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.

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.

The separate distributions are:-

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.

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.