two computations concerning fatigue damage and the power spectral density frank sherratt
Post on 29-Dec-2015
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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.
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.
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.
Rainflow range distributions of Signal A and a Narrow Band model.
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0 1 2 3 4 5 6 7 8
Rainflow range, RMS units
Ran
ge p
rob
ab
ilit
y d
en
sit
y
Signal A
MODEL
The rainflow distributions of a wide-band signal and a model.
PSD of Signal A
0
1
2
3
4
5
6
7
8
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Relative frequency
G(w
), g
^2/
Hz
The PSD whose rainflow distribution was modelled in the previous slide.
Fitting narrow-band signals to a wide-band rainflow count.
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0 1 2 3 4 5 6 7 8
Rainflow range, RMS units.
Pro
bab
ilit
y d
en
sit
y.
Signal A
RMS 1
RMS 1/3
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
PSD's, Signals B and C
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Relative frquency
G(w
), g
^2/
Hz
Signal B
Signal C
Examples of other PSDs examined.
Rainflow range distribution, target and model, Signals B and C
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0 1 2 3 4 5 6 7 8
Rainflow range, RMS units
Ran
ge p
rob
ab
ilit
y d
en
sit
y
Signal B
Model B
Signal C
Model C
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
79
0
11
10
Signal C
93
7
0 0
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.
PSD
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Relative frequency
G(w
)
A PSD to illustrate the technique.
Damage distribution
0
0.0002
0.0004
0.0006
0.0008
0.001
0.0012
0.0014
0.0016
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Relative frequency
Rel
ativ
e d
amag
e in
a s
trip
Damage distribution over
the frequency range.
The method must have acceptable resolution
with reasonable computation times.
A PSD with spikes tests resolution.
PSD
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Relative frequency
G(w
)
A PSD to test
resolution ability.
Damage distribution
0
0.0001
0.0002
0.0003
0.0004
0.0005
0.0006
0.0007
0.0008
0.0009
0.001
0.35 0.37 0.39 0.41 0.43 0.45 0.47 0.49
Relative frequency
Rel
ativ
e d
amag
e in
a s
trip
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:
Summation of damage over a PSD
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Relative frequency
G(w
) an
d d
amag
e
Dam Acu
PSD
Accumulated damage below a
chosen cut-off level, example A.
Summation of damage over a PSD
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Relative frequency
G(w
) a
nd
da
ma
ge
Dam Acu
PSD
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.
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