Mechanical Characterization of Materials and Wave Dispersion V 2

<p>Preface xxi</p>
<p>Acknowledgements xxxi</p>
<p>PART I – MECHANICAL AND ELECTRONIC INSTRUMENTATION 1</p>
<p>Chapter 1. Guidelines for Choosing the Experimental Set–up 3<br /> Jean Tuong VINH</p>
<p>1.1. Choice of matrix coefficient to be evaluated and type of wave to be adopted 4</p>
<p>1.2. Influence of frequency range 8</p>
<p>1.3. Dimensions and shape of the samples 9</p>
<p>1.4. Tests at high and low temperature 10</p>
<p>1.5. Sample holder at high temperature 10</p>
<p>1.6. Visual observation inside the ambient room 11</p>
<p>1.7. Complex moduli of viscoelastic materials and damping capacity measurements 11</p>
<p>1.8. Previsional calculation of composite materials 11</p>
<p>1.9. Bibliography 11</p>
<p>Chapter 2. Review of Industrial Analyzers for Material Characterization 13<br /> Jean Tuong VINH</p>
<p>2.1. Rheovibron and its successive versions 14</p>
<p>2.2. Dynamic mechanical analyzer DMA 01dB Metravib and VHF 104 Metravib analyzer 17</p>
<p>2.3. Bruel and Kjaer complex modulus apparatus (Oberst Apparatus) 18</p>
<p>2.4. Dynamic mechanical analyzer DMA Dupont de Nemours 980 20</p>
<p>2.5. Elasticimeter using progressive wave PPM 5 22</p>
<p>2.6. Bibliography 24</p>
<p>Chapter 3. Mechanical Part of the Vibration Test Bench 25<br /> Jean Tuong VINH</p>
<p>3.1. Clamping end 25</p>
<p>3.2. Length correction 29</p>
<p>3.3. Supported end 33</p>
<p>3.4. Additional weight or additional torsion lever used as a boundary condition 34</p>
<p>3.5. Free end 34</p>
<p>3.6. Pseudo–clamping sample attachment 35</p>
<p>3.7. Sample suspended by taut threads 38</p>
<p>3.8. Sample on foam rubber plate serving as a mattress 41</p>
<p>3.9. Climatic chamber 41</p>
<p>3.10. Vacuum system 41</p>
<p>3.11. Bibliography 42</p>
<p>Chapter 4. Exciters and Excitation Signals 43<br /> Jean Tuong VINH</p>
<p>4.1. Frequency ranges 43</p>
<p>4.2. Power 43</p>
<p>4.3. Nature and performance of various exciters 44</p>
<p>4.4. Room required for exciter installation 47</p>
<p>4.5. Details for electrodynamic shakers 48</p>
<p>4.6. Low cost electromagnetic exciters with permanent magnet 54</p>
<p>4.7. Piezoelectric and ferroelectric exciters 55</p>
<p>4.8. Design of special ferroelectric transducers 67</p>
<p>4.9. Power piezoelectric exciters 69</p>
<p>4.10. Technical details concerning ultrasonic emitters for the measurement of material stiffness coefficients on ultrasonic test benches 70</p>
<p>4.11. Bibliography 74</p>
<p>4.12. Appendix 4A. Example of ferroelectric plates and disks 74</p>
<p>Chapter 5. Transducers 77<br /> Jean Tuong VINH and Michel NUGUES</p>
<p>5.1. Introduction 77</p>
<p>5.2. Transducers and their principal performance 78</p>
<p>5.3. The main classes of fixed reference transducers 79</p>
<p>5.4. Condenser–type transducer 82</p>
<p>5.5. Inductance transducers 89</p>
<p>5.6. Mutual inductance transducer 92</p>
<p>5.7. Differential transformer transducer 93</p>
<p>5.8. Contactless inductance transducer with a permanent magnet 93</p>
<p>5.9. Eddy current transducer 94</p>
<p>5.10. Seismic transducers 97</p>
<p>5.11. Piezoresistive accelerometer 109</p>
<p>5.12. Other transducers 110</p>
<p>5.13. Force transducers 111</p>
<p>5.14. Bibliography 113</p>
<p>5.15. Appendix 5A. Condenser with polarization 113</p>
<p>5.16. Appendix 5B. Eigenfrequencies of some force transducers: Rayleigh and Rayleigh–Ritz&nbsp;upper bound methods 115</p>
<p>5B.1. Rayleigh s method 116</p>
<p>5B.2. Rayleigh–Ritz s method 117</p>
<p>5B.3. Preliminary experimental test on the force transducer 117</p>
<p>Chapter 6. Electronic Instrumentation, Connecting Cautions and Signal Processing 119<br /> Jean Tuong VINH</p>
<p>6.1. Preamplifiers and signal conditioners following the transducers 120</p>
<p>6.2. Cables and wiring considerations 121</p>
<p>6.3. Transducer selection and mountings 123</p>
<p>6.4. Transducer calibration 129</p>
<p>6.5. Digital signal processing systems: an overview 133</p>
<p>6.6. Other signal processing programs 141</p>
<p>6.7. Reasoned choice of excitation signals 142</p>
<p>6.8. Bibliography 146</p>
<p>6.9. Appendix 6A. The Shannon theorem and aliasing phenomenon 147</p>
<p>6.10. Appendix 6B. Time window (or weighting function)150</p>
<p>6B.1. Kaiser–Bessel window 151</p>
<p>6B.2. Hamming window 152</p>
<p>Chapter 7. The Frequency Hilbert Transform and Detection of Hidden Non–linearities in Frequency Responses 155<br /> Jean Tuong VINH</p>
<p>7.1. Introduction 155</p>
<p>7.2. Mathematical expression of the Hilbert transform 157</p>
<p>7.3. Kramer–Kronig s relationships 162</p>
<p>7.4. Causal signal and Fourier transform 163</p>
<p>7.5. Hilbert transform of a truncated transfer function 164</p>
<p>7.6. Impulse response of a system. Non–causality due to measurement defects 172</p>
<p>7.7. Summary of principal result in sections 7.5 and 7.6 174</p>
<p>7.8. Causalized Hilbert transform 175</p>
<p>7.9. Some practical aspects of Hilbert transform computation 176</p>
<p>7.10. Conclusion 181</p>
<p>7.11. Bibliography 181</p>
<p>7.12. Appendix 7A. Line integral of complex function and Cauchy s integral 182</p>
<p>7A.1. Analyticity of a function f(z) of complex variable z 182</p>
<p>7A.2. Expression of Cauchy s integral of the function f(z)/(z– 183</p>
<p>7.13. Appendix 7B. Hilbert transform obtained directly by Guillemin s method 184</p>
<p>Chapter 8. Measurement of Structural Damping 187<br /> Jean Tuong VINH</p>
<p>8.1. Introduction 187</p>
<p>8.2. Overview of various methods used to evaluate damping ratios in structural dynamics 190</p>
<p>8.3. Measurement of structural damping coefficient by multimodal analysis 197</p>
<p>8.4. The Hilbert envelope time domain method 201</p>
<p>8.5. Detection of hidden non–linearities 203</p>
<p>8.6. How to relate material damping to structural damping? 203</p>
<p>8.7. Concluding remarks 207</p>
<p>8.8. Bibliography 208</p>
<p>PART II – REALIZATION OF EXPERIMENTAL SET–UPS AND INTERPRETATION OF MEASUREMENTS 209</p>
<p>Chapter 9. Torsion Test Benches: Instrumentation and Experimental Results 211<br /> Michel NUGUES</p>
<p>9.1. Introduction 211</p>
<p>9.2. Industrial torsion test bench 211</p>
<p>9.3. Parasitic bending vibration of rod 215</p>
<p>9.4. Shear moduli of transverse isotropic materials 215</p>
<p>9.5. Elastic moduli obtained for various materials 220</p>
<p>9.6. Experimental set–up to obtain dispersion curves in a large frequency range 222</p>
<p>9.7. Experimental results obtained on short samples 224</p>
<p>9.8. Experimental wave dispersion curves obtained by torsional vibrations of a rod with rectangular cross–section 227</p>
<p>9.9. Frequency spectrum for isotropic metallic materials (aluminum and steel alloy) 230</p>
<p>9.10. Impact test on viscoelastic high damping material 232</p>
<p>9.11. Concluding remarks 238</p>
<p>9.12. Bibliography 239</p>
<p>9.13. Appendix 9A. Choice of equations of motion 240</p>
<p>9A.1. Circular cross–section 240</p>
<p>9A.2. Square cross–section 241</p>
<p>9A.3. Rectangular cross–section 241</p>
<p>9A.4. Ratio of Young s modulus to shear modulus 241</p>
<p>9A.5. Special experimental studies of wave dispersion phenomenon 242</p>
<p>9.14. Appendix 9B. Complementary information concerning formulae used to interpret torsion tests 242</p>
<p>9B.1. Quick overview of Saint Venant s theory applied to the problem of dynamic Torsion 242</p>
<p>9.15. Appendix 9C. Details concerning the (c) function in the calculation of rod stiffness<br /> CT 245</p>
<p>9.16. Appendix 9D. Compliments concerning the solution of equations of motion with first order theory 246</p>
<p>9D.1. Displacement field 246</p>
<p>9D.2. Relations between two sets of coefficients 246</p>
<p>9D.3. Equations giving the two sets of coefficients Aa, Ba, Ca, Da deduced from the four boundary conditions 248</p>
<p>9D.4. Evaluation of coefficients in [9D.6] 248</p>
<p>9D.5. Equations in Aa, Ba, Ca, Da deduced from the four boundary conditions 249</p>
<p>Chapter 10. Bending Vibration of Rod Instrumentation and Measurements 255<br /> Dominique LE NIZHERY</p>
<p>10.1. Introduction 255</p>
<p>10.2. Realization of an elasticimeter 255</p>
<p>10.3. How to conduct bending tests 262</p>
<p>10.4. Concluding remarks 267</p>
<p>10.5 Bibliography 268</p>
<p>10.6. Appendix 10A. Useful formulae to evaluate the Young s modulus by bending vibration of rods 268</p>
<p>10A.1. Bernoulli–Euler s equation 268</p>
<p>10A.2. Timoshenko–Mindlin s equation 269</p>
<p>10A.3. Boundary conditions and wave number equation 269</p>
<p>10A.4. Important parameters in rod bending vibration 269</p>
<p>10A.5. Expression of the wave number 270</p>
<p>10A.6. Young s modulus (Bernoulli s theory) 270</p>
<p>10A.7. Young s modulus (Timoshenko–Mindlin s equation) 270</p>
<p>Chapter 11. Longitudinal Vibrations of Rods: Material Characterization and Experimental Dispersion Curves 271<br /> Yvon CHEVALIER and Jean Tuong VINH</p>
<p>11.1. Introduction 271</p>
<p>11.2. Mechanical set–up 272</p>
<p>11.3. Electronic set–up 272</p>
<p>11.4. Estimation of phase velocity 274</p>
<p>11.5. Short samples and eigenvalue calculations for various materials 280</p>
<p>11.6. Experimental results interpreted by the two theories 283</p>
<p>11.7. Influence of slenderness ( L = 2L/h) on eigenfrequency 291</p>
<p>11.8. Experimental results obtained with short rod 292</p>
<p>11.9. Concluding remarks 292</p>
<p>11.10. Bibliography 295</p>
<p>11.11. Appendix 11A. Eigenvalue equation for rod of finite length 296</p>
<p>11.12. Appendix 11B. Additional information concerning solutions of Touratier s&nbsp; equations 300</p>
<p>11B.1. Eigenequation with elementary theory of motion 301</p>
<p>Chapter 12. Realization of Le Rolland–Sorin s Double Pendulum and Some Experimental Results 305<br /> Mostefa ARCHI and Jean–Baptiste CASIMIR</p>
<p>12.1. Introduction 305</p>
<p>12.2. Principal mechanical parts of the double pendulum system 305</p>
<p>12.3. Instrumentation 312</p>
<p>12.4. Experimental precautions 315</p>
<p>12.5. Details and characteristics of the elasticimeter 317</p>
<p>12.6. Some experimental results 318</p>
<p>12.7. Damping ratio estimation by logarithmic decrement method 322</p>
<p>12.8. Concluding remarks 324</p>
<p>12.9. Bibliography 325</p>
<p>12.10. Appendix 12A. Equations of motion for the set (pendulums, platform and sample) and Young s modulus calculation deduced from bending tests 326</p>
<p>12A.1. Equations of motion 326</p>
<p>12A.2. Solutions for pendulum oscillations 328</p>
<p>12A.3. Relationship between beating period and sample stiffness k 329</p>
<p>12A.4. Young s modulus calculation 330</p>
<p>12.11. Appendix 12B. Evaluation of shear modulus by torsion tests 331</p>
<p>12B.1. Energy expression 331</p>
<p>Chapter 13. Stationary and Progressive Waves in Rings and Hollow Cylinders 335<br /> Yvon CHEVALIER and Jean Tuong VINH</p>
<p>13.1. Introduction 335</p>
<p>13.2. Choosing the samples based on material symmetry 336</p>
<p>13.3. Practical realization of a special elasticimeter for curved beams and rings: in plane bending vibrations 337</p>
<p>13.4. Ultrasonic benches 342</p>
<p>13.5. Experimental results and interpretation 343</p>
<p>13.6. List of symbols 358</p>
<p>13.7. Bibliography 359</p>
<p>13.8. Appendix 13A. Evaluation of Young s modulus by using in plane bending motion of the ring 359</p>
<p>13.9. Appendix 13B. Determination of inertia moment of a solid by means of a three–string pendulum 360</p>
<p>13B.1. Principle of the method 360</p>
<p>13B.2. Calculations 361</p>
<p>13.10. Appendix 13C. Necessary formulae to evaluate Young s<br /> modulus of a straight beam 364</p>
<p>Chapter 14. Ultrasonic Benches: Characterization of Materials by Wave Propagation Techniques 367<br /> Patrick GARCEAU</p>
<p>14.1. Introduction 367</p>
<p>14.2. Ultrasonic transducers 367</p>
<p>14.3. Pulse generator 369</p>
<p>14.4. Mechanical realization of ultrasonic benches 371</p>
<p>14.5. Experimental interpretation of phase velocity and group velocity 375</p>
<p>14.6. Some experimental results on composite materials 380</p>
<p>14.7. Viscoelastic characterization of materials by ultrasonic waves 383</p>
<p>14.8. Bibliography 388</p>
<p>14.9. Appendix 14A. Oblique incidence and energy propagation direction 389</p>
<p>14.10. Appendix 14B. Water immersion bench, measurement of coefficients of stiffness matrix 392</p>
<p>14B.1. Expression of phase velocity in the sample 393</p>
<p>14B.2. Phase velocity measurement by propagation time (?&middot;?nt ) evaluation 394</p>
<p>14B.3. Phase velocity evaluation without time measurements 394</p>
<p>Chapter 15. Wave Dispersion in Rods with a Rectangular Cross–section: Higher Order Theory and Experimentation 397<br /> Maurice TOURATIER</p>
<p>15.1. Introduction 397</p>
<p>15.2. Summary table of some wave dispersion research 398</p>
<p>15.3. Longitudinal wave dispersion: influence of the material and geometry of the bounded medium 399</p>
<p>15.4. Bending wave dispersion 403</p>
<p>15.5. First order for torsional motion in a transverse isotropic rod 408</p>
<p>15.6. Interest in theories with higher degrees of approximation 414</p>
<p>15.7. Experimental set–ups to visualize stationary waves in rods 416</p>
<p>15.8. Electronic set–up and observed signals on a multi–channel oscilloscope 421</p>
<p>15.9. Presentation of experimental results 424</p>
<p>15.10. Concluding remarks 427</p>
<p>15.11. Bibliography 428</p>
<p>15.12. Appendix 15A. Touratier s theory using Hellinger Reissner s mixed fields 429</p>
<p>15A.1. Outline of Touratier s mixed field theory 429</p>
<p>15A.2. General equations deduced from the two fields principle 432</p>
<p>15A.3. Formulation of the boundary condition problem 432</p>
<p>15A.4. Symmetry considerations concerning the three kinds of motion 433</p>
<p>15A.5. Truncating process for one dimensional theories: extensional waves 437</p>
<p>15A.6. Equations of motion for extensional movement 438</p>
<p>15A.7. Effective front velocity and wave front velocity 439</p>
<p>15A.8. Bending equations of motion 441</p>
<p>15A.9. Equations of motion: torsional vibration 444</p>
<p>15.13. Appendix 15B. Third order Touratier s theory 445</p>
<p>15B.1. Extensional waves with nine evaluated modes 446</p>
<p>15B.2. Geometrical characteristics of displacement components uj mn and physical interpretation 447</p>
<p>15B.3. Bending mode in the direction x geometrical interpretation 448</p>
<p>15B.4. Shear motion around longitudinal rod axis 450</p>
<p>List of Authors 453</p>
<p>Index 455</p>