-8531 13. Type of Report and Period Covered 2. Sponsoring Agency Name and Address Contractor Report National Aeronautics and mace Administration 14. Sponsoring Agency Code Washington, D.C. 20546 5. SupplementaryNotes

6. Abstract

Saturn V structural dynamic experiencein replica modeling, math modeling and dynamic testing was assessed. Major problems encountered in each of these are= and their solutions are discussed. The material is presented in two volumes. Volume I (NASA CR-1539) contains a summary of the material presented inVolume II and is oriented toward the program managers of future structural dynamic programs. Volume 11 contains the methods and procedures used in the Apollo Saturn V structural dynamics programs. The major problems encounteredand their solutions are discussed. Volume E is oriented toward the technicalleaders of future structural dynamics programs.

7. Key Words (Suggested by Author(s)) 18. DistributionStatement Saturn V structural dynamics 1/10 scale Saturn V replicamodels Unclassified - unlimited Saturn V math models Dynamic testing and data reduction 3. Security Classif. (ofreport) this 20. Security Classif. (of this page) 21. No. of Pages 22. Price* unciasszied Unclassified 163 Unclassified unciasszied $3.0

*For sale by the Clearinghouse for Federal Scientific and Technical Information Springfield, Virginia 22151 PREFACE

This documentwas producedunder NASA-Langley Research Center Con- tract NASl-8531.The contentsare intended to carry forward to future programs thestructural dynamics experience gained during theSaturn V programs.

The valuablecontributions of S. A. Leadbetter, H. W. Leonard, and L. D. Pinson, Structural VibrationSection, LangleyResearch Center, are gratefully acknowledged. The major contributors to this document include: M. L. Biggart P. J. Grimes J. E. Henry N. L. Hudson W. H. Lawler L. D. McTigue G. F. Riley D. I. Tilden Questions on thecontents of this document should be addressed to L. D. McTigue,The Boeing Company, P. 0. Box 1680, Mail Stop AG-36, Hunts- vi 11 e ¶ A1 abama 35807.

iii

PARAGRAPH PAGE PREFACE iii CONTENTS V ILLUSTRATIONS AND TABLES viii LIST Ut SYMBOLS ix ABBREVIATIONS xi SECTION 1 - SUMMARYSTRUCTURAL DYNAMIC TECHNOLOGY DEVELOPMENT 1 1 .o INTRODUCTION 1 1.1 DEFINITION OF SATURN V 1 1.2 PROGRAM DESCRIPTION AND HISTORY 3 1.3 SUMMARY OF 1/10 SCALE MODEL TECHNOLOGY 5 1.3.1 Introduction 5 1.3.2 Description and History 5 1.3.3 Scale Model Costand Accuracy 8 l .3.4 1/IO Scale Model Contributions and Areas of Improvement 8 1.4 SWMARY OF MATHEMATICAL MODEL TECHNOLOGY 12 1.4.1 Introduction 12 1.4.2 Technical Approach 15 1.4.3 MathModel Analysis Guidel ines 17 1.4.4 Math Mode1 Costand Accuracy 28 1.5 SUMMARY OF DYNAMIC TEST TECHNOLOGY 30 1.5.1 Introduction 30 1.5.2 Dynamic Test Program Guidel ines 36 -1 .6 CONCLUSIONS 43 REFERENCES 44

CONTENTS OF VOLUME I1 PREFACE iii CONTENTS V ILLUSTRATIONS AND TABLES vi i.i LIST OF SYMBOLS xi ABBREVIATIONS xiii SECTION 2 - INTRODUCTION 1 2.0 GENERAL 1 2.1 DESCRIPTION OF SATURN V VEHICLE 2 2.2 HISTORYOF SATURNv STRUCTURAL DYNAMIC PROGRAMS 5 2.2.1 1/10Scale Mode1 Program 6

V CONTENTS OFVOLUME I1 (Continued)

PARAGRAPH PAGE 2.2.2 FullScale Math Modeling 7 2.2.3 FullScale Dynamic Test and Correlation 10 2.2.4 Saturn V Flight 11 REFERENCES 12 SECTION 3 - 1/10 SCALEMODEL TECHNOLOGY 13 3.0 GENERAL 13 3.1 THE SCALEMODEL TEST PROGRAM 14 3.1.1 Scale Model Description 14 3.1.2 Scale Model Test 18 3.2 CONTRIBUTIONS TO FULLSCALE TEST 18 3.3 CONTRIBUTIONS TO SATURN V MATH MODELING 20 3.3.1 MathematicalAnalysis of the 1/10 Scale Model 21 3.3.2 Scale Model Test and Analysis Correlation 21 3.4 CONTRIBUTIONS TO SATURN V ANOMALYRESOLUTION 24 3.5 COST AND ACCURACY 24 3.5.1 cost 24 3.5.2 Accuracy 24 3.5.3 Scale Model Joint Flexibility 31 REFERENCES 33 SECTION 4 - MATH MODELTECHNOLOGY 35 4.0 GENERAL 35 4.1 TECHNICAL APPROACH 35 4.1.1 Stiffness Analysis 37 4.1.2 Inertia Analysis 40 4.1.3 Eigenvalue Solution 40 4.2 MODELINGPHILOSOPHY 40 4.3 ST1FFNESS MATRIX DEVELOPMENT 43 4.3.1 General Gui del ines 43 4.3.2 Ideal ization Exampl es 57 4.3.3 Shel 1 Ideal izati ons 59 4.3.4 Major Component Idealization 67 4.4 INERTIA MATRIX DEVELOPMENT 71 4.4.1 GeneralGuidelines 71 4.4.2 Inertia Examples 72 4.4.3 Shel 1 Inertia Matrices 72 4.4.4 Propellant Tank Inertia Matrices 75 4.4.5 Rigid Subsection Inertia Matrix 81 4.4.6 'Major Component Inertia Matrices 85

vi CONTENTS OF VOLUME II (Continued)

I PARAGRAPH 4.5 VIBRATIONANALYSIS AND MODAL SYNTHESIS 4.5.1 General 4.5.2 EigenfunctionSolutions and Modal Orthogonality 4.5.3 Modal Synthesis 4.5.4 Evaluation 4.5.5 Establ ish Tolerance 4.5.6 Damping Considerations 4.6 SATURN V MODEL EVOLUTION 4.7 COST AND ACCURACY 4.7.1 cost 4.7.2 Accuracy REFERENCES 107 SECTION 5 - DYNAMIC TEST TECHNOLOGY 109 5.0 GENERAL 5.1 TESTREQUIREMENTS 109 5.1.1 Test Objectives 111 5.1.2 Vehicle Configuration 113 5.1.3 Test Facilities Requirements 116 5.1.4 DataAcquisition and Reduction System 124 5.1.5 Test Conduct 131 5.2 DIGITAL DATA REDUCTION TECHNIQUES 133 5.2.1 Fourier Analysis 136 5.2.2 Point Transfer Functions 7 37 5.2.3 TransferFunction Equations 137 5.2.4 Computation of Modal Parameters 5.3 TEST DATA EVALUATIONPROCEDURES 146 5.3.1 On-Si te Data Eval uation 146 5.3.2 Test Data Val i dation 147 5.3.3 Test Data Eval uati on 148 5.3.4 Test Data Reporting 148 REFERENCES SECTION 6 - CONCLUSIONS 1 51

vi i ILLUSTRATIONS

Apol lo Saturn V Configuration MathModel Evolution Schematic of 1/10 Scale Apol lo Saturn V Model 1/10Scale Model in Test Stand Illustrationof Detail Achieved in Modeling Comparison of1/10 Scale Model and Full Scale- Vehicle Bendin Stiffness Comparison of 9 /IO Scale Longitudinal Test and AnalysisResults - 100Percent Propellant Comparison of1/10 Scale Pitch Test and Analysis Results - 100 Percent Propellant Comparison of1/10 Scale and Full Scale Pitch Test Results - Modes 1 and 2 - 100 Percent Propellant Comparison of1/10 Scale and Full Scale Pitch Test Results - Modes 3 and 4 - 100 Percent Propel 1 ant Comparison of 1/10 Scale and Full Scale Longitudinal Test Results - Modes 1 and 2 - 100 Percent Propellant Comparison of1/10 Scale and Full ScaleLongitudinal Test Results - Modes 3 and 4 - 100 Percent Propellant Load Path Through 1/10Scale Model and Full Scale Joints TypicalSaturn V NodalBreakdown MathModel Structural Elements Transition Technique - Large to Small Plate Elements RedundantDegree of FreedomExample Structural Symnetry Example Quarter She1 1 Analysis Coordinate System Beam Elements for Ring Model ing Merge-Reduce Error Accumulation Classesof Modules for Saturn V Models S-IVB Forward Skirt NodalBreakdown ShortStack NodalBreakdown Instrument Unit Model Honeycomb Geometry Bul khead/Cabl e Analogy S-IC Thrust Structure S-I I Thrust Structure S-IVB Thrust Structure LM AsymmetryExample Influenceof Major .Componentson Vehicle Dynamics Service Module Tanks Ring Shape of the 181th Order (Base1 ine Model Ring Shape of the 78thOrder (Guyan Consistent Mass Model ) Ring Shape of the 78thOrder (RelumpedMass Model Propellant Tank LiquidIdealization Liquid Motion Due to Tank Expansion Geometry of Deformed Bulkhead

vii i ILLUSTRATIONS (Continued) FIGURE

4-27 Beam-She1 1 Interface 4-28 Warped Section of Cylindrical Model 4-29 S-IVB Engine Model 4-30 Control Gyro Location 4-31 Illustration of Modal StackingTechnique 4-32 MathModel Evolution 4-33 Comparison of Full ScalePitch Test and Analysis Results - Modes 1 and 2 - 100 Percent Propellant 4-34 Comparison of Full Scale Pitch Test and Analysis Results - Modes 3 and 4 - 100 Percent Propellant 4-35 Comparison of FullScale Longitudinal Test and Analysis Results - Modes 1 and 2 - 100 Percent Propel lant 4-36 Comparison of FullScale Longitudinal Test and Analysis Results - Modes. 3 and 4 - 100 Percent Propellant 4-37 Longitudinal Frequency Response of Outboard Gimbal - 100 Percent Propel 1ant 5-1 Test-AnalysisPhilosophy 5-2 Hydrodynamic Support 5-3 Statjil ization System 5-4 Full ScaleData Acquisition andReduction System Signal Train 5 -5 Full ScaleAccelerometer Data Error 5 -6 Full Scale Test Data Reduction Flow Chart 5-7 Typical Curve Fit of Full Scale Test Data 5 -8 Typical Curve Fit of Double PeakResponse 5-9 Typical Effect of Force Level on FrequencyResponse

TABLES TABLE 3-1 Correlation of 1/10Scale Model and Full Scale ResponseData 4- I Assessment of Saturn V Models 4-11 MathModel Development P1 anning Estimate 5- I Saturn V SensitiveParameters

ix

LIST OF SYMBOLS

A Area (Cross Sectional ) C Damping Coef f i ci en t E Modul us of El asti ci ty e Error F Force G Gain or Transfer Function GT Gain To1 erance

9 GravitationalConstarit h Height I Moment of Inertia [I1 Identity Matrix j 6 K Stiffness Coefficient L Length

M, m Mass Coefficient - m General ized Mass

q Genera7 i zed Coordinate Rx Rotationabout X Axis

RY Rotationabout Y Axis

RZ Rotationabout 2 Axis r Radius S Prestress Force for Bulkhead [TI CoordinateTransformation Matrix t . Time

xi Volume Work Longitudinal Axis, CartesianCoordinates Mass DistributionMatrix Pitch Axis , Cartesian Coordinates Yaw Axis, CartesianCoordinates Deflection,Displacement Displacement in RadialDirection Displacement in X Direction Displacement in Y Direction Displacement in Z Direction Virtual(Prefix used with various symbols) Damping Factor AngularMeasurement, Spherical Coordinates Dens i ty DimensionlessRadial Coordinate Thickness Matrix of Mode Shapes Mode Shape Function Matrix of Rigid Body DisplacementVectors Frequency ABBREVIATIONS

AS-SON Nth ApolloSaturn V space vehicle CM Comnand Module dB Decibel DO F Degree of Freedom HZ Hertz, cycles per second IU Instrument Unit L ES Launch Escape Sys tern Liquid hydrogen # Lunar Module lox Liqui d oxygen 1RC LangleyResearch Center MSFC MarshallSpace F1 ight Center NASA NationalAeronautics andSpace Administration Pogo A divergent longitudinal osci 1 1 ation produced by regenerative coupling between the vehicle structure and propulsionsystem RP-1 A kerosene-1 ike fuel used in the S-IC stage s- IC First BoostStage of Saturn \I Vehicle s-I I Second BoostStage of Saturn V Vehicle S-IVB ThirdBoost Stage of Saturn V Vehicle SLA Saturn LM Adapter SM Service Module

xiii ADVANCEMENTS IN STRUCTURALDYNAMIC TECHNOLOGY RESULTING FROM SATURN V PROGRAMS ~- By P. J. Grimes, L. D. McTigue, G. F. Riley, and D. I.Tilden The Boeing Company

SECTION 2 INTRODUCTION

2.0 GENERAL The ApolloSaturn V Program was created by NASA toaccomplish the objectiveof a manned 1unar landing by the end of the decade. On July 20, 1969, the program objective was realized. This reportcovers the struc- tural dynamic technologythat was developed tosupport the ApolloSaturn V Program. ApolloSaturn V structural dynamicsprograms considered in this document include the 1/10scale model analysis and test program, the full scale analysis and test program,and the flight data evaluation program. The results of these programs were reviewedto the extent necessary to establish anddocument the following: 1. Technicalcontributions of the 1/10scale model to Apollo Saturn V structural dynamic characteristics prediction. 2. Illustrationsof what scale modelingcan contribute to future programs. 3. Proceduresfor performing stiffness, inertia and vibration analyses of 1arge booster vehicles. 4. Improvements in test techniques and datareduction procedures evolvedduring the experimentalstudies. 5. Problems in mathematicalmodeling and dynamic testing' that require further study. Practicalguidelines for accomplishing structural dynamic analy- sis, dynamic test, and datareduction that were established within the successful Saturn V Program are defined in this document. These guide- 1 ines andrecomnended practices are presented so that major pitfalls and problemsencountered in this programcan be avoided in future programs. The followingprocedure was used to generate the Saturn V flight predictions : 1.Develop mathematical models and methods of solution. lines for thefull scale program. 3. Perform pretestanalysis using baseline math models and pub- 1 ish dynamic characteris tics of the test article prior to the test. 4. Perform fullscale tests and compare results with analytical predictions . 5. Revise the mathematical models as required to makethem correlate with test data. 6. Modify these test-verified math models to representthe exact flight configuration of each flig,ht vehicle. 2.1 DESCRIPTION OF APOLLO SATURN V VEHICLE

The Apollo Saturn V vehicleconsists of thethree stage Saturn V launch vehicle, an instrument unit and the Apollo spacecraft. Schematics of thethree boost configurations and thecoordinate systemused throuqhout thisreport are shown in Figure 2-1. There arethree stages of 1 aunch vehicle powered flight. The first stage boost configurationconsists of the total Apollo Saturn V vehicle. The second stage boost configuration consists of the S-I1 stage, S-IVB stage, IU, and the Apollo spacecraft. The thirdstage boost configurationconsists of the S-IVB stage, IU, and the Apol 1o spacecraft. S-IC Stage - The first stage (S-IC) of the Saturn V launch vehicle has a nominal diameter of396 inches (10.06 m). It has a liquid oxygen (LOX)/kerosene (RP-1) propulsion system and is poweredby five F-1 engines with.atotal thrust of 7.5 million pounds (33,360,000 N). The fuel and oxidizerare in separate pressurized tanks, the LOX tank being forward. The tanks arejoined by an unpressurized intertankstructure. Internal construction,material, and fabrication of the two tanks aresimilar. The cy1 indrical portion of each tank is made up of four quarter-sections joined by longitudinal welds. Integrally milledtee-section stringers, located on theinterior surface, provide additionalstructural rigidity. A1 1 bulkheads are el 1 ipti cal in shape and are constructed from gores welded together. The bulkheads are joined to thecylindrical tank and skirt sections through a Y-section ring welded to theequator of each bulkhead. Ring-type sloshbaffles are fusion welded to theinternal stringers in each tank, and cruciform baffles are located in eachlower bulkhead. Five insulatedtunnels lead through thefuel tank to permit passage of suctionducts which supply LOX to the engines.

2 STA 4240 (107.7 M) LES 26 IN. DIA. A (0.66 M) STA 3890 (98.8 STA 3840 (97.5 154 IN. STA 3594 (91.3 (3.9

STA 3258 (82.8 STA 3222 (81.8 260 IN ,.“.\ (6.6 M)

STA 2519 (64.0 M) Y THIRD STAGEBOOST CONFIGURATION

‘396IN. DIA. (10.1 M)

(39.7 STAGEBOOST CONFIGURATION PITCH PLME I

YAW PLANE

ENGINEARRANGEMENT -115 (BASEVIEW LOOKING FWD) FIRST STAGEBOOST CONFIGURATION FIGURE 2-1 APOLLO SATURN V CONFIGURATIONS 2.1 ( Con ti nued) The four outboard engines areattached to thrust postslocated on theperiphery of the aft skirt section. A cruciform beam supportsthe centerengine. Four holddown posts provide anchor points for mounting thevehicle to thelauncher. Aerodynamic fairings and stabilizingfins are provided at each outboard engine location. With the exception of theintertank structure, which is of corrugated skin-ring frame construc- tion, all unpressurized skirts and fairings have extruded hat-section stringersriveted to theexternal surface.

S-I1 Stage - The second stage (S-11) of the Saturn V vehicle has the same diameteras the first stage. The two stagesare joined by a series of skin-stringer type shells with hat-shaped stringers riveted to the externalsurface. The S-I1propulsion system consists of five 5-2 engines burning liquid hydrogen (LH2) fuel with liquid oxygen (LOX) as the oxi- di zer and having a .total thrust of about 1 mi 11ion pounds (4,448,000 rj) . The fuel tank is forward and theoxidizer tank is aft. An insulated common bulkhead separatesthe two pressurevessels. The cylindrical portion of thefuel tank has integral circum- ferential and longitudinalstiffeners which are machine milled on the insidesurface to form a rectangular grid pattern. Five longitudinal and four circumferentialfusion welds areutilized to assemble the cylinder. The upper fuel tank bulkhead and lower LOX tank bulkhead are fabricated from gores which arefusion welded together to form elliptical diaphragms. Thecommon bulkhead is a sandwich structure consisting of gores,fusion welded to an elliptical shape and bonded to a fiberglas honeycomb core. The thrust structure consists of a truncated cone with hat-shaped stringersriveted along theexternal structure. Thrust longerons, at the four outboard engine locations,transmit the engine force. The center engine is mounted at thecenter of a cruciform beam. All un- pressurizedshell structures are skin-stringer types with extruded hat-shaped stringers riveted to theskin. S-IVB Stage - The third stage (S-IVB) of theSaturn, V vehicle has a liquid hydrogen-liquid oxygen propulsion system utilizing a single 5-2 engine,located on thestage center line, with 200,000-pound (889,600 N) thrust capabil ity. This stage has a nominal diameter of 260 inches (6.60 m) . The oxidizer tank is located aft of thefuel tank. An insulated common bulkhead separatesthe two tanks. A squarewaffle pattern having a 45 degree orientation to thevehicle longitudinal axis is machine milled on theinner surface of thefuel tank cylindricalsection. Longitudinal fusion welds are used to jointhe six sheets forming this cylinder. Construction of the hemispherical bulkheads follows thepattern described for the S-I1 stage. The unpressurized structurefore and aft of the tankage, including the conical interstage, is of skin-stringerconstruction with extruded hat-shaped stringers riveted to theoutside of theskin.

4 2.1 (Con ti nued)

Instrument Unit - The instrument unit (IU) is a shortcylindrical structure having anominal diameter of 260 inches(6.60 m) . Structurally the instrument unit is asandwich shell consisting of aluminum facesheets bonded to analuminum honeycomb core. Instrument packages are mounted to the innerwalls of the structure. Apollo Spacecraft - The Apollo spacecraft is composed of five sub- structures: Lunar Module (LM), Saturn Lunar Module Adapter (SLA), Service Module (Sb!) , ComnandModule (M), andLaunch EscapeSystem (LES).

1. Lunar Module - The LM is a two-stage,soft-landing spacecraft which carries two astronaunts to the 1unar surface from 1unar orbit and subsequentlyreturns these two men to arendezvous with the orbiting CM. In the launch configurationthe LM is attached at four points inside the LM adapter cone. 2. Saturn Lunar Module Adapter - The SLA structure is a conical frustum of aluminum face andhoneycomb core sandwich material with diameters of 154and 260 inches (3.91 and6.60 m) at the fore and aft ends , respectively, The adapter also serves as an interstage structure between the Apollo Saturn V IU and the SM. 3. Service Module - The SM is ana1 uminum honeycomb she1 1 with internalradlal shear web partitions. The SM hasa nominal diameter of 154 inches (3.91 m) . Equipment on this part of the spacecraft supplies the power for mi dcourse corrections, retro-braking into lunar orbit, and return flight propulsion.

4. Comnand Module - The CM is a conical frustum fabricated of steel face-and-core honeycomb with interior accomnodations and instru- mentation for three astronauts. From this section of the space- craft, the crew monitors and controls a1 1 functions throughout launch, translunar flight, lunar orbit, return flight and re-entry. 5. Launch Escape System - The LES consists of a titanium open truss tower supporting a launch escape motor. The motor is a sol id propellant device with a steel case and having a nominal diameter of 26 inches (7.92 m). The tower attaches to the top of the CM cone and is jettisoned 32 seconds after second stage ignition.

2.2 HISTORY OF SATURN V STRUCTURAL DYNAMIC PROGRAp1S The Apol lo Saturn V structural dynamics activity can be sub-divided into four major phases: the 1/10 scaleanalysis and test activity, full scale math modeling, full scale test activity, and actualSaturn V flight. These four phases are discussed in the paragraphs that follow. 2.2.1 1/10 Scale ModelProgram With the commitment to design and fabricate a full scale test and to fly the 1argest space vehicle ever conceived, came the recogni tion that the rep1 ica model testing concept could pi lot the program and resol ve technical problems before they became insurmountable from both a cost and schedule standpoint. A 1/10 Scale Model Program was established to support the Apollo Saturn V Program and to accomplish several important research objectives. These researchobjectives were not directly concerned with the Apol lo Saturn V Program and wi 11 not be discussed in this document.

A 1/10 scale replica model of the Saturn V vehicle was completed nearly 18 months in advance of fullscale Saturn V hardware. A dynamic test program of thescale model was established to provide guidelines for performing the fullscale dynamic test program. As a secondary objective,the program was eFpected to indicatepossible loads, or dynamicsproblems that might be inherent in thefull scale design so that these problems could be resolvedbefore the first Saturn V vehicle was completed. A more fundamental long-range objective of thescale model pro- gram was to supplement the full scale test program in verification of the Saturn V structural dynamic math models. A mathematical model ing program was out1 ined between Langley Research Center (LRC) , Marshal 1 Space Flight Center (MSFC), and TheBoeing Company whereby the Sam basicanalysis models and methodswould be used for both the 1/10 scale and the full scaletest articles. The results of both programs would beused to either verify the math models or to identify areas where the modelswere inadequate and structuralidealization or analysis techniquesrequired updating. Even though the schedule was tight, portions of thescale model test and analysis program preceded the full scale program enough to allow for the more important influences on thefull scale pretest. analysis and the test program. The scale model test data showed that theliquid and structural coupling was not modeled adequately; that thetruncated cones used to make vehicle diameter transitions produced a longitudinal and bending stiffness characteristic which was not being adequately modeled; that structural joint modeling is sensitive to axial loading (i.e. , g-levels, mass, etc.); and that the ring mode activity, assuspected, was an important phenomenon for largeshell structure. Data from thescale model tests were also use- ful in defining and assessinginstrumentation and thruster requirements s for the full scale program. Math modelswere reviewed and improvementsmade in the stiffness representation. Theseimprovements were developed in time to usethem in thepretest analysis of the fullscale dynamic test vehicle. Theremod- eling of theliquid and structural coupling and the cone area that resulted from correlation of the 1/10 scaletest and analysisresults prevented a schedule impact resulting from math modelingproblems during the full scale test program.

6 2.2.1 (Continued)

The scale modeland full scale test programswere well coordinated. For example, the same type ballast simulants and the same liquid fill con- ditions were being studied in both programs. Thesame general thruster locations and the same tank pressures were also used. The intent was to establishcorrelation between thescale model test results and the full scaletests. Conducting thescale model test program in advance of the full scale program answered many technicalquestions that might otherwise have impacted the much costlier full scale program.

2.2.2 FullScale Math Modeling Thedynamics of the flight article differed from the dynamics of the ground testarticle in severalsignificant ways.Hardware substitu- tions had to bemade inthe test article because of the cost of theactual hardware, availability of hardwarefrom vendors, and because theschedule requiredresults well in advanceof flight. The test suspension system did not duplicatethe free-free conditions of flight, and the cryogenic propel 1ants of the flight vehicle had to be replaced with less hazardous simulants for ground test. As aresult, ground test data could not be used directly for flight assessment. Themath model was used for projecting the ground test results tothe flight vehicles. On the Saturn V Program, dynamic testresults wereused primarily to verify mathmodels of the test article. After verification, differences between the test and flight articles were modeled,and the resulting modelswere used to predict flight charac- teristics with a high level of confidence. The frequencies of the bending modes that were detected in flight of the first stage boost configuration were predicted within five percentaccuracy. The frequencies of thedetec- ti blelongitudinal modeswere predicted within threepercent accuracy. The correlation of the first stage flight data and analytical predictions are presented in Reference 2-1. Mathematical analyses of the Saturn V vehicle started with the developmentof basic beam-rodmodels to answer questions in support of dynamic test requirements. These earlyuniaxial modelswere used' to I obtain answers to such questionsas: what is the effect of replacing LOX with water as a test simul ant?, what are the effects of replacing a ~ flight article component with a dynamic simulator?, what size of thruster is required to excite the vehicle to readable levels?, andwhere should the vehicle be instrumented in order to obtain accurate mode shape char- acteristi cs? These early mathmodel s werea1 so used to he1 p establish the regions wheremodeling of 1oca1 she1 1 characteristics was required SO that more detailed idealizations of these sections could bemade in I subsequent mathematical models. A schematic of mathmodel development ~ history is shown in Figure 2-2.

7 - 400 DOF ' 300 DOF

APOLLOSATURN V BEAM-ROD BEM-ROD/QUARTER QUARTER SHELL THREE-UIMENSIONA4 SPACE VEHICLE MOLtEL SHELL t4QML MODEL MOD'EL

FIGURE 2-2 MATH MOlELEVOLUTION 2.2.2 (Continued)

The next step in the math model evolution was to add detailed quarter-shellsections to the basicuniaxial model inregions where local effects were consideredimportant. The primaryobjective of these math models was to predict accurately flight control gain factors between a gimbaledengine and a flightcontrol sensor response. This required shell models of the thrust structure regionsand the flight controlsensor areas. When the dynamic test program was initiated, the forward skirt of the S-IC stage, and the aft skirt of the S-IVB stage were bothpotential locationsfor the flightcontrol sensors, in additionto the primaryloca- tion in the IU. All locations were designedto accomnodate three axis sensors, with one axis always oriented normal to the skin, As a result, she1 1 models were required to represent local out-of-pl ane bending in these regions. The combinationbeam-rod/quqrter-shell models were used in the analysisof the 1/10scale model,

Computer size l imitations at that time (up to 130 degrees of free- dom could be handled in the eigenvalueroutine) required the development ofseparate math models toemphasize detail in the S-IC thrust structure area, the S-IC forward skirt, the S-IWB aft skirt and the Ill. For example, if accurate transfer functions from the S-IC thrust structure to the S-IVB aft skirt were desi red, then oneparticular model would be used. If trans- fer functions between the S-IC thrust structure and the IU were desired, another math model would be used. This approach resulted in slightly dif- ferent modal parameters being predicted by each of the separate models, and required a decision as to which solutionshould be used. To prevent this and still have the abi 1i ty to predict the 7 oca1 anomalies that might become evident from test or from flight, a single shell model was developed. This model included a quarter-shell represen- tation of the total launch vehicle and SLA with a uni axi a1model of the service/comnandmodules and the launchescape tower. Knowledge gained from correlationof 1/10 scale model analysis and test results was included in this mathmodel. The size problem in analyzing this model was resolved by a processof modal stacking. In this approach,cantilevered modes of the S-IVB stage and spacecraft were obtained. These were then used in the analysesof the second flight stage, which consisted of the S-I1stage, the S-IVB stage, andspacecraft. Then cantilevered modes from this com- bination, in turn, were used toanalyze the total vehicle. The modal stacking approach required additionalflow time, but proved to be an accurateand economical means of analyzing the individual configurations and in predicting the characteristics of the dynamic test vehicle. Results from this approach were valuable in monitoring dynamic test results and in evaluating their validity. The math model provedhighly accurate in predicting overall modal propertiesof the dynamic test vehicle. There were, however, severalareas I where the math model provedinadequate. First, the model did not predict

9 2.2.2 (Continued) localslopes of the flight control sensors with the desired degree of accu- racy. Second, asymmetries in thespacecraft introduced coupling between pitch, yaw, longitudinal and torsionalplanes. Third, interplane coupling provedto be evenmore significant in the flight vehicles than in the test article. Fourth, the quarter she1 1 model did not have the capabi 1i ty of predicting this coup1 ing. To eliminatethese problems a three-dimensional model of the total vehicle was developed. Inparallel with this, a computerprogram was devel- oped that had the capabi 1i ty of analyzing sys tems starting with 12000 stiff- nessdegrees of freedom and reducingto 300 dynamicdegrees of freedom. Dynamic test programexperience showed thatthe bending and longitudinal propertiesof the launch vehicle stages could be represented by beam and axisymmetricshell elements, respectively, for frequencies up to 25 Hz. Thisallowed the three-dimensional characterjstics of the Apollo Saturn V vehicleto be representedwith only 300 dynamicdegrees of freedom. This simplified full shell math model was used in Pogo, loads, and flight control predictions ona vehicle-by-vehiclebasis to support the Saturn V Program.

2.2.3 FullScale Dynamic Test and Correlation

Resultsfrom the dynamic tests not only proved invaluable in es- tab1 ishing verified mathmodels but the converse wasa1 so true. The pretest analysis results were thebasic tool used for on-site evaluation and verification of dynamic test data.

The dynamic test program contributedseveral hardware changes to theSaturn V vehicle.First, the test results showed that a longitudinal and lateralcoupling mechanism existedin the CM and SM interface.This mechanism, coupled with the stiffness asymmetryand weak torsional stiff- ness of the CM and SM interfaceproduced large torsional responses of the CM. Acting upon testresults, the spacecraft contractor modified this interfaceto eliminate the torsional weakness.Second, the dynamic tests proved that the location of the flight control sensors was not satisfactory. The flight control package, which was located at the top of a plate in the IU, was susceptible to strong local effects involving bending of the plate itself, as well as shell deformation introduced by spacecraft dynamics , particularlythe dynamics ofthe LM. To correctthis, the flight control package was relocated to the bottom of the plate where local effects were much less pronounced. The dynamic test of this new configuration was then run to provide data for establ ishing verified mathmodels. The local characteristics of the plate and bracketry, including the stiffening effect of the components attachedto the plate, had to be modeled in detailto predict flight control sensor rotations with required accuracy. In addition to the relocation of the flight control sensor, some minor revisions in the flight control filter network were necessary to ensure stability in the secondand thirdvehicle bending modes during first stage boost .

10 2.2.4 Saturn V Flight The base1 ine math model, which was verified by both replica model and fullscale test, was the principaltool used toresolve Saturn V structural dynamic problems and to provide NASA management assurance that each flightvehicle was flight worthy. In March 1968, it was decided to expand the model tocouple in thethree dimensional dynamics of the spacecraft components to improve the accuracy of loadscalculations. At the start of the program it was consideredadequate to ignore crossaxis coupling and verifythe math models up to 10 Hz in frequency.These assumptions were proven incorrect by full scale test and eventsoccurring on Saturn V flights. On the second flight in April1968, the firststage exhibited a Pogo instability in a five Hz, first vehiclelongitudinal mode. Becauseof stiffness asymnetry in the LM, this longitudinal modewas stronglycoupled with a bending mode in the same frequencyrange. The combined longitudinal/pitchenvironment raised concern for the integrity of the structure and the comfortof the crew. On the third and fourth flights, strong Pogo osci 1lations developed in the S-I1 stage in an 18 Hz crossbeam mode. During the fifth flight possible Pogo occurred on the S-IVB stage. The S-IVB oscillations were also in a high frequency mode (18 Hz). These flightexperiences intensified the developmentof math modelscapable of predicting cross axis coupling toapproximately 25 Hz in frequency. In six years, the mathematicalmodels of the Saturn V vehicle have grown from primitive uniaxialmodels used to support test requirement studies todetailed full shell modelscapable of investigating flight anomalies up to 25 Hz in frequency. The technologysupporting this growth is contained in this document.

11 REFERENCE

2-1Document D5-15575-1 , SA-501 PostflightStructural Dynamic Flight Evaluation, The Boeing Company, Huntsvi 1 le, A1 abama, February 16, 1968.

12 SECTION 3 1/10 SCALE MODEL TECHNOLOGY

3.0 GENERAL This sectionpresents the contributions of thescale model to the full scale Saturn V Programand illustrates what scale modeling techniques can contribute to future programs. The 1/10 Scale Apollo Saturn V modelwas conceived by Langley Research Center as an early, economical source of vibration response data. Themodel also supportedthe Saturn V Program by contributing to problem resolution, by providing test planning guidelines, andby early mathmodel evaluation. Themodel was built in advance of the full scalevehicle so that data could be used todefine potential vibration problems and suggest solutions to these problems without impacting scheduled launch. Dynamic test data from the model could a1 so be used to Val idate the methods and procedures developed for both analytical and test investigation of the structural dynamic characteristics of theprototype. The scale model could also be used to investigate problems observed during flight test of thefull scale vehicle. The basic objectives of the scale model program were achieved. Datafrom the scale model verified that the shakerlocation selected for full scaletest could exciteall modes of interest.It also verified that the force capabi 1i ty of the shakers being developed for the full scale test would be adequate. It confirmed thatthe sensorlocations selected for full scale tests were adequate, and that theacceleration levels selected for the full scale test were correct. Datafrom the scale model tests also confirmed that the basic mode shapes predicted by early Saturn V mathmodels were adequate to support 'prel iminary design work. From thecorrelation of scale model analyses and scale model tests results, several shortcomings of the mathmodeling approaches were uncovered. Most significant of these was the manner in which the liquid was represented in thelongitudinal analysis. As a result of this early warning, more accurate mathmodel ing procedures were devel oped in advance of the start ofthe full scale program. Afterthe full scale programwas wellunder way, thescale model still proved useful in supportinginvestigations of problem areas. The modelwas used to demonstrate the integrity of the structure around the LM attachpoints during theinvestigation of a localstructural failure that occurred during flight of the second Saturn V (AS-502) vehicle.Detailed rep1 ica models of the SLA, SM, and LM were built. From the scale model tests and other related studies it was determined that the in-flight failure did not stem from localfailure around LM attachpoints. Themodel was also used to check out a gravitysimulation harness proposed for a full scale test to investigate the AS-502 anomaly.

13 While thescale modelprogram did achieve its basicobjectives, its potential value to the Saturn program was never realized because correlation was not achieved between the model and theprototype. Because of scalingeffects, testing of the 1/10 Scale Model in a 1 g environment is equivalent to testingthe prototype in a 1/10 g environment. Several joints along the modelopened slightly in thisequivalent low g environ- ment resulting in local flexibility being introduced intothe scale model. The joints wouldhave requiredredesign to eliminatethis flexibility. Had this beenaccompl ished, the dynamic characteristics of the scale model would have provided an excellentsimulation of theprototype for' overall vehiclecharacteristics. Followingcomparison of scale model and fullscale results,the scale model should havebeen revised to establishcorrelation. Because thescale modelwas built before all secondary structure and major compomentswere designed on theprototype, the replica modelwas not UP to date. Themodel should havebeen updated as the program progressed to includedefini,tion of this hardware. Had these changes beenmade, the AS-502 anomaly study could havebeen performed without the delayrequired to redesign and fabricate the SLA, LM, and SM models. In retrospect, it is clear that significant advantage would also havebeen obtained if the same data reduction methods had been applied to both thescale model data and thefull scale data. Approximately six months were required to check out the data reduction methods and automate them sufficiently to handle the volume of data being obtained from the full scale tests. This six month training period could havebeen completed in advance of thefull scale test by using scale model data. The scale model program was cost effective and did provide much usefulinformation to thefull scale program. Becauseof thetechnological advances gained from the 1/10 Scale Program, a futurescale model could take advantage of these advances and provide even greater benefits to similar space programs. 3.1 THE SCALE MODEL TEST PROGRAM 3.1.1 Scale Model Description The 1/10 scale Apollo Saturn V dynamic model consisted of five basicunits, representing the S-IC stage,the S-I1 stage,the S-IVB stage, the IU of the Saturn V launch vehicle, and the Apollo spacecraft. The spacecraft wascomposed of the LM, SM, CM and LES systems. Each of these modules isillustrated in Figure 3-1. The complete model is shown supported inthe test facility by a four-cable suspenslon system (designed to provide the proper simulation forfree-free longitudinal vibration response) in Figure 3-2. The interior of the 1/10 scalefuel tank is shown with that of theprototype in Figure 3-3.This figureillustrates the fidelity with which the Saturn V launch vehicle was modeled.

14 0

39 1 .o.

FIGURE 3

75 FIGURE 3-2 1/10 SCALE MODEL IN TEST STAND 16 SATURN V S-IC FUEL TANK ASSEMBLY

77 3.1.1 (Continued)3.1.1 Where state-of-the-artpermitted, the primary load-carrying structure of the 1 aunch vehicle and spacecraft were geometrically scaled in the model . Above the LM adapter, geometric scal ing was not feasible due tothe time and costrequired. However, theexternal dimensions and gross stiffnessproperties were scaled. The major spacecraft component, the LM, wasmass simulated. A comparisonof the 1/10 scale modeland fullscale vehicle bending stiffnessproperties is presented in Figure 3-4. This comparison shows the accuracy with which theoverall stiffness properties were reproduced. A complete description of thescaling and manufacturing techniques is presented in Reference 3-1. 3.1.2 Scale Model Test A special facility was establ ished at LangleyResearch Center to test the 1/10 scale model. A cablesupport system, which simulated free-free boundary conditions, suspended the model inthe pendulum-1 ike mannershown in Figure 3-2. The scope of theactual tests exceeded that of the fullscale test program which followed. In addition to supporting the Saturn V program, the modelwas used to check out a new booster design concept, and to explorethe 1 imits of scale model technology in the areas ofimpedance and dynamic test. Themodel was used to investigatepitch, yaw, and longitudinal properties for each of thethree stages of launch vehicle boost. The longi- tudinal test data are presentedin Reference 3-2. The pitch and yaw test data are being documented in NASA Technical Notes, that will be published later by LangleyResearch Center. The test conditions were selected to parallel those planned forthe full scale test. For example, water wasused to simulatethe LOX and RP-1 propellants in both thescale model and proto- type tests and the same tank pressures were used. This allowed scale model test results to carry forward and directly supportthe prototype test program. 3.2 CONTRIBUTIONS TO FULL SCALE TEST

One of the major objectives of the full scale test was to define thetransfer function between thecontrol engines and the flight sensor locations. To satisfythis objective, the vehicle had to be excited through the gimbal blocks of thecontrol engines. Preliminary math model work indi- cated that the modesof interest probably could be excited to readablelevels from the gimbal block. To add confidence to theanalytical results, the 1/10 scale modelwas excited through the same basic thrust structure location. Resultsindicated that the proposed thrusterlocation was satisfactory to ac- complish basicfull scale test objectives, and that the modes obtained from exciting at this location were essentidl ly normalmodes. 440 f 1 I I I I I l/iO-SCALE MODEL - 400

360

320

280

240

200

‘160

120

80

40

0 0 m 40 60 80 100 120 I El , lb-in2 0 1 2’ 3 4 I El ,kN-m 2 FIGURE 3-4 COMPARISON OF 1/10 SCALE MODELAND FULL SCALE VEHICLE BENDINGSTIFFNESS

79 3.2 (Continued) The ability to use thesingle shaker location throughout the test greatly simplified test conduct and enabled test objectives to be met on an optimum schedule. The evidence gained from thescale model tests was later confirmed by thefull scale results, as all modesof interest were suitablyexcited from the gimbal location. Data from thescale. modelwere also used to increase confidence that thethrusters beingdeveloped for the full scale tests could force thevehicle to readable response levels in all modes of interest. The acceleration per pound of force measured on the 1/10 scale model test was scaled to theprototype. This required that thescale model accelerations per unit of force be reduced by a factor of 1000 to simulatethe prototype condition. A correlation and discussion of the 1/10 scale and fullscale acceleration/forceratios are presented in Section 3.5. A parameter study was performed on thescale model to determine the effects of different propellant tank ull age pressure on the dynamic characteristics. Results of this study indicated that reducing theullage pressure from 30 to 10 psi had very little effect on the dynamic character- istics. The lower pressure wasused for both thefull scale and scale model tests to reduce the hazard to test personnel. The Cali bration levels required on the full scale test program were supported by data from thescale model. For example, ininstrumenting the bending test, accelerometer ranges had to be determined. Information obtained from thescale model confirmed that accelerometers selected were adequate. Data from thescale model also confirmed that thesensor loca- tions were adequate to obtain enough information from the test for accurate plotting of overallvehicle mode shapes. The scale model data showed that the mostcomplex modal deflectionpatterns occurred in thespacecraft region of thevehicle and that it wouldbe necessary to instrument this region in considerably more detail to obtainaccurate modal information. 3.3 CONTRIBUTIONS TO SATURN V MATH MODELING

The early mathematicalmodels of the fullscale Saturn V vehicle used to support major program decisions were primitive, and had no test data to support theirreliability or to indicatetheir weaknesses. Data from thescale model filled the gap between the time theearly modelswere developed and when actual test data were available. Scale model results added confidence that thebasic math models of the Saturn V vehicle were predictingoverall vehicle modal frequencies and shapes with reasonable Val idi ty. Scale model test data helped identify major shortcomings in the mathematicalmodel. At ?he start of theanalysis program, structural drawings of thespacecraft were very difficult to obtain. In the absence of detailed drawings, a crude LM mathematical modelwas developed from results of a vibrationtest that indicatedthe LM had a fundamental pitch and yaw frequency of 4.5 Hz (Reference 3-3).

20 3.3 (Continued) 3.3 Quicklook data from full scale dynamic test of the S-IVB/IU/ spacecraft combination showed a strong mode around 4.5 Hz, which could have beena LM mode, plus two modes around 9 Hz for which the mathmodel had no equivalent. Here the scale model test data helped isolate the mathmodel problem area. The scale model -- whichhad an equivalent LM simulator fre- quency of 14 Hz -- showed that there was a strong vehicle mode around 4 Hz, but no vehicle modes around 9 Hz. The absenceof vehicle modes at 9 Hz led to theconclusion that the 9 Hz resonances of theprototype were probably LM modes.Drawings of the LM were obtainedas required to develop a detailed model. This modelshowed that the fundamental mode of the Lbl was around 9.Hz. It also showed that the twin 9 Hz resonances were produced by pitch and longitudinal coup1 ing caused by stiffness eccentricities in the LM. 3.3.1 Mathematical Analysis of the 1/10 Scale Model A vibration analysis of the scale modelwas performed using the same basic modelsand methods being developed forthe Saturn V vehicle. Thebeam- rod/quartershell modelshown in Figure 2-2 was used in this analysis. Re- sul ts from the analysis were correlated with scale model test data to uncover major weaknesses in the models and/or analysistechniques. A discussionof the results obtained from this correlation follows. 3.3.2 Scale Model Test and AnalysisCorrelation Figure 3-5 presents a30.6 Hz mode obtained during the scale model testing. This mode is the fundamental bulging mode of the S-IC tanks caused by coupling between the propellant and structure. This phenomenon occurs in a large liquid propelled vehicle like the Saturn V, when the shell deforma- tions produced by the liquid exert adominant influence on the longitudinal modes ofthe vehicle. A counterpart to the 30 Hz modewas not obtained from the math model , and subsequent investigation of this problem showed that the coupling between ropellant and structure wasmodeled inadequately. Due to this lack of comeP ationsthe matrix method presented in Section 4.4.3 was developed to simulatethe interaction between the liquid and structure in time to support thefull scale program. The results of the second longitudinal mode obtained from the scale model test and analysisare presented in Figure 3-5.These results show that the scale model is more flexible in the S-IC intertank and IU areas thanpre- dicted by the mathmodel. The test frequency was also six percent lower than theanalysis frequency. The 1/10 scale model test and analysis correlation of the first two pitch modes for the fully filled condition is presented in Figure 3-6. Al- though theoverall correlation of these mode shapes is good, there is still evidence of localflexibility in the scale model data. The test frequencies

27 -1.11/10 SCALE ANALYSIS FREQUENCY 42.72 Hz VEHICLE Hz STATION 1.90 9--=*-*-* 1/10 SCALE TEST FREQUENCY 40.20 Hz

I I I 1 -l:o -0.5 I 0 0.5 1 .O FIRST LONGITUDINAL MODE SHAPE SECOND LONGITUDINAL WESHAPE,

FIGURE 3-5 COMPARISON OF 1/10 SCALELONGITUDINAL TEST AND ANALYSISRESULTS -100 PERCEIJTPROPELLANT VEHICLE- PRETEST ANALYSIS FREQUENCY 10.69 HZ ...... 1/1 o SCALE TEST FREQUENCY 9.1 2 HZ %+

4

FIRSTPITCH MODE SHAPE FIGURE 3-6 COMPARISON OF 1/10 SCALEPITCH TEST AND ANALYSISRESULTS - 100 PERCENTPROPELLANT

N w 3.3.2 (Continued)

are 17 and 26 percentlower, respectively, than the analysisfrequencies.

3.4 CONTRIBUTIONS TO SATURN V ANOMALY RESOLUTION During the second flight (AS-502), Pogo oscillationsdeveloped in the S-IC stage. During the Pogo oscillations, a localfailure of oneof the SLA panelsoccurred. The mostpopular failurehypothesis was that the panel failedaround the LM attachpoints under combined static (4 g) and dynamic (t0.6 g) loads. To explore this hypothesis, a specialscale model of the SLT, LM and SM was built. A specialharness wasa1 so constructedto simulate the 4 g staticload. This modelwas tested aspart of the total firststage boost configuration. Electrodynamicshakers wereused to excite the suspendedvehicle toscaled AS402 oscillation levels. No sign’of a localfailure was observed. The static loads were increasedto the flight condition,but no failure was produced. From this and relatedstudies conducted throughout the aerospace industry, it was firmlyestablished that a localfailure around the LM attachpoints could not haveoccurred. It was later determinedthat moisture had penetrated one of the honeycomb SLA panels. It was hypothe- sized that aerodynamicheating had vaporizedthis moisture and explosively delaminated a section of the honeycomb. 3.5 COST AND ACCURACY 3.5.1 Cost3.5.1 The 1/10scale modelwas built at a costof approximately 1/20 ofthe full scale test vehicle. The dynamic teststhat were conducted on the model required a crewof four engineers and technicians. The cost ofthe scale model test program was roughly1/10 the cost of an equivalentfull scale test program. 3.5.2 Accuracy A correlation between the 1/10scale model test and full scale test results wasmade toassess the accuracy of scale modeling techniques. This comparison showed that the scale model provided a fairprediction of vehicleresponse characteristics. The response per unit force of the model and the prototypeare compared in Table3-1. The scale model responses have to be reduced by a scale factor of 1000 to compare with prototype values. The ratioof the 1/10scale to full scale for the bending response rangedfrom 0.45 to1.85, and the longitudinalresponse ranged from 0.17 to1.50. Most of the differencesare due to: 1.Joint flexibility (this is discussed in Section3.5.3). 2. Configurationdifferences - The SM tank and the CM-SM interface was notmodeled adequately for the 1/10scale

24 TABLE 3-1 CORRELATION OF 1/10 SCALE MODEL AND FULL SCALE RESPONSEDATA 3.5.2 (Continued) model. Also, excessive motionof thescale model LM simulator was produced by localized deformation of the simulator at the support strut locations.

3. Damping differences caused by innatedifferences in the two vehicles, and by exciting the vehicles to different ampl itude levels(full scale testing showed that dampingchanged non- 1 i nearly with ampl i tude). The response per unit force is a difficult parameter to predict. Before test data was available to define modal damping, the responseper unitforce predicted by the math model differed from the full scale results by factors of 0.2 to 1.4 (which is the same type of accuracy obtained from the scale model ). A correlation of the first fourpitch modesof the 1/10 scale and fullscale tests are presented in Figures 3-7 and 3-8. The corre- lation of the first mode shapes is excellent, but the full scale frequency is 22 percenthigher than that of the 1/10 scale model.The second mode shapes show the 1/10 scale model is more flexible in the IU area than the fullscale model. The correlation of the' mode shapes is good, but the fullscale frequency is 14 percenthigher than the 1/10 scale. The third pitch mode frequency of the full scale model is approximately three percenthigher than the 1/10 scale model. Both the third and fourth mode shapes show that the 1/10 scale model is muchmore flexible around the SM-CM interface than thefull scale vehicle. The fullscale frequency for the fourth mode is 12 percenthigher than the 1/10 scale frequency. A correlation of the first andsecond longitudinal modesof the 1/10 scale and fullscale test is presented in Figure 3-9. The first mode is produced bycoup1 ing between the 1 iquid propellant and structure. The 1/10 scale mode shape followsthe general trend of the full scale mode shape, but the amplitude is smaller at most locations. The fullscale frequency of this mode is 23 percent higher than the 1/10 scale. The correlation of the second mode shapes shows the same generaltrend with the 1/10 scale model being more flexible in the IU area. The fullscale frequency for this mode is 11 percenthigher than the 1/10 'scale frequency. The third and fourthlongitudinal modesof the 1/10 scale and fullscale test are correlated in Figure 3-10. The correlation of the third modeshows that the 1/10 scale model is more flexible in the Ill area. The fullscale frequency of this mode is 13 percenthigher than the1/10 scale model frequency. The correlation of thefourth modeshows the 1/10 scale model responding more in the S-I1 stage, S-IVB stage, IU, and payload than the fullscale model. Also, theflexibility of the IU joints is clearly shown by the kink in the 1/10 scale modelmode shape at Station 3250.The fullscale frequency is 6 percenthigher than the 1/10 scale model frequency.

26 VEHICLE- FULL SCALE TEST FREQUENCY 1.11 HZ VEHICLE -". FULL SCALE TEST FREQUENCY ,1.82 Hz STATION ?id 1/10 SCALE TEST FREQUENCY 0.91 Hz ..LCL...I i/I o -SCALE TEST 'FREOUENCY I .60 HZ m*4000

m3000

rn2000

m1000

I -1 .o -0.5 0 0.5 1.0 uL -1 .o -0.5 0 0.5 1 .o FIRSTPITCH MODE SHAPE SECOND PITCH MODE SHAPE

N U vmm~-, FULL SCALE TEST FREQUENCY 2.55 Hz

THIRD PITCH MODE SHAPE FIGURE 3-8 COMPARISON OF 1/10 SCALE AND FULLSCALE PITCH TEST RESULTS .. MODES 3 AND 4 - 100 PERCENTPROPELLANT VEHICLE- FULL SCALE TEST FREQUENCY 3.75 HZ VEHICLE- FU8I.L SCALE TEST FREQUENCY" 4.46 Hz STATION ..*...-.-.1/10 SCALE TEST FREQUEIKY 3.09 Hz ...I.. "U 1/10 SCALE TEST FREUENC 4.02 HZ

"

"

" 1

"

+ 0 -1 .o -0.5 0 0.5 1 .o FIRST LONGITUDINAL MODE SHAPE SECOND LONGITUDINAL MODE SHAPE

FIGURE 3-9 COMPARISON OF 1/1.0 SCALE AND FULLSCALE LONGITUDINAL

N TESTRESULTS - MODES 1 AND 2 - 100 PERCENTPROPELLANT w w 0

FOURTH LONGITUDINAL MODE SHAPE

FIGURE 3-10 COClPARISON OF 1/10 SCALE AND FULLSCALE LONGITUDINAL TESTRESULTS - MODES 3 AND 4 - 100 PERCENTPROPELLANT 3.5.3 Scale Model JointFlexibility While analysis and test correlation uncovered major weaknesses inthe math model, it also revealed a problem with thescale model joints. Themodel jointsare scaled duplications of theprototype joints. Rep1 ica scaling of the joints was used sinceconsiderable engineering time would, be required to properly design an easily manufactured connection with com- parable dynamic properties. The joint flexibilities exhibited by thescale modelcan be explained if the jointsare assumed to behave as illustrated inFigure 3-11. The physical rationale supporting these types of joint behavior is given below: 1. Due to thefabrication tolerances and thedifferences in ratios of stiffness and massbetween the model and fullscale vehicle, some model joints open slightly at thesurface of the vehicle after the two flangesare bolted together. This phenomena is illustrated in Figure 3-11 (a) and is discussed in Reference 3-4. A fullyeffective joint, that is, where the stiffness of the joint approaches the stiffness of theskin, is presented in Figure 3-11 (b). 2. Themodel joints could gapbetween fasteners as shown in Figure 3-11 (c): This type of phenomenon could.occur because the thin- flange ring framesused in thescale model would not hold their planar shape. Also, the number of fasteners used on the modelwas less than the number used on the full scalevehicle. Gappingbetween fasteners was observed along the joint between the S-IVB forward skirt and IU. The stiffness of the joints presented in Figure 3-11 (a) and (c) is nonl inear and tends to increase under 1oad. The major cause of differences between scale modeland full scale test results is the joint flexibility of the model. An additional cause was configurationdifferences between thescale modeland test article spacecrafts. Had the modelbeen redesigned to eliminatethese differences, overall correlation would havebeen excellent for primary vehicle charac- teri stics . 1" 1" .- LOAD PATH-"------.-I

VERTICAL STRINGER

JOINTOPENING

VERTICAL STRINGER

(a) 1/10 SCALE MODEL JOINT (b) FULL SCALE JOINT

(c) SCALE MODEL GAPPINGBETWEEN FASTENERS REFERENCES

3-1 Leadbetter, Sumner A., Leonard, H. Wayne, and Brock,John E. Jr., Designand FabricationConsiderations for a 1/10 ScaleReplica Model of the Apol ld/Saturn V, .NASA TN D-4138, October,1967. 3-2 Pinson,Larry D. and Leonard, H. Wayne, LongitudinalVibration Characteri st? cs of 1/lo Scale Apol lo/Saturn V Rep1 ica Model , NASA TN D-5159, April 1969. 3 -3 Reuort LED-520-8, LTA-2 VibrationSurvey at G.A.E.C., Grumnan Aikraft EngineeringCorporation, BethGge, Long Is1 and, New York, March 2, 1965. 3-4 DocumentD5-15631-AY 1/10 ScaleSaturn V Model Structural Dynamic Analysis, The Boeing Company, Huntsville, Alabama, May 1, 1967.

33

SECTION 4.0 MATHEMATICAL MODEL TECHNOLOGY

4.0 GENERAL This sectionpresents procedures and guidelinesfor formingmathe- matical modelsof large space vehicles such asthe Saturn V. The section includesdiscussions ofmath model philosophy, stiffness and inertia matrix development, and vibrationanalysis techniques. Problems that havebeen encountered arediscussed along with therationale for thesolution; also guide1 ines for mathematical model development are presented. The math modeling techniques used in the Saturn V structural dynamics program were adequate for predictingthe overall vehicle dynamic characteristics. Most of the problems encountered in the program were associated with local deformation or componentdynamics. These problems occurred because : 1. Proper emphasis was not given to math modeling of local and componentdynamic effects. 2. Structural dynamic techniquesrequire testresults to guide the modeling of these effects. Some of the problems associated with modeling local effects are presented in Section 4.3. 4.1 TECHNICAL APPROACH Matrix formulation of structural dynamicsproblems was used extensively in theSaturn V program. Themethods used in thepretest analysisare documented in Reference 4-1. Thesemethods have since been revised anddocumented in internal Boeingdocuments. Each Saturn V 'structure was idealized as a lumped parameter system that satisfied the following matrix equation:

Inertia Dampi ng El astic A,ppl ied Forces Forces Forces Forces

The generalized coordinates , the q e's , used in a lumped para- meter analysis are frequently chosen to be{he Cartesian components of displacementand/or rotation atdiscrete points on the system.These points are known as nodes. A typical nodalbreakdown is shown in Figure 4-1. In the lumped parameter. approach theelastic properties of an actual structure are represented by a network of 1 inear springs coupling the node pointstogether. These springconstants, in matrix array, form thestruc- turalstiffness matrix. A typicalelement, Kij, in this matrixrepresents the stiffness of the effective structural spnng coup1 ing qi and q- motions. A1 1 forces are applied to a lumped parameter model at the nodesana are distributed into the model by the springs which simulatethe load paths in theactual structure. For example, when a forceis applied in the qi

35 S-IVB UPPER LH2 BULKHEAD S-IVB COMMON BULKHEAD

n

S-IVB LOWER LOX BULKHEAD S-IVB THRUSTSTRUCTURE AND ENGINE

S-IC(S-11) TYPICAL LOX AND RP-1 TANKMJLKHEAD S-11 THRUSTSTRUCTURE AND ENGINE

FIGURE 4-1 TYPICALSATllRh V NODAL BREAKDOWN 4.1 (Conti nued) direction, a portion of theforce will be transmitted through thespring Kij to induce motion inthe qj direction. In a dynamicmodel, mass matricesare developed to representthe distributedinertia characteristics of theactual system. These general” ized inertia matrices range in complexity from a simple diagonal array of lumped masses to thestrongly coupled generalizedinertia arrays developed in Section4.4. Energy dissipative elements maybe introduced into the model to simulatethe effects of damping forces on theactual structure. Normally, the damping elements used in the model are 1inear. The characteristics of these damping elements are chosen so that the average rate of energy dissipation in the model over a cycle of oscillation is the same as that of theactual structure. Damping valu$s used in the Saturn V analyses were obtained from the fullscale dynamic test. These damping values areproportional, that is, all sections of thestructure are assumed to have the same damping. Thelumped parameter model is completed by a set of concentrated time dependent loads applied at the node points. These loadsrepresent the distributedforces, temperatures, or prestressesacting on thereal system. Themost difficult part of the process of creating a lumped para- meter model is obtaining a spring network that adequatelydescribes the elasticproperties of the systembeing analyzed. The structuralspring networks maybe obtained from a finite element displacement method, known as the direct stiffness method (Reference 4-2 and 4-3). The direct stiffness method is derived from energy considerations. Ifthe deformedshape of a system canbe adequatelydescribed in terms of a limited number of independent quantities, for example, the deformations of a limited number of points on thestructure, the actual system can be represented by a mathematical model having a finite number of degrees of freedom. The strain energy of the systemcan be expressed in terms of thesegeneralized coordinates. The stiffnesscoefficients are derived from the strain energy quadratic. 4.1.1 Stiffness Analysis Prior to establishing a stiffness model, detailed drawings of the structure are studied in order to determine mechanisms by which loads aretransmitted through the system. Next,nodes arelocated on structural drawings at points along the primary load paths or in regions of critical motion. The actualstructure connecting these nodes is then described in terms of combinations of the following basic elements: 1. beam - stringer - torque tube 2. plate (membrane)

37 4.1.1 (Continued)4.1.1

3. plate(bending) 4. Axisymmetricshell The plate and shellelements may be either isotropic or orthotropic. Once thestructural modelhas been established,the stiffness analysis computerprogram inputdata are compiled. These datainclude a list of nodenumbers, thegeometric coordinates and boundary conditions of eachnode, along with thephysical properties of each structuralelement; i.e.,cross-sectional area, moment ofinertia, thickness, type of material and temperature. Load informationalso may be contained inthe input data. Thisincludes prestresses, external forces, and nodaltemperatures. The stiffness programgenerates stiffness matrices for each structuralelement defined in the input data. Each matrix is generated in its own localcoordinate system.Transformation matrices are computed and applied to obtain the stiffness coefficient components of eachelement in a centralcoordinate system. Thissystem may be Cartesian,cylindrical orspherical at the user’s discretion. The stiffnessmatrix transforms from one coordinatesystem to another as follows:

where [Tpi]is the transformation relating displacements in the original system to the generalized coordinates in the new system, and i, j , p,and q arerow andcolumn indices. Once thestiffness matrices for all elements havebeen generated and transformedto the central coordinate system, the stiffness coefficients are merged. The mergingoperation is basically a process ofmatrix addition in whichcorresponding stiffness coefficients from each structuralelement connectingat a given node are combined to obtain the stiffness properties ofthe structural model at that node. Aftermerging, the boundary conditions for the nodescan be imposed and matrixreductions performed. Boundary conditionsfor a stiffnessanalysis aredetermined by structural synetry or antisymmetry as well as by physical constraints.For example, a typicallaunch vehicle, which has at least two orthogonalplanes of symmetry,can be representedby a quarter she1 1 model . The effects of the missing three quadrants of the structure can be represented byappropriate boundary conditions. Fixed boundary conditions are imposedon a stiffness matrix by deleting the rows and columns associatedwith the degrees of freedom at undeformablepoints on thestructure. A pin joint is introduced byreducing the appropriate rotational freedom prior to merging.

38 4.1.1 (Continued) Imposingboundary conditions is not the only way the size of the stiffness matrix is decreased.Frequently, the degrees of freedom subjected to small inertia loadsare assumed to be unloaded. These degree of freedom can then be eliminated from the force-deformation relationships by a process ofmatrix reduction. Assume thatthe stiffness matrix is ordered so that direct stiffness of the unloaded degrees of freedom appear in the upper left corner of thematrix. The force-deformationmatrix equation can then be parti ti oned as fol 1ows : I- I-

L where the q arethe displacements of the unloadednodes , and the q and F arethe #isplacements and forces,respectively, at the loaded no%es. Tieequation abovecan be solved for the qu in terms of the 9%: r -1 1

Applying thereduction transformation to Eq. 4-3 gives

This same transformationwill beused to develop consistent inertia matrices in Paragraph 4.4.3. The dimensions of the reduced stiffness matrixare R rows and R columns smaller than the original matrix, where R is equal to the number of unloaded degrees of freedom. The size of a stiffness matrix can also be decreased by the con- straints transformation procedure. In the Saturn V models, the motion of many of the nodes was expressed in tenns of polynomialshape functions satisfyingthe appropriate geometric boundary conditions. The shape functions wereused to represent ring modes, simplify bulkhead models, and idealize near rigid substructure. These constrainttransformations were applied to both stiffness andmass matrices so that consistency would be maintained.

39 4.1.2 InertiaAnalysis

Inthe past, it was standardpractice to construct mass matrices bysimple lumping techniques. However, theimproved accuracy obtained by developinginertia matrices and stiffnessmatrices in an identical manner is now widelyrecognized (Reference 4-4 and 4-5). Inertiacoefficients aregenerated for the same fine nodal breakdownused in the stiffness analysis. The inertiamatrices are reduced or constrained using the same transformationsapplied to the stiffness matrix. The inertia characteristics of liquid filled tanks are handled by specialtechniques that represent the dynamics of the liquid as well as interactionwith the elastic tank structure. On largebooster systems, the liquid is usually treated as beingincompressible, and the liquid freesurface is assumed to remainplane. Under theseconditions, the motion of the liquid can be definedby the elastic degrees of freedom ofthe tank. A techniquefor relating liquid motions to tank deforma- tions is developed in Section 4.4.3. 4.1.3 EigenvalueAnalysis

After generating consistent inertia and stiffness matrices, the normal modes and frequenciesof the system are obtained,by solving the equation of undamped freevibration for the eigenvalues and eigenvectors. CWijI Iqjl + CKijI Cqjl = (01 (4.6)

If thesystem is free-free,the equation above wi 11have up to six zero frequencysolutions which are the rigid body modes ofthe system. These zerofrequency solutions can be developed directly from the geometry of thesystem. The eigenvaluesolution of large order systemscan be costly, however.Modern computer techniques still take up to 75 minutes of computer time to obtain all of the mode shapes of a 300thorder system, calculate the generalized mass matrix and plot the results. 4.2 MODELINGPHILOSOPHY

The first step in developing a math model is to understand how resultsfrom the model will be used. The threerime uses forthe Saturn V mathmodels were for fl ight loads analyses, fligEt control analyses and Pogo stability analyses. Each of theseanalyses required special refinement in certainareas. Unfortunately, time and cost does notpermit the use of sophisticatedgeneral purpose models. Consequently, it must be determined in advancewhere detai 1 is neededand where grossrepresentation will be adequate. Inshell structures, such required foresight is often beyond realism. Groundand flighttest data will pointout deficiencies in the engineer'sjudgement. For this reason, there should be room in the model for growth.Growth potential isalso required to allow for changing objectivesof the program.For example, on the Saturn V, an accuratepre- diction of dynamic activity above 10 Hzwas not required early in the program. After 18 Hz osci 1 lations were observed in the flight data, it was necessary to developmathematical models that could produce accurate results up to 25 Hz. 4.2 (Conti nued) Once the use of the analytical results is established,the next step is to define a model that can satisfy user requirements. This requires that a number of technicaldecisions be made, such as: 1. Should the modelbe three-dimensional, two-dimensiona 1 , uniaxial , or some combination of these? 2. Where is shellactivity important? 3. How fine should the nodalnetwork be? 4. What arethe limitations of the computer software andhardware used in the model analysis? The first decision is to determine whether the dynamic character- istics must be described in the coupled (three dimensional)sense or whether a coplanar (two dimensional) or evena uniaxial model willsuffice. Many space vehicles,including the Saturn W, have at least two orthogonalplanes of symetryfor the primary structure. However, theinternal structure and major components areoften not symnetrical. These secondary asymmetries are a mechanism for coupling the coplanar modes of the primary structure together.If two of these modes should coalesce, evena small asymnetry canproduce significant coupling thatrequires a three dimensional model. Another consideration is whether follow-on analyses can use three dimen- sional modelingand also whether theextra refinement is warranted. For example, if the dynamic loadsare only 10 percent of thetotal load, the refinement in using a coupled modelmay not be warranted. A third con- sideration is whether restraining the structure to act in one plane will result in accuratecharacteristics in that plane. Using theSaturn V as an example , the proper in-pl ane characteristics of the LM cannot be pre- dicted without allowing the LH to have out of plane motion. The second decision is to determine where shell action must be included. Initial Saturn V models allowed for shell deformation through- out the complete launch vehicle. However, full scaletests confirmed that a beammodel would represent 1aunch vehicle bending action adequately up to a frequency of twenty times the fundamental bending mode.They also showed that the shell characteristics required a much finer nodalnetwork than was originally provided, both in the instrument unit where flight sensors were located and in the bulkheads thatcarried the longitudinal propellant load. The present solution for Saturn V models is acombination of beamand shell modules that includes a detailed nodal network only in areas of significant localdeformation. The third decision is to determine how fine the nodal network should be. The first consideration here is the numerical accuracy that can be maintained during the solution procedure. The final stiffness mat- rix must behomogeneous, that is, the order of magnitude of like stiffness termsalong the diagonalshould be similar. If term-to-term fluctuations of two or more orders of magnitude occur, the analyst should antlclpate

41 4.2 (Continued)

a large round-off error in hissolution. The stiffness of each element is dependent on its size. Thus the nodesmust be located so that stiffness magnitudechanges occur graduallyrather than abruptly. In many cases , the structure itself dictates the degree of finerless. The engineer must allow sufficient nodes to simulatethe physical load path and to allow efficient use of the elements available in the software program. When a stiffness mathematical model is developed on thisbasis, theengineer looks for sensitiveareas in thestructure from a dynamic standpoint. For example, the payload on the Saturn V is a highly active area compared with the 1 aunch vehicle, Consequently, thedensity of nodes is much higher in the payload than in the launch vehicle. Also, where high load gradientsoccur, such as in the thruststructure, additional model detailis required. The final decision point in establishing a model involvesthe avail- able computer software and hardware capabi 1 i ty. This capabi 1 i ty includes the size of theindividual stiffness matrix modules that canbe generated,the complexity of merging and reducing these modules , the order of the dynamic matrix that can be handled and the flow time and costassociated with each of the above computations. The natural tendency is to have a model as large as the computer capability wi 11 allow. This occurs either because of lack of confidence in theidealization of thestructure or because the mathematical modelmay be required to do many differentthings. Both of thesesituations occurred in the Saturn V analyses. Initial model develop- ment did not include sufficient engineering judgement. The models presently in use, while greatlysimplified in certainareas, are still complex because they handle problems in a1 1 dynamic discipl ines. Unless theresources are available to permit production analyses using large mathematicalmodels, sizelimitations should be established early in the program. Of course , restricting the size of the model wi 11 increaseengineering tolerances that mustbe placed on the math model data. Section 4.6 of this document describesthe evolution of the Saturn V models.Resources required for mathmodeling were controlled to some extent by correlating a basicsophisticated model with test results and then carefullyeliminating detail until a degradation of results became noti ceabl e. In large programs,such as Saturn V, the requirements of the mathe- matical models continually change as the program matures. Because of the constant state of change, the following two ground rules cannot be over- emphasized. 1. Establish a baseline model that where possible, istest verified.

2. Develop new models in para1 le1 with the basel ine model ; do not change the basel ine until the new models are completely verified.

42 4.2 (Continued) Two baseline modelswere used in Saturn V analyses. The first baseline was a beammodel that was useful in full scale pre-test work as well asin providing agross check on the three-dimensional models.. The second baseline modelwas thetest-verified coplanar model.All subsequent models developed were validated by comparisonof results with the basel ine model.Only when these modelshad proven their superiority to the basel ine modelwere they accepted as the new basel ine. Dynamicists were trapped in severalinstances by not adhering to the first ground rulelisted previously. The "man on the moon in this decade" goal required strict adherence to schedules.Pioneering with new models and new software programs should not be attempted when the matrixorder is large and theoutputs of theanalyses are program critical, both from a schedule and qualitystandpoint. New models should be developed in parallel with productionanalyses and phased into theanalysis system only when they arefully checked out. Engineers and programmers habitually underestimatethe cost and flow time for these changes. Thedynamic characteristics generated from the Saturn V mathemat- i cal model s wereused by many government and contractor agencies. S truc- tural dynamic characteristics were developed for each space vehicle and rigorously documented insource data documents.The configuration of each vehicle was tracked andwhen major changeswere made,new structural dynamic characteristics were generated and thesource data document updated. Despite this rigor, the dynamic characteristics wereused improperly on a number of occasions. Experience hasshown that written communication, al- though necessary, is not sufficient. Continuing face-to-face communication between the model developers and theusers of thedata is essential.

4.3 STIFFNESSMATRIX DEVELOPMENT 4.3.1 General Gui del i nes Meeting the accuracyrequirements of the Saturn V programmade necessarya complex finite-elementanalysis. The lineardirect stiffness methodwas used. The stiffness method is based on representingthe actual structure byan assemblage of finite elements such as axial load members , bending membersand plates. From basicstructural theorythe essential characteristics of each type of element are known. The science is well documented in References 4-2, 4-6, 4-7, and 4-8. But idealization of a structure to obtain a satisfactory, we1 1 behaved, mathematical model is an art as well as a science. The art as it applies to Saturn V math mod- el ing wi 11 be covered herein and will involve the documentation of guide- lines for using the finite element methods to solvepractical structural problems. The elements used in the Saturn \I analysesare shown in Figure 4-2 l and are as fol lows :

43 P P

AX I AL TORSION BENDING TRIANGULAR PLATE

!@DE TEMPERATURE QUADRILATERAL PLATE QUADRILATERAL TEMPERATURE !@DE

TEMPERATURE GRADIENT X X X

CROSS-SECTION AREA X

POLARJXMNT OF INERTIA X BENDING WENT OF INERTIA X SHEAR UEE AREA X

~ - WBRRNE THICKNESS X X X

BENDING THICKNESS X X X AXISYWETRIC SFELL

FIGURE-4-2 MTH NOEL STRUCTURAL ELEMENTS 4.3.1 (Continued) Axial members Torsion members Beammembers Isotropic triangular plates Isotropic quadri 1ateral plates Orthotropic triangular plates Orthotropicquadrilateral plates Axisymmetric shell element The following paragraphs present some guidelines for the use of theseelements. While theguidelines in some instancesare a function of The Boeing Company software programs, theyare typical of the types of 1 imi ts imposed by software. One of the prime considerations in defining anodal network is to followprescribed rules for plate geometry. To avoid numerical difficulties thelength-to-width ratio of each plate element should not be greater than two and no anglewithin a triangular plate should be greater than 90 degrees. Triangular plates are generally used in making the transition from large plates to small plates. An example is shown in Figure 4-3. When making a transition from a stiff structure to a relatively soft structure, care must be taken in thetransition zone. The soft structureidealizatioc would normally imply a finer nodal network. The plate geometry rulespreviously cited must be kept in mind in creating the transition nodal pattern. One of the comnon problems that occurs in the idealization of a structure is the generationof redundantdegrees of freedom. Redundant degrees of freedomcan be introduced into the mathmodel unintentionally during the process of coordinatetransformation. Suchmechanisms are not present in the actual structure but arise through reorientation of the structural elements. This may be illustrated bymeans of a simple example. Consider a horizontalbar pinned at one end and supported by a vertical bar at the other end as shown in Figure 4-4 (a). Since only the axial behavior of the bars is of interest, the bending behavior is ignored. Assuming both bars have the same cross-sectionalarea, the stiffness matrix is:

"X1 Ax2 Ay2- -1 0 1 0 (4.7) Lo 0 1 -1

45 L < 2.0 -h

FIGURE 4-3 TRANSITIONTECHNIQUES - LARGETO SMALL PLATE ELEMENTS

F2Y

LL"L-4

FIGURE 4-4 REDUNDANTDEGREE OFFREEDOM EXAMPLE 4.3.1 (Continued) No lateral stiffness terms appear inthe matrix for node 1 because the bending stiffness of the bar hasbeen neglected. The matrix is non- singul ar andmay be inverted to sol ve for the displacements of the nodes in a straight-forward manner. Now considerthe same physical problem but with the bars oriented at an angle of 45 degrees to their former positions as shown in Figure 4-4 (b). The stiffness matrix and load column for therotated structure are obtained by coordinatetransformation of the form.

where [TI is thematrix of directioncosines relating the (x,y) system to the (y,y) system.In this case,the transformed stiffness matrix and load column are, - - AY1 Ax2 *y2 1 -1 /2 1 -1/2 -1"/*l /2 (4.9) -1 /2 7

-1 /2 0 1O-J

(4.10)

The stiffness matrix in Equation (4.9) is obviously singularsince the first two rows areidentical. The process of transformation has intro- duced a redundant equation in the variable ~y7. The singularityarises from introducing two degrees of freedom at node 1 for the transformed structure where only one independent degree of freedom is defined. In this example either AX^ or A~Imay be consideredas a redundant degree of freedom andmay be constrained out. Choosing Ayl asthe redundant degree of freedomand imposing theconstraint Ayl = AXl gives:

47 -1 01 -j1 (4.11) and

The stiffness matrix of Equation (4.11 ) representsthe structure with a supporting roller on a 45 degree plane at node 1 as shown l’n Figure 4-4 (c). This rendersthe stiffness matrix non-singular and it maybe inverted to solve forthe displacements. An a1 ternatesolution is to include bending stiffness in thehorizontal bar. This introduces an inde- pendent AS1 equation which eliminatesthe redundancy. Ifreductions are inadvertently performed at a node having a redun- dant degree of freedom, in theory a rowand column of zeroes wi 11 be generated. These zero rows and columns areoften automatically deleted within stiffness analysis programs so that thesolution canproceed nor- mally. However, because of numerical accuracy, true zero rows and columns are seldom generated in thereduction process. The result then is the retention of highly ill-conditioned nonphysical terms in the matrix. In this simple example, the freedom to be constrained is quite obvious. But in more complicated structures using beams, axial members, torque tubes, and plates, redundant freedoms are easy to miss,especially if theengineer is not intimately familiar with the mathematics of the analysis program he is using. If he is aware of the program limitations, hecan idealizethe structure to eliminate redundant degrees of freedom. There are three ways of avoiding redundant degrees of freedom: 1. Include stiffness elements that resistforces and moments in all directions.

2. Constrain out redundant degrees of freedom, and 3. Choose a coordinate system consistent with the geometry of the problem. 4.3.1 (Continued) The first solution is not always acceptable. Adding additional elements introduces new force/deformationequations that may not be well conditioned. Forexample, a longeron may be stiff axially but weak in bending. If its bending propertiesare included, the resulting ill conditionedequations added to the problem may prevent an accurate solution from being obtained. The optimum solution is a combination of the last two solutions: choose the coordinate system to minimize transformation reorientation, then constrain a1 1 redundantdegrees of freedom that appear. Taking advantage of a plane of structural symmetry to reduce the number of variables in a linear deflection analysis is a technique often used by structuralengineers. The procedure derives from thefact that any part of a linear structure may be considered cut awayand analyzed as a separate entity while retaining the effect of the whole structure by applyingthe proper boundary conditions at thecut. When the cut is made at a plane of structural symnetry, the boundary conditions are simply and uniquely defined bytwo distinct types of displacement modes,symmetric and antisymmetric. Theseboundary conditions may be deduced by consideringthe relative displacement of a point on the struc- ture and its conjugate point, or "mirror image", on theopposite side of theplane of symmetry. For a symnetricdisplacement mode, any movement of the point will be accompanied by a corresponding mirror image behavior of its conjugate point as in Figure 4-5(a). For an antisymmetric mode the movement of the conjugate point will be exactly opposite to that produced in the mirror image. This is ill ustrated in Figure 4-5(b). As the location of point P is chosen closer to the plane of sym- metry, it is seen that points P and P' become coincident in the limit. Symmetric behavior of thecoincident points is contradictoryunless the motion is restricted by imposing constraints against translation out of the planeof symnetry and against rotation about any axis lying in the plane of symmetry. These constraintsare the appropriate boundarycon- ditions for symnetric behavior at a plane of structural symmetry. For antisymmetricbehavior the appropriate boundary conditions are determined by similar reasoning to be just theopposite; i.e., constraints are imposed against translation in any direction lying in the plane of symmetryand against rotation about an axis normal to the plane of symmetry. Apollo Saturn V stiffness andmass asymnetries were identified and assessed. The launch vehicle, below the payload , was found to be essentially symmetrical about the pitch/longitudinal and yaw/longitudinalplanes. This symmetry allowed the launch vehicle to be represented by quarter-shell (coplanar) models. Four cop1 anar models were formed by applying either symnetric or antisymnetric boundary conditions at the quarter-shell cuts. Thesemodels were for pitch, yaw, longitudinal, and torsional analyses. Figure 4-6 shows the coordinate system and the quadrant used in the Saturn V analysis.

49 FIGURE 4-5 STRUCTURALSYMMETRY EXAMPLE 4.3.1 (Continued) For pitchanalysis (motion in the Z-direction),antisymmetric boundary conditions were applied at Boundary A, while syrrrnetricboundary conditions were applied at Boundary 6. For yaw analysis (motion in the Y-direction),symmetric boundary conditions were applied at Boundary A, while antisymetric boundary conditions were appliedat Boundary B. For longitudinalanalysis (motion in X-direction), symmetricboundary con- ditions were applied at both Boundaries A and B. For torsionalanalysis 1 (rotationabout X axis),antisymmetric boundary conditions were applied at Boundaries A and E. The imposed boundary conditionsused abovecon-

~ strainthe vehicle centerline to move insingle a plane. Cross axis couplingcannot be studiedwith such a mode!.

~ The Saturn V vehicleundergoes two temperatureextremes. One extreme is due to the cryogenicpropellants whereas the other is due to inflight aerodynamicheating. There are two methods of accounting for thermal effects in thefinite element software program. The programhas the capability of acceptingdiscrete temperature values for each node plus atemperature gradient across plate sections. Through table look-up features,the program automaticallyaccounts for temperaturechanges in the modulus of elasticity. The alternate method is to scalarmultiply the free-free stiffness matrix of a module of the structure that is affectedbefore merging with the other modules. The first method has theadvantage of accuracy but the disadvantage that a new stiffness matrix must be developed for every flight time point analyzed. The lat- ter method, whichassumes a uniform temperatureacross the entire module, was used in Saturn V analyses to representthe average effect of temperature on the modulus of elasticity . With largeanalysis programs that have criticalschedules attached to the outputs, there is atendency to forego normal engineeringdiscipline in checkingprocedures and documentation of engineering calculations. In other words, the program tends to move faster thanthe documentation. Engineering management must takea firm stand on controllingchecking procedures and insuring the preparation of good engineeringnotes. It is better to have asimple analysis rigorously checked and documented than to have an elegantanalysis that is incompletelysubstantiated. The following guidelines for avoidinga schedule squeeze resulted from Saturn V experience . 1. Scope the model effortallowing a realistic ''pad" for contingencies. 2. Never makemodel changes or improvements on a tight schedule if an existing model canproduce acceptable results. 3. Placethe most experiencedengineers in the checking loop. A check on arl thmetic is only a partial check.

I 57 4.3.1 (Continued) Lapses inthese disciplines result in severepenalties in doing the same work twice. Evenmore important,relaxing the checking discipline increasesthe risk that erroneous data will be released for use. Stiffness calculations mustbe prepared in a formal manner with adequate sketches and referenceto the structural drawing number and thedifferent vehlcles for which the drawing iseffective. A11 calculationsrequire an independent check and should be initialed by both theoriginator and the checker. This is the prime engineeringtask in that it certifies asaccurate the struc- turalidealization and the elasticproperties of the finite elements. The structural geometry input tothe computerprogram should be plotted to scale. Thenodal points and finite elements should be posi- tioned on thissketch. Using this technique, nodes out of position, elements missing, and elements duplicatedwill show up clearly. Theabove procedure should be automated. The accuracy with which the nodal coordinatesare described is highly important. For example, if a cylinder is being described in Carte- siancoordinates, six significant figures should beused to definethe nodal coordinates. The accuracy of thetransformations from local to centralcoordinates depends on the accuracy with which the nodes are 1 ocated. The idealization will be determined moreby thelimitations of the computerprograms (both in size and in numerical accuracy) than by physical properties of thestructure itself. The analyst should investigatethese limitations and be thoroughly familiar with them beforeever starting an analysis. For example, if a stiffstructure connects to a flexiblestruc- ture, thedifference in stiffnessintroduces numericalproblems that may invalidatethe analysis. The stiffstructure will require wider spacing of the nodes, or will have to be represented as a rigid body. On flexible structure, too fine a breakdowncan introduce numericalproblems that de- stroy accuracy. Consider a ring represented by flat beam elements as shown in Figure 4-7. As the elements become smallerthe angle 8 between the elements becomes small. One of thethree independent stiffness terms (AX, Ay, Rz) approaches dependency. Eventually,reduction of one of these terms will introduce significant round-off error into the solution. The computer generated stiffness matrix should be checked both mechanically and automatically. The mechanical check involves checking the stiffness matrix to insure all diagonal terms arepositive and to check the diagonal terms against the physical situation. 111-conditioned degrees of freedom or improper idealization often produce terms of unusually large or small magnitude. The automatic check consists of programming the com- puter to perform the following computations:

[KIT - [K] = [O] (4.13) [K] = 0 (4.14)

52 FIGURE 4-6 QUARTER SHELL ANALYSIS COORDiNATE SYSTEM

I I I ,

L*x

FIGURE 4-7 BEAM ELEMENTS FOR RING MODELING

53 4.3.1 (Continued) where [K] is the stiffness matrix and 5 is a rigid bodyvector. The first computationchecks symmetryand should result in a matrixof zeros.This check can be avoided if through all stagesof stiff- ness manipulation(generation, merge, reduction,constraints), only a sym- metricalhalf of the matrix is used. The secondcomputation, used in a free-free system only, involves multiplication of the stiffness matrix by asmany rigid body displacementvectors as thesystem has kinematicsin- gularities. The resulting column matricesshould show residualforces that approachzero. Due tonumerical round off errors in forming and re- ducingthe stiffness matrix, these residual force terms will not be exactly zero.Checking the output by this method will uncoverimproper restraints on thestructural systemintroduced either in the idealization or the con- straintsoperations. It is a1 so a powerfulcheck on round-offerrors. For eachrow ofthe stiffness matrix, the permissible residual force resulting from a unit rigid body displacementshould be at least five orders of mag- nitudessmaller than the diagonal stiffness term in that row. If the round- off errors are larger than this, the math model should be reviewed and the ideal izati on changed to improve numerical accuracy. Inconducting a computer analysis of acomplex system, it is easy tolose sight of thephysical -realities of theproblem being Solved. Corn- puter programs are an aid to experience and engineeringjudgement, not a substitutefor them. A computer solutionrequires all ofthe ski1 1s dis- pl.ayed in a hand solution , plus an intimate knowledgeof how the computer programworks, what its limitations are , and whatnumerical problems are apt to occur.

At all pointsin,the computeranalysis, checkpoints with physical real ity need to be plannedin. Always precede acomplex analysis with a good simplifiedanalysis. This will provide a grosscheck on the answers acomplex analysisis giving. In checkingout a complex analysis,canti- leverthe system and invertthe stiffness matrix. Multiply the stiffness matrixby its inverse. This product should yield an identitymatrix. Check theproduct matrix. If thediagonal terms are different than unity by more than &0.001, orthe off-diagonal terms are different than zero by more than +0.001, theidealization should be reviewed to eliminate the source of The numericaldifficulty. Plot the force/deflection coefficients Examinethem cqrefully. Does point B deflect to the left when engineering judgementsays it should go tothe right? If so, stop and investigate until either an understandingof the physical mechanism that makes point B deflect to the left is gained or a modelingerror hasbeen identified. Take advantage ofall available test data. The majorstructural assembliessuch as thruststructures, interstages, and tanks, are usually subjected to staticloading. Become familiarwith these tests. Try to influence the tests to obtain data that can be used to check the model. Neverpass up an opportunityto put the model tothe test. Hassomeone else modeled the same structure? Meet andcompare notes; where differences exist,resolve them on thebasis of physical arguments ratherthan size

54 4.3.1 (Continued) and complexity. A 10thorder modelmay reveala major flawin a 1000th order model . Over theperiod of time covered by a majorprogram, continuing computerhardware development will dictate software changes. On the Saturn W program, early workwas done with computer routines developed in the 1950's. The advent of new generation hardware forcedthe conver- sion of virtuallyall structural dynamics routines. In alimited number of cases, the conversion was accomplished directly from onecomputer language to another; however, for the majority of programs the opportunity to open up operationalroutines for updates, improvementsand general cleanup was irresistible. Consequently, the conversionprocess becomes time consumingand costly, since some amount of new programing and error eliminationactivities are required. Even direct conversion requires someamount of programming,such as revising machine 1anguage level routines to FORTRAN compiler 1eve1 s . Due allowance for these types of perturbations and their associated schedule impacts must be made. In the,case of the Saturn V dynamic analysis work, the change from IBM 7094 to the IBM 360/67system required the revision of the direct stiffness finite element program, slosh dynamics program, eigenvaluesolution programs, various dynamic response programs, and associated programssuch asmatrix algebra andmass characteristics. These revisions took place over a period of 18 monthsand required the full time participation of 12 engineers and programers. This was three times theoriginal estimate. The impact was caused by such subtle items as lack of properassessment of the effect of word length differences between the two machines. The IBM 360has ashorter single precision word length than the IBM 7094.The same math models being analyzed success- fully on the IBM 7094 would not provide satisfactory accuracy on the IBN 360. All stiffnessanalysis algorithms had to be reprogramned in double precision arithmetic to give acceptable results. The problem of computersystem accuracy is of immediate concern to thepracticing dynamicist. The finite element approach to stiffness analysis of a complex structure necessarily involves thousands (quite possibly mil- l ions) of mu1 tip1ication-addi tion operations on the computer. Round-off and truncation errors, small differences of large numbers, division by near zero and other similar numerical problems cannot be assumed as auto- matical ly self-cancel 1ing. Indeed, the computation process can become unstableas problem size increases. It is difficult to design the math model Complexity to be optimal from the computation standpoint; however, there is a distinct trade between model size and numerical accuracy. An investigation was conducted to compare the numerical accuracy of the IBM 7094and 360 computers. The model used in this investigation was a uniform straight bar of crosssection A and length L. The baseline model represented this bar as a single finite elementwith an end-to-end stiffness of AE/L. Then thebar was divided into equal length elements, in steps of 50, up to 300 elements as illustrated in Figure 4-8, For each

55 x\ 300 ELEMEWTS ... I......

\ 50 ELEMENTS I I i I . I I! I I I I i

"I 1 ELEMENT

MERGE-REIIUCEtRROR ACCUMULATION THISWDEL CONSISTS OF LONG AXIAL MEMBER I IDEALIZED BY 300 NODESCONNECTED BY STRINGER ELEMENTS

IBM 360 DOUBLE PRECISION (DIAGONAL TECEMS)

IBM 7094 SINGLEPRECISION (DIAGONAL TERMS) e a 0 e

SINGLEPRECISION 1 (DIAGONALTERMS)

40

I I I I I c I 0 50 100 750 200 250 300 NUHBER OF REDUCTIONS FIGURE 4-8 MERGE-REDUCEERROR ACCUMI-JLATION 4.3.1 (Continued) stepthe element stiffness was calculated, merged, and reduced to obtain a single end-to-end stiffness term for the bar. For eachmodel, stiffness error terms were calculated and plotted as a percent of the basel ine stiffnessas shown in Figure 4-8. The error was thedifference between the calculated stiffness and thetheoretical stiffness AE/L. For the 300 element model, the IBM 7094 error was 10 parts per mi 11 ion ~ while the IBM 360 error was 40 parts per mi 11 ion. Both machines were using slngleprecision arithmetic. The maximum error of 40 partsper million mayseem small. However, this is near thethreshold where results obtained from the stiffness matrix start showing accuracy problems. For example, moment coefficient diagrams may fail to closesatisfactorily. The roundoff erroralso occurred after only 300 reductions. In an actual system, tens of thousands of reductionsare often required. The accuracy obtained using single precision arithmetic on the the IBM 360 computer was not adequate for the Saturn V analysis. Conversion of key algorithms to double precision arithmetic solved the IBM 360 accuracy problem, As shown in Figure 4-8, the maximum round-off error in the uniform bar problem was reduced from40 parts per million to four parts per million by using double precision arithmetic. Another caseconsidered was that of inverting a 100th order stiff- ness matrix, with the checkperformed on the KK-1 product. Singleprecision arithmetic on the IBM 360 produced off-diagonal terms(which should be zero), in excess of unity. Double precisionoperations improved theresults such that no off-diagonal terms exceeded 0.1, but still the results could not be used. Th’emodel had to be re-idealized to improve the numerical accuracy. Theseexamples indicate the necessity for the dynamicist to understand the limitations of computer analyses, and to be alert for the symptoms of numerical round-off emor. 4.3.2 Idealization Examples The Saturn V vehicle canbe represented by severalclasses of structures.Specific examples ofthese classes are presented in the fol- lowing sections, providing practicalapplications of thetechniques dis- cussed in the preceding section. Even though the vehicle canbe physically represented by the following classes of structures, the final modelmay be simplified in certainareas. Forexample, the initialidealization of the Saturn V launch vehicle included shell modeling for all tanks,inter- stages, .and thrust structures. Subsequent correlazion with dynami c test results showed that this level of detail was not required to represent bending in the launch vehicle, al though axi symmetri c she1 l modeling was required to representlongitudinal liquid and structural coup1 ing in the tanks. Test results also showed thenecessity for a she1 1 representation in the payload, including a very fine grid in the instrument unit area to allow correlation at flight controlsensors. See Figure 4-9 for decomposition of Saturn V into classes of modules.

57 EXCEPT FOR MoNoCOQUEBULKHEADS

THRUST STRUCTURE

FIGURE 4-9 CLASSES OF MOWLES FOR THE SATURN V MODELS

58 4.3.3 She1 1 Ideal izations

She1 1 structure employed in the Apoll o Saturn W vehicle can be classifiedinto four groups: (1) ring and stringerstiffened shells, (2) honeycomb she7 Is, (3) monocoque bulkheads, and (4) thrust structures. A. Ring and stringer stiffenedshell idealization

Most of the mainline structure of the Saturn W launch vehicle consists of ring and stringerstiffened shells. An example of a ring stiffenedshell is the S-IVB forward skirt shown in Figure 4-10. Note that the 108 hat stiffeners havebeen idealized as only eight stringers having the same totalcross sectional area. The stringers on the Y and Z boundaries in the model have half of the crosssectional area of the others. The Apollo Saturn V shell structure consists of horizontal ring stiffeners,vertical "hat'' stiffeners, and thin plates. In idealizing thebasic ring stiffened shell it is advantageous to locate node lines on the rings. On Saturn V, it was found that a ring of nodes could be located at each ring stiffener without violating the plate geometry limits given in Section 4.3.1 if the nodes were placed at 15 degree intervals around the perimeter. If there hadbeen too many rings to use this technique, then the rings would be "smeared" over thedistance between the nodes. For thecase with nodes at each ring, only the membrane properties of the pl ates are required to define we1 1 conditioned 1oad paths. If more nodes arerequired than thereare physical rings, node lines are positioned between rings. The bending properties of the plates must then be included so that the mathmodel represents a kinematically stable system or the actual rings must be subdivided into a greater number of idealized rings. ~ When a shell is stiffened with verticalstringers, the axial characteristics of the stringerare included. The local bending charac- teristics are usually ignored because they contribute 1 ittle to the overall bending stiffness of the cross-section. An exception to this is when accuratelocal deformations are required. Local discontinuities in the structure (such ascutouts, doors, local stiffeners or protuberances) may or may not be a concern. If localdeformations are not required,the effects of local discontinuities can be ignored. For example, the hatch openings in the boostersare not considered in the idealization. B. Honeycomb shell idealization Much of the payload of the Saturn V is made of honeycomb she1 1s . This includes the spacecraft lunar module adapter/instrument unit area (SLA/IU) in which the flight control sensors arelocated. Results from the dynamic tests showed that a very fine grid was required in this area to allow correlation of dynamic characteristics at flight control sensors. The SLA consists of a honeycomb skin with four ring stiffeners as shown in Figure 4-11. This Figure shows the required nodalbreakdown for the SLA/IU

59 IDEALIZED

FIGURE 4-10 S-IVB FORWARD SKIRT NODAL BREAKDOWN 4.3.3 (Continued)4.3.3 region. Ring stiffenersare located at only five levels of nodes. Conse- quently,the idealized plates in the honeycomb areaare required to include the bending properties as well asthe membrane properties.

The IU is a 36 inch (91.5 cm) high honeycomb section with small ring stiffenersat top and bottom. Within theinstrument unit, "Black Boxes" are mounted on 16 thermal conditioningplates around theinterior circumference. These platesare 30 inches (76.2 cm) square and approxi- mately 1.25inches (3.18 cm) thick(see Figure 4-12). In some modes, local deformationscause achange insign of theslope between the top and bottom of theplate. These local deformations are produced by the way dynamic loads from the spacecraft are carried through the instrument unit. Up to 20 Hz, local IU dynamicshad little influence on thesedeformations. In turn, theselocal effects have little influence on overallvehicle response. Therefore,the local deformations can be obtained by applying dynamic loads from the top of the SLA, bottom of the IU,and the LM and solving an equivalent static problem. The honeycomb plate is idealized as an equivalent monocoque Plate. Equivalent membraneand bending thicknesses are calculated as follows: - Tern - TU f TL (4.15)

(4.1 6) where = equivalent membrane thickness, T = equivalent bending thickness, T and TE are the thicknessesof %e twohoneycomb face sheets, and 8~ and dg are the distances from the honeycomb neutral axis to the centers of the two facesheets. Thehoneycomb geometry is shown in Figure 4-13. The engineer must be carefulto include enough structural members to prevent any redundant freedoms, especially normal to the plate. (See Section 4.3.1 ) . C. Bulkhead idealization When the propellant tanks are full, the lower and common .bulkheads in the Saturn V structure support 90 percent of the total vehicle mass in the longitudinaldirection. Consequently, the stiffnessidealization of this structure is of prime importance in modeling for longitudinal dynamics. The interaction between breathing motion of the shell and longitudinal motion of the liquid must be understood andmodeled. The first longitu- dinal mode of the S-IC boostconfiguration at liftoff is a tank and bulk- head mode characterized by large bulging motion of the S-IC LOX and fuel tanks. Consequently, the bulkhead and tank mathematical model must be compatible with the technique that will be used to represent 1 iquid and tank interaction. The technique used in the Saturn V analyses is presented in Section 4.4.3.

61

4.3.3 (Continued) The Saturn V bulkheads weremodeled in quarter-shell detai 7 with appropriateconstraints at the boundaries. The bulkheads were idealized by a network of triangular plates. The 1 imitations on theplate geometry discussedin Section 4.3.1 were observed in definingthe nodal breakdown. Quiteoften "waffle" skins with milled stiffeners exist in bulkheads. These plates are represented by uniform plates with the same average thickness and bending moment of inertia. Inlower bulkheads, the stiffening effect of the membrane prestress from the static load is important. In ellipsoidalshells, a major section in the bottom of the tank is quite flat. Using flat plates or truncated cones to representthe curved surfaceexaggerates this flatness. Under static load,the shell stretches until the bulk of the load iscarried via membrane action. The load carrying mechanism is analogous to a stretchedcable. (See Figure 4-14.) The higherthe prestress S, the more effectivethe cable is in carryingthe normal load F. If small perturbations, SA about the loaded equilibriumposition are being inves- tigated, the cable action canbe approximated by a linear stiffness relationship

SF = -2s 6A (4.17) L

A similar 1 inear approximation appliesfor prestressed plates. If this prestress mechanism is not modeled, the load carrying capability of the idealization is limited to bending action alone in the bottom of a bulkhead. The idealization will bemuch more flexible than theactual bulkhead. The prestress stiffening effect canbe developed as an incremental stiffness matrix to add to the linear solution. (4.18)

The linear stiffness matrix is independent of tank pressure and propellant level. Only the stretchstiffness terms change for each propellant and pressure condi ti on analyzed. To avoid having to change the stiffness matrix with flight time, theprestress effect was not included in the Saturn V analyses.Instead, polynomial shape functions wereused to simulatethe correct inertia load and to distribute this reactionhigher in the bulkheadwhere the problem just discussed does not occur. This approach representedthe dynamics of the primary structure accurately; however, it did not predict tank bottom fluid dynamics adequately. The lattercharacteristics proved to be important in resolving high frequency Pogo problems. A discussion of the Saturn V tank models, their accuracy and limitations, is presented in Section 4.4.3.

63 I I I 7- -"------"-- d" I -L I' 4 1 tv dR I J

FIGURE 4-13 HONEYCOMB GEOMETRY

FIGURE 4-14 BULKHEAD/CABLE ANALOGY 4.3.3 (Continued) D. Thrust structureidealization The thrust structures for each stage of the Saturn V are different. TheS-IC structurerequires massive rings, crossbeams, and longerons to allow theloads from holddown and thrust to be equally distributed around the circumference in the tank area. The thruststructure is a ring stif- fened she1 1 with four vertical thrust posts and four vertical holddown postsas shown in Figure 4-15. The thrust and holddown loads aredistri- buted intothe ring stiffened shell through .shear in theplates. Early in the Saturn V program, the S-IC thrust ring was found to be so rigid that it was introducing numericalproblems into the model. Theseproblems were eliminated by constrainingthe ring to be rigid. This assumption was adequate for predicting overall dynamic characteris tics, but prevented the modelfrom representingthe local shear flow loads around the thrust andholddown posts. The crossbeam, a deep I-beam attached to the holddown posts, was idealized by three levels of nodes. The S-I1 thrust structure is a ringstiffened, truncated cone shell with four stiff longerons toshear out the thrust loads. The center engine is supported by atapered, pin-ended beam attached at the base of theshell as shown in Figure 4-16. Until theflight of the third Saturn V (AS-503),modeling this area was not considered to be a problem. Longitudinal oscillations observed in AS-503 flight data and reported by theastronauts dictated a detailed review of the model.The oscillations wereproduced when the frequencies of the crossbeam mode and first LOX tank mode coalesced at 18 Hz, producing Pogo in the inboard engine. The crossbeamand thrust structure model s were revised, but cor- relation with flight data indicated that analyses frequencies were still too low and the coupling betweencrossbeam and tank was not being ade- quatelypredicted. The modeling problemswere due to: 1. Uncertain end conditions of the pinnedcrossbeam under thrust, 2. Tank stiffness not representing membrane pre-stress, 3. Lack of enough flexibility in shell-liquid couplingassumptions, 4. Nonlinear crossbeamdamping. Models developed throughout theindustry haveproved inadequate for inves- tigating Pogo stability. The analyses havehad to fall back on test data to identify model inadequacies. The S-IVB thrust structure(Figure 4-17) is a ring stiffened cone attached to the lower bulkhead and consequently it has considerable influ- I ence on the bulkheaddynamics. The first coupled mode of the tank and I thrust cone occurs around 18 HZ.

65 THRUST ’POSTS

THRUSTPOSTS 0 HOLDDOWN POSTS

il LHoLDDoWNPOSTS FIGURE 4-15 S-IC THRUSTSTRUCTURE

LONGERON (TYPICAL 4 PLACES) TAPERED pu CROSSBEAM

FIGURE 4-16 S-I1 THRUSTSTRUCTURE FIGURE 4-17 S-IVB THRUSTSTRUCTURE

I 66 4.3.4 Major Component Idealization The 1unar module is a goodexample of the importance of model ing major components adequately. The LM is the two stagevehicle illustrated in Figure 4-78. Both theascent and descentstages are nearly symmetrical in mass.However, the connection between the two stages is asymmetrical in stiffness. Because of this asymmetry, longitudinaloscillation of the lunar module will induce both longitudinal and pitchresponse, as shown in Figure 4-19. The efficiency with which the LM couples pitch and longitudinal responses in the vehicle was demonstrated during the second Apoll o Saturn V flight. The frequency of thevehicle first longitudinal mode coalesced with the 5 Hz first pitch mode of the LM and spacecraft after 130 seconds of first stageboost. As coalescence was approached, the coupling became so strong that longitudinal oscillation of the LM produced more pitch than longitudinal response inthe spacecraft (see Figure 4-19). Usually cross-plane couplingcauses a decrease in thein-plane re- sponse. But, this was not thecase In this particular mode.The strong pitch and longitudinal coupling increasedthe gain.of the first longitu- dinal mode. As used here,structural gain is the dynamic response produced by a unit longitudinalsinusoidal force applied at the engine thrust pad. Themath models used prior to the secondApollo Saturn V flight neglected the LM coupling mechanism.These models predicted the vehicle would not experiencea Pogo instability as shown by the lower (coplanar) curve in Figure 4-19. During the flight, a Pogo instability did occur between100 seconds and 135 seconds of flight time. Afterthe math model was revisedto include the pitch and longitudinal coupling mechanism, a 30 percentincrease in first longitudinal mode gain was obtained. This coupled model closely reproduced the Pogo characteristics as shown by the upper curve in Figure 4-19. There are four basicquestions that require an answer before thelevel of detail in mathmodeling of componentscan be assessed: 1. Are the dynamicsof the component as a separateentity of interest from a loadsstandpoint? 2. Will the dynamics of the component affectthe primary modes of the vehicle? 3. Does the componenthave inherent mass and stiffness asym- metries that wi71 providea coupling mechanismbetween primary vehicle modes? 4. Will the component generate important localreactions on thevehicle, or affectthe local deformation in thecontrols area?

67 CONNECTION

VEHICLE ATTACH POINTS (TYPICPA FOUR PLACES)

xDESCENT STAGE FIGURE 4-18 LM ASYMMETRY EXAMPLE

0

FLIGHT TIME (SEC) FLIGHT TIME (SEC)TIME FLIGHT (SEC) TIME FLIGHT

FIGURE 4-19 INFLUENCE OF MAJOR COMPONENTS ON VEHICLE DYNAMICS

68 4.3.4 (Continued) The LM, mainstage engines, and service module tanks aretypical examples of majorcomponents. Actuallythe lunar module contains pro- pel 1ant tanks that themselves qualify as majorcomponents. The cri terfa above must be applied to the sub-components. The LM is attached to flexible structure (SLA shell) which also requires detailed modeling. The LM presents a realchallenge in s.tructura1 idealization. Correlation with ground test data is essential for this type of structure. There are no basic ground rules that canbe established for this type of structure other than to spend as much preparation time as necessary to thoroughlyunderstand the loadpaths under a17 loading conditions. The engines are often quite rigid andhave primitivefrequencies above the range of interest. However, theyshould be analyzed to determine this. Control engines and theiractuator assemblies have a low rotational frequency that is of interest and is accounted for bytwo methods. When only the effects on the primary resonances are of concern, the engine is treatedas anundamped elastic model. When the engine and its servo- actuator system are consideredas a closed loop system, the englne lateral characteristics are deleted from the dynamic characteristics analysis and included in the dynamic stability and response analyses. Ifthe engines are removed in the dynamic characteristic analysis, some meansmust be employed to includethe local elastic deformation the engines produce on the thrust structure. When the engines are removed, this deformation comes from high frequency modes that areusually not included in the analy- sis (see Paragraph 4.5.3 for an explanation of this effect). On Saturn V analyses, when the engine elastic degreesof freedomwere deleted, the engine masswas lumped on theappropriate thrust structure node to repre- sentlocal thrust structure deformation produced by theengines.

The service module tanks (Fiqure 4-20) were considered rigid because their bending frequenciesare about 20 Hz. tiowever, they are mounted on a soft, flat bulkhead that results in basicCantilever reSOnances around 6 Hz. Consequently; the bulkheadwhere the tanks attach was modeled in considerable detail to a1 low the proper inertia loads to be transmitted into thevehicle shell at the properfrequency. The service module is unsymnetrical in both massand stiffness, and is a significant coup1 ing mechanismbetween pitch andyaw vehicle modes.

Smallcomponents a1 so may requirespecial attention. For example, each thermal conditioningplate that supports flight controlsensors jn the Ill can be considered a component.Although this componentdoes not flt the criteria heretofore establ ished, the sensitivity of the local structure deformation warrantsspecial modeling consideration.

69 t +z

tLPROPELLANT TANK(TYPICAL FOURPLACES)

PROPELLANTTANKS BULKHEAD FIGURE 4-20 SERVICE MODULE TANKS

70 4.4 INERTIAMATRIX DEVELOPMENT 4.4.1 GeneralGuide1 ines In general , muchmore attention hasbeen given to the development of stiffness model technology than has been directed toward mass modeling. This is partially justified in thatthe lower frequency system character- istics can generally be representedadequately by fairly crude means. However, ordinary masslumping techniques may not suffice for higher frequency modeswhen thesehigher frequencies are near a local resonance of thestructure. As an example, four masses lumped at the quadrant points around a cy1 indrical shell structure maybe very adequate for predicting the lower frequency body bending modes but very inadequate for predictinglocal ring mode characteristics. Consequently, it is most important to keep the detail of the mathematical model consistent with thegoals of the analysis. The Saturn V dynamic analysis program has provided many examples which support the necessity for recognizing problem goals prior to model formulation. Deciding what is required to attainthese goals without excessive detail calls for sound engineering judgementbacked by adequate prel iminary analysis.Preliminary analysis includes estimating unreal istic local resonances which arise due to mass lumping techniques,simplified analyses to establish basic system properties, and estimating probable s.l;aticeffects in criticalareas of thestructure. Static effects which can perturb the basic linearity assumptions includethe effects of com- pressive loads on localpanels andbeam-column effects on lateral frequencies. These phenomena should be assessed prior to forming the dynamicmodel, to ensure that a realistic representation of the system characteristics is maintained. Establishment of the goals of the analysis will requireclose coordination with theusers of the dynamic characteristics such asthe flight control system designers,the vehicle loads analysts, and the Pogo stability analysts, to insure that their requirements wi 11 be met by the proposed model.Once the model hasbeen established, it isessential that clear comnunication be established with theengineers responsible for determining thestructural mass distribution. Inmany casesthey will not be famil iar with the dynamic requirements of the problemand will not be ableto employsound judgement in interpreting incomplete requests for mgss data. In addition to theclear communication required, it is essential that the mass data be carefully checked by the dynamic analyst prior to the performance of the dynamic analysis. Most of the engineering errors made in the Saturn V dynamic analyses involved mass data. In its most basic form the mass matrix [MI is a diagonal matrix with the systemmasses concentrated at discrete points IX) of the structure. This procedure does not account for the distributed mass effects of the structure. Caremust be taken to avoid theintroduction of artificial resonances in the frequency range of interest due solely to mass lumping procedures.

71 Forpurposes of developing inertia matrices, a large space vehicle canbe represented as some combination of the fol1 owingmodules:

1. Lightweight She1 1 Structure 2. Propellant Tanksand ContainedLiquids 3. RigidSubsections

4. Major Components

Methods of treating eachmodule, such as Guyan'smethod, (Reference 4- 4) constraintsprocedures, assumed shape functions, andmodal synthes is are presented and illustrated with examples fromthe Saturn Program. The necessity of treating the stiffness and inertia matrices consisten tly is a1 so covered.

4.4.3 ShellInertia Matrices

The primaryproblem informulating the shell mathematical model is to allow proper description of inertia loads without generating a model that is too large'from a standpointof both software and hardware capability and cost. The engineermust use judgment as to how theshell wi11 act and then manipulate the model to permit this activity without degradation ofresults. The followingconsiderations are involved in determiningthe detail of the model required: 1. Representation of ring dynamics with minimumnumber of degrees of freedom.

2. Representationof liquid and structuralinteraction. 3. Transition between she1 1 and beam modules.

4. Prevent ion of artificial local resonances due to mass 1 umpi ng

5. Control of local distortion at nodesdue to gross mass 1 umpi ng Consistentinertia matrix techniques (Reference 4-5) wereused to allow representation of ring dynamicsand eliminate unrealistic distor- tion at nodeswhere localdeformation are important (i.e. flight gyro locations).Although this technique is fully covered in the literature, examples will be given here to show theaccuracy of thetechnique. The consistent mass reductiontechnique uses the same coordinate transformation used to reduce the stiffness matrix in Section 4.1.1.

72 I 4.4.3 (Continued) The reduced stiffness in equation 4.5 is:

Using the same transformationthe reduced inertia matrix maybe expressed as : (4.19)

This consistentreduction of both the mass and stiffness matrices is known as Guyan's method (Reference 4-4). The transformationmatrix [TJ which is given by equation 4.4 is derived directly from the stiffness matrix. The assumption is thatthe real inertia loads at the reduced degrees of freedomcan be replaced by their effects at the loaded degrees of freedom in the form of inertial couplingterms. Application of this technique for the Saturn V has signiffcant advantages for the IU and SLA model, the lunar modulemodel and the comnand-service module models. A study was conducted on the short stack section (S-IVB forward skirt, IU and SLA panels) of the Apollo Saturn V vehicle. The structural idealization of theshort stack section is shown in Figure 4-11.The purpose of the study was to compare the accuracy of the consistent mass reduction method with that of the lumpedmass method. To accomplish this, a baseline model of theshort stack section was developed. This model contained 181 degrees of freedom. Thenormal modes and frequencies of this modelwere obtained andused to determine the accuracy of two dif- ferent 78th order models. I The first 78th order modelwas obtained from the181st order basel ine modelby a process of Guyan reduction. This model will be referred to asthe "Guyan Consistent MassModel". The second 78th order modelwas obtained from the 18lst order baseline mdel'by reducing the stiffness matrix and then independently relumping the mass matrix. The relumping maintained the same total massand center of gravity. This model wi 11be referred to as the "RelumpedMass Model 'I. Characteristics of the first free-free mode obtained from the 181st order baseline.mode1, the Guyan consistent massmodel, and the relumped massmodel are shown in Figures 4-21 , 4-22, and 4-23, respectively. Each figure indicates the predicted frequency and ring mode shapes at the bottom and top of theinstrument unit. The frequency of the first free-free mode of the Guyan consistent massmodel is 6.5 percenthigher than that of thebaseline model.The ring mode shapes andnode points are similar to those of the baseline model . The frequency of the first free-free mode of the re1 umped mass model is 20 percent lower than thebaseline model. Thetwo lobe ring mode shapes show 1ittle resemblance to the three lobe ring modes predicted by the base1 ine model .

73 FIGURE 4-21 RINGSHAPE OF THE 181ST ORDER(BASELINE MODEL)

FIGURE 4-22 RINGSHAPE OF THE78TH ORDER (GUYAN CONSISTENT MASS MODEL)

FIGURE 4-23 RINGSHAPE OF THE 78TH ORDER (RELUMPED MASS MODEL)

74 4.4.3 (Continued)

The frequency variation in the two 78th order models is caused by thedifference in the mass matrices. The relumped mass causes aconcen- tration of mass that lowers frequencies, while the consistent mass reduc- tionintroduces a small constraint on the system that produces slightly higher frequencies. 4.4.4 Propellant Tank Inertia Matrices Initial longitudinal math models of theSaturn V structure did not adequate7y represent interaction between the liquid and the structure in thepropellant tanks. Correlation with 1/10 scale modeldynamic test results showed a serious deficiency in mathematical modeling for the longitudinalcase. A method was developed tocorrect this deficiency. This involved relating radial and vertical motion of the contained liquid to breathing motion of the tank. When the tank walls deform, motion is produced in thecontained liquid. Formost practical problems, the change in tank volumedue to elastic deformation is large compared with volume changes within the liquid due to liquid compressibility. As a result, the liquid canbe assumed to be incompressible. In the Saturntanks, whichhave a diameter of 33 feet (10.06 m) , the frequencies of the fundamental surface waves are well below thefrequencies of the tank modes themselves. Hence, theeffects of surface wave motion can also be neglected. The liquid surface can beassumed to remain flat. Under the above conditions, a simple relationship between vertical liquid motions and shell motions can be derived.Referring to Figure 4-24, thevertical motion of all liquid particles in a given crosssection will be equal, i.e.:

AX(r, 0, x) = AX(x) (4.20) The vertical motion in any plane X = h will be equal to the volume of deformation Y(h)below X = h divided by the cross sectional area A(h).

(4.27)

To simplifythe finite element solution, the liquid canbe divided into n laminae centering about the n elastic nodal circles. The radialvelocity distribution can be derived from the conditions of incompressibility and uniform verticalvelocity (Equation 4.20). Referring to Figure 4-25 , if the liquid is incompressible, the volume of liquid contained in a cylinder originally of radius r and height X must be constant, i.e. PROPELLANT TANK

LIQUIDLAMINA

.. I \ \

I I TANKBULGING MODE ' I I LOWER BULKHEAD

FIGURE 4-,24 PROPELLANTTANK LIQUID IDEALIZATION

7- +- .. 1 -It" dr -Il- i

FIGURE 4-25 LIQUIDMOTION DUE TO TANKEXPANSION

76 4.4.4(Continued)

- dV = 0 = 2rrXdr + nr2dX (4.22a)

dr = - I-dX (4.22b) 2x

dX = - dr (4.22~) r

As dX is independentof r (Equation4.20), it can be evaluatedwithout loss of generality on the she1 1 boundary where

r = ri, X = hi, dr = Ari, dX = Axi

AXi = - -2hj Ari (4.23) ri Using Equation (4.23) to evaluateEquation (4.22b) gives

Inderiving Equation (4.24), it was assumed that the radial velocity distribution in eachliquid lamina could be satisfactorily approxi- mated by the radialvelocity at the verticalmidpoint of the 1amina. A sufficient number oflaminae were used in the Saturn V analysis to make this agood assumption. An effective radial mass mie will now be defined by equating ki neti c energi es .

(4.25)

where the intergral is taken over the volume of the ith liquidlamina, p is the mass density, and Mi is the total mass of that lamitla. From Equation (4.25), it can be deduced that the effective mass is equal one-ha1 f the total

1 mie = 2 Mi (4.26) ~~ .~~ ~ .~ . in the bulkheads , the elastic defoimations of the ‘shell were represented by power series. The radial and vertical deformations wereeach repre- sented by a power series in the dimensionless radialcoordinate ‘ij = r/rT (See Figure 4-26). N ar(7) = c anan (4.27) n=O ti Ax(T) = c bnzn (4.28) n=O The coefficients an and bn werechosen to satisfythe conditions of ax i syme try

Ar(0) = 0 (4.29) ax ap(0) = 0 (4.30) plus the displacement matching conditions arpi ) = Arj i = 1, 3, 5 (4.31)

AX(Ti) = AX1 i = 1, 3, 5 (4.32)

The equations above are satisfied by third order polynomials (N = 3 in Equations (4.27) and (4.28)). The solution for the bulkhead shell displacement components will have the form Ars(P) = (4.33)

(4.34)

78 4.4.4 (Conti nued) where the aij and bi . coefficients are determined by making the third o'rde r polynomials sahsfy Equations (4.27) through (4.32). The volume of deformation up to the midplaneof each of the bulk- head 1liauid laminae ca Ln be obtained by integrationof Equations (4.33) and (4.34). ' The vertical displacementsof these laminae can then be evaluated from Equation (4.21 ) . These displacements will have the form: F' .. Axl AX? AX; AXs 3 I S (4-35) I "i j Ax5 I Ars AX F 1 .-6 Ar; Ar; ,-I LIQUID SHELL DISPLACEMENTS DISPLA ,CEMENTS

The radialdisplacements in the midplaneof each bulkhead lamina areobtained by combiningEquations (4.24) and (4.33):

SHELLDISPLACEMENTS J

The liauid disD1acements in the c.yl indricalsection of tank above the bulkhead will be the sum of the vertica1 displacement at the topof the bul khead, AX!, plus the displacement due to radialexpansion of the cylindricalsections. From Equation (4.23)

79 I Equations (4.35), (4.36), and (4.37) linearly relate the deflection components of all the liquid laminae to the shell generalized coordinates. This linear transformationwill have the form:

k:}=Arf [ Ail] {AX!} Arj (4.38)

2nxl 2nx2m 2mx 1

The liquid mass matrix in the original liquid coordinates system has the form

CMFijI] = (4.39)

The factorof l/Z applied to the radial masses is the effective mass reduction derived in Equation (4.26). The generalized coordinates asso- ciated with these effective mass terms are the radial displacementsof the shell at the midplane of each liquid lamina. Transformation of the 1 iquid mass matrix to the shell coordinate system is accomplished using Equation(4.38).

80 4.4.4 (Continued)

(4.40)

The tank mathmodel is completed by using Equations (4.33) and (4.34) to perform an equivalenttransformation on the shell stiffness matrix. This methodproved highlyaccurate in predictingthe first two tank modes of the S-IC stage. The frequency of the first tank mode - 3.75 Hz - was predicted exactly by this methQd. Excel lent mode shape correl a- tion was alsoobtained. Themethod requires no special computerprogram, other than a standard matrixmanipulation package. The coefficientmatrices for all six Saturn V tanks werecomputed in two man-weeks. Different pro- pellantlevel conditions can be analyzed rapidly by this method.The method just derived was selected because of its simplicity and ease of application. This method is not applicable where higher tank modes, or local liquid effects such as tank bottom pressures, must be predicted accurately. However, the method could be extended to these cases by retaining more terms in the power series approximations to theshell displacements. A more accuratetechnique has been derived(Reference 4-9) and appl ied to theanalysis of S-IC tank modes. Frequencies from this solution havebeen correlated with measured flight data through the first six modes. 4.4.5 Riqid Subsection Inertia Matrices Rigid cross-sectiontransformations are based on the assumption that the plane cross-section remains plane and rigid during deformation. These transformations canbe used where shells are locally stiffened to maintain rigidity. They are veryuseful in coupling beam and shell models together. I I Figure .4-27 is a sketch of a beam-shell model interface. The shell I model interface has 24 degrees of freedomwhich consists of X, Y, and Z I displacements at each of the eight nodes.These 24 degrees of freedomcan I be expressed in terms of the 6 degrees of freedom on the beam by employing thefollowing rigid body relationships between coordinates:

Let Axc, AYc, Azc, kc, Ryc, Rzc representthe motion of the plane section assumed positive as shown in Figure 4-27 and Axn , Ayny Azn represent the motion of mass Mn. X

I I 1 I 2 \

FIGURE 4-26 GEOMETRYOFDEFORMED BULKHEAD

SHELL MODEL

FIGURE 4-27 BEAM-SHELLMODEL INTERFACE

82 4.4.5 (Continued)4.4.5

AXn = AxC + (r sin 8,) RyC - (r COS en) Rz, (4.41a)

AYn = AYc - (r sin en) Rx, (4.41b)

AZn = AZc -+ (r cos e,) kc (4.41~)

Or expressed as - 0 r sin el r cos el r sin 81 0 0

r cos 81 0 0

I

1

I 0 r sin e8 -r cos e8 r sin e8 0 0 r cos e8 0 0 - (4.43)

Transforming the mass matrix usingEquation (4.43) 1. Sums the eight masses, ml .... m8, as the effective mass on eachof the Axc, ayc anddzC coordinates, 2. Sums the mr2 terms as the effective rotationalinertia about the X axis. 3. Sums the m(r sin e)2 terms as the effective rotationalinertia about the Y axis . 4. Sums the m(r cos e)' terms as the effectiverotational inertia about the Z axis. 5. Cal cul ates the appropriate inertial coup1 i ng terms in the event the masses arenot equal. linear constraint relationships can still be established through the use of polynomial shape functions. This technique requires that all co- ordinates involved in thetransformation be on a curve or surface which is described by polynomial functions thatsatisfy appropriate boundary condi ti ons . The following calculationsillustrate the steps which lead to the "warped crosssection" transformation. , Thisapproach permits thecross section to warp either anti-symmetricallyacross the neutral axis or act as a plane section, whichever most closely approximates thenatural struc- turalaction. This example is concerned with vertical displacements (AX) due to rotation about thepitch axis ,only; however, the method is equally adaptable to other displacements.Application of the following trans- formation to the mass matrixwill yield inertial coupling terms which representthe effective rotational inertia of thecross section. These calculations weremade to develop verticalcross sectional constraints used inthe original S-IVB/IU/spacecraft fullshell model. Themodel crosssection was given freedom to rotate in both pitch and yaw. The constraint equations were writtenin terms of two freedoms for eachmotion so a1 1 "AX" motions at a particular level were writtenas functions of fourgeneralized coordinates. The constraintequations were flexible enough to allow a plane section or a warped plane symmetrical across one axis and anti-symmetric acrossthe other. If it appears desir- able, a fifth freedomwhich allows theentire cross section to translate vertically can easily be incorporated.

The following example is for vertical displacement AX, due to rotation about thepitch axis only. The generalexpression forthe cross section deformation using thenotation in Figure 4-28 is:

84 (4.45c)

The constants Bo, AI, and A3 arefunctions of the two independent variables AI and 82. 9nce theyare solved from the Sivenbomdary con- ditions, the displacement AX can be written as: (4.46)

The relationship between thevariable; AI ,A~and a typical dependent variable AX^ at coordinate acan be solved by substituting the an value into the polynomial . fhis process leads to a 1 inear coordinate transformation of the form:

=[ (4.47)

4.4.6Major Component Inertia Matrices Inertiamatrices are formed for componentssuch asthe lunar' module just as they are for primary structure. When the component is rigid compared with the structure it attaches to, a simpletransformation is used to express the motion of the component center of gravity motion in terms of the motion of its attach points. An example of how the S-IVB engine was represented in the final mathmodel follows. The thrust structure and engine structure wereassumed rigid. The actua- tor was flexible and theengine could pivot about the gimbal. Figure 4-29 shows theessential elements involved. The redundant freedom Ry2 was expressed in terms of the retained freedoms q, Ryl and A22 in matrix form in Equation (4.48).

85

. 100 010 (4.48) 001

-7- 0-1 a a

This transformation was used totransform the engine.massmatrix. 4.5 VIBRATIONANALYSIS AND MODAL SYNTHESIS 4.5. 1 General This sectiondiscusses how the math models of stiffness and inertia characteristicsare combined to form adynamic model of the vehicle.It alsocovers special techniques, suchas modal synthesis, used to simplify this model.Additional material is presented on damping considerations used in modeling . 4.5.2 EigenfunctionSolutions and Modal Orthogonal i ty Once the stiffness and mass matricesare generated, the system canbe solved for its characteristicroots and vectors. The undamped systemequation of free vibration is:

NxN Nxl NxM NxlNxl There are N independentsolutions to this homogeneous systemof equations. Becauseof the speedand the accuracyof the matrixtransformation technique, it is numericallyand economically feasible to obtain all of the eigenvalues from even largeorder matrices. As a result, a direct formu- 1 ationof the eigenvalue problemcan bemade. In general , the inertia matrix will be nonsingular, even for a free-free system.Consequently, Equation (4.49) can be premul tiplied by the inverse of the inertia matrix togive (4.50) 4.5.2 (Continued)

If theequation above is rooted,the eigenvalue routines will converge tothe highest frequency solutions first. This means thatthe roots Of greatestinterest, the low frequency solutions, cannot be ob- taineduntil after all the higher frequency solutions havebeen converged to and eliminated. However, thematrix transformation technique can rapidly and accuratelyobtain all the modes involved. It isnot neces- sary to eliminate the rigid body modes prior to rooting the matrix, as it is when matrixiteration methods are used. If rigid body (zerofrequency) SO1 UtiOns are included in thesystem, these wi 11 a1 so be solved for by thistechnique.

A modifiedHouseholder routine is used to obtain the eigenvalues and eigenvectors Of theSaturn V vehicle.This routine is programed for the IBM 360 computerusing double precision arithmetic (Reference 4-10). The routine will obtain all the roots of a 300thorder system in 75 minutes ofcomputer time. This includes calculation of natural frequencies, mode shapes, generalized masses,and generation of theplot tapes.

Another methodused to obtain the mode shapesand frequenciesof systems Up to 130th order is the iterative QR transformation scheme derived in References 4-11and 4-12. The dynamic matrixis first converted to the upperHessenberg form (zero in all positions i, j for i > j + 1 ) using elementaryrow and column manipulations. A sequence of mathematicaltrans- formationsare then applied to the dynamic matrixwhich causes thediagonal elements to converge tothe desired eigenvalues. The largesteigenvalue, whichoccupies the Nth diagonal position in an Nthorder matrix, will con- vergethe most rapidly. If theformulation of Equation (430) is rooted, the first eigenvalue will correspond to the largest value of wzwhich will be thehighest frequency root. This routine will obtain mode shapes, fre- quencies,generalized masses and generatethe plot tape for a 130thorder system in 15 minutes of computertime. The requirementsfor a compositestructural dynamicmodel and a corresponding vibration analysis arise from the necessity to ensurethe integrity ofthe vehicle. The natural and inducedenvironment to which thestructure is subjected contain some vibrational power COntent~VerSus frequency. The fundamentaltask ofthe dynamicist is to ensure that vehicleresponse to these environments .does not become CataStrOPtlic. A detailed knowledgeof thestructural dynamic characteristics is thus neces- sary to define frequency separation and/or is01 ation required to el iminate near-resonancy f ai1 ures . Structural dynamic characteristicsperform a second usefulfunction: that of providing a generalizedcoordinate system within which the dynamic problemcomplexity is reduced.This normal mode methodcan be ViSua1 i zed as 8 processof representing the loaded deflection shape (static and dynamic) byuse of a truncatedseries. The similaritytransformation from Structural togeneralized coordinates decouples the equations of motion, which greatly facilitatestheir solution. Additionally, the order of thesystem of

88 4.5.2 (Continued) differentialequations can be substantially reduced by retaining only those modes that contribute to the response problem being solved. The number of modes which are required to simulate the dynamic behavior of the structure depends on three major factors. 1. The frequencycontent of the imposed forcing function; a1 1 modes within the frequency spectrum must be candidates for inclusion in the problem. 2. The quasi-steadydeflected shape of thevehicle may dictate that similar appearing modes be included in the problem to obtain the proper static solutions. 3. Particular modes may be re,quired in the analysis simply because these modes,a1 though lightly excited, produce unusually largeloads and/or acceleration at particular vehicle stations. Examples of this 1ast case did occur on the Saturn V. The 1 iquid and structure coup1 ing in the propel I ant tanks produces a characteristic tankbreathing or bulging mode, which obviously contributes significantly to local bulkhead and Y-ring loads.Inclusion of this mode in the liftoff and rebound response analysis is thus mandatory if bulkhead pressures are required; however, this mode contributes little to accelerations in the payload of the vehicle. Determining which modes canproduce large local effects is accom- plished by inspection-of modal gain characteristics. Modal gain is obtained directly from the dynamic characteristics and is by definition the (static) ratio of output to input. Depending on the nature of the problem, output can be an acceleration, internal load or liquid pressure; input may be a generalized force or displacement. During the Saturn V program severaltypes of gain were used to identify dynamicmodes of interest; these included the instrument unit control system atti tude/attitude rate gyroscope gains (modal slope per control thrust force), the comnand/service module interfaceload gains (modal bending moment per unit thrust) and lower bulkhead pressure gains for thrustoscillations. Calculation of these types ofgain factors is incorporated directly in the computer routine for vibration analysis. Frequency response plots are a convenient way to evaluate the composite modal gain characteristics. Early in the Saturn V program, it was judged advantageous to include thepropellant slosh behavior in the structural dynamic analysis. Subse- quent eigensolutions of the system revealed slosh modes which were simply linear combinations of rigid bodyand flexible modes. Defining slosh stability margins was extremely difficult using modes from the structural dynamic analysis. Consequently, in the latter stages of Saturn V design

89 4.5.2 (Continued) assurance studies,slosh masseswere treated as a frozen solid inthe structural model.Spring-mass slosh modelswere then included inthe dynamic response and stabil i ty analyses.

The slosh modes of thesix Saturn V mainstage tanks are obtained from thedigital computer routinedescribed in Reference 4-13. This pro- gram was developed from variationalprinciples. It is based on using the sum of shallow tank and deep tank model solutionsas a set of velocity potentials. The unknown coefficientsassociated with these modal functions are solved for by a Rayleigh-Ritz procedure. The tanks are assumed to be rigid, which isvalid for the Saturn V, sincethe ring mode frequencies are two orders of magnitude higher than theslosh frequencies. For each slosh mode retained in theanalysis, an equivalentspring mass system is constructed. The mass, mn, and attachstation x,, for each of these mechanical oscillators is obtained by equating the totalforce and moment of the mechanical and 1iquid systems. Step-by-step procedures for devel op- ing the parameters associated with the spring-mass analogy are presented in Reference 4-14. The entire operation of the dynamic characteristics analysis hinges on the usage of theresults. Requisite accuracy in sensitiveareas mustbe definedas early in the development as feasible: for the Saturn V program, thesesensitive regions included thruststructure/gimbal block areas; IU localdeformations; and precisedefinition of modal bending moment and shearcontributions in the forward portions of thevehicle. Synthesis of the dynamicmodel is accomplished within the frameworkof these requirements. 4.5.3 Modal Synthesis Modal synthesis is a process whereby a complex structureis divided intosections and the dynamic properties ofeach sectionrepre- sented by selected modesfrom thatsection. Themodes from each section mustbe chosen carefully to make sure that they can represent deformations produced by reactions from theadjoining sections. The following example i 1 lustrates the importance of these reaction effects. One of the prime goals of the Saturn V test program was to develop mathemagicalmodels which could accurately predict the structural gainrequired to perform control system analysis. One of theessential parameters required was thelocal modal slope at the flight control gyro which is locatedin the Saturn V instrument unit. The gyro location is shown on Figure 4-30. A detailed math modelwas developed for theshort- stack,consisting of the S-IVB forward skirt, IU, and SLA (see Figure 4-11 ). A free-free modal analysis of this modelwas made. Modes up to 45 Hz in frequency wereused to represent this model in theanalysis of thetotal vehicle. This model accuratelypredicted the vehicle centerline slope, but failed to predictthe local control gyro slopeobtained from test (see Figure 4-31). An investigation showed thedifference in centerline and control gyro slopes wasdue to local deformations inthe IU. These

90 FIGURE 4-30 CONTROL GYRO LOCATION

91 ducted through the IU. When a modal analysis wasmade of the IU and SLA with the CSM removed, theselocal deformations did not appear in the modes below 45 Hz. Consequently, the modal synthesissolution effectively eliminatedthe local effects being sought. An interimsolution for the local deformations was obtained by: 1. Obtaining the modal displacements at the top of the SLA and at the bottom of the S-IVB forward skirt from the dynamicmodel of thetotal vehicle. 2. Determining the inertia loads forthe short stack model, using theaccelerations from the total model. 3. Applying the boundary displacements and inertia loads from steps 1 and 2 to theshort stack stiffness model. 4. Solving the static deformation problem toobtain the local deformations of the IU. Themodal synthesissolution was later used successfully by cantilevering the shortstack model at the bottom of the S-IVB forward skirt and accounting forthe CSM reactions at the top of the SLA. This approachproduced thedesired local deformations in all modes of interest. Cantilever modes are better than free-free modes for mostmodal synthesis applications. When using modal synthesis,the modesmust be chosen carefully so that deformations produced by theattaching structure are represented. Themodal models should be loaded with static forces at the interface to make surethe resulting deflections are nearly identical to theoriginal model. When the size of the dynamicmodel exceeds the capabil i ty of availableeigenvalue routines, modal synthesis techniques canbe used to obtain a solution. Given an eigenvalueroutine whichcan solve an "n" degree offreedom problem, thestructure canbe divided into secti,ons which contain less than n degrees of freedom as illustrated in Figure 4-31. The flexiblecantilevered and appropriaterigid body modes are then determined. In determining theflexible modes of thecantilevered sections, all of the inertia effects of the system above thecantilevered location should be includedin that section. For example, theinertia effects of Sections 2 and 3 should belumped at the top node of Section 1 , before thecantilevered modes of Section 1 are determined. A1 so, the inertia effects of Section 3 should belumped into the top nodeof Section 2 beforethe cantilevered modesof that section are determined. This enables the modes of interest of thetotal coupled structure to be obtained using a small number of cantilevered modes. Before the mass matrides are coupled togetherthe inertia effects that wereadded to the top nodes of sections 1 and 2 should be accounted for. The generalized mass and

92 FREE-FREE SYSTEM CANTILEVERED SYSTEMS / / / / / / / / / SECTION I 3

m

SECTION 2

m

SECTION 1

> II I I \ \ \ RIGID BODY RIGID BODY FLEXIBLE MODES ROTATI:; ION TRANSl.ATION L FIGURE 4-31 ILLUSTRATION OF MODAL STACKING TECHNIQUES from the stiffness matrices and thesections coupled together using modal synthesis. If the number of modesused to couple thethree sections together is less than' n onecoup1 ing is required, Present Saturn V analyses do not requirethe use of modal synthesis techniques because accuratelarge order eigenfunction routines havebeen developed to handle the total vehicle coupled dynamic matrix. 4.5.4 Evaluation Evaluation of the analysis results relies heavily on experience and judgement. Rough calculations to determine approximately the first modal frequency for each axis analyzed and for large components should bemade before theanalysis starts. These frequencies help in evaluatingthe results of theanalysis. Ifthe structure is unrestrained,the numerical accuracy of thesolutions canbe partially checked by evaluatingthe system dynamic shear andmoment closure. With no externalforces on the system or any grounded springs, both the shear and moment values should sum to zepo in each mode if the structure is in equilibrium.If closure is not achieved the problem will be either a poor eigenvalue solution or an incorrect matrix. A lumped parameter model will give resonances which are not realistic but arisesolely from the lumping procedure. The analysisresults must be evaluated to be sure that these lumping resonances either lie above the frequency range of interest or their presence does not adversely affect the dynamic characteristics being sought. The eigenvalue solution is checked by the orthogonal i ty condition of the modes. This is accomplished by checking the off-diagonal terms of thegeneralized mass [$]T[M][$] and stiffness [4]T[K][+]. The off-diagonal terms aretheoretically zero when the modes are orthogonal. In realitythe off-diagonal terms are never zero because of numerical error. A good rule of thumb is that sixorders of magnitude difference between diagonal and off-diagonal termsshould be a minimum objective for the mass matrix, with threeorders of magnitude required for thestiffness matrix.Plots of the mode shapes are necessary for evaluating whether, the modes satisfy Physical considerations. The eigenvalue solution of a large free-free system canproduce small negativeeigenvalues for therigid body (zero frequency) modes. This is caused by numerical errorsassociated with thelarge number of operations required to develop the stiffness matrix and to solvethe eigenvalue problem. The eigenvectorsassociated with these rigid body modes should beexamined to make sure they are acceptable rigid body displacement vectors. The off-diagonal terms in the rigid body rows of thegeneralized mass matrix should be at least six orders of magnitude smaller than the diagonal terms in those samerows. Normally, if the absolute value of thenegative eigen-

94 4.5.4 (Continued) values in frequency (w) units is less thanradians/second, the eigen- vectors should be Val id rigid bodymodes. 4.5.5 Establish Tolerance Tolerances are determined for two parameters of the analytical data, frequency and gain. The frequency tolerances for theSaturn V flight vehicles are assumed to be the same as the percentdifference between the analysis and fullscale test data. The gaintolerances are obtained by determining the differences between the analytical and test gains and ex- pressing them aspercents of the maximum analytical gain. For example:

GT = Gt - Ga (4.53) Gmax a where GT = Gain tolerance

Gt = Gain from test data Ga = Gain from analyticaldata

Gmaxa = Maximum gain from analytical data

Gain is defined in Section 4.4.2. 4.5.6 Damping Considerations On the Saturn V program the dampingwas assumed to be proportional. This allowed the damping to be introducedas an equivalentviscous damping factorapplied to each mode. These equivalent modaldamping factors were obtained from full scale tests. Damping is not only dependent oncomponent .construction, it is alsostrongly influenced by load,reference Section 5.2.4. In the' Saturn V vehicle,typical equivalent viscous damping ranges from 0.4 percent in the S-I1 tanks to three percent in the spacecraft and to 70 percent in theengine servoactuator system. Where local damping differences of this magnitude exist, the normal assumption of modal (uniform) damping is invalid. In this case localamplitude and phase characteristics predicted bymodal damping techniques willnot give acceptable accuracy. Readily usabletechniques need to be developed for estimating and includingnonproportional damping in structural dynamic analyses. Math modeling of the S-I1 crossbeam and LOX tank is not accurate enough to support Pogo analyses. Much of the predictiontolerance is asso- ciated with the damping uncertainty. There is no advantage to developing

95 is handled by a-gross estimate that may be in error by 1000 percent. Struc- tural damping characteristics need to be investigated to where material Properties, geometry, type of joints, loads, and temperatures could be the input that would allow a damping matrix to be generated in much the sameway a stiffness matrix is generated now. 4.6 SATURN V MODEL EVOLUTION Early in the Saturn V dynamic analysis program, it was decided that localstructural and inertial effects could have a significanteffect on the response of controlinstrumentation. This decision precluded the use of beam type mathematical modelswhich would give centerline motions but would not be able to predict motions on the surface of the structure where thecontrol instrumentation was to be located. The importance of localeffects set the requirement for a three-dimensional shellrepresen- tation of the Saturn V structure.Initially, it was not possible to model thestructure in three-dimensional shelldetail throughout because the computer computational capacity was not available. Therefore , the original modelswere a combination of beams and three-dimensional shells whichwere intended to model both overall dynamic effects and local effects at the flight sensorlocations. As the program progressed, both theengineers and the computer programswere uprated through experience and the model evolved accord- ingly. All the mathmodels used in thepre-test analysis were coplanar models (with the exception of a full shell S-IVB/IU/spacecraft model which incorporatedpitch-longitudinal coupling capability). Available data in the early stages of the program indicated that thevehicle was symmetrical in both mass and stiffness. This meant that thestructure could be represented by modelingone quadrant of theshell andimposing symmetrical -antisymetrical boundary conditions to represent the effect of theother three quadrants. These boundary conditions did not allow any cross plane coupling and thevehicle was effectively constrained to planar motions. As more and more information became available and as testing progressed, it became apparent that significant asymmetries did exist in the Saturn V structure. Theseasymmetries caused coupling between all planes of motion which could not be predicted by thequarter shell models then in use. The AS-501 and AS-502 fl ights brought to 1 ightstrong pitch and longitudinal coupling with thelunar module actingas the primary coupling mechanism. As a result of these two flights, along with the introduction of a heavier and more flexible payload on AS-503 and sub- sequent vehicles, it was decided to develop a math model with freedom in all six planes which could fullyincorporate cross plane coupling capability. When the first coupled modelwas initiated theeigenvalue capabil i ty was 1 imi ted to 130th order problems so it was necessary to use the modal synthesis approach discussed in Section 4.5.

96 4.6 (Continued) TheAS-503 and AS-504 vehicles were analyzed using modal synthesis techniques. The effective model size was 291st order. Development of a 300th order eigenvalue capability was completed prior to the AS-505 ana1ysi.s. This capability proved to bemore efficient and less subject to engineering. error so it was adopted and is thetool presently being used for Saturn V dynamic analysis.Figure 4-32 shows the evolution of these models. Advan- tages and disadvantages of each model are shown in Table 4-1. 4.7 COST AND ACCURACY 4.7.1 Cost Cost and flow time are prime considerations in mode7 development. As an aide for future Governmentand Industryengineering groups involved in math modeling of complex structures, a0 estimate has been made of the costs of developinq thepresently used Saturn V model.In forming this estimate, the following assumptions have been made: 1. Adequate software and hardwareprogram capabil ity exists. 2. The current three-dimensional model is the base 1 ine. 3. No iterations havebeen considered due to ideal ization errors. 4. Structural drawings are avail ab1 e. 5. No evaluation of data is involvedexcept the mechanical checks. 6. Mass data breakdowns of each stage andpayload module are available, but mass data requires redistribution to nodal network. 7. No sl lowance has beenmade for the learning curve. 8. Computer hours are based on the IBM-360 system. 9. NQ documentation of data has been included. 10. Manpower does not include anyal7owance for clerical work or supervision. The planning estimate is shown in Table 4-11. Computer hours, manpower resources in manmonths,and flow time are shown for the stiffness matrix development, the merge of componentmodules and the generation of dynamic characteristics for one time. As previouslystated, this cost

97 +- 30 DOF 300 - 400 DOF DOF + I t

+e

APOLLOSATURN V BEAM-ROD BEAI.l-ROD/QUARTER QUARTERSHELL THREE-LJIMEHSIONAL SPACEVEHICLE D1ObEL SHELL MODEL MODEL MOUEL

FIGURE 4-32 MATH MQDEL EVOLUTION MODEL ADVANTAGE DISADVANTAGE Beam-RodModel Good centerline charac- No interp1 ane coupl ing teri s ti cs , through first a1 1owed. Inaccurate four modes. slopes at F1ight Gyro Locations . Poordynamic character istics above 5 Hz.

Beam-Rod/Quarter Simp1 i ci ty Different model re- Shel 1 Model quired for each flight sensorlocation.

Quarter Shel 1 Good centerline, com- Model 1imited due to pay- Model ponen t characteri s ti cs loadrepresentation, no to 10 Hz, efficient use interplane coupli'ng of freedoms , minimum representation. Different cost, good reprrsenta- models required for pitch, tion of instrument unit yaw, 1ongi tudi nal , and region. torsionalanalysis.

Three-Dimensi onal Good characteristics to Configurationchanges are Model 20 Hz, interplace costly to make due to coupl i ng represented , large size. efficient use of degrees offreedoms, goodpay- 1oad representation , good representationof i nstrument uni t region. Coupled pitch, yaw, 1ongi tudi nal , and tor- si onal characteri s ti cs obtained from single analysis.

TABLE 4-1 ASSESSMENT OF SATURN V MATH MODELS

99 SATURN V COMPUTER FLOW TIME MODULE HOURS MANMONTHS (MO.1 Launch Escape Sys tem 1 .o 1 .o 1 .o Command Modu 1 e 2.0 2.0 2.0 Servi ce Module 4.0 6.0 3.0 Saturn LM Adapter Instrument Unit 20 .o 30.0 6.0 Lunar Module 6.0 12.0 4.0 S-IVB Stage 5 .O 4.0 2.0 S-I1 Stage 6.0 5 .O 2.5 S-IC Stage 6.0 5.0 2.5 TOTAL 50.0 65 .O 13.0 *

Sys tem Merge 2.0 0.5 0.5 Mass Distribution 6 .O 8.0 4.0 Dynamic Characteristics Analysis (1 case) 3.0 0.5 0.5

* Considering work done in series

TABLE 4-11 MATH MODEL DEVELOPMENT PLANNINGESTIMATE 4.7.1 (Continued) breakdown is based on experienced personnel and with theprovision that the model nodal network is defined. For theengineering group with average talent faced with a new program such asthe Saturn V, factors should be appl ied to the estimates shown. It is suggested that a factor of 2 .O be applied to manpower,computer hours and flow time to bring an organization up tothe plateau of the learningcurve. Even experiencedengineers in this type of analysis must iterate solutions many times to determine the proper structural idealization with a practical number of degrees of freedom. A factor 2.5 is recommended to be applied to account for the formul ation of the optimum nodalnetwork. Checkout of new computerprograms is not normally a large expense, however, the flow time is considerable even with comparatively minor program changes. 4.7.2 Accuracv The mathmodel was accurate in predictingthe overall modal properties of the full scale test vehicle for the modes of interest. The frequencies of the first 4 pitch modeswere predicted within four percent, and the mode shapes were predicted with equal accuracy. A correlation of the full scale analysis and test results of the first four pitch modes for 100 percentpropellant condition is presented in Figures 4-33 and 4-34. The first threelongitudinsl modes for 100 percent propellant condition were predicted with the same accuracy as thepitch modes. A correlation of these modes are shown in Figures 4-35and 4-36. The predicted frequency of thefourth longitudinal mode was seven percent lower than that from the full scaletest. The corre- lation of the mode shapes was not very good for this mode.The corre- lation is shown in Figure 4-36. The fourthlongitudinal mode is extremely weakand has little effect on the response characteristics of thevehicle. A frequency response plot of the S-IC outboard gimbal is shown in Figure 4-37. The fourth mode contribution is shownby the small peak near 7.5 Hz. An extensivecorrelation of pretestanalysts and test results is contained in Reference 4-15.

101 FIRST PITCHWDE SHAPE FIGURE 4-33 COMPARISON OF FULLSCALE PITCH TEST AND ANALYSISRESULTS MODES 1 AND 2 - 100 PERCENTPROPELLANT VEHICLE - FULLSCALE TEST FREQUENCY 2.55 Hz ':~~T{o" -1- FULL SCALE ANALYSIS FREQUENCY 2.65 Hz

-1 .o -0.5 0 0.5 THIRD PITCH MODESHAPE FIGURE 4-34 COIIPARISON OF FULLSCALE PITCH TEST AND ANALYSISRESULTS - MODES 3 AIiD 4 - 100 PERCENT PROPELLANT

4 0 CJ VEHICLE TEST VEHICLI FULL SWE TEST FREQUENCY 4.46 hz -FULL SCALE FREOUENCY 3.75 HZ STATIG - STATIW flN.\ FULL w- -9- FULL SALE ANALYSIS FREQUENCY 3.75 HZ - SCALE ANALYSIS FREOUENCY 4.39 Hz 4Cm L- 6 I

2

2OOo I 8

lo00 I 4

a- c I . -1.a 4. I FIRST LONGITUDINALMODE SHAPE StCONU LONGITUUINALMoUt SHAPE FIGURE 4-35 Cot-lPARISONOF FULL SCALE LONGITUDINAL TEST AND ANALYSIS RESULTS - HODES 1 AND 2 - 100 PERCENTPROPELLANT -1 .o -0.5 0 0.5 1.0 UCJ -2.0 -1 .o 0 1 .o 2.0 THIRD LONGITUDINAL MODESHAPE FOURTH LONGITUDINAL MODESHAPE FIGURE 4-36 COI4PARISOFI OF FULLSCALE LONGITUDIrjAL TEST AND ANALYSIS RESULTS - MODES 3 AND 4 - 100 PERCEI4T PROPELLANT -I 0 Ul FREQUENCY (Hz)

FIGURE 4-37 LONGITUDINAL FREQUENCY RESPONSE OF OUTBOARD GIMBAL - 100 PERCENT PROPELLANT

106 REFERENCES 4- 1 DocumentD5-15204, Saturn V DTV Pre-Test Analysis Methods, The Boeing Company, Huntsvi 1 le, Alabama, February 26, 1965. 4-2 Martin, H., Introductionto Matrix Methods ofStructural Analysis, McGraw-Hi 11 , New York, 1966. 4-3 Greene, B. , Strome, D. , and Weikel , R., Application of the Stiff- ness Method to the Analysisof Shell Structures, Paper presented at the AviationConference, American Societv of Mechanical Engineers,January, 1967. 1 4-4 Guyan, R. J., Reduction of Stiffness and Mass Matrices,AIM Journal ,- Vol. 111, P. 380, 1965. 4- 5 Archer, J., Consistent Mass Matrix for Distributed lvlass Systems, Journal of the StructuralDivision, American Societyof Civil Engineers, Volume 89, August,1963. 4-6 Przemieniecki , J. , Theory of Matrix Structural Analysis, McGraw- Hill , New York, 1968. 4- 7 Pestel, E. and Leckie, F., Matrix Methods in Elastomechanics, McGraw-Hill, New York, 1963. 4- a Hurty, W. and Rubinstein, M. , Dynamics of Structures,Prentice- Hall, Inc. , Englewood Cliffs, New Jersey, 1965. 4- 9 Palmer, J., and Asher, G., Calculation of Axisynmetric Lcngitu- dinal Modes for Fluid-Elastic Tank41 1age Gas Systems and Comparison with Model Test Results, AIM Symposium on Structural Dynamics and Aeroel astici ty, Boston, September 1 , 1965. 4-1 0 Wilkinson, J. H., Householder's Method for the Solution of the AlgebraicEigenproblem, Computer Journal, Vol. 3, 1960. 4-1 1 Francis, J. , The QR Transformation -- A Unitary Analogue to the LR Transformam October, 1961. 4-1 2 Francis, J., The QR Transformation -- Part 2, The Computer Journal, Volume IV, January, 1962. 4-1 3 Lomen, D., DigitalAnalysis of Liquid Propellant Sloshing in Mobile Tanks with Rotational Symnetry. NASA CR-230, 1965. 4-1 4 Lomen, D., Liquid PropellantSloshing in Mobile Tanks of Arbitrary ShaDe. NASA CR-222. 1965.

107 REFERENCES (CONTINUED)

4-1 5 Document D5-15722 a Dynamic Test Vehi cl e Tes t-Analysi s Correl ation, The Boei ng Company a/

L 1 08 SECTION 5 DYNAMIC TEST TECHNOLOGY 5.0 GENERAL The purpose of this section is to presentthe experience and know- ledge gained from the Saturn V full scale dynamic test program on the technology of dynamic testing. Beginning with theestablishment of test requirements,the material presented includes a discussion of digital data reductiontechniques and concludes with test dataevaluation procedures. The major objective of the dynamic test program was to verify Saturn V math models.These mathmodels were then used to predict charac- teristics of each flight vehicle with a high degree of confidence. This test-analysiscorrelation cycle isillustrated in Figure 5-1. This figure is a flow chart showing that the full scale math modelwas developed using the 1/10 scaleanalysis and testresults. This modelwas used to predict the dynamic characteristics of the full,scale test vehicle. Thedynamic characteristics were correlated with test results and the math model revised until good correlation was achieved. The mathmodel was then considered testverified. This verified model then became thebaseline model for the Saturn V family of flightvehicles. Tolerances were derived from the correlation. The variousstructural andmass differences between the test and flight article were then included in the flight model and the test system restraints removed. Since thedifferences noted above repre- sented a change in input data rather than a mathmodel change, the flight article model could also be considered testverified. The basiccriterion was that as long asthe article being tested could be described analytically, then defineddeviations from the flight article wouldbe allowed. However, thedifferences between pretest analysis and flight analysis mustbe kept to a minimum. A considerabletechnical judgment is required to determine what differences can be bridged analytically. This judgment must be tempered with thepractical aspects of cost and schedule. For example, thelocalized response that occurred in the IU during full scale testing showed that the flight configuration IU should havebeen simulatedas closely as possible. This is due to the extreme sensitivity of the dynamics at the flight control gyro location to local massand stiffness changes. TheIU simulation on the full scaletest vehicle proved to be adequate for the job. However, the simulated LM (which represented the massand C.G. locations, but not the dynamicsof the LM) proved to bean inadequatesimulation of flight hardware. Using this simulator,the bridge between the full scale test and analysiscorre- 1 ation mathmodels and flight modelswas too great. As a result the required confidence level of the flight analysis was degraded where LM activity was significant. 5.1 TEST REQUIREMENTS A most important item in any major dynamic test is the definition of the test requirements. Test requirements are not a pattern copied from another similar test, but must be carefully considered in light of the objectives. Every facet must be considered to determine whether theobjec-

7 09 I TEST-ANALYSIS

FLIGHTVEHICLE 1 ANALYSI; I -/- ”“ \ \ 0 I /’0

REV I SE MATH DTV TEST MOD TEST DTV EL S / ! -/ I I f t I /

0 , I , 1/10 SCALE MODEL BUILD SATURN V ANALYSIS - TEST -Ir MATH MODELS I I 1 I

FIGURE 5-1 TEST ANALYSIS PHILOSOPHY

110 5.1 (Continued) tiveswill be met, consideringadequate technical output, cost, schedule andvolume of datacollected. Experience with thefull scale test indicates thit theuser organization mustbe responsible for certifying the technical adequacy of the test in terms of quality and quantity of data. User require- mentsmust be clearly defined and understood before test requirements can be written. In particular,the dynamic parameters thatare important to each user mustbe identified, and allowabletolerances specified. The sensitive parameters forthe Saturn V vehicle are listed in Table 5-1 for Pogo, loads, and flight controlstudies. The userorganization must specify the accuracy with which each of these parameters must be determined before requirements for math models and dynamic testing canbe defined. For example, if a1 1 accuracy requirements fa1 1 within the mathmodel confidence band, thereis no requirement for dynamic test. All affectedorganizations must have insight into the'requirements and be able to critique requirements that havebeen initiated by the prime organization. This is a basic considerationin insuring that the test will satisfyall affected organiza- tions. Test requirementsshould be a top levelcontrol of every important aspect of the test. The fullscale test requirements covered testobjectives, test configuration, test facility, data acquisition and reductionsystems, tests that were to be conducted, pretest analysis, data correlation, and reporting of test results. All of.these requirements weremerged into a single source document that controlled the test program (Reference 5-1 ) . Requirements could not bechanged without a revision to this document being made. Consequently, only the major items were includedin thissingle source document. This documentwas alsothe basis for contractual actions in deter- mining thenegotiated cost of the total test. Program level requirements mustbe identified early inthe program. If a program covers severalcontractors and/or government agencies, a mechanism for maintainingrapid data flow needs to be established at the program management level. One of the major problems encountered initially on the full scale program was the difficulty of getting spacecraft structural drawings. Normal data acquisition channels fromone NASA center to another required supplemental action to support therequired schedule.

5.1.1 TestObjectives I_ The basic objective of the full scale test program was to verify mathematical models.These mathematical models could then beused to obtain structural dynamic sourcedata for use. in flight control system analyses, flight and ground' loadsanalyses, and Pogo stability analyses. Some secondary objectives were required-tosupport the mathematical model.These objectives i ncl uded: 1 . Determination ofdynamic characteristics for major components such asengines, first stage engine fins and fairings, and the first stage propellant lines. TABLE 5-1 SATURN V SENSITIVE PARAHETERS

POGO LOADS FLIGHT CONTROL

1. Frequency Range 1. Frequency Range 1. Frequency Range

2. Longitudinal Mode FrequenciesP, Pitch (Yaw) ModeFrequencies 2. Pitch/Yaw/Torsional Mode Frequencies 3. PropellantLine Fluid Mode 3. Bending Mode Slopes Frequencies 3. Pitch/Yaw/Torsional Mode 4, Bending ModeShapes Damp ing 4. Longitudinal ModeShapes 5. ModalDamping 4. Frequency Resp. of the 5. ModalDamping following to a Unit Pitch/ 6, Frequency Response of the Yaw/TorsionalForce Applied 6. Frequency Response of the following to a Unit Pitch at the EngineThrust Pad: Followingto a UnitLongi- (Yaw) ForceApplied at the tudinalForce Applied at EngineThrust Pad: a.Pitch/Yaw/Torsional Slope the EngineThrust Pad: at Control Gyro Brackets a. Pitch (Yaw) Bending a. Thrust Pad Longitudinal Moment at Key Vehicle b. Pitch/Yaw/Torsional Acceleration Stations Accelerations at Thrust Pad b. Tank Pressure at b. Pitch (Yaw) Slope at Propel lant Line Thrust Pad 5. Slosh Mode Frequency Outlet c.Reactions at Major 6. Slosh ModeDamping c. Pump InletLongitudinal Component AttachPoints Acceleration 7. Pitch/Yaw/Torsional Response d. Pump Inlet Pressure at Key Vehicle Stations Produced by Unit Longitudinal ForceApplied at Thrust Pad 5.1.1 (Continued) 2. Uefinition of the thrust vectorcontrol systemdynamics using live hydraulics with proper feedbacks in the actuator system.

3. Determination of modal damping factorsto support the flight control and Pogo stabi 1i ty analyses. At the time of the full scale test, the objectives were considered to be essentially satisfied if the characteristics through the first four vehicle modes (0 to 10 Hz) were adequatelydefined. As the Saturn V pro- gram evolved, the number of modes required to support the problems of the fl i ght vehi cles increased. Forexample on the second stage , a strong Pogo instability occurred at 18 Hz when the frequencies of the LOX tank mode and center engine crossbeam mode coalesced. This local mode corresponds to the 56th elastic mode obtained from the math model. Bending test results were obtained to 11 Hz and 1ongi tudinal test results were obtained to 30 Hz on the fullscale vehicle. Since there wereno well defined userorganization requirements above10 Hz, the long- itudinaldata were notextensively reduced and correlated.If a user organ- ization required that all high gain modes be explored toobtain flight control andPogo parameters, the second stage Pogomodes could havebeen identi- fiedprior to the first flight. This illustratesthe importance of identify- ing in the requirements all parameters that are of importance to the user organizations and following through in the measurement and correlation phases. 5.1.2 VehicleConfiguration There are two basicvehicle configuration guidelines which the full scale experienceindicates are essential requirements f0.r successful test-analysiscorrelation. The first is to know completely and accurately theconfiguration at a1 1 times. The second is toexercise a stringent technical review of each and everyconfiguration deviation. These are ordinaryconfiguration control functions onany test, but they take on added significance in adynamic test and analysistask. The following paragraphs will discussconfiguration requirements in two parts, struc- ture and ballast. The basisfor configuration decisions includethe follOWing considerations: 1. Will a special testarticle be built? I 2. If not, on what schedule will a flight article be available? 3. What restraints does the schedule impose on the tests? 4. Are simulationstechnically acceptable? The philosophy of the full scale test programwas to have as few structuraldifferences as possible between the test specimen and the flight

113 ware sincethe ground testobjectives are to support flight tests Father than to be paralleleffort. The deviations which are allowed should be assessedmathematically to insure that they canbe modeled accurately. The full scale test requirements placed the following controls on deviations: 1. All deviations from flight hardwarewhere the item weight is more than 20 pounds (9.1 KG) shall be simulated. 2. Simulation of flight hardware for weight, stiffness, and moment of inertiashall be within 5 percent. 3. Simulated componentsmust be mounted on theirrespective fl ight specification brackets. The above requirements were found to be unrealistically stringent. It would navebeen better to establish a reviewsystem for thosedevia- tions from flight hardware which exceeded the allowable mass or stiffness changes. For example, a masschange of 1,000 pounds (453.6 KG) inthe first stage tank might not have any effect on the ability to meet the testobjectives. However, a masschange of 50 pounds (22.7 KG) inthe region of the IU flight gyros wouldhave a considerable effect on the flight gyro response and wouldbe a seriousdetriment to meeting the testobjectives. These decisions should bebased on engineering judgment and math simulation. They require an advancedknowledge of user accuracy requi rements. A good example of the importance of using actual hardware or accurately simulated hardware was the LM. An excellent LM simulator, fabricatedlargely from flight article hardware, was originally slated for use on thefull scale test. However, as the production of LM hardware fell. behind schedule, substitutions had to bemade to keep thetotal program moving. A simple mass simulator was constructed using flightarticle hardware only in thebracketry that attachedthe simulator to thevehicle. This simulator was designed to be easy to model and to introduce no unaccountable dvnamic characteristics into test results. The cost of this substitution was not recognized until after the second Saturn V flight. A first stage Pogo instability developed in a strongly coupled longitudinal and pitch mode.The predominantCoupling mechanismwas traced to stiffness asymnetry inthe LM. A spec.ia1 dynamic test had to be conducted to establish confidence inthe mathmodels devel oped to represent this coup1 ing and to provide verification of Pogo suppression hardware. It is essenti a1 to know the configu'ration being tested to the extent of being able to define at any time what the structure of the various stages is, know what deviations havebeen made, and be able to

114 5.1.2(Continued)

describe the deviations in the pre-testanalyses such that the math model beinganalyzed and the test vehicle are identical.This is a normalcon- figurationcontrol task on most tests, but in dynamic testing it is essentialthat the configurationdoes not change from the flight con- figuration unless the requirementspermit this change.Therefore, every deviation from flight hardware that exceedsallowable tolerances in certain areas of the vehicle must be passedthrough the technicalorganization for approvalbefore the configurationchange is allowed. The vehicle propel 1 ants were simulated in a1 1 areas of the vehicle for the testing.Simulants were usedbecause of the logistics problemsand costassociated with storingliquid cryogenics (LH2 and LOX) forlong periods of time, and the potentialhazard to personnel working near the vehicle duringtesting. A study to determine a propellant simulant considered densities, viscosity,homogeneity, flammability, compatibility, contamination, corro- sion,along with handling facilities, storage facilities, dumping facilities and cleaning facilities. The results ofthe study indicated a solution of water andsodium dichromate would fulfill mostrequirements for a fluid simulant. This solution was used in all casesto simulate the LOX and RP-I. A separateanalysis determined that for bending tests the LH2 tankscould remain empty with no degradationof results and for longitudinal tests the LH2 could be simulated by an equalweight of the water solution. To minimize corrosion, the tanks of the first stage were filled with watercontaining 100 parts per million by weightof sodiumdichromate. Since the secondstage would beused for other testing andcould not be contaminated,deionized water with a minimum of100 parts per million by weight o-f sodiumdichromate was used as ballast. Thirdstage andpay'load propellant masses were also simulated with a ballast of deionized water andsodium dichromate. In all cases where there was liquidpropellant, the ballast was simulated by weightas opposed to a volume simulation. Where there was solidpropellant, the ballast was simulated by leadrings. In the oxidizer (LOX) tanks, the water levelsextended further up into the bulkheadsthan in the normal flightcondition. Therefore, special hardware was required toextend the LOX ventvalves above their flight level. No hardware changes were necessary on the S-IC fuel (RP-1) tank since the waterlevel was lower than the normal flightcondition. Anotherstudy was performed to determine the tolerance required in establishing the levels ofpropellants in the tanks. The following to1 erances resulted: fi rst and second stage tank to1 erances were k2,OOO pounds (907.2 KG) per tank; third stage tank tolerance was *1 ,000 pounds (453.6 KG) per tank. A sightgage'was used to indicate fill level and the quoted tolerances came from the maximum resolutionobtainable from the verticalsight gage coupled with the tankcalibration error. The

7 15 acceptable. The SM and LM tank tolerances were one-tenth of one percent. These tolerances were a combinationof the a1 lowable error in tank cali- bration and the allowable error in measurement. Propellantlevels were prescribedin the requirements document, and then establishedin test within theprescribed tolerances. The liquidlevels were recorded in test and usedin thepost-test analysis (whenany significant changes occurred). The requirement forpropellant levels was supported by math analysis. Both theactual condition and theequivalent weight water simulation were analyzed. The results were similar. Gaseous nitrogen (GN2) wasused to simulatethe ullage pressure in a1 1 stage tanks. The ul1age pressure used on the 1 aunch vehicle stages was 10 psi (68,948 N/M2) (about one-third of the flightpressure). Posi- tivepressure in all stage propellant tanks was maintained at all times, but ullagepressure was not used in payload propellanttanks. The require- ment for the reduced ullagepressure was based on: 1. Flightpressures could not beused sincesafety requirements wouldhave barred all personnel from the test area and this was not acceptable. 2. Some ul1 age pressure was necessary to eliminatelocal bulk- head resonances. 3. Ten psi was shown to be satisfactory on the 1/10 scale model test and compatibility between the two tests was desirable.

4. The possi bi li tv of neqatiye Dressures due to temDerature changes needed to be el imlnated. Pretestanalysis and early 1/10 scale model testsresults indicated that the reduced ullage pressure would not significantly affect modal characteristics of thevehicle structure. 5.1.3Test Facilities Requirements The major items considered in defining requirements for the full scale facility were the housing structure, suspension system, stabilization system, excitation system, hydraulic system and propellant ballast system. The experience gained with these items on the full scale test program led to theestablishment of the following general guidelines for defining test faci1 i ty requirements : 1. The effects of the suspension system on the dynamic characteristics of the test article mustbe assessed analytically before the suspension system requirements canbe defined. 5.1.3 (Continued)

2. When research and development i tems are required for a test program theschedule must include adequate time for acceptance tests and modification. Design criteria for such items must be stringently evaluated to determine compatibility with testobjectives, structural capabilities, and overalltest requirements. 3. In a test program involvinglarge pressure vessels or tanks such asthe Saturn V vehicle, it should be a test require- ment that a fail-safe system be installed to preventnegative pressuresoccurring in thesetanks. The Saturn V tanks are such that draining the tank with no open vents wi11 buck1 e the upper bul kheads.

4. The stabilization system must provide static stability and protectthe vehicle fromdamage during test.Stabilization system effects on test article dynamic characteristics must be assessed analytically in order to definefinal require- ments.

5. Test faci 1 i ty components designed to commerical criteria mustbe thoroughly evaluated to insure that their perfor- mance meets the established stage criteria. The major components comprising the test facility are discussed indepen- dently in the following paragraphs. A. Housing structure The test tower structure was designed prior to theestablishment of the test requirements due to the verylong lead time required. The test tower was also designed to handle possible growth vehicles such as the Nova,which was consideredas a continuation of the Saturn family at that time. The test tower protectedthe vehicle from the environment, such as windand rain, allowed access bymeans of platforms adjacent to thevehicle at various levels, carried and supported thepropellant trans- fer lines, instrumentationcables and pneumatic pressurization and control system lines that go to different stages on the vehicle, and provided a means ofattachment of thestabilization and suspension systems. The tower included a mobile, overhead crane for lifting thevehicle stages insidethe building and stacking them on thesupport facility. B. Suspension system The requirement for simulating a free-free condition in the vertical position posed a major vehicle suspension system design problemdue to the physical size of the test specimen. Prior to designing the suspension system, the test tower had been constructed;therefore, the suspension system desig- natea would have to fit into the existing structure without creating major

117 para1 le1 . The cablesystem, beingbased on a proven design , was the primary support system. The cable system supported thevehicle as a giant pendulum withinthe tower. The cables were attached to the tower through coilsprings to give thevehicle as little vertical and rotationalres- traint as possible. This system turned out to be bulky, with large effective masses near the gimbal plane .of thevehicle. The cable vibration modes wouldhave added effective damping to the test system. Themore revolutionary hydrodynamicsystem was to be checked out during the program to determine its application to future dynamic testing. The original concept devised by George Von Pragenauof NASA is described in Reference 5-2. Fortunately,the hydrodynamic suspension systemproved to be highlysuccessful and thespring cable concept was abandoned. The hydrodynamic support system was a tremendous asset to the full scale program. Itseffects on the dynamic characteristics of thevehicle wereminimal due to a very low effective mass and very lowdamping. For example, rigid body rocking tests wereperformed by two men hand exciting the system by pushing on the first stagefins until the 6,000,000 pound (2,721,600 kg) systemreached two inch (5.1 cm) amp1 i tudes. The develop- ment and successfulapplication of this design concept represented a significant advancementin the field of dynamic support systems.

The hydrodynamic support system was a combination of oil bearings and vertical gas springs whichgave thevehicle essentially no restraint inthe horizontal direction and a very soft springsupport in thevertical direction. A major advantage of theoil/gas support system over thecable/ spring system was therelatively little massadded to thevehicle structure. This system was much morecompact khan thecable/spring system and did not sharethe inherent cable dynamic problems. Figure 5-2 is an illustration of oneof thefour hydrodynamic (oil/gas)supports. The upper plate, upon which thevehicle rides, has a milledspherical surface on its underside which floats on an oilfilm pumped through in continuous high pressure flow. The upper plateis free to rotate and offers no localrotational restraint to thevehicle. The upper plate in turn rides on a second plate with a flat undersurfacesupported by a high pressureoil film. This flat surface provides thelateral freedom to thevehicle. The lower plate is supported by a cylinder/piston assembly which entraps a preset volume of nitrogen gasused to givethe softvertical spring support to thevehicle. Raising and lowering thevehicle is accomplished by changing thelevel of oil upon which the entrappednitrogen rideswithin the piston. The vertical springrate is set by controllingthe initial nitrogen precharge volume. After installation in the test tower the system was checked out using the first stagevehicle and operated very satisfactorily, passing all of thespecification requirements. Therewere two major concerns with this system. The first was whether there wouldbe any metal -to-metal

118 SIGHT-GLASS

DRAIN - OIL RETURN P

FIGURE 5-2 HYDRODYNAMIC SUPPORT 5.1.3 (Continued) contact of thebearings auring test, how this could be iaentified, and what theeffects on testresults would be. The second was whether the system was a fail-safe mechanism, i.e., what wouldhappen if one of the pressure 1 ines should fail and oneof the supports' should drop from the vehicle.Testing proved that neither concern was warranted. However, a failure analysis was performed to determine what structural damage to the vehicle would result fromhydrodynamic support failure. This analysis indicatedpossible catastrophic damage to thevehicle if one support failed while the system was operating at the planned test height of 1.5 inches (3.8 cm) above the parked position. Rather than redesign the support system to prevent thiscatastrophic failure, analysis showed that changes inoperation methods could effectivelyeliminate the problem. Operating height was reduced to 0.7 inch (1.8 cm) above the parked position and shimswere installed under thesupport system to lower deceleration loading that would occur if one support failed. C. Stabilization system The stabilization systemaccomplished the following two objectives: 1. Maintained staticstability of the vehicle. 2. Kept thevehicle centered on thesupport systef, during test. The vehicle under some propellant loading conditions was statically un- stable. The criterionfor stability was that the stabilityratio (the restoring moment divided by theoverturning moment)must be equal to or greater than 1.5. As the hydrodynamic support system itself provided no lateralrestraint, thevehicle also had a translationalinstability. To eliminatethese kinematic instabilities, an elastic restoring system was added. Lightweight collars wereused to attachcoil springs to thevehicle at the S-IC thruststructure and at the top of the second stage. Figure 5-3 is a schematic of the stabilization system. An upper and lower snubber system shown inFigure 5-3 was designed to restrain the vehicle in the event of excessive tilt resulting from a failure of therestoration system, hydrodynamic support system, vehicle structure or force excitation system. The design of therestoration system had to satisfy the following dynamic criteria: 1. The rigid body lateral or rotational frequency was 1 imited to one-sixth of the fundamental vehicle bending frequency. 2. The rigid body torsional frequency was limited to one-sixth of the fundamental vehicletorsional frequency.

3. The system was required to have small damping ,

120 UPPERSPRING VEHICLE STA. VEHICLE 2519 SNUBBER AND

LOWER SPRING LOWER SNUBBER VEHICLE STA. 256

SNUBBER AND RESTORATIONSYSTEM

LONERSPRING (STATION 256)

"* bJ "_ """" UPPERSPRING AND SNUBBER(STATION 2519)

FIGURE 5-3 STABILIZATIONSYSTEM

727 5.1.3 (Continued) I 4. The location of therestoration system attachment to the vehicle should result in negligible effects on the mode shapes and frequencies. 5. The restoration system itself was not a1lawed to havea resonant frequency inthe test range of frequencies (0.1 to 38.0 Hz). A parameter study was initiated to determine the effects of the suspension on theflexible mode shapes and frequencies.Results indicated that the upper and lowersystems could be positioned at points that were near the nodal pointsfor the first fourflexible vehicle modes. Beam analyses were conducted to compare the dynamic characteristics with and without therestoration system. The results of thestudy showed negligible differences in the modal characteristics. A1 1 of the restoration springs were pretensioned to a1 1 ow a residualtension stress in allsprings when thevehicle was at its limit of travel, i .e., all springs werealways fully effedtive up to maximum excursion of thevehicle. The fatigue life of the system was designed for 107 cycles. The snubber system was required to restrainthe vehicle in such a manner that no structural failure would occur at the points ofsnubber contact and thatthe vehicle would not tilt sufficiently to contactthe work platforms or tower structure. The snubbersystem was equipped with micro-switches to senseexcessive vehicle motion and turn off the thrusters as quickly as possible such that:

1. The oscillations would bereduced beforethe snubber loads become excessive. 2. Timewas available to takecorrective action to rightthe vehicle in event of excessive tilt. 3. The test data would not be affected by undetected snubber contact. D. Exci tati on system As was the case for the suspension system, the excitation sys tem was a projectrequiring research and development.Based on a preliminary projection of requirements, NASA decided that thethruster systemsthen on the market would not be-sufficient to do the job. Consequen t’lY specifications weredeveloped to procure a thruster system with a cap- ability of four-inch (10.2 cm) single amplitude linear stroke w ith a 20,000 pound (88,965 N) force output. An independent study was made to determine requirements for an excitation system. These studies general ly concurred with the NASA requirements for the four-inch(10.2 cm ) stroke.

122 5.1 .3 (Continued)

This stroking requirement was based on an excitation of sufficient ampli- tuue at thesensor points to allow an accuratedetermination of mode shape using theavailable instrumentation. It was determined subsequently that thefour-inch (10.2 cm) stroke was not a requirement and that a 0.5 inch (1.2 cm) stroke wouldhave been sufficient. This difference wasdue to improvements ininstrumentation sensitivity and to modal damping that was near 0.5 percent rather than theestimated 2 percent. Also, the 1 imited structural capability at the CSM interface and at the upper Y rings of the S-IVB stage restricted the excitation force and stroke. Acceptance testing showed that the initially procured thrusters were total ly unacceptable. The sine wave was distorted, and a 1arge amount of third harmonic content was evident in thesignal, It appeared for some time that the thrusters wouldhave to be rejected and the test scheduleextended. However, modification to the equipment and changes in theoperating procedures produced acceptable thruster characteristics. The major problems encountered and their solutions are: 1. The output of the thruster was unstable and tended to enter a divergent 120 Hz oscillation that was driven by noise in the power amplifier. It was found that the armature and stinger connecting it to thevehicle had a 120 Hz component resonance. The amplifier feedback circuit coupling the output signal back to theamplifier input had a large enough gain at 120 Hz to induce the instabi 1 i ty . The stinger was redesigned to eliminate the 120 Hz resonance , and a notch filter wasadded to the feedback circuit to reduce gain at 120 Hz. 2. Harmonic distortion in the thruster force output was caused by back e .m.f. generated by the thruster. The distortion was eliminated by operating with maximum armature current and minimum field current necessary to obtain the desi red force level . The tnrusters were connected to the vehicle by stinger assemblies that consisted of flexures, a load cell, and apiece of 'pipe. Dynamic analysis was performed on the assemblies to insure that no resonances occurred in the test frequency, range. The primary purpose of theflexures was to limitthe moment and shear loads imposed on thethrusters. At the beginning of testing a 1arge number of flexures and 1oad cells failed. Analytica7study indicatedthat the flexures were inadequate from afatigue standpoint. This problemwas eliminated by using flexures from a different manufacturer. A ground hydraulic system was provided to support the frequency and transient response testing of the first stage thrust vectorcontrol (TVC) servoactuators. These servoactuatorscontrol the thrust alignment by gimballing thefour outboard engines. The ground hydraulic system was designed to commercial system criteria. The supplied filtration proved inadequate to maintain the combined ground-stage system at the cleanl iness 1 evels required by the stage. These requirements were necessary to obtainrepresentative performancefrom theservovalves. The addition of a filter solved this problem.System cleanliness was checked by particle counts of supply and return fluid samples prior to test and every four hours during test. Evidence of servovalve silting waschecked for by examining servoactuator static stability as evidenced by limitcycling. A pretestanalysis of thestage TVC system and ground hydraulic unit was accomplished to predict sys tem performance during test and to determine groundsystem accumulator design require- men ts .

5.1 -4 Data Acquisition and Reduction System The experience from previous dynamic tests indicated that a program of the magnitude of the full scale test would require a completely automated system for data acquisition and reduction. Thevolume of data to be acquired and reduced was such that the more completely automated the system could be,the more engineering time could be spent on data validation and evaluation. The system used for the fullscale test,was completely automated. This allowed thedata to be evaluated to a high 1 eve1 of confidencewithin 24 hours ofwhen the data were obtained. A1 so, rapiddissemination of reduced data to user organizations was possible. The following guidelines pertain to the es tab1 ishment of require- ments for the data acquisition and reduction systems : 1. Large scaletests with extensiveinstrumentation require that the data acquisition and reduction systems be auto- mated.Automatic controlincreases consistency and decreases error incidence. In addi tion, automati c data reduction techniques increasethe amount of data that canbe effectively reduced. Further, it is important to reduce, Val idate and evaluate a1 1 test data. Acquiring more data than canbe effectively reduced and properly evaluated is an expens4 Ye waste ofequipment andmanpower.

124 5.1.4 (Continued)

2. The end i tern accuracy of the i nstrumentation train required to meet theobjectives must be determined. This will permit accuracy tolerances to be assigned to each instrumentation component,such asaccelerometers andampl ifiers.

3. Acceptance test requirements for test equipmentmust be rigorous enough to detect any deviation from specifications. Also, the requirements for acceptance test should be defined in light of the test objectives rather than the theoretically possible performanceof the equipment. 4. The reputation of the vendor is of equal importance to the specification in theselection of critical instrumentation. 5. Data acquisition system requirements mustbe closely matched to those of the datareduction system to insuretotal system compati bi1 i ty. 6. Graphic display systems should be considered when establishing datareduction system requirements for futuretests. Espe- cially valuable for on-site evaluation, such systems can provide real-time visibility of test parameters. Graphic displaysare also valuable tools for final data validation and editing functi om. It should be a test requirement that graph? calrecords bemade of a1 1 test data, and that theserecords be displayed on-site as the data are obtained.

A. Data acquisition system The basic component of anydynamic test data acquisition system is the sensorinstrument. Requirements forthe sensinginstruments used on DTV werebased on the avai 1able accuracy and sensitivity in the frequency andampl itude ranges of interest. Stringent accuracy requirements were im- posed on the full scale test instrumentation because the test data would be used to forcethe mathematical model to conform to the measured structural dynamic response of the test article. Theend i tern accuracy of the enti re Instrumentation trainis governed by the testobjectives. From theoverall accuracy required to satisfy test objectives, individual component accuracies canbe determined. Allowable percentage error canbe allotted to thesensor thesignal train, the data reduction equi pent, etc. The dynamicist has to be concerned with the amplitude and phase characteristics of his instrumentation system over the amplitude and frequency range of the test. An instrumentation specialist has to convertthese requirements into the specifications to be used in the selection of particular instruments. on magnetictape (126 channels) and analog signals on oscillographchart recorders (64 channels) . The factthat only one-ha1 f thetotal channels could be disprayed on oscillographrecorders wasa disadvantagefor on- site review. Forexample, the automaticgain control system increased or decreasedthe gain by a factorof 10. During initial testing it was noted that in isolatedinstances the gain was really a random number for some Channels. This gainchange error was very difficultto identify since the changenormally occurred approaching a resonant peak where a factor of 10 change would not be noticed except on the oscillographrecords. Future data acquisitionsystem requirements should include on-linedisplay of all recordeddata. Automationof the acquisitionsystem required that the datacol- lection and recordingfunctions be computercontrolled. This imposed a requirementfor significant amounts of softwaredevelopment or preprogram- ming of the computersystem. This was illustrated on the full scale test by the automaticfrequency interval selection and steady-state ampl i- tude determinationfeatures of the acquisitionsystem. The acquisition computervaried the thruster frequency increments such thatrelatively large increments were produced at the low responseportions where definition was non-criticalwhile increments assmall as 0.002 Hz were chosen at the peaks for maximum definitionof resonant frequencies and responsepeaks. The steady-state ampl i tude feature a1 lowed the computer to automatical ly decide when the vehicle had settled out at each test frequency and the datacollec- tion would commence. A minordisadvantage to automatic computer control of the data acquisitionsystem is the total dependence upon the state of the equipment. Failureof the computercore or peripheral equipment cancause considerable down time whilewaiting replacements. A preliminaryrequirement that the responsible vendor have rep1 acements avail ab1 e on short notice would reduce this problem significantly. B. Data reduction The purposeof the datareduction system is to reduce the data acquiredand present it in the form required forproper analysis and evaluation. On-site reductionof data is necessary to permit a quick evaluation of whether the test canproceed to the nextcondition. In addition,real time datareduction and displayare essential to insure that response levels detrimentalto the test article are not produced during test. Considerableadvance planning of how the datawill be processed is required for these reasons: 1.Validation of a large volume of data is a time consuming task without automated data reduction techniques.

126 5.1.4 (Con ti nued) 2. Data must be reduced and presented in the formatrequired by the user organizations in order to beof maximum benefit to them. 3. The datareduction methods must be selected; complexreduc- tiontechniques such as the Levy curve fit procedureare especially tailored to automated processes. The on-site mode shape displayconcept employed on the full scale test would be ofsingular value to exploratory testing. In this system a Fourieranalysis was made on up to 30 significantsensors. The normalized output was displayed on an oscilloscope with the vehicle outline sketched on the screen. The mode shapecharacteristics were monitored on site and were alsorecorded on film to obtain a permanentrecord of mode shape development. Centralcontrol should be exercised over all structural testing and data from a majorprogram such as the Saturn V. Several different Government agencies and their contractors will be involved in structural dynamic testing of programhardware. .For example, on Saturn V, MSFC was responsible for dynamic tests of the Saturn V performed by Chrysler and Boeing, ofmajor component tests performed by NorthAmerican, and of an S-I1 mini-stage test performedinternally. On the same program, MSG was responsible for a spacecraft test performedinternally, a spacecraft test performed by Wyle Labs,and tests of the LM performed byGrumman. LRC was responsible for testing the 1/10 scale model. Ineach case, different dataacquisition and reduction techniques were used. Where contractors were involved, they were each givenresponsi- bility for setting up and maintaining the library of their test data. Each agency established its own system with varyingdegrees of rigorous documen- tation. Theend result is a collection of data in different formats that impedesflow ofinformation across a programand insures thatcarryover from oneprogram to the next will be negligible. An integrated technical test planshould be developed for a1 1 structuraltesting within a program. Part of this planshould be the developmentof a single structural dynamic data reduction programhaving variousoutput options that are all in consistentformat. This program should be used by the Government and their contractors. General criteria should be developed to insure consistency in the dataobtained from the various tests. A central libraryof test experienceshould be established and maintained. Data from this library should be readilyaccessible to anyagency having a legitimate need-to-know. This will promotedata cross flowand insure preservation of data for future programs. C. Calibration There are four basic cal i brations that each sensor should be subjected to:

127 2. Pretestcalibration 3. On-site calibration 4. Post-testcalibration The acceptance test is nothing mre than a check on the vendors product to insure that it meets specifications. The importance of this step is i 11us trated by the servo-force-bal ance accelerometers i ni ti a1 ly procured forthe full scale test. Eighty-fivepercent of theseacceler- ometers were rejected as received from the vendor because they did not pass the acceptance test. The primaryproblem was that thebearings in theaccelerometer would malfunction and cause a considerable shift in the sensor output. This wasof real concern sincethe test schedule was impacted considerably dnd obtaining special purpose accelerometersrequires a long leadtime. Theproblem was resolved by going to another vendor who, at considerableextra program cost, produced sufficient accelerometers to maintain the fullscale test schedule. This experience not only points out the importance of the acceptance test but also emphasizes the importance of thevendor’s reputation. The second calibration required for each instrument is thepre- testcalibration. This is a sophisticatedcalibration that determines DC output factors, amplitude and phase output deviations as a function of frequency and linearity. The linearitycharacteristics are determined by performing the cal i bration at several different levels, i .e., 10 per- cent fullscale, 50 percent fullscale, and 100 percent full Scale. In addition to verifyingthe vendors calibration, the pretest calibration a1 so: 1. Provides thecorrection factor that must beused in data reduction. For example, 1.0 g may equal 9.985 voltsas opposed to 10 volts nominal. 2. Provides thescale factors for use in theon-site calibrations. 3. Provides the phase and amp1 i tude deviationsas a function of frequency to beused in the data reduction program for sensor output correction. This correctionis combined with the phase and amplitude deviations of other elements in the acquisition system. 4. Provides thebaseline calibration to beused to spot check calibrationsafter individual tests. The on-sitecalibration performs two basicfunctions. It corrects amplifier settings to assure that no drift has occurred in the system. Secondly, it provides verification that noneof theinstruments havebecome 5.1.4 (Continued)5.1.4 invalidduring the test. On the full scale test thiscalibration was performed every four hours byan automatic procedure programmed through the control computer.

D. analysisError

An error analysis was performed to include the effects of the complete signal train, beginning with the transducer and terminatingwith the processeddata. This was accomplished in three steps. First, the transducer error was determined;second, the error of the transducerplus the linear amplifier was determined;and third, the error of the total system was determined. No attempt wasmade to perfoy a purely theoretical error analysis because a large amount of test data was availableto work with. This pro- cesseddata contained the actual errors which hadaccumulated as the data passedthrough the variousanalog and digital systems usedduring the full scale program. Figure 5-4shows the typicaldata signal path with the type oferrors introduced in the dataas it changedfrom raw analog signals to processeddata. Each block of the Figure 5-4 diagramrepresents a separate component functionin the signaltrain. In addition a rigid body error analysis was made in order to check the results obtained bya statistical and probabilityanalysis. Rigid body mode shapes offer a unique method of determining relative errors of a large number oftransducers simultaneously. Rigid body frequency is definedas thatresonance at which the dynamic test vehicle is vibrating as a single mass with the suspension system acting as the effective spring and with no flexibledeformation of the vehicle introduced. The rigid body frequencies on the full scale test were an order of magnitudelower than the 1 owest flexible bodymodes. The acceleration rigid body mode shape is characterized bya straight line and the rate gyro rigid body mode slope is characterized bya constant rate throughout the vehicle. Longitudinalrigid body mode shapes shouldconsist of vertical line displaced from the origin by an amount equal to the g-level; pitch or yaw rigid body mode shapesshould consist of aninclined straight line intercepting the zero g-axis at the center ofrota- tion of the vehicle. The error is defined as the difference between the best straight line fit between all the accelerometerreadings at the rigid body frequencyand the deviation from this line of an individualaccelero- meter. For a rate gyro, the error is definedas the difference between the calculated mode slope(from the accelerometer mode shape)and the indicated slope. The chief advantages in usingthis technique to determine relative errors betweentransducers are that many instruments canbe compared I simultaneously and that the highestacceleration magnitudes, which are

7 29 ERRORS SIGNALTRAIN

I- 1 CALIBRATION ROLL-OFF OFF SITE PHASE LAG PHASE SITE OFF * L -J 1

CAL I BRAT I ON CALIBRATION ACCURACY

RESOLUTION RESPONSE ON HYSTERESIS SENSITIVITY I- f NO I SE ANALOG SIGNAL SCALARERRORS TRAIN

RESOLUTION t SCALARERRORS DATA ACQUISITION NOISE

1

ROUND OFF DATAREDUCTION d ERRORS 1 r-7PROCESSED DATA

FIGURE 5-4 FULLSCALE DATA ACQUISITION AND REDUCTIONSYSTEM SIGNAL TRAIN

130 5.1.4 (Continued) generally the most accurate, canbe used to set the rigid bodymode shape line. The rigid body accelerations at the top of thevehicle are signifi- cantlyhigher than at othervehicle stations. A1 though the rigid body error check offered a re1 atively quick method of determiningthe error band of the full scale test acceleroneters and rate gyros , the maximum response amplitudes were usually only five per- cent of fullscale. Because of the extremely low natural frequency of the suspensionsystem, a two inch displacement produced only 0.01 g accelera- tions. In orderto overcome theselimitations, a technique was develcped to compare the output of two sets of accelerometers and rate Gyros located at the same station on the testvehicle. When comparing thesesensor out- puts two factors mustbe included in order to buildup an error curve: 1. Any errors in two samples, if assumedrandom and non-scalar, will have an equal chance of occurringsimultaneously as either positive, or negative or of the same sign. 2. The standarddeviation of thedifference between two samples from the same population is equal to thestandard deviation of that population (from statistical theory). At least 100 data points wereused at each test time point incre- mental frequency sweep . In all casessensor comparisons weremade at exactlythe same frequency. Figure 5-5 shows theerror measured by the rigid body comparisonmethod. The dashed 1ine represents the L one sigma deviations from the average error distribution represented by the solidline. The error curves in Figure 5-5have been extrapolated from 10 per- cent to 100 percent full scale by assuming that the error values are depen- dent only on thedata acquisition system inputsignal level. The gain change of a factor of 10 has therefore beenassumed to reproduce the same error curve between 10 and 100 percent full scale as was found betweenone and 10 percent fullscale. This figure shows thepercent erroris reduced when theamplification factor is increased. The ability to change theamplifica- tion factor enabled the system to receive data onemagnitude lower and maintainthe saw accuracy. The effectivity of theFourier analysis, in removing noise and harmonic distortion from the data, canbe seen by the relatively low aver- age errors contained within the test data. At signallevels of 0.1 percent fullscale the analog signal to noise ratio is 1:1, whereas the average error is only 5 22 percent. The Fourieranalysis enabled thesignal level to reduce to 0.01 percent full scale before.the signal to noise ratio became 1:l. 5.1.5 Test Conduct Experience with ces-c programs prior to the full scale test pointed out the importance of having detailed test procedures written prior to actual testing. This guideline was adhered to on the fullscale program and resulted c

1

01

PERCENT ERROR (+)-

FIGURE 5-5 FULLSCALE TEST ACCELEROMETER DATA ERROR

132 5.1 -5 (Continued)

in a minimum of test conduct problems. The test procedures delineatedthe vehicle and test supportconditions necessary to satisfy theestablished test requirements. Included in the test procedures were the bal last and pressurelevels, restoration system preload, shaker orientation and a detailedstep-by-step operational procedure to followin the conducting of tests. An additionalfactor in the successful accomplishmentof the tests was the test readiness review. Imnediately prior to the start of each test a meeting washe1 d betweena1 1 responsible test engineers and technicians to review the test procedures and to ascertain that all representatives were cognizant of and in agreement with any deviations from the procedures. The foregoing stepsare considered essential in conducting a smooth and orderly dynamic test. The actual test sequence began with a manual excitation test which consisted of two technicians manually shaking the vehicle at each ,rigid body resonantfrequency. This was performed to check the phasing of the out-of-planeinstruments to verify that each sensor was alignedproperly and connected to a specific channel. Then a manually control 1 ed vibration sweepwas performed to gain advanceknowledge of the vehicle resonances and maximum forcelevels that could beused during test. During the resonance searchthe in-phase sensors werechecked for both amplitude and phasing at therigid body resonance. Once theresonant frequencies were known, a frequency sweep program wasmade for the control computerwhich contained the minimum and maximum frequency increments to be taken through- out the test frequencyrange. Thedynamic testing wasbegun with the pre- programmed control computer providing the incremental frequency sweep to the excitation system and recording theoutput data. Additional tests performed included theforce linearity test, ring- out damping test and ring mode test. The forcelinearity test consisted of excitingthe vehicle at threedifferent force levels at each resonant frequency to determine thenonlinear characteristics of thevehicle. The ring-out damping test consisted of exciting the vehicle at each of the first four flexible mode resonantfrequencies and measuring thelogarithmic decay of the response to determine the damping at each sensor when the force was suddenly removed. The ring mode test consisted of an incremental frequency sweep to determine IU ring mode activity. In the.beginning of the test program twenty-three (23) men per shift were required to conduct the test previouslyoutlined. As the test program progressed fewer personnel were required due to constant review and streamlining of procedures and increasedefficiency. The required manning level was reduced to fifteen (15) men per shift in the latter stager of testing. 5.2 DIGITAL DATA REDUCTION TECHNIQUES

The following paragraphs review the techniques used in the full scale digital data reduction program and place particular emphasis on the

133 methods used to curve fit the test data and compute the vehicle modal parameters. The actualdata reduction problems encounteredwill be pre- sented along with the necessary workarounds used. A criticalevaluation of the accuracy and limitationsof the techniques used on the full scale program wi 11 also be given. A chart showing the generaloperations of the digital data acquisi - tion and datareduction system is given in Figure 5-6. The dataacquisition system was designed to digitize and record on magnetictape 900 datapoints per second per channel for 128channels of data. The principaloperations of the digitaldata reduction program were to: 1. Readraw datatapes and check gain settings, 2. Performa Fourier analysis for eachchannel of sinusoidal data, 3. Calculatea transfer function by dividing each instrument response by the forcingfunction of each incremental frequency, 4. Convertselected instrument responses to displacement units prior to curve-fitting,

5. Curve-fit the frequencyresponse data u.sing a modified Levy compl ex curve-f i t routi ne, 6. Reduce the equationsobtained from the Levy routine by a partialfractions scheme to represent each modal resonant peak as a singledegree of freedom system for computation of modal parameters, 7. Plotfrequency response transfer functions,curve-fi t transfer functions, mode shapes, and mode slopes. The automateddata reduction techniques used on the fullscale test proved highlysuccessful in: 1.Obtaining modal parameters from test data. 2. Providinganalytical expressions for test resultsthat could be compared directly with mathematicalpredictions. 3. Eliminatingnoise and harmonic distortion from the excita- tion and responsesignals. 4. Providingrapid display and evaluationof large volumes of data and rapid distribution to user organizations. Similarprocedures are highly recommended for future programs.

134 I I I -7- I

I

64 CHANNEL STRIP RECORDER I

COMPUTER

t

FREQUENCY DAMPING GENERALIZED MASS MOD5 SHAPES

ON SITE -I CFF SITE "1 W ul FIGURE 5-6 FULLSCALE TEST DATAREDUCTION FLOW CHART 5.2.1Fourier Analysis For sinusoidal dwell excitation, the response measured by each instrument is the sum of a periodicfunction plus noise. Thistype of data is conveniently reduced by Fourieranalysis. Three significant ad- vantages arerealized from a Fourieranalysis: 1. It is an effective narrowband filter to remove the effects of thruster wave distortion, 2. It permits analysis of the harmonic content of the output wave, 3. It represents each cycle of data by only twonumbers , therebysimplifying calibration corrections. Using the correct procedure for determining the sampling rate willresult in accuracy andmore efficient computation. This was a basic deficiencyin the fullscale test data reduction system. The dataacqui- sition system was limited to a fixed rate of 900 datapoints per second. This would provideonly 20 samples per cycle for 45 Hz ring modes , but would provide manymore samples per cycle thannecessary for the basic vehicle modes,which were in the 1-10 Hz range. This caused extra expense for the low frequency Fourieranalysis program since each cycle of data had to be analyzed in segments. Also, the fullscale test constant sampling rate never produced cycles of data that started and ended on datapoints, thus requiring that linear interpolation be employed to yield exactly one cycle of data. To circumvent these problems, it is recommended that the sampling rate be madean exactmultiple of the excitation frequency. The multiplier would be the number of datapoints per cycle required to attain the desired accuracy. By always dividing each cycleinto exactly N equal parts the sine and cosine sample functionsare numerically orthogonal. This ortho- gonality reduces the least squaressolution for the Fouriercoefficients to the followingsimple form:

-1 1 1 ..... 1 YO N+ 1 N+1 m Nil 1 cos & cos 4'IT .. . , . N N 1 sin sinsin 4n Lo N 3- ..". 0

= constant term = coefficientof fundamental cosine term ;; = coefficient of fundamental sine term N+ 1 = nuher of samples per cycle of data yo, y, ,---yn = sampled response function 5.2.1 (Continued)5.2.1 This method isreadily extendable to includehigher harmonics. Calculations of this type canbe performed rapidly on a digital computer. Themethod conserves both tape storage andcomputer time. If the sampling rate is frequency dependent, the sine and cosine Fourier coefficients for each cycle of dataare automatically orthogonal and theorthogonality is not affected by the useof a numerical integration routine. Thus, a significant computational advantage is achieved by eliminatingcycle interpolation and by reducing the sampling rate. The fixed sampling rate of 900 samples per secondused on the full scale program made calculation of higher harmonics unacceptably expensive. Studies made using DTW data show that twelve samples per cycle would havebeen suficient to analyze thedata accurately through the third harmonic. 5.2.2 Point Transfer Functions The next step in data reduction is to determine the amplitude and prlaseof each sensor on the test article relative to the input force. Before this canbe done,however, each channel of data must have certain correctionsincorporated andhave the voltages converted to engineering units. The correctionsinclude curvesof phase lag and rol loffcharacter- istics for each instrument and similar data for the data acquisition system. Full scaletest datareduction equations were obtained for each correction curve through the use of a curve fit routine and then these curves were applied to eachchannel of data after completion of the Fourier analysis. The transferfunction for each data channel is the ratio(accel- eration per unit force) of thecorrected response divided by theforcing function. The transferfunction and the corresponding phase angle ofeach instrument relative to that of the forcingfunction wascomputed through- out the test frequency range. 5.2.3 Transfer Function Equations The structural responseof a system is the sum of the responses in several modes. Separation of the test data into modalcomponents is recognized to requirespecial techniques; some of these techniques are cited in References 5-3and 5-4. The solution chosen forseparation of full scale test data into modalcomponents was a modifiation (Reference 5-5) to a complex curve fitting method developed by E. C. Levy in Reference 5-6. Thecomplex curve fit techniquF used is explained in detail in Reference 5-7.

7 37 a ratio of two frequency-dependent complex polynomials.’

* where G(jw)k = Curve fit transfer function

k = The kth sensor transfer function

w = Excitation frequency ao, al. ..., bo, bl . . . . are constants to be determined n = Numbermodesof being curve fit The error at each discreteexcitation frequency is thedifference between theabsolute magnitudesof theactual transfer function and the po lynomi a1 rati 0. * ebIk= G(jwIk - G(jwIk (5.3) where G(jw)k = transfer function measured during test

e(jw)k = error The polynomial coefficientsare evaluatedas the result of /minimizing the sum of the squares of the above error at the experimental points. The least squares solution iscarried out by an iterative pro- cedure. To isolatethe modal parameters thepartial fractions expansion of thetransfer function Equation (5.2) is carried out by well known techniques. The resulting expression is :

n = thetotal numberof modes inthe transfer function

k = the kth transfer function A,, Buy Cu, & Du areconstants determined from partial fractions expansion.

1 38 5.2.3 (Continued) The procedure forevaluating these parameters is explained indetail in Reference 5.7. An example of the curve fit achieved on full scale test data is shown in Figure 5-7. The sol id linerepresents results of the curve fitting routine for both the ampl itude and phase plots. The dashed 7 ine shows the weakmode that was not obtained by the curve fit routine. The pointsplotted are the test transfer function data whichwere always plotted with the curve fit results to evaluate the accuracy of the curve fit. The quality of the curve-fit was found to be dependent on the following: 1. Relative peakampl i tudes 2. The units of the data 3. The distribution of the data points 4. The frequency range 5. Closeness of the modal resonantfrequencies The preceding factors affected the resulting curve-fi t to such an extent that the curve fit for a transfer function could not be obtained in a single computer run. Instead, it was require:! that an engineer evaluatethe transfer function data considering the above parameters and submit the routinefor fitting. Then theresulting curve fit was evaluated and changeswere made in the parameters to improve the fit. The computer time associated with curve fitting decreases with experience. This is particularly true when, as in full scale testing, several tests are to be run which produce data having similarcharacteristics. The addition of an engineeringgraphics computational system for evaluation of the curve fit while thedata is in the computerwould greatly shorten both flow and computational time required for curve fitting. The relative peakampl i tudes for each peak curve fitted within a selected frequency range should be fairly consistent since the nature of a the curve fit routine gives more accuracy to the larger peaks inthe data, while some smaller peaks may not be fit. This is evident.in Figure 5-7. Thepeak amplitudesobtained for the suspension system modes in full scale testing were much larger than amplitudesobtained in the flexible modes. To obtain satisfactory solutions for flexible body resonances, it was necessary to eliminate the low frequency range containingthe suspension modes. As a rule of thumb, modes whosepeak ampl itudes are at least one fifth of the amplitude of the strongest mode in the array should be curve fit accurately. Most of the modes of interest occur in the low frequency range (above the frequency range of the suspension modes). Displacement units emphasize the low frequency peaks, thus improving the accuracy with which these peaks are curve fit. The accuracy of the curve fit is affected significantly by the distribution of thedata points over each resonant peak. Smaller frequency increments should beused around the resonances. Full scale test experience

139 1 50

100

50

0

T 50

-1 00

-1 50

FREQUENCY (Hz)

FIGURE 5-7 TYPICAL CURVE FIT OFFULL SCALE TEST DATA

140 5.2.3 (Continued) has shown that 8-10 frequency increments are sufficient to define a reso- nanceabove its half ampl i tude point; the accuracy of the curve fit will not be greatly increased by adding more points. Limiting the frequency range alsoincreases the accuracy. This is because theequations that have to be solved to obtainthe curve fit contain powers of the frequency. The highest power is equal to the number ofterms retained in the denominator of Equation (5.2). As the frequency range is extended, the magnitude spread between terms in the coefficient matrix rapidlydiverges for two reasons. First,increasing the frequency range increasesthe number of modes included and requires higher order term in theleast squares solution. Second, o itselfgets bigger. Experience has shown that theorder of the polynomialcan be as high as 12 (6 resonances) and cover a frequency range of 20w0, where w0 is the lowest sample frequency, provided double precision is used. If more mdes, or a larger frequency range must be covered, the curve fit canbe carried out in segments. Two modesof nearlyidentical frequency aredifficu?t to curve fit. For example, double resonant peaks occurred in some pitch and yaw data. These nearly identical resonances wereproduced by the small pitch and yaw asymmetries inthe test article. An exampleof a double peak obtained during the test is show in Figure 5-8. In this casethe peaks arenearly equal in ampl i tude. Usually oneof the peaks was larser than the other. Thesepeaks were fitted as onemode (Figure 5-8 shows a typical fit) because of the smal 1 frequency difference in the modes. A7 though the curve fit did not separate the two modes, the single mode approximation did represent the composite response characteristics of the two modes accurately. Modal resonances closer than one percent in frequency could not be separatedsuccessfully. However,modes as closeas two percent in frequency could be separated accurately if they wereof nearly equal strength. 5.2.4 Computation of llodal Parameters

The partial fraction expansion formof the transfer function equation(Paragraph 5.2 -3) is recognized to have the general form of the equation for the transfer function of a linear viscously damped system, which is represented by: I I I I 1 I 1 I I I 1 I 1.821.83 1.84 1.851.861.87 1.88 1.89 1.90 1.91 1.92 FREQUENCY (Hz)

FIGURE 5-8 TYPICAL CURVE FIT OF DOUBLE PEAK RESONANCE 5.2.4 (Continued) where

$1 u = -Normalized modal displacement at the forcingpoints

@ku = Normalized modal displacement at the kth sensorstation - mu = Generalized mass for the uth mode

WU = Frequency of the uth mode G(jw)k = Transferfunction (displacement per unit force)

5 = Modal damping factor This equation was used for full scale test data to compute for each mode (u) and mode shapesensor transfer function (k), the modal frequency and damping, the gain of the mode, and the generalized mass. The workable techniquesdeveloped to extract these data from test results are discussed in the followingparagraphs. A more detaileddescription of these tech- niques is given in Reference 5-7. Experienceindicates that a close evaluationof the partialfractions is requiredto assure that they represent the actual data to an acceptabledegree of accuracy before proceedinginto the computationof modal parameters. A. Modal frequency The modal frequency was calculated for each mode (u)of each sensortransfer function (k). It was determined by extracting the square rootof from Equation(5.5). The naturalfrequency values derived from the varioussensors was amazinglyconsistent. The values seldom deviated by as much as 1/2 percent between sensors showing well defined response. For modes too weak to curve fit, the frequency at which the imaginary component ofresponse peaked was takenas the modal resonant frequency. B. Modal damping The modal damping was determined from the transfer function equation by equating like terms in Equations(5.4) and (5.5).

= cu 2wu (5.6)

743 5.2.4 (Continued) The degree of linearity of the test article is of prime importance in accuratelydefining the system damping. Generally in fullscale test data,it was observed that the response per pound did not remain constant but tended to decrease with an increase in force. This nonlinear response isillustrated in Figure 5-9. This figurepresents the frequency response data of the LES forthree different force levels. This non- linearity made it essential to excite the test article to levels expected during flight. Linearized damping estimatesobtained from thesensors seldom differed bymore than fivepercent. This established a high confi- dence that the damping values were valid.

Originally, it was intended to obtain mode shapes by normalizing like mode componentsfrom the curve fit routine. A computer subroutine was written to calculate the single mode amp1 i tudes of selected sensors at the modal frequencies. This subroutine used thepartial fractions (one for eachmode). Some of the problems encountered here were: 1. Some of the mode shape sensors would not have a peak of sufficient amplitude to curve-fit. 2. In the case of a double peak it was difficult to display a mode shape because thecurve-fit equations gaveone peak for some sensors and two peaks for othersensors. This distorted the mode shapes insections of the vehicle. Generally the amplitude subroutine was of insufficient accuracy to present a good plot of the mode shape, and another run was required to eliminatethe erroneous values after evaluation by theengineer. For this reason thepoint transfer function data of the mode shape instru- mentswere used to hand plotthe mode shape. It was found that the normalized accelerometerreadings produced mode shapes of acceptable accuracy, although the phase angle was difficult to interpret for sensors showing small response. Usually thesign was determined by establishing a band ofphase such as 0 degrees to 180 degrees as positive and 0 degrees to -180 degrees as negative.

144 FREQUENCY (Hz)

FIGUR'E 5-9 TYPICALEFFECT OF FORCELEVEL ON FREQUENCYRESPONSE 5.2.4 (Continued) D. Mode slopedetermination

Themode slope is defined as therate of change of the mode shape with vehiclelength. Two methodswere used to determine the mode slopes for the fullscale test. The first was direct measurement by rate gyros. The other methodof determining the mode slope involved using a polynomial to curve fit the mode shape data (Reference 5-7). The mode slope was obtained by differentiatingthis polynomial solution. This technique has the advantage of determining theslope of any specific station along thevehicle. I n using this technique,considerable error may occur i f the slope changes appreciably from interval to interval or if theinter- vals are too 1 arge. These limitations mustbe recosni- zed in spacing the accelerometers along the vehicle. E. Generalized mass The general ized mass values were also obtained from curve fit results. In practicethe coefficient Bu of the first order term in the numeratorproved to be negligible. Consequently, 8,was set to zero so that a direct comparison of theoretical and test results cou Id bemade.

Rearranging, theexpression for generalized mass is:

Generalized mass values obtained from thesensors that could be curve fitted accurately, normally agreed within 10 percent. 5.3 TEST DATA EVALUATION PROCEDURES 5.3.1 On-Site Data Evaluation

A qualifiedstructural dynamicist was on-site at all times to evaluate data as it was taken. This is an essential requirement. During each test sweep, the raw data as provided from the "quick-look" system wascompared with the pretest analysis to ensure that a1 1 thepredicted

146 5.3.1 (Continued) modeshad been excited and that sufficient data hadbeen recorded to analyze thestructure completely at the test time point. The output signal ofeach transducer wasexamined at the rigid b0d.y resonant frequency to ensure that thesensors werephased properly and that the Signal level was in agreement with thevehicle response. All sensors were monitored for noise content andwave shape distortion by use of an analog chartrecorder. Oscilloscopes wereused to determine if trans- ducer output wasbecoming erratic or faulty. In addition to monitoring theindividual sensor output signals, the overall vehicle response as portrayed by the mode display , and 12 channels of transfer plots were closely watched in order to insure that the test data was realisti c and that no anmal ies were present. If data correlation wasgood between the pretest analysis and raw test data, the vehicle was then made ready for the next time point and thethrusters reposjtioned for excitation in a different axis.If data correlation was not good, then an overall check was performed on thoseinstruments which deviated to the greatest extent from the pre- di cted response. Furthermore , a manually control led frequency sweep wasmade for excessively weakmodes until the structural dynamicists were satisfiedthat the pretest analysis was in error.If the on-site evaluationindicated no problems, it was considered that there was an 80 percent confidence level in thedata validation. The value of a rigorous pretest analysis to on-site evaluation cannot be overemphasized. emphasized. 5.3.2 Test Data Validation While the test data were being processed,the analog recordings were being annotated. The processing of the 128 channels of digital datausually took about eight hours. Afterthe printouts and plots were received, the modal frequencies were approximately determined from the plots and then accurately determined from theprintout. Data from the mode shape sensors were recorded and normalized to the same stations used in thepre-test analysis. The comparison of test and analytical mode shapes and frequencies was the first step in datavalidation. The second step in validation was extractingdata for all flight sensors. This data was compared at the rigid body frequency to insure that all amplitudes were within instrumentationtolerances. Slopes of flight gyros at flexible frequencies were also canpared with the mode shapes and with thenon-flight backup instrumentation. The rest of the instrumentation was then checked by scanning the transfer function plots to determine if any unexplainable activity was evident. Where sensoroutput was questioned,a comparison wasmade with the analog recorder data and the transfer function plots.

7 47 hours. When sufficient data were acquiied to allow a-satisfactorydef- inition of the dynamic characteristics of thevehicle, a phone call was made to test site indicating a "go" decision on validation, and testing proceeded to the next time point. 5.3.3 Test Data Evaluation The validateddata were then evaluated more carefully by experienced structural dynamicistsbefore the data were considered acceptable for release in the testreport. This evaluationconsisted of a detailed examination of each sensor output to determine whether it was consistent with othersensors and represented a condition that was physicallypossible. Component characteristics and out-of-plane motion wereexamined inthe eval uation.Results of thisevaluation are con- tained in thediscussion section of all testreports. A common mistake which should be avoided is the recording of such a volume of data that complete evaluation is impractical. 5.3.4 Test Data Reporting The data reporting for the full scale test program consisted of two reports: Flash Report and Test Data Report. A FlashReport was sub- mitted for each time point tested and contained the reduced data obtained from the "quick-look" analysis. These data included modal frequencies, mode shapes, excitationforces, modal damping values,transfer functions, and fl ight gyro response. These reports were released within two days after the conclusion ofeach test. A Test Data Report was submitted after the completion of each configurationtested. Each report contained a description of the test objectives,configuration, support equipment, data acquisition system, instrumentation,data reduction techniques, pretest analysis and models, and data evaluation. The reportalso contained transferfunctions, mode shapes, frequencies, and damping Val ues .

148 SECTION 5 - REFERENCES 5-7 Document05-71 093, Saturn V Dynamic Test Program Requirements , The Boeing Company, Huntsville, Alabama, October26, 1967.

5-2

5-3 Woodcock, D., On the Interpretationof the Vector Plots of Forced Vibrationsof a Linear System with Viscous Damping, The AeronauticalQuarterly, February, 1963. 5-4 Kennedy, C. and Pancu, C. , Use ofVectors in Vibration Measure- ment and Analysis,Journal of the AeronauticalSciences, Vol. 14, p. 603,1947. 5-5 Sanathanan, C. andKoerner , J ., Transfer Function Synthesis as A Ratioof Two Complex Polynomials, IEEE Transactions on Automatic Control , January , 1963. 5-6 Levy, E., Complex Curve Fittings, IRE, Professional Group Transactions on Automatic Control , Vol . AC-4, May , 1959. 5-7 DocumentD5-1521OCY Saturn V Dynamic Test VehicleData Reduction Techniques, The Boeing Company, Huntsville, Alabama, October 26, 1967.

SECTION 6 CONCLUSIONS This document has presented practical guide1 ines for accomplishing structural dynamic analysis, dynamic test, and data reduction that were establishedduring the successful Apol loSaturn V programs. These guide- lines and recommended practices were presented so major pitfalls andproblems encounteredduring this program could be avoided in future programs. The materialpresented in this document is orientedtowards the technical managers of future structural dynamic programs. The followingpoints are emphasized based on the experience gained during the Saturn V programs : 1. Replica models can bean effectivetool to pilot structural dynamicprograms of future spacevehicles, 2. Math modelscan be used topredict the overallvehicle dynamic characteristicsaccurately, provided the guide1 ines presented in this document arefollowed, 3. Local deformations and major component dynamicscan be pre- dicted with limited confidence; static or dynamic testing ofmajor assemblies or stages is requiredto guide math model development, 4. Automated dataacquisition and reduction systems should be used ona17 major dynamic test programs. 5. A singletechnical test planshould be developed for each space program. Consistentdata acquisition, data reduction, and data 1ibrary procedures should be used for all major dynamic tests within a program.