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Mechanical Vibration and Shock Analysis, Second Edition Volume 2Mechanical Shock This volume considers the shock response spectrum, its various definitions, its properties, and the assumptions involved in its calculation. In developing the practical application of these concepts, the shock shapes or profiles most often used in test facilities are presented, together with their characteristics and indications of how to establish test configurations comparable with those of the real-world, measured environment. Following this analysis there is a case study of how to meet these specifications using standard laboratory equipment, shock machines, electrodynamic exciters driven by a time signal or a response spectrum. Discussion of the limitations, advantages and disadvantages of each method is presented. The Mechanical Vibration and Shock Analysis five-volume series has been written with both the professional engineer and the academic in mind. Christian Lalanne explores every aspect of vibration and shock, two fundamental and extremely significant areas of mechanical engineering, from both a theoretical and practical point of view.The five volumes cover all the necessary issues in this area of mechanical engineering. The theoretical analyses are placed in the context of both the real world and the laboratory, which is essential for the development of specifications.
Contents
Foreword to Series xi Introduction xv List of Symbols xvii Chapter 1. Shock Analysis 1 1.1. Definitions 1 1.1.1. Shock 1 1.1.2. Transient signal 2 1.1.3. Jerk 3 1.1.4. Simple (or perfect) shock 3 1.1.5. Half-sine shock 3 1.1.6. Versed sine (or haversine) shock 4 1.1.7. Terminal peak sawtooth (TPS) shock (or final peak sawtooth (FPS)) 5 1.1.8. Initial peak sawtooth (IPS) shock 6 1.1.9. Square shock 7 1.1.10. Trapezoidal shock 8 1.1.11. Decaying sinusoidal pulse 8 1.1.12. Bump test 9 1.1.13. Pyroshock 9 1.2. Analysis in the time domain 12 1.3. Fourier transform 12 1.3.1. Definition 12 1.3.2. Reduced Fourier transform 14 1.3.3. Fourier transforms of simple shocks. 14 1.3.4. What represents the Fourier transform of a shock? 25 1.3.5. Importance of the Fourier transform 27 1.4. Energy spectrum 28 1.4.1. Energy according to frequency 28 1.4.2. Average energy spectrum 29 1.5. Practical calculations of the Fourier transform 29 1.5.1. General 29 1.5.2. Case: signal not yet digitized 29 1.5.3. Case: signal already digitized 32 1.5.4. Adding zeros to the shock signal before the calculation of its Fourier transform 32 1.6. The interest of time-frequency analysis 36 1.6.1. Limit of the Fourier transform 36 1.6.2. Short term Fourier transform (STFT) 39 1.6.3. Wavelet transform 44 Chapter 2. Shock Response Spectrum 51 2.1. Main principles 51 2.2. Response of a linear one-degree-of-freedom system 55 2.2.1. Shock defined by a force 55 2.2.2. Shock defined by an acceleration 56 2.2.3. Generalization 56 2.2.4. Response of a one-degree-of-freedom system to simple shocks 61 2.3. Definitions 65 2.3.1. Response spectrum 65 2.3.2. Absolute acceleration SRS 65 2.3.3. Relative displacement shock spectrum 65 2.3.4. Primary (or initial) positive SRS 66 2.3.5. Primary (or initial) negative SRS 66 2.3.6. Secondary (or residual) SRS 66 2.3.7. Positive (or maximum positive) SRS 67 2.3.8. Negative (or maximum negative) SRS 67 2.3.9. Maximax SRS 68 2.4. Standardized response spectra 69 2.4.1. Definition 69 2.4.2. Half-sine pulse 71 2.4.3. Versed sine pulse 72 2.4.4. Terminal peak sawtooth pulse 74 2.4.5. Initial peak sawtooth pulse 75 2.4.6. Square pulse 77 2.4.7. Trapezoidal pulse77 2.5. Choice of the type of SRS 78 2.6. Comparison of the SRS of the usual simple shapes 79 2.7. SRS of a shock defined by an absolute displacement of the support 80 2.8. Influence of the amplitude and the duration of the shock on its SRS 81 2.9. Difference between SRS and extreme response spectrum (ERS) 82 2.10. Algorithms for calculation of the SRS 82 2.11. Subroutine for the calculation of the SRS 83 2.12. Choice of the sampling frequency of the signal 86 2.13. Example of use of the SRS 90 2.14. Use of SRS for the study of systems with several degrees of freedom 92 Chapter 3. Properties of Shock Response Spectra 95 3.1. Shock response spectra domains 95 3.2. Properties of SRS at low frequencies 96 3.2.1. General properties 96 3.2.2. Shocks with zero velocity change 96 3.2.3. Shocks with V n0 and D n0 at the end of a pulse 105 3.2.4. Shocks with V n0 and D n0 at the end of a pulse 108 3.2.5. Notes on residual spectrum 110 3.3. Properties of SRS at high frequencies 111 3.4. Damping influence 114 3.5. Choice of damping 114 3.6. Choice of frequency range 118 3.7. Choice of the number of points and their distribution 118 3.8. Charts 118 3.9. Relation of SRS with Fourier spectrum 120 3.9.1. Primary SRS and Fourier transform. 120 3.9.2. Residual SRS and Fourier transform 122 3.9.3. Comparison of the relative severity of several shocks using their Fourier spectra and their shock response spectra 125 3.10. Care to be taken in the calculation of the spectra 129 3.10.1. Main sources of errors 129 3.10.2. Influence of background noise of the measuring equipment 130 3.10.3. Influence of zero shift 132 3.11. Use of the SRS for pyroshocks 135 Chapter 4. Development of Shock Test Specifications 139 4.1. Introduction 139 4.2. Simplification of the measured signal 140 4.3. Use of shock response spectra 142 4.3.1. Synthesis of spectra 142 4.3.2. Nature of the specification 144 4.3.3. Choice of shape 144 4.3.4. Amplitude 146 4.3.5. Duration 146 4.3.6. Difficulties 150 4.4. Other methods 151 4.4.1. Use of a swept sine 152 4.4.2. Simulation of SRS using a fast swept sine 153 4.4.3. Simulation by modulated random noise 157 4.4.4. Simulation of a shock using random vibration 158 4.4.5. Least favorable response technique 159 4.4.6. Restitution of an SRS by a series of modulated sine pulses 160 4.5. Interest behind simulation of shocks on shaker using a shock spectrum 162 Chapter 5. Kinematics of Simple Shocks 167 5.1. Introduction 167 5.2. Half-sine pulse 167 5.2.1. General expressions of the shock motion 167 5.2.2. Impulse mode 170 5.2.3. Impact mode 171 5.3. Versed sine pulse 181 5.4. Square pulse 183 5.5. Terminal peak sawtooth pulse 186 5.6. Initial peak sawtooth pulse 188 Chapter 6. Standard Shock Machines 191 6.1. Main types 191 6.2. Impact shock machines 193 6.3. High impact shock machines 203 6.3.1. Lightweight high impact shock machine 203 6.3.2. Medium weight high impact shock machine 204 6.4. Pneumatic machines 205 6.5. Specific testing facilities 207 6.6. Programmers 208 6.6.1. Half-sine pulse 208 6.6.2. TPS shock pulse 216 6.6.3. Square pulse ntrapezoidal pulse 223 6.6.4. Universal shock programmer 224 Chapter 7. Generation of Shocks Using Shakers 233 7.1. Principle behind the generation of a signal with a simple shape versus time 233 7.2. Main advantages of the generation of shock using shakers 234 7.3. Limitations of electrodynamic shakers 235 7.3.1. Mechanical limitations 235 7.3.2. Electronic limitations 236 7.4. Remarks on the use of electrohydraulic shakers 237 7.5. Pre- and post-shocks 237 7.5.1. Requirements 237 7.5.2. Pre-shock or post-shock 238 7.5.3. Kinematics of the movement for symmetric pre- and post-shock 242 7.5.4. Kinematics of the movement for a pre-shock or post-shock alone 253 7.5.5. Abacuses 255 7.5.6. Influence of the shape of pre- and post-pulses 256 7.5.7. Optimized pre- and post-shocks 259 7.6. Incidence of pre- and post-shocks on the quality of simulation 264 7.6.1. General 264 7.6.2. Influence of the pre- and post-shocks on the time history response of a one- degree-of-freedom system 264 7.6.3. Incidence on the shock response spectrum 266 Chapter 8. Control of a Shaker Using a Shock Response Spectrum 271 8.1. Principle of control using a shock response spectrum 271 8.1.1. Problems 271 8.1.2. Parallel filter method 272 8.1.3. Current numerical methods 273 8.2. Decaying sinusoid 278 8.2.1. Definition 278 8.2.2. Response spectrum 279 8.2.3. Velocity and displacement 282 8.2.4. Constitution of the total signal 283 8.2.5. Methods of signal compensation 284 8.2.6. Iterations 290 8.3. D.L. Kern and C.D. Hayes' function 292 8.3.1. Definition 292 8.3.2. Velocity and displacement 292 8.4. ZERD function 294 8.4.1. Definition 294 8.4.2. Velocity and displacement 295 8.4.3. Comparison of ZERD waveform with standard decaying sinusoid 297 8.4.4. Reduced response spectra 298 8.5. WAVSIN waveform 299 8.5.1. Definition 299 8.5.2. Velocity and displacement 300 8.5.3. Response of a one-degree-of-freedom system 302 8.5.4. Response spectrum 305 8.5.5. Time history synthesis from shock spectrum 306 8.6. SHOC waveform 307 8.6.1. Definition 307 8.6.2. Velocity and displacement 310 8.6.3. Response spectrum 311 8.6.4. Time history synthesis from shock spectrum 312 8.7. Comparison of WAVSIN, SHOC waveforms and decaying sinusoid 313 8.8. Use of a fast swept sine 314 8.9. Problems encountered during the synthesis of the waveforms 317 8.10. Criticism of control by SRS 319 8.11. Possible improvements 323 8.11.1. IES proposal 323 8.11.2. Specification of a complementary parameter 324 8.11.3. Remarks on the properties of the response spectrum 329 8.12. Estimate of the feasibility of a shock specified by its SRS 329 8.12.1. C.D. Robbins and E.P. Vaughan's method 329 8.12.2. Evaluation of the necessary force, power and stroke 331 Chapter 9. Simulation of Pyroshocks 337 9.1. Simulations using pyrotechnic facilities 337 9.2. Simulation using metal to metal impact 341 9.3. Simulation using electrodynamic shakers 342 9.4. Simulation using conventional shock machines 342 Appendix: Similitude in Mechanics 345 Mechanical Shock Tests: A Brief Historical Background 349 Bibliography 351 Index 365 Summary of other Volumes in the Series 369
