MINER’S CUMULATIVE DAMAGE VIA RAINFLOW CYCLE COUNTING
By Tom Irvine
Email: tom@vibrationdata.com
January 26, 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 3
(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.
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
Table 3.
Fatigue Damage Results for Various Input Levels,
180 second Duration
Input Overall
Margin (dB)
R
Level
6.1
0
4.92E-09
8.6
3
4.70E-08
12.2
6
2.24E-06
17.2
9
0.00114
19.3
10
0.0158
21.6
11
0.351
24.3
12
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.
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.
14
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. V. Adams and A. Askenazi, Building Better Products with Finite Element Analysis,
OnWord Press, Santa Fe, N.M., 1999.
15
APPENDIX A
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.
16