Mechanical Characterization of Materials and Wave Dispersion, Volume 2
Instrumentation and Experiment Interpretation
Will be published approx. on 16. July 2010
Book
Hardback
490 pages
978-1-84821-193-3 (ISBN)
Description
Over the last 50 years, the methods of investigating dynamic properties have resulted in significant advances. This book explores dynamic testing, the methods used, and the experiments performed, placing a particular emphasis on the context of bounded medium elastodynamics. Dynamic tests have proven to be as efficient as static tests and are often easier to use at lower frequency. The discussion is divided into four parts. Part A focuses on the complements of continuum mechanics. Part B concerns the various types of rod vibrations: extensional, bending, and torsional. Part C is devoted to mechanical and electronic instrumentation, and guidelines for which experimental set-up should be used are given. Part D concentrates on experiments and experimental interpretations of elastic or viscolelastic moduli. In addition, several chapters contain practical examples alongside theoretical discussion to facilitate the readers understanding. The results presented are the culmination of over 30 years of research by the authors and as such will be of great interest to anyone involved in this field.
More details
Product info
gebunden
Edition
1. Auflage
Language
English
Place of publication
London
United Kingdom
Target group
Professional and scholarly
Product notice
Unsewn / adhesive bound
Paper over boards
Dimensions
Height: 234 mm
Width: 155 mm
Thickness: 33 mm
Weight
862 gr
ISBN-13
978-1-84821-193-3 (9781848211933)
Schweitzer Classification
Other editions
Additional editions
Yvon Chevalier | Jean Vinh Tuong
Mechanical Characterization of Materials and Wave Dispersion
Instrumentation and Experiment Interpretation
E-Book
03/2013
Wiley-ISTE
€243.99
Available for download
Yvon Chevalier | Jean Vinh Tuong
Mechanical Characterization of Materials and Wave Dispersion
Instrumentation and Experiment Interpretation
E-Book
03/2013
Wiley-ISTE
€243.99
Available for download
Persons
Yvon Chevalier is Emeritus Professor at the Institut Superieur de Mecanique de Paris (SUPMECA), France. Since 2000 he has been coeditor-in-chief of Mecanique et Industries journal, supported by the French Association of Mechanics. He is a well-known expert in the dynamics of composite materials and propagation of waves in heterogeneous materials. He also has extensive experience in the areas of hyper-elasticity and non-linear viscoelasticity of rubber materials.
Jean Tuong Vinh is Emeritus University Professor of Mechanical Engineering at the University of Paris VI in France. He carries out research into theoretical viscoelasticity, non-linear functional Volterra series, computer algorithms in signal processing, frequency Hilbert transform, special impact testing, wave dispersion in rods and continuous elements and solution of related inverse problems.
Jean Tuong Vinh is Emeritus University Professor of Mechanical Engineering at the University of Paris VI in France. He carries out research into theoretical viscoelasticity, non-linear functional Volterra series, computer algorithms in signal processing, frequency Hilbert transform, special impact testing, wave dispersion in rods and continuous elements and solution of related inverse problems.
Content
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 ??(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 (?L = 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 ? 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
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 ??(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 (?L = 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 ? 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