MINER’S CUMULATIVE DAMAGE VIA RAINFLOW CYCLE COUNTING
Revision B
By Tom Irvine
Email: tom@vibrationdata.com
January 29, 2012
Introduction
This example is an innovation upon a similar problem in References 1 and 2. It uses a more
conservative method than that in Reference 2.
Consider a power supply mounted on a bracket as shown in Figure 1.
Power Supply
Solder
Terminal
0.006 m
0.050 m
0.120 m
0.140 m
Aluminum
Bracket
Figure 1.
1
The model parameters are
Power Supply Mass
M=0.20 kg
Bracket Material
Aluminum alloy 6061-T4
Mass Density
ρ=2700 kg/m^3
Elastic Modulus
E= 7.0e+10 N/m^2
Viscous Damping Ratio
0.05
The area moment of inertia of the beam cross-section I is
I
1
b h3
12
(1)
I
1
0.050 m0.006 m3
12
(2)
I  9.0 (10 10 ) m 4
(3)
The stiffness EI is


EI  7.0 (10 10 ) N / m 2 9.0 (10 10 ) m 4

EI  63.0 N m 2
(4)
(5)
The mass per length of the beam, excluding the power supply, is


  2700 kg / m3 0.050 m0.006 m
(6)
  0.810 kg / m
(7)
2
The beam mass is
L  0.810 kg / m0.14 m
(8)
L  0.113 kg
(9)
Model the system as a single-degree-of-freedom system subjected to base input as shown in
Figure 2.

x
m
c
k
y
Figure 2.
The natural frequency of the beam, from Reference 3, is given by
fn 
1
2
1
fn 
2
3EI
0.2235  L  m  L3
3  63.0 N m 2 


0.2235 0.113 kg   (20.0 kg) 0.14 m 3
f n  88.0 Hz
(10)
(11)
(12)
3
POWER SPECTRAL DENSITY
6.1 GRMS OVERALL
2
ACCEL (G /Hz)
0.1
0.01
0.001
10
100
1000
2000
FREQUENCY (Hz)
Figure 3.
Table 1. Base Input PSD, 6.1 GRMS
Frequency (Hz)
Accel (G^2/Hz)
20
0.0053
150
0.04
600
0.04
2000
0.0036
Now consider that the bracket assembly is subjected to the random vibration base input level
shown in Figure 3 and in Table 1. The duration is 3 minutes.
4
Synthesized Time History
Figure 4.
An acceleration time history is synthesized to satisfy the PSD specification from Figure 3. The
resulting time history is shown in Figure 4. The synthesis method is given in Reference 4.
The corresponding histogram has a normal distribution, but the plot is omitted for brevity.
Note that the synthesized time history is not unique. For rigor, the analysis in this paper could
be repeated using a number of suitable time histories.
5
Figure 5.
Verification that the synthesized time history meets the specification is given in Figure 5.
6
Acceleration Response
Figure 6.
The response acceleration in Figure 6 was calculated via the method in Reference 5. The
response is narrowband. The oscillation frequency tends to be near the natural frequency of 88
Hz. The histogram has a normal distribution due to the randomly-varying amplitude
modulation.
The overall response level is 5.66 GRMS. This is also the standard deviation given that the
mean is zero.
The absolute peak is 24.5 G, which respresents a 4.3-sigma peak.
Note that some fatigue methods assume that the peak response is 3-sigma and may thus
underpredict fatigue damage.
7
Stress and Moment Calculation
x
L
MR
R
F
Figure 7.
The following approach is a simplification. A rigorous method would calculate the stress from
the strain at the fixed end.
A free-body diagram of the beam is shown in Figure 7.
The reaction moment MR at the fixed-boundary is
MR  F L
(13)
The force F is equal to the effect mass of the bracket system multiplied by the acceleration
level. The effective mass me is
m e  0.2235  L  m 
(14)
m e  0.2235 0.113 kg   (0.20 kg) 
(15)
m e  0.225 kg
(16)
The bending moment M̂ at a given distance L̂ from the force application point is
M̂  m e A L̂
(17)
where A is the acceleration at the force point.
8
The bending stress Sb is given by
Sb  K M̂ C / I
(18)
The variable K is the stress concentration factor. Assume that the stress concentration factor is
3.0 for the solder lug mounting hole.
The variable C is the distance from the neutral axis to the outer fiber of the beam. The crosssection is uniform in the sample problem. Thus C is equal to one-half the thickness, or 0.003 m.
Sb (t )  (K me L̂ C / I) A(t)
(19)

(K m e C / I)  (3.0)(0.225 kg)0.003 m (0.120 m) / 9.0 (10 10 )m 4
(K m e C / I)  270000 kg/m 2

(20)
(21)
Apply an acceleration unit conversion factor.


(K m e C / I) 9.81 m/sec^2/G   2.649E  06 kg/m 2 m / sec^ 2 / G 
(22)
9
Figure 8.
The standard deviation is equivalent to 15 MPa. The highest peak is 65 MPa, which is 4.3sigma. The 4.3 multiplier is also referred to as the “crest factor.”
Next, a rainflow cycle count was performed on the stress time history using the method in
Reference 6. The binned results are shown in Table 2.
The binned results are shown mainly for reference, given that this is a common presentation
format in the aerospace industry. The binned results could be inserted into a Miner’s
cumulative fatigue calculation.
The method in this analysis, however, will use the raw rainflow results consisting of cycle-bycycle amplitude levels, including half-cycles. This brute-force method is more precise than
using binned data.
10
Table 2. Stress Results from Rainflow Cycle Counting, Bin Format, Stress Unit: MPa, Base Input Overall
Level = 6.1 GRMS
Range
Upper
Limit
Lower
Limit
Cycle
Counts
Average
Amplitude
Max Amp
Min Amp
Average
Mean
Max
Mean
Min
Valley
Max
Peak
115.0
128.0
7.5
60.0
64.0
-2.5
0.2
3.6
-63.0
65.0
102.0
115.0
37.5
54.0
57.0
-3.1
0.1
3.8
-58.0
60.0
89.6
102.0
120
47.0
51.0
-3.4
0.3
5.6
-53.0
54.0
76.8
89.6
455.5
41.0
45.0
-5.1
0.1
4.7
-48.0
48.0
64.0
76.8
1087
35.0
38.0
-6.6
-0.1
6.7
-44.0
44.0
51.2
64.0
2149
28.0
32.0
-6.6
0.0
6.5
-36.0
38.0
38.4
51.2
3344
22.0
26.0
-7.6
-0.1
7.0
-32.0
31.0
25.6
38.4
4200
16.0
19.0
-8.4
0.0
6.7
-25.0
24.0
19.2
25.6
1980
11.0
13.0
-6.0
-0.1
5.2
-18.0
17.0
12.8
19.2
1468.5
8.1
9.6
-7.0
0.0
6.1
-14.0
14.0
6.4
12.8
987.5
4.9
6.4
-8.9
0.0
11.0
-13.0
15.0
3.2
6.4
659.5
2.3
3.2
-9.4
0.1
13.0
-12.0
15.0
0.0
3.2
10820
0.2
1.6
-40.0
0.1
40.0
-41.0
40.0
Note that: Amplitude = (peak-valley)/2
11
Miner’s Cumulative Fatigue
Let n be the number of stress cycles accumulated during the vibration testing at a given level
stress level represented by index i.
Let N be the number of cycles to produce a fatigue failure at the stress level limit for the
corresponding index.
Miner’s cumulative damage index Rn is given by
m n
R i
N
i 1 i
(40)
where m is the total number of cycles or bins depending on the analysis type.
In theory, the part should fail when
Rn (theory) = 1.0
(41)
For aerospace electronic structures, however, a more conservative limit is used
Rn(aero) = 0.7
(42)
The number of allowable cycles for a given stress level are determined from an S-N fatigue
curve for the 6061-T4 aluminum bracket in the sample problem.
12
S-N FATIGUE CURVE FOR ALUMINUM 6061-T4
FOR REFERENCE ONLY
250
STRESS (MPa)
200
150
100
50
0
0
10
10
1
10
2
10
3
10
4
10
5
10
6
10
7
10
8
10
9
NUMBER OF CYCLES TO FAIL, N
Figure 9.
The fatigue curve for aluminum 6061-T4 is shown in Figure 9, as adapted from Reference 7.
(Note that published S-N curves all tend to be somewhat tenuous.)
Note that a "less conservative" fatigue curve for this same alloy is given in Reference 1.
The curve in Figure 9 is characterized by the following two equations, where S is the bending
stress in MPa.
S 17 log N 240 MPa
(43)
 240  S 
N  10^ 

 17 
(44)
13
Input Level Study
Table 3. SDOF System, Solder Terminal Location, Fatigue
Damage Results for Various Input Levels, 180 second Duration,
Crest Factor = 4.33
Input Overall
Level
(GRMS)
Input Margin
(dB)
Response
Stress Std Dev
(MPa)
R
6.1
0
15.0
4.92E-09
8.6
3
21.1
4.70E-08
12.2
6
30.0
2.24E-06
17.2
9
42.3
0.00114
19.3
10
47.5
0.0158
21.6
11
53.1
0.351
24.3
12
59.8
Fail
The accumulated fatigue damage was calculated for a family of cases as shown in Table 3.
Each case used the base input PSD from Figure 3 with the indicated added margin.
Furthermore, each used a scaled version of the same synthesized time history.
Each full and half-cycle from the rainflow results was accounted for. An allowable N value was
calculated for each stress amplitude S using equation (44) for each cycle or half-cycle.
A running summation was made using equation (40).
Again, the success criterion was R < 0.7.
The 12-dB margin case produced an outright failure because several stress cycles exceeded an
amplitude of 240 MPa which is the maximum allowable amplitude for one cycle per Figure 9.
The data shows that the fatigue damage is highly sensitive to the base input and resulting stress
levels.
14
Duration Study
A new, 720-second signal was synthesized for the 6 dB margin case. The time history plot is
omitted for brevity.
A fatigue analysis was then performed using the SDOF system in Figure 1. The analysis was
then repeated using the 0 to 360 sec and 0 to 180 sec segments of the new synthesized time
history. The R values for these three cases are shown in Table 4.
Table 4. SDOF System, Solder Terminal Location,
Fatigue Damage Results for Various Durations, 12.2
GRMS Input, with Stress Response = 30 MPa RMS,
Crest Factor = 4.73
Duration (sec)
R
180
3.29e-06
360
6.51e-06
720
1.26e-05
The R value is approximately directly proportional to the duration, such that a doubling of
duration nearly yields a doubling of R.
Note that the R value for the 180 second case is 3.29e-06 in Table 4. The stress response crest
factor was 4.73. This is the ratio of the peak-to-standard deviation, or peak-to-rms assuming
zero mean.
In contrast, the R value for the 180 second case at the same input level was 2.24E-06 in Table 3,
with a crest factor of 4.33.
The R value in Table 4 is thus nearly 50% higher than the corresponding value in Table 3.
The R value is thus very sensitive to the crest factor and may be sensitive to other underlying
statistical parameters as well.
Note that Rayleigh distribution predicts a 4.5 crest factor for an 88 Hz oscillator over a 180second duration. The formula is given in Appendix A. This theoretical prediction is about
halfway between the crest values for the two 6-dB margin cases. Each numerical value was
approximately either 5% higher or lower than theory.
15
An important point is that the crest value can vary from theory for both numerical experiments
as well as for shaker table test. Even a 5% variation can have a significant effect on the fatigue
R value.
Time History Synthesis Variation Study
A set of ten time histories was synthesized to meet the base input PSD + 6 dB. The response of
the SDOF system in Figure 1 was calculated. The results are given in Table 5.
Table 5. SDOF System, Solder Terminal Location, Fatigue Damage Results for
Various Time History Cases, 180-second Duration, 12.2 GRMS Input
Stress RMS (MPa)
Crest Factor
Kurtosis
R
30.0
4.76
3.09
4.96e-06
30.0
4.66
3.03
4.05e-06
29.6
4.69
3.07
3.81e-06
30.3
4.72
3.05
6.03e-06
30.3
4.93
3.00
6.72e-06
29.8
4.48
3.04
2.60e-06
29.8
4.64
2.99
3.11e-06
30.0
4.69
3.04
3.79e-06
29.7
4.30
2.98
1.61e-06
29.9
4.68
3.00
2.96e-06
The overall stress level remains nearly constant across the set of time histories. The crest factor
varies between 4.30 and 4.93.
The R value was a strong function of both the Stress RMS and the Crest Factor, increasing with
each of these independent variables. See Figures 10 and 11, which include a linear curve-fit.
The R value was a weak function of kurtosis.
16
FATIGUE DAMAGE R, 180 SECOND DURATION, 12.2 GRMS INPUT
7x10
-6
6x10
-6
5x10
-6
4x10
-6
3x10
-6
2x10
-6
1x10
-6
1
2
R
y = +5.70E-6x -1.67E-4, max dev:1.78E-6, r =0.713
0
29.5
29.6
29.7
29.8
29.9
30.0
30.1
30.2
30.3
30.4
30.5
STRESS RMS (MPa)
Figure 10.
FATIGUE DAMAGE R, 180 SECOND DURATION, 12.2 GRMS INPUT
7x10
-6
6x10
-6
5x10
-6
4x10
-6
3x10
-6
2x10
-6
1x10
-6
1
2
R
y = +8.16E-6x -3.40E-5, max dev:1.54E-6, r =0.758
0
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
5.0
CREST FACTOR
Figure 11.
17
Conclusion
This is a “work-in-progress.”
Fatigue analysis already carries uncertainty given that S-N curves and stress concentration
factors are both tenuous.
The crest factor is very important.
contribution to fatigue damage.
Response peaks above 3-sigma make a significant
The results in Figure 5 show variation in the fatigue damage R with a set of independent base
input time histories each of which satisfies the same base input PSD specification. The
difference is partly due to the response crest factor variation. A conservative approach would
thus be to take the maximum R for ten different synthesis cases.
Note that response crest factor variation would also occur for a shaker table test, even if the
base input is peak-limited.
There are addition concerns arising from modeling simplifications.
The analysis used pure bending stress. A better approach would have been to use the
maximum principal stress or von Mises stress which would have included the shear stress. The
stress contributions of higher modes should also be considered.
There are additional material concerns as given in Appendix B.
Idealizing a system as a single-degree-of-freedom system may yield to an under-prediction of
the fatigue damage as shown in Appendix C. The inclusion of higher modes may not increase
the stress level much but will increase the fatigue damage, because higher modes add relatively
high-frequency stress reversal cycles.
Additional Applications
The fatigue method is applied to a continuous beam model in Appendix C.
18
References
1. Dave Steinberg, Vibration Analysis for Electronic Equipment, Second Edition, Wiley, New
York, 1988.
2. T. Irvine, Random Vibration Fatigue, Revision B, Vibrationdata, 2003.
3. T. Irvine, Bending Frequencies of Beams, Rods, and Pipes, Revision S, Vibrationdata,
2012.
4. T. Irvine, A Method for Power Spectral Density Synthesis, Rev B, Vibrationdata, 2000.
5. David O. Smallwood, An Improved Recursive Formula for Calculating Shock Response
Spectra, Shock and Vibration Bulletin, No. 51, May 1981.
6. ASTM E 1049-85 (2005) Rainflow Counting Method, 1987.
7. Shigley, Mechanical Engineering Design, Third Ed, Table A-18, McGraw-Hill, New York,
1977.
8. K. Ahlin, Comparison of Test Specifications and Measured Field Data, Sound & Vibration,
September 2006.
9. V. Adams and A. Askenazi, Building Better Products with Finite Element Analysis,
OnWord Press, Santa Fe, N.M., 1999.
10. T. Irvine, Modal Transient Vibration Response of a Cantilever Beam Subjected to Base
Excitation, Vibrationdata, 2013.
19
APPENDIX A
Rayleigh Distribution Crest Factor for an SDOF System Response
The formula is for the maximum predicated crest factor C is
2 ln fn T  
C
0.5772
2 ln fn T 
(A-1)
where
fn
is the natural frequency
T
is the duration
ln
is the natural logarithm function
Equation (A-1) is taken from Reference 8.
20
APPENDIX B
Fatigue Cracks
A ductile material subjected to fatigue loading experiences basic structural changes. The
changes occur in the following order:
1. Crack Initiation. A crack begins to form within the material.
2. Localized crack growth. Local extrusions and intrusions occur at the surface of
the part because plastic deformations are not completely reversible.
3. Crack growth on planes of high tensile stress. The crack propagates across the
section at those points of greatest tensile stress.
4. Ultimate ductile failure. The sample ruptures by ductile failure when the crack
reduces the effective cross section to a size that cannot sustain the applied
loads.
Design and Environmental Variables affecting Fatigue Life
The following factors decrease fatigue life.
1. Stress concentrators. Holes, notches, fillets, steps, grooves, and other irregular
features will cause highly localized regions of concentrated stress, and thus
reduce fatigue life.
2. Surface roughness. Smooth surfaces are more crack resistant because
roughness creates stress concentrators.
3. Surface conditioning. Hardening processes tend to increase fatigue strength,
while plating and corrosion protection tend to diminish fatigue strength.
4. Environment. A corrosive environment greatly reduces fatigue strength. A
combination of corrosion and cyclical stresses is called corrosion fatigue.
21
APPENDIX C
Continuous Beam Subjected to Base Excitation
The response of a continuous beam to an arbitrary base input can be calculated via
Reference 10. This allows the bending stress to be calculated from the strain.
Consider a beam with the following properties:
Cross-Section
Rectangular
Boundary Conditions
Fixed-Free
Material
Aluminum
Width
w
=
0.050 m
Thickness
t
=
0.006 m
Length
L
=
0.30 m
Elastic Modulus
E
=
7.0e+10 N/m^2
Area Moment of Inertia
I
=
9.0e-10 m^4
Mass per Volume
v
=
2700 kg/m^3
Mass per Length

=
0.810 kg/m
Viscous Damping Ratio

=
0.05 for all modes
Calculate the bending stress at the fixed boundary. Omit the stress concentration factor.
22
Figure C-1.
The base input time history is shown in Figure C-1. It satisfies the PSD in Figure 3 with 6 dB
margin. The PSD verification is omitted for brevity.
23
Figure C-2.
The response analysis is performed using Matlab script: continuous_beam_base_accel.m. The
normal modes results are given in Table C-1. A typical response is shown in Figure C-2.
24
Table C-1. Natural Frequency Results, Fixed-Free Beam
Mode
fn (Hz)
Participation
Factor
Effective Modal
Mass (kg)
1
53.7
0.391
0.153
2
336.7
0.217
0.047
3
942.9
0.127
0.016
4
1847.7
0.091
0.008
5
3054.4
0.071
0.005
Again, both the mode shape and participation factor are considered as dimensionless, but they
must be consistent with respect to one another.
Table C-2. Continuous Beam, Stress at Fixed Boundary, Fatigue Damage Results, 180second Duration, 12.2 GRMS Input
Modes
Included
Stress RMS
(MPa)
Crest Factor
Kurtosis
R
1
7.40
4.98
3.02
4.18e-10
2
7.84
5.01
3.02
1.06e-09
3
7.84
4.84
3.02
1.40e-09
4
7.84
4.85
3.02
1.53e-09
5
7.84
4.85
3.02
1.53e-09
The results show that the stress RMS can be accurately calculated using only two
modes.
The fatigue damage R continues to rise as higher modes are included, however.
Note that fifth modal frequency is well above the maximum frequency of the base
input, so its contribution to the overall fatigue damage response is negligible.
A more thorough investigation would involve repeating this analysis for a family of time
history inputs, either with the same or varying overall levels.
25