AeroHydroPLUS1

End-to-end statistical modelfor maximum expected vibration response

under aero-acoustic loading

Paul Bremner

AeroHydroPLUS

Outline

• Motivation• Variation in flight test data• Ensemble Mean vibration response • Ensemble Variance in modal response• Total Variance of vibration response• Contributions to Total variance

2

AeroHydroPLUS

MotivationMaximum expected vibration under aero-acoustic loading

I. Vibration test specification for space flight vehicle equipment

II. Sonic fatigue of aircraft empennage

3

AeroHydroPLUS

Outline

• Motivation• Variation in flight test data• Ensemble Mean vibration response • Ensemble variance in vibration modes• Total variance of vibration response• Contributions to Total variance

4

AeroHydroPLUS

Variation in launch vehicle skin vibration levelsPredict maximum expected from Mean, Variance and PDF

5

NASA HDBK-7005 (A. Piersol Ch.6)• Different points on skin panel• Data from multiple flights

1.E-03

1.E-02

1.E-01

1.E+00

1.E+01

10 100 1000

Rel

ativ

e Va

rianc

e

Frequency (Hz)

RELATIVE VARIANCE

2

22 2 2Rev

lVar v v

0.2

52 297.5 1.96Log v Var Log vP Log v

97.5%

MEAN 2 2

, SpatialFlt fltMean v v

AeroHydroPLUS

Outline

• Variance [uncertainty] in flight test data• Ensemble Mean vibration response • Ensemble variance in vibration modes• Total variance of vibration response• Contributions to Total variance

6

AeroHydroPLUS

Vibration response to general random FSP load

7

– Temporal term: Modal receptance

– Modal joint acceptance

– Homog. cross spectrum of aeroacoustic loading

2 22 2

22, r o pp r

vv or r r

G jG A

m Y

xx

22

1( ) ( )G ( , ; ) ( )G ( )r r pp r

pp A

j d dA

x x x x x x

( , '; )pG x x

,vv oG x

22 rrrr iY

G ( , ; ) G ( ) cosy cx c c k yc k xpp pp ce e k x x x

Gpp (x,ω)

Gvv (x,ω)

AeroHydroPLUS

Mean vibration Energy levelOver ensemble of similar structures & uncertainty in loading

8

1. Surface integral (≡ average) for total energy

2. 2nd average over uncertain input pressure distribution and spatial correlation

3. Integrate (3rd average) over frequency band Δω

2 2 2

22 2 2 4i

pp r

xr

r r r r

A G jE

m

2 2

, 2i

pp r

xr

G j A rEm

Total Energy E 2ˆ ,

A

E m v d x x x

2 2 * *

*2 2 2 2

, ,

1 1

r r s s

r s r r r s s s

A P j P j d dE m

m j m j

x x x x x x

AeroHydroPLUS

Mean Power InputOver ensemble of similar structures & uncertainty in loading

9

1. Power input from modal expansion

2. Average over uncertain input pressure distribution and spatial correlation

3. Integrate (2nd average) over frequency band Δω

1. Surface integral (≡ average) for total energy

2. 2nd average over uncertain input pressure distribution and spatial correlation

3. Integrate (3rd average) over frequency band Δω

* *

2 2

, ,Re

1r r r rin

r r r r

j P j P j d d

m j

x x x x x x

2 2 * *

*2 2 2 2

, ,

1 1

r r s s

r s r r r s s s

A P j P j d dE m

m j m j

x x x x x x

2 2 2

22 2 2 4i

pp r

xr

r r r r

A G jE

m

2 2

, 2i

pp r

xr

G j A rEm

2 2

22 2 2 4

r pp rin

inr

r r r r

A G j

m

2 2

2 2,

Re2i

pp rpp r iix

G j A r A G j Mm

Power Input ΠinTotal Energy E 2ˆ ,

A

E m v d x x x *Re , ,ini i i

A

P j v j d x x x

AeroHydroPLUS

This is familiar SEA !

Power Input Πin

10

1. Power input from modal expansion

2. Average over uncertain input pressure distribution and spatial correlation

3. Integrate (2nd average) over frequency band Δω

* *

2 2

, ,Re

1r r r rin

r r r r

j P j P j d d

m j

x x x x x x

Total Energy E

1. Surface integral (≡ average) for total energy

2. 2nd average over uncertain input pressure distribution and spatial correlation

3. Integrate (3rd average) over frequency band Δω

2ˆ ,A

E m v d x x x

2 2 * *

*2 2 2 2

, ,

1 1

r r s s

r s r r r s s s

A P j P j d dE m

m j m j

x x x x x x

2 2 2

22 2 2 4i

pp r

xr

r r r r

A G jE

m

2 2

, 2ix

pp rG j A rm

E

rω η

*Re , ,ini i i

A

P j v j d x x x

2 2

22 2 2 4

r pp rin

inr

r r r r

A G j

m

2

,

2

2i

pp r

x

G j A rm

SEA mean response of ANY modal subsystem

2inΠ = ω η E = ω η m v

AeroHydroPLUS

Outline

• Variance [uncertainty] in flight test data• Ensemble Mean vibration response • Ensemble Variance in modal response• Total Variance of vibration energy• Contributions to Total variance

11

AeroHydroPLUS

2nd statistical moment - Variance

12

2

22

2 Re2r

iipp r rj A

Var E Var QG

pV Mm

ar j

2 1 2 222 2

1 1 1 1 1, , 2 tan ln 1 ln 12

r B B B BB B

mm m B

modal densityQm

Lyon 1969, Weaver 1989 and Langley 2004

• Modal overlap

• Relative bandwidth

• Loading spatial correlation variance

2

22 2, 0

2 22 2 2

Re

Re , ,

iiVar p Var Q

ii

Var E p Q Var j M

p Q j M r m B

/B Q

24 2r rE j E j

AeroHydroPLUS

Relative Variance of ensemble modal dynamicsr2 [E] = Var[E] / <E>2

13

MEANVibration energy

REL. VARIANCEVibration energyMonte Carlo SIMULATION

with random masses

Laboratory TESTINGwith random masses

AeroHydroPLUS

Variance of distributed random loading

14

21 22

1 1, ,mm

r B F B F Bm

Monte Carlo SIMULATIONdistrib. load & random masses

Dirac Delta spatial correlation

Diffuse acoustics, TBL spatial correlation

4

22

11E j K

NE j

AeroHydroPLUS

Outline

• Variance [uncertainty] in flight test data• Ensemble Mean vibration response • Ensemble variance in vibration modes• Total variance of vibration response• Contributions to Total variance

15

AeroHydroPLUS

TOTAL Variance of vibration Energy

16

2 2

2 2 2 22 22 2 2 2 2 2

2 222 2 2 2 2 2

Re

Re Re Re

Re Re Re

s ii

s ii s ii ii s

s ii s ii ii s

Var E Var p Var Q Var j M

p Q Var j M p j M Var Q Q j M Var p

p Var Q Var j M Q Var p Var j M j M Var p Var Q

2 2 2 2 2 2

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

Re

Re Re Re

s ii

s ii s ii s ii

r E r p r Q r j M

r p r Q r Q r j M r p r j M r p r Q r j M

2 2 2 2 2 2j Res iir E r p r Q r M

2 1ir X 2 1ir X

2 2 Re iiE p j M Q Product of 3 statistically uncorrelated variables

AeroHydroPLUS

Spatial varianceDiscrete point response versus Energy level

• Energy

• Vibration at discrete point [Langley 2005]

17

2

2

ˆA

sp

E m v d

m v

x x x

x

2

2 2

2 2 2 2 2

2

2 Re

sp

s ii

v E m

Var v F B r E

F B r p r j M r Q

x

x 22r v

x

2r E

AeroHydroPLUS

Outline

• Variance [uncertainty] in flight test data• Ensemble Mean vibration response • Ensemble variance in vibration modes• Total variance of vibration response• Contributions to Total variance

18

AeroHydroPLUS

Measured Variance – FLUCT SURFACE PRESSUREAt different AoA and Azimuthal station

Transonic FSP

19 AeroHydroPLUS166th Meeting: Acoustical Society of America

1.E‐03

1.E‐02

1.E‐01

1.E+00

1.E+01

1 10 100 1000 10000

Relative

 Variance

Frequency (Hz)

1.E+02

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1 10 100 1000 10000

Mea

n Sq

. Pressure [Pa^

2]

Frequency (Hz)

Sta 14 M=0.9 Run 70 0°Sta 14 M=0.9 Run 70 45°Sta 14 M=0.9 Run 70 90°Sta 14 M=0.9 Run 70 135°Sta 14 M=0.9 Run 70 180°Sta 14 M=0.9 Run 70 225°Sta 14 M=0.9 Run 70 270°Sta 14 M=0.9 Run 70 315°Sta 14 M=0.9 Run 72 0°Sta 14 M=0.9 Run 72 45°Sta 14 M=0.9 Run 72 90°Sta 14 M=0.9 Run 72 135°Sta 14 M=0.9 Run 72 180°Sta 14 M=0.9 Run 72 225°Sta 14 M=0.9 Run 72 270°Sta 14 M=0.9 Run 72 315°Sta 14 M=0.9 Run 67 0°Sta 14 M=0.9 Run 67 45°Sta 14 M=0.9 Run 67 90°Sta 14 M=0.9 Run 67 135°Sta 14 M=0.9 Run 67 180°

AeroHydroPLUS

Measured Variance – DAMPING LOSS FACTORDLF = 1/Q

20 AeroHydroPLUS166th Meeting: Acoustical Society of America

e

PIAAV Test Panele

0.001

0.01

0.1

10 100 1000 10000

Damping

 Loss Fa

ctor

Frequency (Hz)

Oper TFn FitModal TestMESAM 22‐25 secMESAM 37‐40 secT60 High DLFT60 Low DLFMEAN

0.001

0.01

0.1

1

10

10 100 1000 10000

Relative Va

rian

ce

Frequency (Hz)

AeroHydroPLUS

Predicted versus Measured Total VarianceModel also ranks the multiple sources of uncertainty

21

A. Piersol NASA HDBK-7005

Vibr

atio

nPS

D

(g^2

/Hz)

2 2 22 22Re2 iisV r p rar v j QB r MF

x

0.001

0.010

0.100

1.000

10.000

10 100 1000

Relative

 Varianc

e

Frequency, Hz

RelVar_MEASRelVar_MODEL2r^2[p^2]2r^2[Q]2r^2[j^2 ReM]F(B)

AeroHydroPLUS

Prediction of P95 maximum expected PSDUsing Total Relative Variance model + LogNormal PDF

22

A. Piersol NASA HDBK-7005

Vibr

atio

nPS

D

(g^2

/Hz)

1.E‐05

1.E‐04

1.E‐03

1.E‐02

1.E‐01

1.E+00

10 100 1000

Autospectral Den

sity,  g^

2/Hz

Frequency Hz

Mean_MEAS

P95_MEAS

P95_MODEL

2

10 102 2

22

10 2

10 5 1

43 1

vv

vv

v

Gvvv

ref vv

GL

vv

GL Log Logv G

LogG

AeroHydroPLUS

Summary

• End-to-end statistical model– Predicts Total variance of vibration response

• Relative contributions to Total varianceI. Variance in fluct. surface pressure (FSP) levelII. Uncertainty in FSP spatial correlation & modal parameters (BCs, etc.)III. Variance in dampingIV. Bandwidth of random RMS response

• LogNormal PDF assumption– Predicts P95 max. expected vibration response– P99.5 ? … [tbc]

23

AeroHydroPLUS

BACK-UP MATERIAL

pbremner@aerohydroplus.comPh: 619 977 4048

24

AeroHydroPLUS

MotivationCorrect uncertainty margins for model-based design ?

Because:i. All three uncertainties are statistically independent ; RSS (3 variances) < SUM ( 3 variances)ii. Fatigue Life is a product of 3 uncertain (random) variables -> LOG Normal distribution

III. Uncertainty inFatigue failure models

I. Uncertainty in Aero & Acoustic loading

1 10 100 1000 10000Frequency (Hz)

Sta 14 M=0.9 Run 72 90°Sta 14 M=0.9 Run 72 135°Sta 14 M=0.9 Run 72 180°Sta 14 M=0.9 Run 72 225°Sta 14 M=0.9 Run 72 270°Sta 14 M=0.9 Run 72 315°Sta 14 M=0.9 Run 67 0°Sta 14 M=0.9 Run 67 45°Sta 14 M=0.9 Run 67 90°Sta 14 M=0.9 Run 67 135°Sta 14 M=0.9 Run 67 180°

Sta 14 M=0.9 Run 70 0°Sta 14 M=0.9 Run 70 45°Sta 14 M=0.9 Run 70 90°Sta 14 M=0.9 Run 70 135°Sta 14 M=0.9 Run 70 180°Sta 14 M=0.9 Run 70 225°Sta 14 M=0.9 Run 70 270°Sta 14 M=0.9 Run 70 315°Sta 14 M=0.9 Run 72 0°Sta 14 M=0.9 Run 72 45°Fl

uct.

Sur

face

Pre

ssur

e P

SD

(P

a^2/

Hz)

RM

S S

tress

(M

N/m

^2)

N Cycles to failure

9 7 6. Loading VA model SN dataMIN EXPECTED Fatigue Life MEAN Fatigue L dB dB dBife

II. Uncertainty inVibro-acoustics model

AeroHydroPLUS26

Maximum expected levelPredictable from PDF, Mean and Variance

Energy Schroeder Pressure (Vibration) Field

2

2

2

v

2 299.5 10.6P v

P P

0.597.5 1.96P Log E Log E Var Log E

243 1Var Log E Log r E

97.5%

Exponential PDF Log Normal PDF

2 2 ReE p j M Q

AeroHydroPLUS

Measured Variance – FLUCT SURFACE PRESSUREAt different AoA and Azimuthal station

Supersonic FSP

27 AeroHydroPLUS166th Meeting: Acoustical Society of America

1.E‐03

1.E‐02

1.E‐01

1.E+00

1.E+01

1 10 100 1000 10000Re

lative

 Variance

Frequency (Hz)

1.E+02

1.E+03

1.E+04

1.E+05

1.E+06

1.E+07

1 10 100 1000 10000

Mea

n Sq

. Pressure [Pa^

2]

Frequency (Hz)

Sta 14 M=0.95 Run 57 45°Sta 14 M=0.95 Run 57 90°Sta 14 M=0.95 Run 57 135°Sta 14 M=0.95 Run 57 180°Sta 14 M=0.95 Run 57 225°Sta 14 M=0.95 Run 57 270°Sta 14 M=0.95 Run 57 315°Sta 14 M=0.95 Run 57 0°Sta 14 M=0.95 Run 57 45°Sta 14 M=0.95 Run 57 90°Sta 14 M=0.95 Run 57 135°Sta 14 M=0.95 Run 57 180°Sta 14 M=0.95 Run 57 225°Sta 14 M=0.95 Run 57 270°Sta 14 M=0.95 Run 57 315°Sta 14 M=0.95 Run 57 0°Sta 14 M=0.95 Run 57 45°Sta 14 M=0.95 Run 57 90°Sta 14 M=0.95 Run 57 135°Sta 14 M=0.95 Run 57 180°Sta 14 M=0.95 Run 57 225°

AeroHydroPLUS28

AeroHydroPLUS

Nasa NESC Mach 8 plume impingement aero-acousic & vibration test (PIAAV)

Tap Test - Linear

PIAAV Test – Non-Linear

AeroHydroPLUS

FE (Abaqus) evaluation – with & without non-linear elasticity at 171dB FPL loading

0 1.0 2.0 3.0 4.0 5.0

PIAAV “equivalent beam” center ACCELERATION

LINEAR model NON-LINEAR model

10E10

10E9

10E6

10E5

PSD (G2/Hz)

0 500 1000 1500 2000 2500 Frequency (Hz)

10E10

10E9

10E6

10E5

PSD (G2/Hz)

0 500 1000 1500 2000 2500 Frequency (Hz)

1.5E6

1.0E6

0 1.0 2.0 3.0 4.0 5.0 Time (s)

G-0.5E6

0.5E6

-1.0E6

-1.5E6

1.0E6

Time (s)

G

-0.5E6

0.5E6

-1.0E6