Two computations concerning fatigue damage and the Power Spectral Density Frank Sherratt

When using frequency domain fatigue analysis fast empirical formulae like the Dirlik expression for the distribution of rainflow ranges may be used to provide estimates of other parameters which are equally empirical but which may be useful in testing or in design.

One is:- (a)Computation of a non-stationary time history made up of short periods of narrow-band signal whose variance is changed from time to time to generate a specified rainflow distribution. This may match the distribution of a stationary wide-band history.

Another is:- (b) Computation of the distribution of damage within the frequency range of a PSD.

A; Simulating a wide-band test using narrow-band excitation. Acceptance or proving tests for vibration resistance often specify a PSD which must be achieved within certain limits, often a wide-band PSD which represents service. If the aim is to simulate the fatigue damage potential of this service a better parameter may be the rainflow count.

Fig. 1 shows the rainflow range distribution for the wide-band PSD shown in Fig. 2 (Signal A). Superimposed on this plot is one made up by adding two Rayleigh distributions.

The rainflow distributions of a wide-band signal and a model.

The PSD whose rainflow distribution was modelled in the previous slide.

The separate distributions are:-

Illustration of fitting procedure using two Rayleigh distributions.

To test the validity of the approach matching was attempted on about thirty different PSD’s,including some derived from field measurements.Irregularity factors down to 0.53 were present. Different numbers of RMS levels were used in different trials, and various mixes of RMS level were examined.

The results showed that simple methods are adequate. Matching similar to that shown in Fig. 1 was achieved for all the signals using only four levels of RMS. Levels in the ratio 1, 2/3, 1/2 and 1/3 gave very good results. Examples of performance are shown in the next slides

Examples of other PSDs examined.

Rainflow range distributions of PSDs B and C, and models.

RMS level % at 1 % at 2/3 % at 1/2 % at 1/3 Signal B Signal C Proportion of time at RMS values of 1,2/3, 1/2 and 1/3 used to model signals B and C.

Conclusion from Section A The rainflow range probability density distribution, P(rr), of a time history having a wide-band Power Spectral Density can be reproduced by summing Rayleigh distributions.

Practical implication A physical test could achieve this by applying narrow-band loading and changing the RMS at controlled intervals. Tests using this approach could use machines of the resonance type. These will use less power and run at higher speeds than conventional servo- hydraulic machines.

B; Computation of the density distribution of damage within the frequency range of a PSD. Ways of estimating fatigue life under a loading history prescribed by a PSD are now well established. A useful extension would be to compute how damage potential is distributed within the PSD.

Problem Damage per Hz at a particular point on the frequency axis depends on the overall shape of the PSD as well as on the local value of G(  ). A PSD with unit width at the frequency point being investigated would just give a narrow band history in the time domain.

Solution An estimate can be made by removing a narrow strip from the PSD at a chosen location, and calculating the difference in damage between the total PSD and the PSD with this strip removed.Scanning the removed strip gives the required distribution.

A PSD to illustrate the technique.

Damage distribution over the frequency range.

The method must have acceptable resolution with reasonable computation times. A PSD with spikes tests resolution.

A PSD to test resolution ability.

Showing adequate resolution of spikes in the PSD

Damage contributed by all frequencies below a certain level may be computed. This has applications in testing. Examples are:

Accumulated damage below a chosen cut-off level, example A.

Accumulated damage below a chosen cut-off level, example B.

If the driver signal is going to be edited, e.g. by removing high-frequency components because of test machine limitations, a plot like this gives information about the damage removed.

Conclusion from Section B It is possible to compute how different parts of a PSD contribute to fatigue damage. The information has uses in testing and design.