Transcript SAVE_conference_2014_Irvine_fatigue_revAx

85th Shock and Vibration Symposium 2014
NESC Academy
Rainflow Cycle Counting for
Random Vibration Fatigue Analysis
Revision A
By Tom Irvine
1
This presentation is sponsored by
NASA Engineering &
Safety Center (NESC)
Dynamic Concepts, Inc.
Huntsville, Alabama
Vibrationdata
2
Contact Information
Tom Irvine
Email: [email protected]
Phone: (256) 922-9888 x343
http://vibrationdata.com/
http://vibrationdata.wordpress.com/
3
Introduction
 Structures & components must be designed and tested to withstand
vibration environments
 Components may fail due to yielding, ultimate limit, buckling, loss of sway
space, etc.
 Fatigue is often the leading failure mode of interest for vibration
environments, especially for random vibration
 Dave Steinberg wrote:
The most obvious characteristic of random vibration is that it is
nonperiodic. A knowledge of the past history of random motion is
adequate to predict the probability of occurrence of various acceleration
and displacement magnitudes, but it is not sufficient to predict the
precise magnitude at a specific instant.
4
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.
5
Some Caveats
Vibration fatigue calculations are “ballpark” calculations given
uncertainties in S-N curves, stress concentration factors, non-linearity,
temperature and other variables.
Perhaps the best that can be expected is to calculate the accumulated
fatigue to the correct “order-of-magnitude.”
6
Rainflow Fatigue Cycles
Endo & Matsuishi 1968 developed the
Rainflow Counting method by relating
stress reversal cycles to streams of
rainwater flowing down a Pagoda.
ASTM E 1049-85 (2005) Rainflow
Counting Method
Goju-no-to Pagoda, Miyajima Island, Japan
7
Sample Time History
STRESS TIME HISTORY
6
5
4
3
STRESS
2
1
0
-1
-2
-3
-4
-5
-6
0
1
2
3
4
5
6
7
8
TIME
8
RAINFLOW PLOT
0
A
Rainflow Cycle
Counting
B
1
C
Rotate time history plot
90 degrees clockwise
2
D
3
TIME
E
Rainflow Cycles by Path
4
F
5
G
6
H
7
I
8
-6
-5
-4
-3
-2
-1
0
STRESS
1
2
3
4
5
6
Path
Cycles
A-B
0.5
Stress
Range
3
B-C
0.5
4
C-D
0.5
8
D-G
0.5
9
E-F
1.0
4
G-H
0.5
8
H-I
0.5
6
9
RAINFLOW PLOT
0
A
Rainflow Cycle
Counting
B
1
C
Rotate time history plot
90 degrees clockwise
2
D
3
TIME
E
Rainflow Cycles by Path
4
F
5
G
6
H
7
I
8
-6
-5
-4
-3
-2
-1
0
STRESS
1
2
3
4
5
6
Path
Cycles
A-B
0.5
Stress
Range
3
B-C
0.5
4
C-D
0.5
8
D-G
0.5
9
E-F
1.0
4
G-H
0.5
8
H-I
0.5
6
10
Rainflow Results in Table Format - Binned Data
Range = (peak-valley)
Amplitude = (peak-valley)/2
(But I prefer to have the results in simple amplitude & cycle format for further calculations)
11
Use of Rainflow Cycle Counting
 Can be performed on sine, random, sine-on-random, transient, steadystate, stationary, non-stationary or on any oscillating signal whatsoever
 Evaluate a structure’s or component’s failure potential using Miner’s rule
& S-N curve
 Compare the relative damage potential of two different vibration
environments for a given component
 Derive maximum predicted environment (MPE) levels for nonstationary
vibration inputs
 Derive equivalent PSDs for sine-on-random specifications
 Derive equivalent time-scaling techniques so that a component can be
tested at a higher level for a shorter duration
 And more!
12
Rainflow Cycle Counting – Time History Amplitude Metric
 Rainflow cycle counting is performed on stress time histories for the case
where Miner’s rule is used with traditional S-N curves
 Can be used on response acceleration, relative displacement or some
other metric for comparing two environments
13
For Relative Comparisons between Environments . . .
 The metric of interest is the response acceleration or relative displacement
 Not the base input!
 If the accelerometer is mounted on the mass, then we are good-to-go!
 If the accelerometer is mounted on the base, then we need to perform
intermediate calculations
14
Reference
Vibrationdata
Steinberg’s text is used in the
following example and elsewhere
in this presentations
15
Bracket Example, Variation on a Steinberg Example
Aluminum
Bracket
Power Supply
Solder
Terminal
0.25 in
2.0 in
4.7 in
5.5 in
6.0 in
Power Supply Mass
M = 0.44 lbm= 0.00114 lbf sec^2/in
Bracket Material
Aluminum alloy 6061-T6
Mass Density
ρ=0.1 lbm/in^3
Elastic Modulus
E= 1.0e+07 lbf/in^2
Viscous Damping Ratio
0.05
16
Bracket Natural Frequency via Rayleigh Method
17
Bracket Response via SDOF Model
fn  94.76 Hz
Treat bracket-mass system as a SDOF system for the response to base
excitation analysis. Assume Q=10.
18
Base Input PSD
POWER SPECTRAL DENSITY
6.1 GRMS OVERALL
2
ACCEL (G /Hz)
0.1
Base Input PSD, 6.1 GRMS
0.01
0.001
10
100
1000
2000
Frequency
(Hz)
Accel
(G^2/Hz)
20
0.0053
150
0.04
600
0.04
2000
0.0036
FREQUENCY (Hz)
Now consider that the bracket assembly is subjected to the random vibration
base input level. The duration is 3 minutes.
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Base Input PSD
The PSD on the previous slide is library array: MIL-STD1540B ATP PSD
20
Time History Synthesis
21
Base Input Time History
Save Time History as:
synth
 An acceleration time history is synthesized to satisfy the PSD specification
 The corresponding histogram has a normal distribution, but the plot is omitted for
brevity
 Note that the synthesized time history is not unique
22
PSD Verification
23
SDOF Response
24
Acceleration
Response
Save as:
accel_resp
 The response is narrowband
 The oscillation frequency tends to be near the natural frequency of 94.76 Hz
 The overall response level is 6.1 GRMS
 This is also the standard deviation given that the mean is zero
 The absolute peak is 27.49 G, which represents a 4.53-sigma peak
 Some fatigue methods assume that the peak response is 3-sigma and may
thus under-predict fatigue damage
25
Stress & Moment Calculation, Free-body Diagram
x
L
MR
R
F
The reaction moment M R at the fixed-boundary is:
MR  F L
The force F is equal to the effect mass of the bracket system multiplied by the
acceleration level.
The effective mass m e is:
me   0.2235  L  m 
me  0.0013 lbf sec^2/in
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Stress & Moment Calculation, Free-body Diagram
ˆ at a given distance from the force application point
The bending moment M
is
ˆ  m AL
ˆ
M
e
where A is the acceleration at the force point.
The bending stress S b is given by
ˆ C/ I
Sb  K M
The variable K is the stress concentration factor.
The variable C is the distance from the neutral axis to the outer fiber of the beam.
Assume that the stress concentration factor is 3.0 for the solder lug mounting hole.


Sb  K me Lˆ C / I A
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Stress Scale Factor


Sb  K me Lˆ C / I A
I=
1
w t3
12
= 0.0026 in^4
ˆ  4.7 in
L
(Terminal to Power Supply)
 K meLˆ C / I 
= ( 3.0 )( 0.0013 lbf sec^2/in ) (4.7 in) (0.125 in) /(0.0026 in^4)
= 0.881 lbf sec^2/in^3
= 0.881 psi sec^2/in
= 340 psi / G
386 in/sec^2 = 1 G
 0.34 ksi / G
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Convert Acceleration to Stress
vibrationdata > Signal Editing Utilities > Trend Removal & Amplitude Scaling
29
Stress Time History at Solder Terminal
Apply Rainflow Counting on
the Stress time history and
then Miner’s Rule in the
following slides
Save as: stress
 The standard deviation is 2.06 ksi
 The highest absolute peak is 9.3 ksi, which is 4.53-sigma
 The 4.53 multiplier is also referred to as the “crest factor.”
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Rainflow Count, Part 1 - Calculate & Save
vibrationdata > Rainflow Cycle Counting
31
Stress Rainflow Cycle Count
Range = (Peak – Valley)
Amplitude = (Peak – Valley )/2
But use amplitude-cycle data directly in Miner’s rule, rather than binned data!
32
S-N CURVE ALUMINUM 6061-T6 KT=1 STRESS RATIO= -1
FOR REFERENCE ONLY
S-N Curve
50
45
MAX STRESS (KSI)
40
35
30
For N>1538 and S < 39.7
25
log10 (S) = -0.108 log10 (N) +1.95
20
log10 (N) = -9.25 log10 (S) + 17.99
15
10
5
0
0
10
10
1
10
2
10
3
10
4
10
5
10
6
10
7
10
8
CYCLES
 The curve can be roughly divided into two segments
 The first is the low-cycle fatigue portion from 1 to 1000 cycles, which is
concave as viewed from the origin
 The second portion is the high-cycle curve beginning at 1000, which is
convex as viewed from the origin
 The stress level for one-half cycle is the ultimate stress limit
33
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 R is given by
m n
i
R
i 1
Ni
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
For aerospace electronic structures, however, a more conservative limit is used
Rn(aero) = 0.7
34
Miner’s Cumulative Fatigue, Alternate Form
Here is a simplified form which assume a “one-segment” S-N curve.
It is okay as long as the stress is below the ultimate limit with “some
margin” to spare.
1
R
A
m
 i b
i 1
A is the fatigue strength coefficient
(stress limit for one-half cycle for the one-segment S-N curve)
b is the fatigue exponent
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Rainflow Count, Part 2
vibrationdata > Rainflow Cycle Counting > Miners Cumulative Damage
36
SDOF System, Solder Terminal Location, Fatigue Damage Results for
Various Input Levels, 180 second Duration, Crest Factor = 4.53
Input Overall
Level
(GRMS)
Input Margin
(dB)
Response Stress
Std Dev (ksi)
R
6.1
0
2.06
2.39E-08
8.7
3
2.9
5.90E-07
12.3
6
4.1
1.46E-05
17.3
9
5.8
3.59E-04
24.5
12
8.2
8.87E-03
34.5
15
11.7
0.219
Cumulative Fatigue
Results
 Again, the success criterion was R < 0.7
 The fatigue failure threshold is just above the 12 dB margin
 The data shows that the fatigue damage is highly sensitive to the base input and
resulting stress levels
37
Vibrationdata
Continuous Beam Subjected
to Base Excitation Example
Use the same base input PSD & time history as the
previous example.
(The time history named accel in this exercise is the
same as synth from previous one.
38
Continuous Beam Subjected to Base Excitation
EI, 
Cross-Section
Boundary
Conditions
Material
Rectangular
Fixed-Free
Aluminum
L
y(x, t)
Width
= 2.0 in
Thickness
= 0.25 in
Length
= 8 in
Elastic Modulus
= 1.0e+07 lbf/in^2
Area Moment of
Inertia
= 0.0026 in^4
Mass per Volume
= 0.1 lbm/in^3
Mass per Length
= 0.05 lbm/in
Viscous Damping Ratio =
w(t)
0.05 for all
modes
39
Vibrationdata
vibrationdata > Structural Dynamics > Beam Bending > General Beam Bending
40
Continuous Beam Natural Frequencies
Mode
1
2
3
4
Vibrationdata
Natural
Frequency
Participation
Factor
Effective
Modal Mass
124 Hz
776.9 Hz
2175 Hz
4263 Hz
0.02521
0.01397
0.00819
0.005856
0.0006353
0.0001951
6.708e-05
3.429e-05
modal mass sum = 0.0009318 lbf sec^2/in = 0.36 lbm
41
Vibrationdata
Press Apply Base Input in Previous Dialog and then enter Q=10 and Save Damping Values
42
Vibrationdata
Apply Arbitrary Base Input Pulse. Include 4 Modes. Save Bending Stress and go to Rainflow
Analysis.
43
Bending Stress at Fixed End
Vibrationdata
44
Vibrationdata
45
Cantilever Beam, Fixed Boundary, Fatigue Damage Results for
Various Input Levels, 180 second Duration
Input Overall
Level
(GRMS)
Input Margin
(dB)
Response
Stress Std Dev
(ksi)
R
6.1
0
0.542
1.783e-13
12.2
6
1.08
1.09E-10
24.2
12
2.16
6.61E-08
48.4
18
4.3
4.02E-05
Cumulative Fatigue
Results
Vibrationdata
The beam could withstand 36 days at +18 dB level based on R=0.7
( (0.7/4.02e-05)*180 sec) / (86400 sec / days) = 36 days
46
Frequency Domain Fatigue Methods
Vibrationdata
Rainflow can also be calculated approximately from a stress response PSD
using any of these methods:
•
•
•
•
•
•
•
•
Narrowband
Alpha 0.75
Benasciutti
Dirlik
Ortiz Chen
Lutes Larsen (Single Moment)
Wirsching Light
Zhao Baker
47
Vibrationdata
Spectral Moments
The eight frequency domain methods on the previous slides are based
on spectral moments.
The nth spectral moment
m n for a PSD is

0
m n   f n G(f ) df
where
f
G(f)
is frequency
is the one-sided PSD
Additional formulas are given in the fatigue papers at the Vibrationdata blog:
http://vibrationdata.wordpress.com/
48
Spectral Moments (cont)
Vibrationdata
The expected peak rate E[P]
E[P] 
m4 m2
The eight frequency domain methods “mix and match” spectral moments to
estimate fatigue damage.
Additional formulas are given in the fatigue papers at the Vibrationdata blog:
http://vibrationdata.wordpress.com/
49
Return to Previous Beam Example, Select PSD
Vibrationdata
50
Apply mil_std_1540b PSD. Calculate stress at fixed boundary.
Vibrationdata
51
Bending Stress PSD at fixed boundary
Vibrationdata
Overall level is the same as that from the time domain analysis.
52
Vibrationdata
Save Bending Stress PSD and to Rainflow Analysis.
53
Vibrationdata
54
Rate of Zero Crossings = 186.4 per sec
Rate of Peaks = 608.5 per sec
Irregularity Factor alpha = 0.3063
Spectral Width Parameter = 0.9519
Vanmarckes Parameter = 0.475
Vibrationdata
Lambda Values
Wirsching Light = 0.6208
Ortiz Chen = 1.097
Lutes & Larsen = 0.7027
Cumulative Damage
(1/sec)
Damage Rate
A*rate
((psi^9.25)/sec)
Narrowband DNB = 1.9e-13,
Dirlik DDK = 1.26e-13,
Alpha 0.75 DAL = 1.53e-13,
Ortiz Chen DOC = 2.09e-13,
Zhao Baker DZB = 1.12e-13,
Lutes Larsen DLL = 1.34e-13,
Wirsching Light DWL = 1.18e-13,
Benasciutti Tovo DBT = 1.48e-13,
1.0573e-15,
7.0141e-16,
8.4808e-16,
1.1602e-15,
6.2029e-16,
7.4303e-16,
6.5634e-16,
8.2304e-16,
5.8100e+30
3.8543e+30
4.6602e+30
6.3754e+30
3.4085e+30
4.0829e+30
3.6066e+30
4.5226e+30
Average of DAL,DOC,DLL,DBT,DZB,DDK
average=1.469e-13
55
Bending Stress Damage Comparison
Vibrationdata
Method
Time History
Synthesis
PSD
Average
Damage R
1.78e-13
1.47e-13
56
Vibrationdata
Plate Response to Acoustic Pressure
57
Objective
Vibrationdata
•
Use frequency domain damage methods to assess acoustic fatigue damage
•
Demonstrated for a rectangular plate subjected to a uniform acoustic
pressure field
•
Consider a plate with dimensions 18 x 16 x 0.063 inches
•
The material is aluminum 6061-T6
•
The plate is simply-supported on all four edges
•
Assume 3% damping for all modes
58
Applied Pressure
Vibrationdata
•
The plate is subjected to the Boeing 737 Aft Mach 0.78 sound pressure level
•
Assume that the pressure is uniformly distributed across the plate
•
The sound pressure level and its corresponding power spectral density are
shown in the following figures
•
Calculate the stress and cumulative fatigue damage at the center of the plate
with a stress concentration factor of 3
•
Determine the time until failure at the nominal level and at 6 dB increments
59
Boeing 737 Mach 0.78 SPL, Aft External Fuselage
Vibrationdata
60
Boeing 737 Mach 0.78 , Equivalent PSD, Aft External Fuselage
Vibrationdata
61
Center of
the Plate
Vibrationdata
The stress concentration factor is applied separately by multiply the magnitude by 3.
The magnitude is then squared prior to multiplying by the force PSD.
62
Center of the Plate Stress Response PSD
Vibrationdata
63
Damage Results
Vibrationdata
Cumulative Damage, Simply-Supported Rectangular Plate, Center, Stress
Concentration=3
Margin
Displacement
Damage Rate
Time to Failure
(dB)
(inch RMS)
(1/sec)
(sec)
(Days)
0
0.0126
1.808e-15
5.53e+14
6.40E+09
6
0.0252
1.076e-12
9.29e+11
1.08E+07
12
0.0504
6.324e-10
1.58e+09
18302
18
0.1008
3.822e-07
2.62e+06
30
64
Vibrationdata
Circuit Board Fatigue Response
to Random Vibration
65
Vibrationdata
• Electronic components in vehicles are subjected to shock and vibration
environments.
• The components must be designed and tested accordingly
• Dave S. Steinberg’s Vibration Analysis for Electronic Equipment is a widely
used reference in the aerospace and automotive industries.
66
Vibrationdata
• Steinberg’s text gives practical empirical formulas for determining the fatigue
limits for electronics piece parts mounted on circuit boards
• The concern is the bending stress experienced by solder joints and lead wires
• The fatigue limits are given in terms of the maximum allowable 3-sigma relative
displacement of the circuit boards for the case of 20 million stress reversal
cycles at the circuit board’s natural frequency
• The vibration is assumed to be steady-state with a Gaussian distribution
67
Circuit Board and Component
Lead Diagram
L
h
Vibrationdata
Relative
Motion
Component
Z
B
Relative
Motion
Component
68
Fatigue Introduction
Vibrationdata
The following method is taken from Steinberg:
•
•
•
Consider a circuit board that is simply supported about its perimeter
A concern is that repetitive bending of the circuit board will result in cracked
solder joints or broken lead wires
Let Z be the single-amplitude displacement at the center of the board that
will give a fatigue life of about 20 million stress reversals in a randomvibration environment, based upon the 3 circuit board relative
displacement
69
Vibrationdata
Empirical Fatigue Formula
The allowable limit for the 3-sigma relative displacement
0.00022B
Z 3 limit 
Ch r L
Z
is
(20 million cycles)
B =
length of the circuit board edge parallel to the component, inches
L =
length of the electronic component, inches
h = circuit board thickness, inches
r =
relative position factor for the component mounted on the board
C =
Constant for different types of electronic components
0.75 < C < 2.25
70
Relative Position Factors for
Components on Circuit Boards
r
Component Location
(Board supported on all sides)
1
When component is at center of PCB
(half point X and Y).
0.707
0.5
Vibrationdata
When component is at half point X and quarter point Y.
When component is at quarter point X and quarter point Y.
71
Conclusions
Relative Position
Factor r
.
0.707
0.5
1.0
0.707
72
Component Constants
C=0.75
Axial leaded through hole or
surface mounted components,
resistors, capacitors, diodes
C=1.0
Standard dual inline package
(DIP)
Vibrationdata
73
Component Constants
C=1.26
DIP with side-brazed lead wires
C=1.0
Through-hole Pin grid array
(PGA) with many wires extending
from the bottom surface of the
PGA
Vibrationdata
74
Component Constants
C=2.25
Vibrationdata
Surface-mounted leadless ceramic chip
carrier (LCCC).
A hermetically sealed ceramic package.
Instead of metal prongs, LCCCs have
metallic semicircles (called castellations) on
their edges that solder to the pads.
C=1.26
Surface-mounted leaded ceramic chip carriers
with thermal compression bonded J wires or
gull wing wires.
75
Component Constants
C=1.75
Vibrationdata
Surface-mounted ball grid array
(BGA).
BGA is a surface mount chip carrier
that connects to a printed circuit board
through a bottom side array of solder
balls.
76
Component Constants
Vibrationdata
C = 0.75
Fine-pitch surface mounted axial leads around perimeter of
component with four corners bonded to the circuit board to
prevent bouncing
C = 1.26
Any component with two parallel rows of wires extending from
the bottom surface, hybrid, PGA, very large scale integrated
(VLSI), application specific integrated circuit (ASIC), very high
scale integrated circuit (VHSIC), and multichip module
(MCM).
77
Circuit Board Maximum Predicted Relative
Displacement
Vibrationdata
•
Calculating the allowable limit is the first step
•
The second step is to calculate the circuit board’s actual displacement
•
Circuit boards typically behave as multi-degree-of-freedom systems
•
•
•
Thus, a finite element analysis is required to calculate a board’s relative
displacement
The formula on the following page is a simplified approach for an idealized board
which behaves as a single-degree-of-freedom system
It is derived from the Miles equation, which was covered in a previous unit
78
SDOF Relative Displacement
 1   
   Q A
Z 3   29.4  
 f n1.5   2 
Vibrationdata
inches
f n is the natural frequency (Hz)
Q
is the amplification factor
A
is the input power spectral density amplitude (G^2 / Hz),
assuming a constant input level.
79
Exercise 1
Vibrationdata
A DIP is mounted to the center of a circuit board.
Thus, C = 1.0
and r = 1.0
The board thickness is h = 0.100 inch
The length of the DIP is L =0.75 inch
The length of the circuit board edge parallel to the component is
B = 4.0 inch
Calculate the relative displacement limit
0.00022B
Z 3 limit 
Ch r L
(20 million cycles)
80
Vibrationdata
vibrationdata > Miscellaneous > Steinberg Circuit Board Fatigue
81
Exercise 2
Vibrationdata
A circuit board has a natural frequency of fn = 200 Hz and an amplification
factor of Q=10.
It will be exposed to a base input of A = 0.04 G^2/Hz.
What is the board’s 3-sigma displacement?
82
Vibrationdata
vibrationdata > Miscellaneous > SDOF Response: Sine, Random & Miles equation >
Miles Equation
83
Exercise 3
Vibrationdata
Assume that the circuit board in exercise 1 is the same as the board in
exercise 2.
Will the DIP at the center of the board survive 20 million cycles?
Assume that the stress reversal cycles take place at the natural frequency
which is 200 Hz. What is the duration equivalent to 20 million cycles ?
Answer: about 28 hours
84
Extending Steinberg’s Fatigue Analysis of
Electronics Equipment to a Full Relative
Displacement vs. Cycles Curve
Tom Irvine
Dynamic Concepts, Inc.
NASA Engineering & Safety Center (NESC)
4-6 June 2013
Vibrationdata
© The Aerospace Corporation 2010
© The Aerospace Corporation 2012
Introduction
Project Goals
Develop a method for . . .
• Predicting whether an electronic component will fail due to vibration fatigue during a
test or field service
• Explaining observed component vibration test failures
• Comparing the relative damage potential for various test and field environments
• Justifying that a component’s previous qualification vibration test covers a new test
or field environment
86
Fatigue Curves
Conclusions
.
• Note that classical fatigue methods use stress as the response metric of interest
• But Steinberg’s approach works in an approximate, empirical sense because the
bending stress is proportional to strain, which is in turn proportional to relative
displacement
• The user then calculates the expected 3-sigma relative displacement for the
component of interest and then compares this displacement to the Steinberg limit
value
87
Conclusions
.
• An electronic component’s service life may be well below or well above 20 million
cycles
• A component may undergo nonstationary or non-Gaussian random vibration such
that its expected 3-sigma relative displacement does not adequately characterize
its response to its service environments
• The component’s circuit board will likely behave as a multi-degree-of-freedom
system, with higher modes contributing non-negligible bending stress
88
•
Develop two-segment RD-N curve for electronic parts (relative
displacement)
•
Steinberg provides pieces for this curve, but “some assembly is required”
•
Steinberg gives an exponent b = 6.4 for PCB-component lead wires, for both
sine and random vibration
•
He also gave the allow relative displacement at 20 million cycles
•
The low cycle portion will be based on another Steinberg equation that the
maximum allowable relative displacement for shock is six times the 3-sigma
limit value at 20 million cycles for random vibration
89
RD-N Equation for High-Cycle Fatigue
Vibrationdata
The final RD-N equation for high-cycle fatigue is

RD  6.05- log10 (N)
log10 

6.4
 Z 3 limit 
Will add to Vibrationdata Matlab GUI package soon.
90
RD-N CURVE
ELECTRONIC COMPONENTS
RD / Z 3- limit
10
Conclusions
1
0.1
0
10
1
10
2
10
3
10
4
5
10
10
6
10
7
10
8
10
CYCLES
The derived high-cycle equation is plotted in along with the low-cycle fatigue limit.
91
RD is the zero-to-peak relative displacement.
Vibrationdata
Fatigue Damage Spectra
Can be calculated from either a response time history or a response PSD.
92
Fatigue Damage Spectra
 Develop fatigue damage spectra concept similar to shock response
spectrum
 Natural frequency is an independent variable
 Calculate acceleration or relative displacement response for each natural
frequency of interest for selected amplification factor Q
 Perform Rainflow cycle counting for each natural frequency case
 Calculate damage sum from rainflow cycles for selected fatigue exponent
b for each natural frequency case
m
D   Ai n i
b
i 1
 Repeat by varying Q and b for each natural frequency case for desired
conservatism, parametric studies, etc.
Response Spectrum Review
..
M1
K
3
C1
fn
2
..
L
Y (Base Input)
C
C3
<
fn
3
<
....
L
ML
K
C2
<
X
....
M3
K
2
1
..
X3
X2
M2
K
1
fn
..
..
X1
<
fn
L
L
• The shock response spectrum is a calculated function based on the acceleration
time history.
• It applies an acceleration time history as a base excitation to an array of singledegree-of-freedom (SDOF) systems.
• Each system is assumed to have no mass-loading effect on the base input.
RESPONSE (fn = 30 Hz, Q=10)
100
100
50
50
ACCEL (G)
ACCEL (G)
Base Input: Half-Sine Pulse (11 msec, 50 G)
0
-50
-50
SRS
Example
-100
0
-100
0
0.01
0.02
0.03
0.04
0.05
0.06
0
0.01
0.02
0.05
0.06
RESPONSE (fn = 80 Hz, Q=10)
RESPONSE (fn = 140 Hz, Q=10)
100
100
50
ACCEL (G)
50
ACCEL (G)
0.04
TIME (SEC)
TIME (SEC)
0
0
-50
-50
-100
0.03
-100
0
0.01
0.02
0.03
TIME (SEC)
0.04
0.05
0.06
0
0.01
0.02
0.03
TIME (SEC)
0.04
0.05
0.06
Response Spectrum Review (cont)
SRS Q=10 BASE INPUT: HALF-SINE PULSE (11 msec, 50 G)
200
( 80 Hz, 82 G )
100
( 140 Hz, 70 G )
PEAK ACCEL (G)
( 30 Hz, 55 G )
50
20
10
5
10
100
NATURAL FREQUENCY (Hz)
1000
Nonstationary Random Vibration
FLIGHT ACCELEROMETER DATA - SUBORBITAL LAUNCH VEHICLE
10
ACCEL (G)
5
0
-5
-10
-5
0
5
10
15
20
25
30
35
40
45
50
55
60
65
TIME (SEC)
Liftoff
Transonic
Max-Q
Attitude Control
Thrusters
Rainflow counting can be applied to accelerometer data.
70
Flight Accelerometer Data, Fatigue Damage from Acceleration
FATIGUE DAMAGE SPECTRA
10
b=6.4
14
Q=50
Q=10
DAMAGE INDEX
10
11
10
8
10
5
10
2
10
-1
10
100
1000
2000
NATURAL FREQUENCY (Hz)
The fatigue exponent is fixed at 6.4. The Q=50 curve Damage Index is 2 to 3
orders-of-magnitude greater than that of the Q=10 curve.
Flight Accelerometer Data, Fatigue Damage from Acceleration
FATIGUE DAMAGE SPECTRA
10
Q=10
15
b=9.0
b=6.4
DAMAGE INDEX
10
11
10
7
10
3
10
-1
10
100
1000
2000
NATURAL FREQUENCY (Hz)
The amplification factor is fixed at Q=10. The b=9.0 curve Damage Index is 3 to 4
orders-of-magnitude greater than that of the b=6.4 curve above 150 Hz.
Optimized PSD Envelope for
Nonstationary Vibration
Tom Irvine
Dynamic Concepts, Inc.
NASA Engineering & Safety Center (NESC)
3-5 June 2014
Vibrationdata
© The Aerospace Corporation 2010
© The Aerospace Corporation 2012
Introduction - Nonstationary Flight Data
ARES 1-X FLIGHT ACCELEROMETER DATA IAD601A
2
ACCEL (G)
1
0
-1
-2
0
20
40
60
80
TIME (SEC)
• Liftoff Vibroacoustics
• Transonic Shock Waves
Ares 1-X
• Fluctuating Pressure at Max-Q
101
100
120
References by Year
• Endo & Matsuishi, Rainflow Cycle Counting Method, 1968
Conclusions
• T. Dirlik, Application of Computers in Fatigue Analysis (Ph.D.), University of Warwick, 1985
.
• ASTM E 1049-85 (2005) Rainflow Counting Method, 1987
• S. J. DiMaggio, B. H. Sako, and S. Rubin, Analysis of Nonstationary Vibroacoustic Flight Data Using
a Damage-Potential Basis, Journal of Spacecraft and Rockets, Vol, 40, No. 5. September-October
2003
• K. Ahlin, Comparison of Test Specifications and Measured Field Data, Sound & Vibration, 2006
• Scot McNeill, Implementing the Fatigue Damage Spectrum and Fatigue Damage Equivalent
Vibration Testing, SAVIAC Conference, 2008
• A. Halfpenny & F. Kihm, Rainflow Cycle Counting and Acoustic Fatigue Analysis Techniques for
Random Loading, RASD Conference, 2010
• T. Irvine, An Alternate Damage Potential Method for Enveloping Nonstationary Random
Vibration, Aerospace/JPL Spacecraft and Launch Vehicle Dynamic Environments Workshop, 2012
- Time Domain Method
102
SDOF Model
• Assume component behaves as single-degree-of-freedom
Conclusions (SDOF) system
• Avionics are typically black boxes for mechanical engineering
purposes!
• Unknowns
Component natural frequency
Amplification factor Q
Fatigue exponent b
• Perform fatigue damage calculation on each response for
permutations of the three unknowns
• This adds conservatism to the final PSD envelope
• The fatigue calculation can be performed starting with either
a time history or PSD base input
103
Relative Damage Index
• A relative fatigue
damage index can be calculated from the rainflow cycles using a
Conclusions
Miners-type summation
m
D   Ai n i
b
i 1
where
Ai
is the acceleration response amplitude from
the rainflow analysis
ni
is the corresponding number of cycles
b
is the fatigue exponent
• The damage index D becomes the Fatigue Damage Spectrum (FDS) metric as a function
of: natural frequency, amplification factor Q and fatigue exponent b
104
Enveloping Approach
Conclusions
• A PSD envelope can be derived for nonstationary flight data using rainflow cycling
counting and the relative fatigue damage index
• The enveloping is justified using a comparison of Fatigue Damage Spectra between
the candidate PSD and the measured time history
• The derivation process can be performed in a trial-and-error manner in order to
obtain the PSD with the least overall GRMS level which still envelops the flight data
in terms of fatigue damage spectra
• Could also seek to minimize overall displacement, velocity, peak G2/Hz level, etc.
• Or minimize weighted average of these metrics
105
Enveloping Approach (cont)
• The DirlikConclusions
semi-empirical method can be used to calculate the FDS for each candidate
PSD in the frequency domain
• The immediate output of the Dirlik method is a “rainflow cycle probability density
function (PDF)”
• The rainflow PDF can be converted to a cumulative histogram
• The cumulative histogram can be converted into individual cycles with their
respective amplitudes
• Compare the fatigue spectra of the candidate PSD to that of the flight data for each
Q & b case of interest
• Scale candidate PSD so that it barely envelops the flight data in terms of FDS
• Include some convergence option along the way
• Select the candidate which has the least overall GRMS level, or some other criteria
106
Dirlik Method
POWER SPECTRAL DENSITY
10
fn=200 Hz Q=10
Conclusions
Response 11.2 GRMS
Input 6.1 GRMS
0.1
The Dirlik equation is based
on the weighted sum of the
Rayleigh, Gaussian and
exponential probability
distributions.
2
ACCEL (G /Hz)
1
0.01
0.001
0.0001
20
Dirlik method calculates
rainflow cycle cumulative
histogram from response PSD.
100
1000
FREQUENCY (Hz)
• Sample base input and SDOF response
107
2000
Uses area moments of the
response PSD as weights.
SDOF Response Time Domain
Conclusions
Response Acceleration
.
Base Acceleration
• The response analysis for the nonstationary time history is performed using the
Smallwood, ramp invariant digital recursive filtering relationship, for each fn & Q
• Perform rainflow cycle count on response time history
• Calculate the damage index D for each fn, Q & b
• The damage for each permutation is then plotted as function of natural frequency, as
an FDS
108
Sample Flight Data
Conclusions
.
• Derive a 60-second PSD to envelope the flight data
• Consider 800 candidate PSDs formed by random number generation, with four
coordinates each
109
Permutations
Q & b Values for Fatigue Damage Spectra
Conclusions
.
Case
Q
b
1
10
4
2
10
9
3
30
4
4
30
9
For Reference Only
Natural Frequencies: 20 to 2000 Hz
All cases will be analyzed for each successive trial.
110
Optimized PSD
POWER SPECTRAL DENSITY ENVELOPE
0.1
3.3 GRMS OVERALL
Conclusions
PSD Envelope,
3.3 GRMS, 60 sec
2
ACCEL (G /Hz)
.
0.01
0.001
20
100
1000
2000
Freq
(Hz)
Accel
(G^2/Hz)
20
0.0018
31
0.0019
211
0.0168
2000
0.0024
FREQUENCY (Hz)
The PSD with the least overall GRMS which envelops the flight data via fatigue
damage spectra
111
FDS Comparison 1
FATIGUE DAMAGE SPECTRA
Conclusions
10
Q=10 b=4
10
PSD Envelope
Measured Data
DAMAGE INDEX
.
10
8
10
6
10
4
10
2
20
100
1000
NATURAL FREQUENCY (Hz)
112
2000
FDS Comparison 1
FATIGUE DAMAGE SPECTRA
Conclusions
10
Q=10 b=4
10
PSD Envelope
Measured Data
DAMAGE INDEX
.
10
8
10
6
10
4
10
2
20
100
1000
NATURAL FREQUENCY (Hz)
113
2000
FDS Comparison 2
FATIGUE DAMAGE SPECTRA
Conclusions
10
Q=30 b=4
11
PSD Envelope
Measured Data
DAMAGE INDEX
.
10
9
10
7
10
5
10
3
20
100
1000
NATURAL FREQUENCY (Hz)
114
2000
FDS Comparison 2
FATIGUE DAMAGE SPECTRA
Conclusions
10
Q=30 b=4
11
PSD Envelope
Measured Data
DAMAGE INDEX
.
10
9
10
7
10
5
10
3
20
100
1000
NATURAL FREQUENCY (Hz)
115
2000
FDS Comparison 3
ConclusionsFATIGUE DAMAGE SPECTRA
10
Q=10 b=9
17
PSD Envelope
Measured Data
DAMAGE INDEX
.
10
14
10
11
10
8
10
5
10
2
20
100
1000
NATURAL FREQUENCY (Hz)
116
2000
FDS Comparison 4
FATIGUE DAMAGE SPECTRA
Q=30 b=9
Conclusions
10
19
PSD Envelope
Measured Data
DAMAGE INDEX
.
10
16
10
13
10
10
10
7
10
4
20
100
1000
NATURAL FREQUENCY (Hz)
117
2000
PSD Comparison
POWER SPECTRAL DENSITY
1
Conclusions
Maximum Envelope of 2.5-sec Segments, 2.0 GRMS
Fatigue Damage Spectrum, Optimized, 3.3 GRMS
.
2
ACCEL (G /Hz)
0.1
0.01
0.001
0.0001
20
100
1000
2000
FREQUENCY (Hz)
Maximum Envelope is traditional piecewise stationary method, but its PSD need
further simplification.
118
Conclusions
• An optimized
PSD envelope was derived for nonstationary flight data using the fatigue
Conclusions
damage spectrum method
• The. FDS case with both the highest Q & b values drove the PSD derivation for the sample
flight data
• Still recommend using permutations because other cases may be the driver for a given time
history
• The method can be used more effectively if the natural frequency, amplification factor, and
fatigue exponent are known
• The method is flexible
• The PSD duration can be longer or shorter than the flight vibration duration
• Could require the candidate PSDs to each have a ramp-plateau-ramp shape
• A similar method could be used for deriving force & pressure PSDs
119