1

Optimized PSD Envelope for Nonstationary Vibration Revision A

By Tom Irvine Email: tom@vibrationdata.com

July 2, 2014

____________________________________________________________________________

Figure 1.

Introduction

There is a need to derive a PSD envelope for nonstationary acceleration time histories, including launch vehicle data, which may be similar to that shown in Figure 1.

A PSD can be derived using rainflow fatigue cycle counting along with a Miners-type relative fatigue damage index as shown in Appendix A. The enveloping is then justified using a comparison of fatigue damage spectra between the candidate PSD and the measured time history, using the methods in References 1 through 4. The fatigue damage spectrum gives the cumulative damage index as a function of natural frequency for duly-noted Q and b values.

-10

-5

0

5

10

-5 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70

TIME (SEC)

AC

CE

L (

G)

FLIGHT ACCELEROMETER DATA - SUBORBITAL LAUNCH VEHICLE

2

The derivation process can be performed in a trial-and-error manner in order to obtain the PSD with the least overall GRMS level, or least PSD level, which still envelops the flight data in terms of fatigue damage spectra. The semi-empirical Dirlik method can be used to calculate the fatigue damage spectrum of each candidate PSD in the frequency domain, instead of using the longer, time domain synthesis approach.

Furthermore, this can be done for a number of amplification factor Q and fatigue exponent b permutations for the case where these values are unknown. This adds conservatism to the final PSD envelope.

Again, the goal is to derive the minimum PSD which envelopes the measured data in terms of fatigue. The PSD’s duration can be selected by the user. It may or may not be the same as that of the measured data. The method will scale the PSD to compensate for either a shorter or longer duration.

Statistical uncertainty factors or safety margins can then be added to the PSD envelope as a separate, post-processing step, with consideration that sufficient conservatism may have already been added via a family of Q and b values.

Example

Derive an optimized PSD with a 60-second duration to envelope the nonstationary, base input

acceleration time history in Figure 1. This is needed for a hypothetical avionics component

which is to be mounted near the measurement location on a future flight. The component

must be designed and tested accordingly.

Measured Time History Fatigue Damage Spectra

Figure 2. Single-degree-of-freedom System Subject to Base Acceleration

m

k c

x

y

m = mass

c = damping coefficient

k = stiffness

3

The first step is to calculate the time domain responses of an array of single-degree-of-freedom

(SDOF) systems to the base input, using the method in Reference 5.

An individual SDOF system is shown in Figure 2. The natural frequency fn is an independent

variable. The amplification factor Q and the fatigue exponent b are fixed for a given case but

are varied between cases. The fn, Q and b are unknown for the avionics component, so

assume the permutations shown in Table 1. Only the fn and Q values are needed for this

intermediate calculation. The fatigue exponent b will be applied in a later calculation. The fn

values will be varied in steps from 20 to 2000 Hz.

Table 1. Q & b Values for Fatigue Damage Spectra

Case Q B

1 10 4

2 10 9

3 30 4

4 30 9

A rainflow cycle count is made for each response time history as a function of fn and Q, using

the method in Reference 1.

Next the fatigue damage spectrum is calculated for the set of response time histories using the

b values and the method in Appendix A.

These calculations yield fatigue damage spectra for each of the cases in Table 1, as shown later

in this report.

Optimized PSD Envelope Derivation

The next step is to generate a candidate base input PSD over the frequency domain from 20 to

2000 Hz. The PSD may have an arbitrary number of breakpoints, but simplicity is better. So

generate a PSD with four points. The PSD coordinates may be arbitrary using random numbers,

as long as the starting and ending frequencies are 20 and 2000 Hz, respectively.

Then the response PSD can be calculated for each candidate input PSD. This is done for each

natural frequency fn of interest and for each Q value. Again, the fn values will be varied in

steps from 20 to 2000 Hz.

4

The Dirlik rainflow cycle counting method is then performed for each response PSD using the

method in References 2 through 4.

The fatigue damage spectrum is calculated for each of the cases using the method in Appendix

A.

The candidate base input PSD is then scaled so that each of its four fatigue damage spectra

envelops those of the measured acceleration time history, while keeping the PSD levels as small

as possible.

This process can be repeated for a few hundred candidate PSDs in order to find the least one

which still satisfies the fatigue damage enveloping requirement for the four cases.

Results

The base input time history in Figure 1 was read into to a C++ program: envelope_fds.cpp,

version 1.3. The program then calculated fatigue damage spectra for this time history for the

four permutations in Table 1.

Next the program generated 800 candidate PSDs then performed the fatigue damage spectra

and scaling steps previously outlined.

The run-time on a desktop PC with Windows 7, 64-bit, and 8192 MB Ram was about 75

minutes.

The PSD envelope with the least overall displacement, velocity and acceleration overall RMS

levels is given in Figure 3.

The fatigue damage spectra comparison is given in Figures 4 through 7.

5

PSD Envelope, 3.3 GRMS, 60 sec

Freq (Hz)

Accel (G^2/Hz)

20 0.0018

31 0.0019

211 0.0168

2000 0.0024

Figure 3.

0.001

0.01

0.1

100 100020 2000

FREQUENCY (Hz)

AC

CE

L (

G2/H

z)

POWER SPECTRAL DENSITY ENVELOPE 3.3 GRMS OVERALL

6

Figure 4.

102

104

106

108

1010

100 100020 2000

PSD EnvelopeMeasured Data

NATURAL FREQUENCY (Hz)

DA

MA

GE

IN

DE

X

FATIGUE DAMAGE SPECTRA Q=10 b=4

7

Figure 5.

103

105

107

109

1011

100 100020 2000

PSD EnvelopeMeasured Data

NATURAL FREQUENCY (Hz)

DA

MA

GE

IN

DE

X

FATIGUE DAMAGE SPECTRA Q=30 b=4

8

Figure 6.

102

105

108

1011

1014

1017

100 100020 2000

PSD EnvelopeMeasured Data

NATURAL FREQUENCY (Hz)

DA

MA

GE

IN

DE

X

FATIGUE DAMAGE SPECTRA Q=10 b=9

9

Figure 7.

Conclusions

An optimized PSD envelope was successfully derived for the measured data using the method

outlined in the main text. The PSD is very conservative for three of the four cases.

The results show that higher values for Q and b lead to a higher PSD envelope for the sample

data in Figure 1. This would probably be true for most real-world measured time histories, but

there could be some exceptions. Thus varying both Q and b is advised. Both of these values

are huge uncertainty factors for “black boxes.”

104

107

1010

1013

1016

1019

100 100020 2000

PSD EnvelopeMeasured Data

NATURAL FREQUENCY (Hz)

DA

MA

GE

IN

DE

X

FATIGUE DAMAGE SPECTRA Q=30 b=9

10

References

1. ASTM E 1049-85 (2005) Rainflow Counting Method, 1987.

2. Halfpenny & Kim, Rainflow Cycle Counting and Acoustic Fatigue Analysis Techniques for Random Loading, RASD 2010 Conference, Southampton, UK.

3. Halfpenny, A frequency domain approach for fatigue life estimation from Finite Element Analysis, nCode International Ltd., Sheffield, UK.

4. T. Irvine, Experimental Verification of the Dirlik Fatigue Cycle Method, Vibrationdata, 2013.

5. David O. Smallwood, An Improved Recursive Formula for Calculating Shock Response Spectra, Shock and Vibration Bulletin, No. 51, May 1981.

6. T. Irvine, An Introduction to the Vibration Response Spectrum, Revision D, Vibrationdata, 2009.

7. T. Irvine, A Method for Power Spectral Density Synthesis, Rev B, Vibrationdata, 2000.

APPENDIX A

A relative damage index D can be calculated using

i

m

1i

bi

nAD

(A-1)

where

iA is the response amplitude from the rainflow analysis

in is the corresponding number of cycles

b is the fatigue exponent

Note that the amplitude convention for this paper is (peak-valley)/2.

11

APPENDIX B

Dirlik Method

The nth spectral moment nm for a PSD is

0

nn df)f(Gfm (B-1)

where

f is frequency

G(f) is the one-sided PSD

The expected peak rate ]P[E is

24 mm]P[E (B-2)

The Dirlik histogram formula N(S) for stress cycles ranges is

)S(pT]P[E)S(N (B-3)

where

T is the duration

S is the stress cycle range (peak-to-peak)

The function p(S) is

0

2

32

2

221

m2

2

ZexpZD

R2

Zexp

R

ZD

Q

Zexp

Q

D

)S(p

(B-4)

12

The coefficients and variables are

2

2m

11

x2D

(B-5)

R1

DD1D

211

2

(B-6)

213 DD1D (B-7)

0m2

SZ (B-8)

1

23

D

RDD25.1Q

(B-9)

211

21m

DD1

DxR

(B-10)

40

2

mm

m (B-11)

4

2

0

1m

m

m

m

mx (B-12)

A cumulative histogram of the peaks can then be calculated from equation (B-3).

The stress range of individual cycles can then be obtained by interpolating the cumulative histogram.