Jaap Wijker

Mechanical Vibrations in Spacecraft Design

EngineeringONLINE LIBRARY

http://www.springer.de/engine/

Springer-Verlag Berlin Heidelberg GmbH

Jaap Wijker

Mechanical Vibrationsin Spacecraft Design

With120Figuresand 29Tables

• Springer

Jaap Wijker

DutchSpaceBYP.O. Box320702303 DBLeidenThe Netherlands

E-mail: j.wijker@dutchspace.nl

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© Springer-Verlag Berlin Heidelberg 2004Originally published by Springer-VerlagBerlin Heidelberg New York in 2004

Softcover reprint ofthe hardcover 1st edition 2004

The use of general descriptive names, registered names, trademarks, etc. in this publication doesnot imply, even in the absence of a specific statement, that such names are exempt from therelevant protective laws and regulations and therefore free for general use.

Typesetting: Dataconversion by authorCover-design: medio, BerlinPrinted on acid-free paper 62 13020 hu - 5 4 3 2 I 0

ISBN 978-3-642-07354-0 ISBN 978-3-662-08587-5 (eBook)DOI 10.1007/978-3-662-08587-5

Dedicated to my mother

Maartje Wijker-Gravemaker

and to the memory of my father

Job Wijker

Preface

This book about mechanical vibrations focuses on spacecraft structuresdesign and reflects my experiences gained at Dutch Space B.Y., formerlyFokker Space B.Y., Fokker Space & Systems B.Y. and the Space Divisionof Fokker Aircraft B.Y., over a period of about 30 years.

Many books about mechanical vibrations have been published, however,in spacecraft structures design, many vibration topics are applied but can beread in different books. I have collected in this book most of the topicsabout mechanical vibrations techniques usually applied in spacecraftstructures design .

I work as a part-time associate professor at the Chair AerospaceStructures & Computational Mechanics, Faculty of Aerospace Engineering,Delft University of Technology, and lecture "Spacecraft Structures" in theMaster's program. The scientific environment at the university, incombination with my work in the aerospace industry, has amplified thewish to write a book about mechanical vibrations with focus on spacecraftstructures design. To bring together most of the techniques of modal anddynamic response analysis is my greatest motivation to write this book.

I would like to express my admiration for the patient attitude of my wifeWil during the time I was preparing this book.

I would also like to acknowledge my colleagues at Dutch Space and theDelft University of Technology in general, but in particular I would like tothank my collegue John Tyrrell at Dutch Space, for all the discussions wehad about vibration problems within the framework of spacecraft structuresprojects, and Gillian Saunders-Smits at the Delft University of Technologyfor reading the English text. Also, I would like to thank Bas Franssen forreading the sections on the Mode Acceleration Technique and LoadTransformation Matrices.

Jaap WijkerVelserbroek 2003

Table ofContents

Introduct ion 1

1.1 Why Another Book about Mechanical Vibrations ? 1

1.2 A Short Overview of Theory 5

1.2.1 Single Degree of Freedom (sdot) Systems 5

1.2.2 Damped Vibrations 6

1.2.3 Multi-Degrees of Freedom (mdot) Dynamic Systems 7

1.2.4 Modal Analysis 8

1.2.5 Modal Effective Mass 9

1.2.6 Response Analysis 9

1.2.7 Transient Response Analysis 10

1.2.8 Random Vibrations 11

1.2.9 Shock-Response Spectrum 12

1.2.10 Acoustic Loads, Structural Responses 13

1.2.11 Statistical Energy Analysis 13

1.2.12 Inertia-Relief 14

1.2.13 Mode Acceleration Method 141.2.14 Residual Vectors 14

1.2.15 Dynamic Model Reduction 151.2.16 Component Model Synthesis 16

1.2.17 Load Transformation Matrice s 17

1.3 Problems 17

1.3.1 Problem 1 17

2 Single Degree of Freedom System 19

2.1 Introduction 19

2.2 Undamped Sdof System 20

2.2.1 Solution of an Sdof System with Initial Conditions 23

x Table of Contents

2.2.2 Solution of an Sdof System with Applied Forces 24

2.3 Damped Vibration and the Damping Ratio 27

2.3.1 Solut ion of the Sdof System in the Time Domain 29

2.3.2 Solution of the Damped Sdof System with Applied Forces 32

2.3.3 Solut ion in the Frequency Domain 34

2.3.4 State Space Representation of the Sdof System .42

2.4 Problems 45

2.4.1 Problem I 45

2.4.2 Problem 2 46

2.4.3 Problem 3 46

2.4.4 Problem 4 46

2.4.5 Problem 5 47

2.4.6 Problem 6 47

2.4.7 Problem 7 47

2.4.8 Problem 8 48

3 Damping Models 49

3.1 Introduction 49

3.2 Damped Vibration 50

3.2.1 Linear Damping 50

3.2.2 Viscous Damping 51

3.2.3 Structural Damping 51

3.2.4 Loss Factor 52

3.3 Amplification Factor 53

3.3.1 Modal Viscous Damping 53

3.3.2 Modal Structural Damping 54

3.3.3 Discussion of Modal Damping 55

3.4 Method of Determining Damping from Measurements 56

3.4.1 The Half-Power Point Method 56

3.5 Problems 57

3.5.1 Problem 1 57

3.5.2 Problem 2 58

4 Multi-Degrees of Freedom Linear Dynamic Systems 59

4.1 Introduction 59

4.2 Derivation of the Equations of Motion 60

4.2.1 Undamped Equations of Motion with Newton's Law 60

4.2.2 Undamped Equations of Motion using Energies 62

4.2.3 Undamped Equations of Motion using Lagrange's Equations 63

Table of Contents

4.2.4 Damped Equations of Motion using Lagrange's Equations 65

4.3 Finite Element Method 70

4.4 Problems 70

4.4.1 Problem I 70

4.4.2 Problem 2 71

5 Modal Analysis 73

5.1 Introduction 73

5.2 Undamped Linear Dynamic Systems 735.2.1 Natural Frequencies and Mode Shapes 74

5.2.2 Orthogonality Relations of Modes 77

5.2.3 Rigid-Body Modes 80

5.2.4 Left Eigenvectors 84

5.3 Damped Linear Dynamic Systems 86

5.3.1 The State Vector 86

5.3.2 Eigenvalue Problem 88

5.3.3 Eigenvectors 89

5.4 Problems 92

5.4.1 Problem I 92

5.4.2 Problem 2 93

6 Natural Frequencies , an Approximation 95

6.1 Introduction 95

6.2 Static Displacement Method 95

6.3 Rayleigh's Quotient 98

6.4 Dunkerley 's Method 101

6.5 Problems 107

6.5.1 Problem I 107

6.5.2 Problem 2 107

6.5.3 Problem 3 107

6.5.4 Problem 4 108

6.5.5 Problem 5 109

7 Modal Effective Mass III

7.1 Introduction III

7.2 Enforced Acceleration III

7.3 Modal Effective Masses of an Mdof System 114

7.4 Problems 122

xi

xli Tableof Contents

7.4.1 Problem I 122

7.4.2 Problem 2 123

8 Response Analysis 125

8.1 Introduction 125

8.2 Forces and Enforced Acceleration 125

8.2.1 Relative Motions 126

8.2.2 Absolute Motions 138

8.2.3 Large-Mass Approach 140

8.3 Problems 146

8.3.1 Problem 1 146

8.3.2 Problem 2 147

8.3.3 Problem 3 148

9 Transient-Response Analysis 149

9.1 Introduction 149

9.2 Numerical Time Integration 151

9.2.1 Discrete Solution Convolution Integral... 151

9.2.2 Explicit Time-Integration Method 153

9.2.3 Implicit Time-Integration Methods 153

9.2.4 Stability 153

9.3 Explicit Time-Integration 154

9.3.1 Central Difference Method 154

9.3.2 Runge-Kutta Formulae for First-Order Differential Equations 156

9.3.3 Runge-Kutta-Nystrom Method for S-O Differential Equations 159

9.4 Implicit Time Integration 159

9.4.1 HouboItMethod 160

9.4.2 Wilson-theta Method 162

9.4.3 Newmark-beta Method 164

9.4.4 The Hughes, Hilber and Taylor (HHT) alpha-Method 166

9.4.5 The Wood, Bossak and Zienkiewicz (WBZ) alpha-Method 167

9.4.6 The Generalised-alpha Algorithm 168

9.5 Piecewise Linear Method 169

9.6 Problems 170

9.6.1 Problem 1 170

9.6.2 Problem 2 172

9.6.3 Problem 3 172

Table of Contents xiii

10 Shock-Response Spectrum 173

10.1 Introduction 173

10.2 Enforced Acceleration174

10.3 Numerical Calculation of the SRS, the Piecewise Exact Method 176

10.4 Response Analysis in Combination with Shock-Response Spectra 181

10.5 Matching Shock Spectra with Synthesised Time Histories 190

10.6 Problems 199

10.6.1 Problem 1 199

10.6.2 Problem2 200

11 Random Vibration of Linear Dynamic Systems 201

11.1 Introduction 201

11.2 Random Process 201

11 .3 Power-Spectral Density 207

11.4 Deterministic Linear Dynamic System 212

11.4.1 Force-Loaded Sdof System 214

11.4.2 Enforced Acceleration 216

11.4.3 Multi-Inputs and SingleOutput(MISO) 223

11.5 Deterministic Mdof Linear Dynamic System 224

11.5.1 RandomForces 224

11.5.2 RandomBase Excitation 227

11.5.3 RandomStresses and Forces 229

11.6 Analysis of Narrow-Band Processes 234

11 .6.1 Crossings 23411.6.2 Fatigue Damagedue to Random Excitation 238

11.7 Some Practical Aspects 241

11.8 Problems 24411 .8.1 Problem 1 24411 .8.2 Problem 2 24411 .8.3 Problem 3 245

11 .8.4 Problem 4 245

11 .8.5 Problem 5 246

12 Low-Frequency Acoustic Loads, Structural Responses 247

12.1 Introduction 247

12.2 Acoustic Loads 247

12.3 Equations of Motion 249

12.4 Problems 260

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12.4.1 Problem 1 26012.4.2 Problem 2 261

13 Statistical Energy Analysis 263

13.1 Introduction 263

13.2 Some Basics about Averaged Quantities 264

13.3 Two Coupled Oscillators 270

13.4 Multimode Subsystems 277

13.5 SEA Parameters 28313.5.1 Subsystem Modal Densities 28313.5.2 Source Power Input 28813.5.3 Subsystem Energies 28913.5.4 Damping Loss Factor 29413.5.5 Coupling Loss Factor 295

13.6 Stresses and Strains 298

13.7 Problems 29913.7.1 Problem 1 29913.7.2 Problem 2 29913.7.3 Problem 3 30013.7.4 Problem 4 30013.7.5 Problem 5 30013.7.6 Problem 6 30 1

13.7.7 Problem 7 30113.7.8 Problem 8 302

14 Free-free Dynamic Systems, Inertia Relief 303

14.1 Introduction 303

14.2 Relative Motion 303

14.3 Relative Forces 304

14.4 Flexibility Matrix 307

14.5 Problems 31014.5.1 Problem 1 310

15 Mode Acceleration Method 313

15.1 Introduction 313

15.2 Decomposition of Flexibility and Mass Matrix 31315.2.1 Decomposition of the Flexibility Matrix 31315.2.2 Decomposition of the Mass Matrix 315

Table of Contents xv

15.2.3 ConvergenceProperties of ReconstructedMatrices 316

15.3 Mode Acceleration Method 318

15.4 Problems 325

15.4.1 Problem 1 32515.4.2 Problem2 325

15.4.3 Problem3 326

15.4.4 Problem 4 326

15.4.5 Problem5 327

16 Residual Vectors 331

16.1 Introduction 331

16.2 Residual Vectors 33116.2.1 Dickens Method 331

16.2.2 Rose Method 334

16.3 Problems 34016.3.1 Problem 1 340

17 Dynamic Model Reduction Methods 343

17.1 Introduction 343

17.2 Static Condensation Method 344

17.2.1 Improved Calculation of Eliminated Dofs 350

17.3 Dynamic Reduction 351

17.4 Improved Reduced System (IRS) 352

17.5 Craig-Bampton Reduced Models 355

17.6 Generalised Dynamic Reduction 358

17.7 System Equivalent Reduction Expansion Process (SEREP) 362

17.8 Ritz Vectors 365

17.9 Conclusion 367

18 Component Mode Synthesis 369

18.1 Introduction 369

18.2 The Unified CMS Method 370

18.2.1 Modal Truncation 371

18.2.2 General Synthesis of TwoComponents 372

18.2.3 General Example 374

18.3 Special CMS Methods 379

18.3.1 Craig-Bampton Fixed-InterfaceMethod 379

18.3.2 Free-Interface Method 384

xvi Table of Contents

18.3.3 General-Purpose eMS Method 391

18.4 Problems 39618.4.1 Problem 1 396

18.4.2 Problem 2 397

19 Load Transformation Matrices 399

19.1 Introduction 399

19.2 Reduced Model with Boundary Conditions 400

19.3 Reduced Free-Free Dynamic Model 404

19.4 Continuous Dynamic Systems .409

19.5 Problems 41319.5.1 Problem I 41319.5.2 Problem 2 414

19.5.3 Problem 3 41519.5.4 Problem 4 415

References 417

Author Index 427

Subject Index 431