Mechanical Characterization of Materials and Wave Dispersion: Instrumentation and Experiment Interpretation

Preface xxi

Acknowledgements xxxi

PART I - MECHANICAL AND ELECTRONIC INSTRUMENTATION 1

Chapter 1. Guidelines for Choosing the Experimental Set-up 3
Jean Tuong VINH

1.1. Choice of matrix coefficient to be evaluated and type of wave to be adopted 4

1.2. Influence of frequency range 8

1.3. Dimensions and shape of the samples 9

1.4. Tests at high and low temperature 10

1.5. Sample holder at high temperature 10

1.6. Visual observation inside the ambient room 11

1.7. Complex moduli of viscoelastic materials and damping capacity measurements 11

1.8. Previsional calculation of composite materials 11

1.9. Bibliography 11

Chapter 2. Review of Industrial Analyzers for Material Characterization 13
Jean Tuong VINH

2.1. Rheovibron and its successive versions 14

2.2. Dynamic mechanical analyzer DMA 01dB–Metravib and VHF 104 Metravib analyzer 17

2.3. Bruel and Kjaer complex modulus apparatus (Oberst Apparatus) 18

2.4. Dynamic mechanical analyzer DMA – Dupont de Nemours 980 20

2.5. Elasticimeter using progressive wave PPM 5 22

2.6. Bibliography 24

Chapter 3. Mechanical Part of the Vibration Test Bench 25
Jean Tuong VINH

3.1. Clamping end 25

3.2. Length correction 29

3.3. Supported end 33

3.4. Additional weight or additional torsion lever used as a boundary condition 34

3.5. Free end 34

3.6. Pseudo-clamping sample attachment 35

3.7. Sample suspended by taut threads 38

3.8. Sample on foam rubber plate serving as a mattress 41

3.9. Climatic chamber 41

3.10. Vacuum system 41

3.11. Bibliography 42

Chapter 4. Exciters and Excitation Signals 43
Jean Tuong VINH

4.1. Frequency ranges 43

4.2. Power 43

4.3. Nature and performance of various exciters 44

4.4. Room required for exciter installation 47

4.5. Details for electrodynamic shakers 48

4.6. Low cost electromagnetic exciters with permanent magnet 54

4.7. Piezoelectric and ferroelectric exciters 55

4.8. Design of special ferroelectric transducers 67

4.9. Power piezoelectric exciters 69

4.10. Technical details concerning ultrasonic emitters for the measurement of material stiffness coefficients on ultrasonic test benches 70

4.11. Bibliography 74

4.12. Appendix 4A. Example of ferroelectric plates and disks 74

Chapter 5. Transducers 77
Jean Tuong VINH and Michel NUGUES

5.1. Introduction 77

5.2. Transducers and their principal performance 78

5.3. The main classes of fixed reference transducers 79

5.4. Condenser-type transducer 82

5.5. Inductance transducers 89

5.6. Mutual inductance transducer 92

5.7. Differential transformer transducer 93

5.8. Contactless inductance transducer with a permanent magnet 93

5.9. Eddy current transducer 94

5.10. Seismic transducers 97

5.11. Piezoresistive accelerometer 109

5.12. Other transducers 110

5.13. Force transducers 111

5.14. Bibliography 113

5.15. Appendix 5A. Condenser with polarization 113

5.16. Appendix 5B. Eigenfrequencies of some force transducers: Rayleigh and Rayleigh-Ritz upper bound methods 115

5B.1. Rayleigh’s method 116

5B.2. Rayleigh-Ritz’s method 117

5B.3. Preliminary experimental test on the force transducer 117

Chapter 6. Electronic Instrumentation, Connecting Cautions and Signal Processing 119
Jean Tuong VINH

6.1. Preamplifiers and signal conditioners following the transducers 120

6.2. Cables and wiring considerations 121

6.3. Transducer selection and mountings 123

6.4. Transducer calibration 129

6.5. Digital signal processing systems: an overview 133

6.6. Other signal processing programs 141

6.7. Reasoned choice of excitation signals 142

6.8. Bibliography 146

6.9. Appendix 6A. The Shannon theorem and aliasing phenomenon 147

6.10. Appendix 6B. Time window (or weighting function)150

6B.1. Kaiser-Bessel window 151

6B.2. Hamming window 152

Chapter 7. The Frequency Hilbert Transform and Detection of Hidden Non-linearities in Frequency Responses 155
Jean Tuong VINH

7.1. Introduction 155

7.2. Mathematical expression of the Hilbert transform 157

7.3. Kramer-Kronig’s relationships 162

7.4. Causal signal and Fourier transform 163

7.5. Hilbert transform of a truncated transfer function 164

7.6. Impulse response of a system. Non-causality due to measurement defects 172

7.7. Summary of principal result in sections 7.5 and 7.6 174

7.8. Causalized Hilbert transform 175

7.9. Some practical aspects of Hilbert transform computation 176

7.10. Conclusion 181

7.11. Bibliography 181

7.12. Appendix 7A. Line integral of complex function and Cauchy’s integral 182

7A.1. Analyticity of a function f(z) of complex variable z 182

7A.2. Expression of Cauchy’s integral of the function f(z)/(z-􀢻􁈻 183

7.13. Appendix 7B. Hilbert transform obtained directly by Guillemin’s method 184

Chapter 8. Measurement of Structural Damping 187
Jean Tuong VINH

8.1. Introduction 187

8.2. Overview of various methods used to evaluate damping ratios in structural dynamics 190

8.3. Measurement of structural damping coefficient by multimodal analysis 197

8.4. The Hilbert envelope time domain method 201

8.5. Detection of hidden non-linearities 203

8.6. How to relate material damping to structural damping? 203

8.7. Concluding remarks 207

8.8. Bibliography 208

PART II - REALIZATION OF EXPERIMENTAL SET-UPS AND INTERPRETATION OF MEASUREMENTS 209

Chapter 9. Torsion Test Benches: Instrumentation and Experimental Results 211
Michel NUGUES

9.1. Introduction 211

9.2. Industrial torsion test bench 211

9.3. Parasitic bending vibration of rod 215

9.4. Shear moduli of transverse isotropic materials 215

9.5. Elastic moduli obtained for various materials 220

9.6. Experimental set-up to obtain dispersion curves in a large frequency range 222

9.7. Experimental results obtained on short samples 224

9.8. Experimental wave dispersion curves obtained by torsional vibrations of a rod with rectangular cross-section 227

9.9. Frequency spectrum for isotropic metallic materials (aluminum and steel alloy) 230

9.10. Impact test on viscoelastic high damping material 232

9.11. Concluding remarks 238

9.12. Bibliography 239

9.13. Appendix 9A. Choice of equations of motion 240

9A.1. Circular cross-section 240

9A.2. Square cross-section 241

9A.3. Rectangular cross-section 241

9A.4. Ratio of Young’s modulus to shear modulus 241

9A.5. Special experimental studies of wave dispersion phenomenon 242

9.14. Appendix 9B. Complementary information concerning formulae used to interpret torsion tests 242

9B.1. Quick overview of Saint Venant’s theory applied to the problem of dynamic Torsion 242

9.15. Appendix 9C. Details concerning the βΤ(c) function in the calculation of rod stiffness
CT 245

9.16. Appendix 9D. Compliments concerning the solution of equations of motion with first order theory 246

9D.1. Displacement field 246

9D.2. Relations between two sets of coefficients 246

9D.3. Equations giving the two sets of coefficients Aa, Ba, Ca, Da deduced from the four boundary conditions 248

9D.4. Evaluation of coefficients in [9D.6] 248

9D.5. Equations in Aa, Ba, Ca, Da deduced from the four boundary conditions 249

Chapter 10. Bending Vibration of Rod Instrumentation and Measurements 255
Dominique LE NIZHERY

10.1. Introduction 255

10.2. Realization of an elasticimeter 255

10.3. How to conduct bending tests 262

10.4. Concluding remarks 267

10.5 Bibliography 268

10.6. Appendix 10A. Useful formulae to evaluate the Young’s modulus by bending vibration of rods 268

10A.1. Bernoulli-Euler’s equation 268

10A.2. Timoshenko-Mindlin’s equation 269

10A.3. Boundary conditions and wave number equation 269

10A.4. Important parameters in rod bending vibration 269

10A.5. Expression of the wave number 270

10A.6. Young’s modulus (Bernoulli’s theory) 270

10A.7. Young’s modulus (Timoshenko-Mindlin’s equation) 270

Chapter 11. Longitudinal Vibrations of Rods: Material Characterization and Experimental Dispersion Curves 271
Yvon CHEVALIER and Jean Tuong VINH

11.1. Introduction 271

11.2. Mechanical set-up 272

11.3. Electronic set-up 272

11.4. Estimation of phase velocity 274

11.5. Short samples and eigenvalue calculations for various materials 280

11.6. Experimental results interpreted by the two theories 283

11.7. Influence of slenderness (δL = 2L/h) on eigenfrequency 291

11.8. Experimental results obtained with short rod 292

11.9. Concluding remarks 292

11.10. Bibliography 295

11.11. Appendix 11A. Eigenvalue equation for rod of finite length 296

11.12. Appendix 11B. Additional information concerning solutions of Touratier’s  equations 300

11B.1. Eigenequation with elementary theory of motion 301

Chapter 12. Realization of Le Rolland-Sorin’s Double Pendulum and Some Experimental Results 305
Mostefa ARCHI and Jean-Baptiste CASIMIR

12.1. Introduction 305

12.2. Principal mechanical parts of the double pendulum system 305

12.3. Instrumentation 312

12.4. Experimental precautions 315

12.5. Details and characteristics of the elasticimeter 317

12.6. Some experimental results 318

12.7. Damping ratio estimation by logarithmic decrement method 322

12.8. Concluding remarks 324

12.9. Bibliography 325

12.10. Appendix 12A. Equations of motion for the set (pendulums, platform and sample) and Young’s modulus calculation deduced from bending tests 326

12A.1. Equations of motion 326

12A.2. Solutions for pendulum oscillations 328

12A.3. Relationship between beating period τ and sample stiffness k 329

12A.4. Young’s modulus calculation 330

12.11. Appendix 12B. Evaluation of shear modulus by torsion tests 331

12B.1. Energy expression 331

Chapter 13. Stationary and Progressive Waves in Rings and Hollow Cylinders 335
Yvon CHEVALIER and Jean Tuong VINH

13.1. Introduction 335

13.2. Choosing the samples based on material symmetry 336

13.3. Practical realization of a special elasticimeter for curved beams and rings: in plane bending vibrations 337

13.4. Ultrasonic benches 342

13.5. Experimental results and interpretation 343

13.6. List of symbols 358

13.7. Bibliography 359

13.8. Appendix 13A. Evaluation of Young’s modulus by using in plane bending motion of the ring 359

13.9. Appendix 13B. Determination of inertia moment of a solid by means of a three-string pendulum 360

13B.1. Principle of the method 360

13B.2. Calculations 361

13.10. Appendix 13C. Necessary formulae to evaluate Young’s
modulus of a straight beam 364

Chapter 14. Ultrasonic Benches: Characterization of Materials by Wave Propagation Techniques 367
Patrick GARCEAU

14.1. Introduction 367

14.2. Ultrasonic transducers 367

14.3. Pulse generator 369

14.4. Mechanical realization of ultrasonic benches 371

14.5. Experimental interpretation of phase velocity and group velocity 375

14.6. Some experimental results on composite materials 380

14.7. Viscoelastic characterization of materials by ultrasonic waves 383

14.8. Bibliography 388

14.9. Appendix 14A. Oblique incidence and energy propagation direction 389

14.10. Appendix 14B. Water immersion bench, measurement of coefficients of stiffness matrix 392

14B.1. Expression of phase velocity in the sample 393

14B.2. Phase velocity measurement by propagation time (?·?nt ) evaluation 394

14B.3. Phase velocity evaluation without time measurements 394

Chapter 15. Wave Dispersion in Rods with a Rectangular Cross-section: Higher Order Theory and Experimentation 397
Maurice TOURATIER

15.1. Introduction 397

15.2. Summary table of some wave dispersion research 398

15.3. Longitudinal wave dispersion: influence of the material and geometry of the bounded medium 399

15.4. Bending wave dispersion 403

15.5. First order for torsional motion in a transverse isotropic rod 408

15.6. Interest in theories with higher degrees of approximation 414

15.7. Experimental set-ups to visualize stationary waves in rods 416

15.8. Electronic set-up and observed signals on a multi-channel oscilloscope 421

15.9. Presentation of experimental results 424

15.10. Concluding remarks 427

15.11. Bibliography 428

15.12. Appendix 15A. Touratier’s theory using Hellinger–Reissner’s mixed fields 429

15A.1. Outline of Touratier’s mixed field theory 429

15A.2. General equations deduced from the two fields principle 432

15A.3. Formulation of the boundary condition problem 432

15A.4. Symmetry considerations concerning the three kinds of motion 433

15A.5. Truncating process for one dimensional theories: extensional waves 437

15A.6. Equations of motion for extensional movement 438

15A.7. Effective front velocity and wave front velocity 439

15A.8. Bending equations of motion 441

15A.9. Equations of motion: torsional vibration 444

15.13. Appendix 15B. Third order Touratier’s theory 445

15B.1. Extensional waves with nine evaluated modes 446

15B.2. Geometrical characteristics of displacement components uj mn and physical interpretation 447

15B.3. Bending mode in the direction x􁈬􁈬􁈬􀬶􀔦 – geometrical interpretation 448

15B.4. Shear motion around longitudinal rod axis 450

List of Authors 453

Index 455