Mechanical Vibration and Shock Analysis, Mechanical Shock

Christian Lalanne is a Consultant Engineer who previously worked as an expert at the French Atomic Energy Authority and who has specialized in the study of vibration and shock for more than 40 years. He has been associated with the new methods of drafting testing specifications and associated informatic tools.

• Foreword to Series xiiiIntroduction xviiList of Symbols xixChapter 1. Shock Analysis 11.1. Definitions  11.1.1. Shock 11.1.2. Transient signal 21.1.3. Jerk 31.1.4. Simple (or perfect) shock 31.1.5. Half-sine shock 31.1.6. Versed sine (or haversine) shock 41.1.7. Terminal peak sawtooth (TPS) shock (or final peak sawtooth FPS)) 51.1.8. Initial peak sawtooth (IPS) shock 61.1.9. Square shock 71.1.10. Trapezoidal shock  81.1.11. Decaying sinusoidal pulse  81.1.12. Bump test 91.1.13. Pyroshock 91.2. Analysis in the time domain 121.3. Temporal moments 121.4. Fourier transform 151.4.1. Definition 151.4.2. Reduced Fourier transform  171.4.3. Fourier transforms of simple shocks  171.4.4. What represents the Fourier transform of a shock? 291.4.5. Importance of the Fourier transform  311.5. Energy spectrum 321.5.1. Energy according to frequency 321.5.2. Average energy spectrum 331.6. Practical calculations of the Fourier transform 331.6.1. General  331.6.2. Case: signal not yet digitized 331.6.3. Case: signal already digitized 361.6.4. Adding zeros to the shock signal before the calculation of its Fourier transform 371.6.5. Windowing 401.7. The interest of time-frequency analysis 411.7.1. Limit of the Fourier transform 411.7.2. Short term Fourier transform (STFT) 441.7.3. Wavelet transform 49Chapter 2. Shock Response Spectrum 552.1. Main principles 552.2. Response of a linear one-degree-of-freedom system 592.2.1. Shock defined by a force 592.2.2. Shock defined by an acceleration 602.2.3. Generalization 602.2.4. Response of a one-degree-of-freedom system to simple shocks  652.3. Definitions  692.3.1. Response spectrum 692.3.2. Absolute acceleration SRS  692.3.3. Relative displacement shock spectrum  702.3.4. Primary (or initial) positive SRS 702.3.5. Primary (or initial) negative SRS 702.3.6. Secondary (or residual) SRS 712.3.7. Positive (or maximum positive) SRS 712.3.8. Negative (or maximum negative) SRS  712.3.9. Maximax SRS 722.4. Standardized response spectra 732.4.1. Definition 732.4.2. Half-sine pulse 752.4.3. Versed sine pulse  762.4.4. Terminal peak sawtooth pulse 782.4.5. Initial peak sawtooth pulse  792.4.6. Square pulse  812.4.7. Trapezoidal pulse  812.5. Choice of the type of SRS  822.6. Comparison of the SRS of the usual simple shapes 832.7. SRS of a shock defined by an absolute displacement of the support  842.8. Influence of the amplitude and the duration of the shock on its SRS  842.9. Difference between SRS and extreme response spectrum (ERS) 862.10. Algorithms for calculation of the SRS 862.11. Subroutine for the calculation of the SRS  862.12. Choice of the sampling frequency of the signal  902.13. Example of use of the SRS 942.14. Use of SRS for the study of systems with several degrees of freedom 962.15. Damage boundary curve  100Chapter 3. Properties of Shock Response Spectra 1033.1. Shock response spectra domains 1033.2. Properties of SRS at low frequencies 1043.2.1. General properties 1043.2.2. Shocks with zero velocity change 1043.2.3. Shocks with ΔV = 0 and ΔD ≠ 0 at the end of a pulse 1153.2.4. Shocks with ΔV = 0 and ΔD = 0 at the end of a pulse 1173.2.5. Notes on residual spectrum  1203.3. Properties of SRS at high frequencies 1213.4. Damping influence 1243.5. Choice of damping 1243.6. Choice of frequency range  1273.7. Choice of the number of points and their distribution  1283.8. Charts  1313.9. Relation of SRS with Fourier spectrum 1343.9.1. Primary SRS and Fourier transform 1343.9.2. Residual SRS and Fourier transform 1363.9.3. Comparison of the relative severity of several shocks using their Fourier spectra and their shock response spectra 1393.10. Care to be taken in the calculation of the spectra 1433.10.1. Main sources of errors 1433.10.2. Influence of background noise of the measuring equipment 1433.10.3. Influence of zero shift 1453.11. Specific case of pyroshocks 1523.11.1. Acquisition of the measurements 1523.11.2. Examination of the signal before calculation of the SRS 1543.11.3. Examination of the SRS 1553.12. Pseudo-velocity shock spectrum 1563.12.1. Hunt’s relationship 1563.12.2. Interest of PVSS 1603.13. Use of the SRS for pyroshocks 1623.14. Other propositions of spectra1653.14.1. Pseudo-velocity calculated from the energy transmitted 1653.14.2. Pseudo-velocity from the “input” energy at the end of a shock  1653.14.3. Pseudo-velocity from the unit “input” energy 1673.14.4. SRS of the “total” energy  167Chapter 4. Development of Shock Test Specifications 1754.1. Introduction 1754.2. Simplification of the measured signal 1764.3. Use of shock response spectra 1784.3.1. Synthesis of spectra 1784.3.2. Nature of the specification  1804.3.3. Choice of shape 1814.3.4. Amplitude 1824.3.5. Duration 1824.3.6. Difficulties 1864.4. Other methods 1874.4.1. Use of a swept sine 1884.4.2. Simulation of SRS using a fast swept sine 1894.4.3. Simulation by modulated random noise 1934.4.4. Simulation of a shock using random vibration  1944.4.5. Least favorable response technique 1954.4.6. Restitution of an SRS by a series of modulated sine pulses  1964.5. Interest behind simulation of shocks on shaker using a shock spectrum 198Chapter 5. Kinematics of Simple Shocks 2035.1. Introduction 2035.2. Half-sine pulse 2035.2.1. General expressions of the shock motion 2035.2.2. Impulse mode 2065.2.3. Impact mode 2075.3. Versed sine pulse 2165.4. Square pulse 2185.5. Terminal peak sawtooth pulse 2215.6. Initial peak sawtooth pulse  223Chapter 6. Standard Shock Machines 2256.1. Main types  2256.2. Impact shock machines 2276.3. High impact shock machines 2376.3.1. Lightweight high impact shock machine 2376.3.2. Medium weight high impact shock machine  2386.4. Pneumatic machines 2396.5. Specific testing facilities 2416.6. Programmers 2426.6.1. Half-sine pulse 2426.6.2. TPS shock pulse 2506.6.3. Square pulse − trapezoidal pulse 2586.6.4. Universal shock programmer 258Chapter 7. Generation of Shocks Using Shakers  2677.1. Principle behind the generation of a signal with a simple shape versus time 2677.2. Main advantages of the generation of shock using shakers  2687.3. Limitations of electrodynamic shakers 2697.3.1. Mechanical limitations 2697.3.2. Electronic limitations  2717.4. Remarks on the use of electrohydraulic shakers  2717.5. Pre- and post-shocks  2717.5.1. Requirements 2717.5.2. Pre-shock or post-shock 2737.5.3. Kinematics of the movement for symmetric pre- and post-shock 2767.5.4. Kinematics of the movement for a pre-shock or a post-shock alone 2867.5.5. Abacuses 2887.5.6. Influence of the shape of pre- and post-pulses 2897.5.7. Optimized pre- and post-shocks 2927.6. Incidence of pre- and post-shocks on the quality of simulation  2977.6.1. General  2977.6.2. Influence of the pre- and post-shocks on the time history response of a one-degree-of-freedom system  2977.6.3. Incidence on the shock response spectrum 300Chapter 8. Control of a Shaker Using a Shock Response Spectrum 3038.1. Principle of control using a shock response spectrum  3038.1.1. Problems 3038.1.2. Parallel filter method 3048.1.3. Current numerical methods  3058.2. Decaying sinusoid 3108.2.1. Definition 3108.2.2. Response spectrum 3118.2.3. Velocity and displacement  3148.2.4. Constitution of the total signal 3158.2.5. Methods of signal compensation 3168.2.6. Iterations 3238.3. D.L. Kern and C.D. Hayes’ function 3248.3.1. Definition 3248.3.2. Velocity and displacement  3258.4. ZERD function 3268.4.1. Definition 3268.4.2. Velocity and displacement  3288.4.3. Comparison of ZERD waveform with standard decaying sinusoid 3308.4.4. Reduced response spectra 3308.5. WAVSIN waveform  3328.5.1. Definition 3328.5.2. Velocity and displacement  3338.5.3. Response of a one-degree-of-freedom system 3358.5.4. Response spectrum 3388.5.5. Time history synthesis from shock spectrum 3398.6. SHOC waveform 3408.6.1. Definition 3408.6.2. Velocity and displacement  3428.6.3. Response spectrum 3438.6.4. Time history synthesis from shock spectrum 3458.7. Comparison of WAVSIN, SHOC waveforms and decaying sinusoid 3468.8. Waveforms based on the cosm(x) window 3468.9. Use of a fast swept sine  3488.10. Problems encountered during the synthesis of the waveforms  3518.11. Criticism of control by SRS 3538.12. Possible improvements 3578.12.1. IES proposal 3578.12.2. Specification of a complementary parameter 3588.12.3. Remarks on the properties of the response spectrum  3638.13. Estimate of the feasibility of a shock specified by its SRS3638.13.1. C.D. Robbins and E.P. Vaughan’s method  3638.13.2. Evaluation of the necessary force, power and stroke  365Chapter 9. Simulation of Pyroshocks  3719.1. Simulations using pyrotechnic facilities 3719.2. Simulation using metal to metal impact 3759.3. Simulation using electrodynamic shakers 3779.4. Simulation using conventional shock machines  378Appendix. Similitude in Mechanics 381A1. Conservation of materials  381A2. Conservation of acceleration and stress 383Mechanical Shock Tests: A Brief Historical Background  385Bibliography  387Index 407Summary of other Volumes in the series 413