Mechanical Vibration and Shock Analysis Volume 5: Specification Development

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Title Page......Page 3
Copyright......Page 4
Table of Contents......Page 5
Foreword to Series......Page 12
Introduction......Page 16
List of Symbols......Page 19
1.1. The effects of vibration......Page 23
1.2.2. Case of a single sinusoid......Page 24
1.2.2.1. Excitation defined by acceleration......Page 25
1.2.3. General case......Page 29
1.2.4. Case of a periodic signal......Page 30
1.2.5. Case of n harmonic sinusoids......Page 31
1.2.6. Influence of the dephasing between the sinusoids......Page 33
1.3.1.1. General case......Page 35
1.3.1.3. Sweep with constant displacement......Page 37
1.3.1.4. General expression for extreme response......Page 38
1.3.2. Swept sine composed of several constant levels......Page 39
Chapter 2: Extreme Response Spectrum of a Random Vibration......Page 42
2.1. Unspecified vibratory signal......Page 43
2.2.1.1. General case......Page 44
2.2.1.2. Case of a narrowband response......Page 50
2.2.1.3. ERS for a duration larger than that of the analyzed signal......Page 51
2.2.2. Use of the largest peak distribution law......Page 52
2.2.3.1. General expression......Page 55
2.2.3.2. Approximate expressions......Page 56
2.3. Limit of the ERS at the high frequencies......Page 70
2.4.1. Complete expression......Page 71
2.4.2. Approximate relation......Page 75
2.4.3. Approximate relation URS – PSD......Page 77
2.4.4. Calculation in a hypothesis of independence of threshold overshoot......Page 79
2.4.5. Use of URS......Page 82
2.5. Comparison of the various formulae......Page 83
2.6.2. Extreme response spectra calculated from the power spectral densities......Page 87
2.6.3. Comparison of extreme response spectra calculated from time history signals and power spectral densities......Page 88
2.7.1. Real environment......Page 89
2.7.2. Case of a single sinusoid superimposed to a wideband noise......Page 91
2.7.2.1. Probability density of peaks......Page 92
2.7.2.2. Distribution function of peaks......Page 93
2.7.2.3. Extreme response......Page 96
2.7.3.1. Approximate relation......Page 100
2.7.3.2. Exact formulation......Page 101
2.7.3.4. Influence of a dephasing between the sinusoids......Page 103
2.8.1. Real environment......Page 104
2.8.2. Case of a single swept sine superimposed to a wideband noise......Page 105
2.9.1. Real environment......Page 106
2.9.2. Extreme response spectrum......Page 107
3.1. Fatigue damage spectrum definition......Page 109
3.2. Fatigue damage spectrum of a single sinusoid......Page 112
3.3. Fatigue damage spectrum of a periodic signal......Page 116
3.5.1. Taking account of fatigue limit......Page 118
3.5.2. Cases where the S–N curve is approximated by a straight line in log–lin scales......Page 119
3.5.3. Comparison of the damage when the S–N curves are linear in either log–log or log–lin scales......Page 120
3.6.1. General case......Page 122
3.6.2.1. General case......Page 123
3.6.2.2. Linear sweep at constant acceleration......Page 124
3.6.2.3. Linear sweep at constant displacement......Page 133
3.6.3.1. General case......Page 134
3.6.3.3. Logarithmic sweep at constant displacement......Page 135
3.6.4.1. General case......Page 137
3.6.4.2. Hyperbolic sweep at constant acceleration......Page 138
3.6.4.3. Hyperbolic sweep at constant displacement......Page 139
3.6.5. General expressions for fatigue damage......Page 140
3.7.1. Fatigue damage equivalence in the case of a linear system......Page 141
3.7.2. Method based on fatigue damage equivalence according to Basquin’s relationship......Page 142
3.8. Notes on the design assumptions of the ERS and FDS......Page 144
4.1. Fatigue damage spectrum from the signal as function of time......Page 145
4.2. Fatigue damage spectrum derived from a power spectral density......Page 147
4.3. Simplified hypothesis of Rayleigh’s law......Page 152
4.4. Calculation of the fatigue damage spectrum with Dirlik’s probability density......Page 158
4.5. Up-crossing risk fatigue damage spectrum......Page 160
4.6.1. Fatigue damage equivalence in the case of a linear system......Page 164
4.6.2. Method based on a fatigue damage equivalence according to Basquin’s relationship taking account of variation of natural damping as a function of stress level......Page 165
4.7.1. Fatigue damage spectra calculated from a signal as a function of time......Page 169
4.7.2. Fatigue damage spectra calculated from power spectral densities......Page 170
4.7.3. Comparison of fatigue damage spectra calculated from signals as a function of time and power spectral densities......Page 171
4.8.1. Case of a single sinusoidal vibration superimposed on broadband random vibration......Page 172
4.8.2.2. Exact formulation......Page 179
4.8.2.3. Calculation from a signal as a function of time......Page 180
4.9.1. Case of one swept sine superimposed on a broadband random vibration......Page 181
4.10. Swept narrowbands on a broadband random vibration......Page 182
5.1. General relationship of fatigue damage......Page 184
5.2. Use of shock response spectrum in the impulse zone......Page 186
5.3. Damage created by simple shocks in static zone of the response spectrum......Page 188
6.1. Variation of the ERS with amplitude and vibration duration......Page 190
6.3. Should ERSs and FDSs be drawn with a linear or logarithmic frequency step?......Page 194
6.4. With how many points must ERSs and FDSs be calculated?......Page 196
6.5. Difference between ERSs and FDSs calculated from a vibratory signal according to time and from its PSD......Page 199
6.6. Influence of the number of PSD calculation points on ERS and FDS......Page 206
6.7. Influence of the PSD statistical error on ERS and FDS......Page 211
6.8. Influence of the sampling frequency during ERS and FDS calculation from a signal based on time......Page 212
6.9. Influence of the peak counting method......Page 221
6.10. Influence of a non-zero mean stress on FDS......Page 225
7.1.2. Specification......Page 236
7.2.3. Evaluation test......Page 237
7.2.6. Pre-qualification (or evaluation) test......Page 238
7.2.8. Qualification test......Page 239
7.2.13. Reception test......Page 240
7.2.16. Sampling test......Page 241
7.3. What can be expected from a test specification?......Page 242
7.4.1. Specification requiring in situ testing......Page 243
7.4.2.1. History......Page 244
7.4.2.2. Major standards......Page 248
7.4.3. Current trend......Page 250
7.4.4.1. Interest......Page 251
7.4.4.3. Exact duplication of the real or synthesized environment......Page 252
7.5. Standards specifying test tailoring......Page 254
7.5.1. The MIL–STD 810 standard......Page 255
7.5.2. The GAM EG 13 standard......Page 257
7.5.3. STANAG 4370......Page 258
7.5.4. The AFNOR X50–410 standard......Page 259
8.1. Need – definitions......Page 261
8.2. Sources of uncertainty......Page 265
8.3.1. Real environment......Page 267
8.3.1.1. Distribution functions......Page 268
8.3.1.2. Dispersions – variation coefficients observed in practice......Page 269
8.3.1.3. Estimate of the variation coefficient – calculation of its maximum value......Page 274
8.3.2.1. Source of dispersion......Page 286
8.3.2.2. Distribution laws......Page 287
8.3.2.3. A few values of the variation coefficient......Page 288
8.4.1.1. Ratio of the smallest strength to the largest load......Page 290
8.4.1.2. Definition using reliability considerations......Page 291
8.4.2.1. Case of Normal distributions......Page 292
8.4.2.2. Case of log–normal distributions......Page 296
8.4.2.3. General case......Page 301
8.4.2.4. Influence of the choice of distribution laws......Page 305
8.4.3. Calculation of an uncertainty factor when the real environment is only characterized by a single value......Page 310
9.2. Aging functions used in reliability......Page 311
9.3. Method for calculating the aging factor......Page 314
9.4. Influence of the aging law’s standard deviation......Page 317
9.5. Influence of the aging law mean......Page 318
10.1. Philosophy......Page 319
10.2.1. Calculation of test factor from the estimation of the confidence interval of the mean......Page 321
10.2.2. Calculation of test factor from the estimation of the probability density of the mean strength with a sample of size n......Page 328
10.3.1. Calculation of test factor from the estimation of the confidence interval of the average......Page 333
10.3.2. Calculation of test factor from the estimation of the probability density of the mean of the strength with a sample of size n......Page 335
10.4. Weibull distributions......Page 336
10.5. Choice of confidence level......Page 338
11.1. Test tailoring......Page 339
11.2. Step 1: analysis of the lifecycle profile. Review of the situations......Page 340
11.3. Step 2: determination of the real environmental data associated with each situation......Page 342
11.4. Step 3: determination of the environment to be simulated......Page 343
11.4.1. Need......Page 344
11.4.3. The need for a reliable method......Page 345
11.4.4. Synthesis method using PSD envelope......Page 346
11.4.5. Equivalence method of extreme response and fatigue damage......Page 349
11.4.6.1. Shock synthesis......Page 351
11.4.6.2. Random vibrations......Page 352
11.4.6.3. Calculation parameters......Page 353
11.4.6.4. Application of uncertainty factor......Page 354
11.4.8.1. Parallel situations......Page 358
11.4.9.1. Method by matrix inversion......Page 360
11.4.9.2. Method by iteration......Page 365
11.4.10. Validation of duration reduction......Page 366
11.5.1. Application of a test factor......Page 374
11.5.2. Choice of the test chronology......Page 375
11.6. Applying this method to the example of the “round robin” comparative study......Page 381
11.7. Taking environment into account in project management......Page 384
12.1. Choice of the number of points in the specification (PSD)......Page 393
12.2. Influence of the Q factor on specification (outside of time reduction)......Page 396
12.3. Influence of the Q factor on specification when duration is reduced......Page 401
12.4. Validity of a specification established for a Q factor equal to 10 when the real structure has another value......Page 405
12.5. Advantage in the consideration of a variable Q factor for the calculation of ERSs and FDSs......Page 406
12.6.1. Case where test duration is equal to real environment duration......Page 408
12.6.2. Case where duration is reduced......Page 410
12.7. Choice of the value of parameter b in the case of material made up of several components......Page 412
12.8. Influence of temperature on parameter b and constant C......Page 413
12.9. Importance of a factor of 10 between the specification FDS and the reference FDS (real environment) in a small frequency band......Page 414
12.10. Validity of a specification established by reference to a one-degree-of-freedomsystem when real structures are multi-degree-of- freedom system when real structures are multi-degree-of-freedom systems......Page 416
13.1. Comparisons of the severity of different vibrations......Page 417
13.1.1. Comparisons of the relative severity of several real environments......Page 418
13.1.2. Comparison of the severity of several standards......Page 419
13.2. Swept sine excitation – random vibration transformation......Page 421
13.3. Definition of a random vibration with the same severity as a series of shocks......Page 426
13.4.1.1. Search for a specification from an ERS......Page 431
13.4.1.2. Search for a specification from a URS......Page 435
13.5. Establishment of a swept sine vibration specification......Page 436
A1.1. Power spectral density envelope method......Page 439
A1.2. Method using ERSs and FDSs......Page 442
A1.3. Comparison of hypotheses......Page 443
A2.1. Specification defined as white noise respecting the rms value of the real environment PSD......Page 444
A2.2. Specification established by drawing several line segments enveloping PSDs of the real environment......Page 446
A2.3. “Raw” envelope of two (or more) PSD with very close frequency content, but with different amplitudes and durations.......Page 453
A2.4. Two (or more) PSD envelopes with different frequency contents and different durations.......Page 457
A2.5. The real environment is not stationary: its rms value varies according to time......Page 458
A2.6. The real environment is not stationary: its frequency content varies over time, rms value being equal......Page 461
A3. Direct generation of a random signal from an FDS......Page 468
A4.1. Bessel function of the first kind of order zero......Page 472
A4.2. Modified Bessel function of the first kind of order zero......Page 473
Formulae......Page 475
Bibliography......Page 498
Index......Page 513
Summary of Other Volumes in the Series......Page 517
Summary of Volume 1: Sinusoidal Vibration......Page 518
Summary of Volume 2: Mechanical Shock......Page 524
Summary of Volume 3: Random Vibration......Page 530
Summary of Volume 4: Fatigue Damage......Page 538