85th Shock and Vibration Symposium 2014 NESC Academy

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 Stress Path Cycles Range A-B 0.5 3 4 F 5 G

6 H 7 I 8 -6 -5 -4 -3 -2 -1 0 STRESS 1 2

3 4 5 6 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 Stress Path Cycles Range A-B

0.5 3 4 F 5 G 6 H 7 I 8 -6 -5 -4 -3 -2 -1

0 STRESS 1 2 3 4 5 6 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 structures or components failure potential using Miners 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 Miners 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 Steinbergs text is used in the

following example and elsewhere in this presentations 15 Bracket Example, Variation on a Steinberg Example Power Supply Aluminum Bracket 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 =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 f n 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 0.01 0.001 10 Base Input PSD, 6.1 GRMS 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. 19 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: M R 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 m e 0.0013 lbf sec^2/in 26

Stress & Moment Calculation, Free-body Diagram The bending moment M at a given distance from the force application point is M m e A L where A is the acceleration at the force point. The bending stress S b is given by S b K M C / I 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 L C / I A 27 Stress Scale Factor Sb K me L C / I A

1 I = w t3 12 = 0.0026 in^4 L 4.7 in (Terminal to Power Supply) K meL 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 28

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 Miners 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. 30 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 Miners 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 15 log10 (N) = -9.25 log10 (S) + 17.99 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 Miners 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. Miners cumulative damage index R is given by m n R i i 1 N i 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

Miners 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 35 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 = 0.05 for all modes w(t) 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 Spectral Moments Vibrationdata The eight frequency domain methods on the previous slides are based on spectral moments. The nth spectral moment mn for a PSD is

m n f n G (f ) df 0 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] m 4 m 2 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, 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 Benasciutti Tovo DBT = 1.48e-13,

8.2304e-16, 4.5226e+30 Average of DAL,DOC,DLL,DBT,DZB,DDK average=1.469e-13 55 Bending Stress Damage Comparison Method Damage R Time History Synthesis 1.78e-13 Vibrationdata PSD Average 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. Steinbergs Vibration Analysis for Electronic Equipment is a widely used reference in the aerospace and automotive industries. 66 Vibrationdata

Steinbergs 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 boards 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 Empirical Fatigue Formula Vibrationdata The allowable limit for the 3-sigma relative displacement 0.00022 B Z 3 limit Ch r L B L h r C Z is (20 million cycles)

= length of the circuit board edge parallel to the component, inches = length of the electronic component, inches = circuit board thickness, inches = relative position factor for the component mounted on the board = 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 boards actual displacement Circuit boards typically behave as multi-degree-of-freedom systems Thus, a finite element analysis is required to calculate a boards 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 1 . 5 f n 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 0.00022 B 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 boards 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 Steinbergs 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 components 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 Steinbergs 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 components 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 components 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 log10 Z 3 limit 6.05 - log10 (N) 6.4 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 b D Ai n i 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 1 K C2 < fn 2 XL ....

M3 K 2 .. X3 X2 M2 K 1 .. .. X1 ML

Y (Base Input) CL C3 < fn 3 .. L < .... < fn 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 Q=10 15 10 b=9.0 b=6.4 11 DAMAGE INDEX 10 10 7

10 3 -1 10 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) Ares 1-X Liftoff Vibroacoustics Transonic Shock Waves 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 b

D Ai n i 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 G 2/Hz level, etc. Or minimize weighted average of these metrics 105 Enveloping Approach (cont) The Dirlik semi-empirical method can be used to calculate the FDS for each Conclusions 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 Conclusions fn=200 Hz Q=10 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 Conclusions 3.3 GRMS OVERALL 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 FATIGUE DAMAGE SPECTRA Conclusions 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 . 16 DAMAGE INDEX 10 13

10 10 10 7 10 4 10 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

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