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85th Shock and Vibration Symposium 2014. Rainflow Cycle Counting for Random Vibration Fatigue Analysis Revision A By Tom Irvine. This presentation is sponsored by. NASA Engineering & Safety Center (NESC ). Vibrationdata. Dynamic Concepts, Inc. Huntsville, Alabama. Contact Information.

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85th Shock and Vibration Symposium 2014

Rainflow Cycle Counting for Random Vibration Fatigue Analysis Revision A

By Tom Irvine

this presentation is sponsored by
This presentation is sponsored by

NASA Engineering & Safety Center (NESC)


Dynamic Concepts, Inc.Huntsville, Alabama

contact information
Contact Information

Tom Irvine

Email: [email protected]

Phone: (256) 922-9888 x343



  • 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.

Fatigue Cracks

A ductile material subjected to fatigue loading experiences basic structural changes. The changes occur in the following order:

Crack Initiation. A crack begins to form within the material.

Localized crack growth. Local extrusions and intrusions occur at the surface of the part because plastic deformations are not completely reversible.

Crack growth on planes of high tensile stress. The crack propagates across the section at those points of greatest tensile stress.

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.


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.”


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


Rainflow Cycle Counting

Rotate time history plot 90 degrees clockwise


Rainflow Cycle Counting

Rotate time history plot 90 degrees clockwise


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)


Use of Rainflow Cycle Counting

  • Can be performed on sine, random, sine-on-random, transient, steady-state, 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!

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

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

Steinberg’s text is used in the following example and elsewhere in this presentations


Power Supply

Aluminum Bracket

Solder Terminal

0.25 in

2.0 in

4.7 in

5.5 in

Bracket Example, Variation on a Steinberg Example

6.0 in


Bracket Response via SDOF Model

Treat bracket-mass system as a SDOF system for the response to base excitation analysis. Assume Q=10.


Base Input PSD

Now consider that the bracket assembly is subjected to the random vibration base input level. The duration is 3 minutes.


Base Input PSD

The PSD on the previous slide is library array: MIL-STD1540B ATP PSD


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

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






Stress & Moment Calculation, Free-body Diagram

The reaction moment MR at the fixed-boundary is:

The force F is equal to the effect mass of the bracket system multiplied by the acceleration level.

The effective mass me is:


Stress & Moment Calculation, Free-body Diagram

The bending moment at a given distance from the force application point is

where A is the acceleration at the force point.

The bending stress Sb is given by

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.


Stress Scale Factor

(Terminal to Power Supply)

= 0.0026 in^4

= ( 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


Convert Acceleration to Stress

vibrationdata > Signal Editing Utilities > Trend Removal & Amplitude Scaling


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.”

Rainflow Count, Part 1 - Calculate & Save

vibrationdata > Rainflow Cycle Counting


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!


S-N Curve

For N>1538 and S < 39.7

log10 (S) = -0.108 log10 (N) +1.95

log10 (N) = -9.25 log10 (S) + 17.99

  • 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

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

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 usedRn(aero) = 0.7


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.

A is the fatigue strength coefficient

(stress limit for one-half cycle for the one-segment S-N curve)

b is the fatigue exponent


Rainflow Count, Part 2

vibrationdata > Rainflow Cycle Counting > Miners Cumulative Damage


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

Continuous Beam Subjected

to Base Excitation Example

Use the same base input PSD & time history as the previous example.

(The time history named accelin this exercise is the same as synthfrom previous one.



EI, 

y(x, t)


Continuous Beam Subjected to Base Excitation


Continuous Beam Natural Frequencies

Natural Participation Effective

Mode Frequency Factor Modal Mass

1 124 Hz 0.02521 0.0006353

2 776.9 Hz 0.01397 0.0001951

3 2175 Hz 0.00819 6.708e-05

4 4263 Hz 0.005856 3.429e-05

modal mass sum = 0.0009318 lbf sec^2/in = 0.36 lbm


Apply Arbitrary Base Input Pulse. Include 4 Modes. Save Bending Stress and go to Rainflow Analysis.


Cumulative Fatigue Results

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

frequency domain fatigue methods
Frequency Domain Fatigue Methods

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
spectral moments
Spectral Moments

The eight frequency domain methods on the previous slides are based on spectral moments.

The nth spectral moment for a PSD is


Additional formulas are given in the fatigue papers at the Vibrationdata blog:

spectral moments cont
Spectral Moments (cont)

The expected peak rate E[P]

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:

bending stress psd at fixed boundary
Bending Stress PSD at fixed boundary

Overall level is the same as that from the time domain analysis.


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

Lambda Values

Wirsching Light = 0.6208

Ortiz Chen = 1.097

Lutes & Larsen = 0.7027

Cumulative Damage Damage Rate A*rate

(1/sec) ((psi^9.25)/sec)

Narrowband DNB = 1.9e-13, 1.0573e-15, 5.8100e+30

Dirlik DDK = 1.26e-13, 7.0141e-16, 3.8543e+30

Alpha 0.75 DAL = 1.53e-13, 8.4808e-16, 4.6602e+30

Ortiz Chen DOC = 2.09e-13, 1.1602e-15, 6.3754e+30

Zhao Baker DZB = 1.12e-13, 6.2029e-16, 3.4085e+30

Lutes Larsen DLL = 1.34e-13, 7.4303e-16, 4.0829e+30

Wirsching Light DWL = 1.18e-13, 6.5634e-16, 3.6066e+30

BenasciuttiTovo DBT = 1.48e-13, 8.2304e-16, 4.5226e+30





  • 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

Applied Pressure

  • 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

Center of

the Plate

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.


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.

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
Circuit Board and Component Lead Diagram


Relative Motion





Relative Motion


fatigue introduction
Fatigue Introduction

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 random-vibration environment, based upon the 3 circuit board relative displacement
empirical fatigue formula
Empirical Fatigue Formula

The allowable limit for the 3-sigma relative displacement Zis

(20 million cycles)


Relative Position Factor r






component constants2
Component Constants

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.

component constants3
Component Constants

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.

circuit board maximum predicted relative displacement
Circuit Board Maximum Predicted Relative Displacement
  • 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
sdof relative displacement
SDOF Relative Displacement


f nis the natural frequency (Hz)

Qis the amplification factor

Ais the input power spectral density amplitude (G^2 / Hz),

assuming a constant input level.

exercise 1
Exercise 1

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

(20 million cycles)

exercise 2
Exercise 2

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?

exercise 3
Exercise 3

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


Extending Steinberg’s Fatigue Analysis of Electronics Equipment to a Full Relative Displacement vs. Cycles Curve

Tom IrvineDynamic Concepts, Inc.

NASA Engineering & Safety Center (NESC)

4-6 June 2013



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

Fatigue Curves



  • 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


  • 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

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
rd n equation for high cycle fatigue
RD-N Equation for High-Cycle Fatigue

The final RD-N equation for high-cycle fatigue is

Will add to Vibrationdata Matlab GUI package soon.


The derived high-cycle equation is plotted in along with the low-cycle fatigue limit.

RD is the zero-to-peak relative displacement.


Fatigue Damage Spectra

Can be calculated from either a response time history or a response PSD.


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
  • Repeat by varying Q and b for each natural frequency case for desired conservatism, parametric studies, etc.

Response Spectrum Review

  • 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 single-degree-of-freedom (SDOF) systems.
  • Each system is assumed to have no mass-loading effect on the base input.

Base Input: Half-Sine Pulse (11 msec, 50 G)

RESPONSE (fn = 30 Hz, Q=10)

SRS Example

RESPONSE (fn = 80 Hz, Q=10)

RESPONSE (fn = 140 Hz, Q=10)


Nonstationary Random Vibration

  • Liftoff Transonic Attitude Control
    • Max-Q Thrusters

Rainflow counting can be applied to accelerometer data.


Flight Accelerometer Data, Fatigue Damage from Acceleration

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

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

Optimized PSD Envelope for Nonstationary Vibration

Tom IrvineDynamic Concepts, Inc.

NASA Engineering & Safety Center (NESC)

3-5 June 2014


introduction nonstationary flight data
Introduction - Nonstationary Flight Data
  • Liftoff Vibroacoustics
  • Transonic Shock Waves
  • Fluctuating Pressure at Max-Q

Ares 1-X


References by Year

  • Endo & Matsuishi, Rainflow Cycle Counting Method, 1968
  • 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



SDOF Model

  • Assume component behaves as single-degree-of-freedom (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

Relative Damage Index

  • A relative fatigue damage index  can be calculated from the rainflow cycles using a Miners-type summation
  • The damage index D becomes the Fatigue Damage Spectrum (FDS) metric as a function of: natural frequency, amplification factor Q and fatigue exponent b



Enveloping Approach

  • 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

Enveloping Approach (cont)

  • The Dirlik 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

Dirlik Method


Dirlik method calculates rainflow cycle cumulative histogram from response PSD.

The Dirlik equation is based

on the weighted sum of the Rayleigh, Gaussian and exponential probability distributions.

Uses area moments of the response PSD as weights.

  • Sample base input and SDOF response

SDOF Response Time Domain


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

Sample Flight Data



  • Derive a 60-second PSD to envelope the flight data
  • Consider 800 candidate PSDs formed by random number generation, with four coordinates each



Q & b Values for Fatigue Damage Spectra

For Reference Only


Natural Frequencies: 20 to 2000 Hz

All cases will be analyzed for each successive trial.


Optimized PSD



PSD Envelope, 3.3 GRMS, 60 sec

The PSD with the least overall GRMS which envelops the flight data via fatigue damage spectra


PSD Comparison



Maximum Envelope is traditional piecewise stationary method, but its PSD need further simplification.



  • An optimized PSD envelope was derived for nonstationary flight data using the fatigue 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