Two computations concerning fatigue damage and the Power Spectral Density

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Two computations concerning fatigue damage and
the Power Spectral Density

• Frank Sherratt

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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.
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• One is-
• 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.

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Another is- (b) Computation of the
distribution of damage within the frequency
range of a PSD.
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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.  
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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.
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The rainflow distributions of a wide-band signal
and a model.
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The PSD whose rainflow distribution was modelled
in the previous slide.
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The separate distributions are-
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Illustration of fitting procedure using two
Rayleigh distributions.
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To test the validity of the approach matching
was attempted on about thirty different
PSDs,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.  
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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
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Examples of other PSDs examined.
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Rainflow range distributions of PSDs B and C, and
models.
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Proportion of time at RMS values of 1,2/3, 1/2
and 1/3 used to model signals B and C.
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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.  
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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.
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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.
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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(w). A PSD with unit width at the frequency
point being investigated would just give a
narrow band history in the time domain.
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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.
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A PSD to illustrate the technique.
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Damage distribution over the frequency range.
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The method must have acceptable resolution with
reasonable computation times. A PSD with spikes
tests resolution.
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A PSD to test resolution ability.
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Showing adequate resolution of spikes in the PSD
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Damage contributed by all frequencies below a
certain level may be computed. This has
applications in testing. Examples are
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Accumulated damage below a chosen cut-off level,
example A.
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Accumulated damage below a chosen cut-off level,
example B.
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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.
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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.