pseudo velocity shock analysis velocity shock spectrum … · shock analysis using the pseudo...

VelSSComparison 1

PSEUDO VELOCITY SHOCK ANALYSISVelocity Shock Spectrum and Analyses

Comparison

•Howard A. Gaberson, Ph.D., P.E.•Consultant

•234 Corsicana Drive•Oxnard, CA 93036-1300

•(805)485-5307•hagaberson@att.net

Mechanical Shock Test Techniques and Data Analysis

SAVIAC

VelSSComparison 2

SHOCK ANALYSIS USING THE PSEUDO VELOCITY SHOCK SPECTRUM

PART 3• Pseudo velocity is compared to relative velocity shock

spectra on 4CP and we find problems with relative velocity.

• Relative velocity has low frequency problem, and doesn't show the maximum acceleration.

• Review tests to determine which transient motion analyses method is the best indicator of damage potential.

• The best damage potential analysis is the damped PVSS on 4CP.

• Shaker Shock and Miscellaneous Issues

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All kinds of shocks: package drops, collisions, transportation bumps, but face it; explosions are where its at

• Wait and see; I’ve got one.• Late 60’s; Prairie Flat

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PRARIE FLAT ALBERTA

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TNT PILE PHOTO

9 Aug 1968

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Explosion I

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Explosion II

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This first set of slides tests relative velocity vs. PV as a spectrum ordinate

• Now we look at four shocks, and their integrals to velocity and displacement.

• Then I calculated their relative and pseudo velocity shock spectra and overlaid them.

• Relative velocity is about equal to pseudo velocity in the severe zone, but has no useful asymptotes or other meaning on 4CP.

• Peak relative velocity doesn't seem helpful.

VelSSComparison 9

Peak accel: 130

Velocity change and peak velocity are 120 ips

Max displacement 2.4 in

Reed’s time history

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Very Low Frequency SDOF Relative Velocity

• Relative velocity sees peak shock (bogey) velocity.

• Pseudo Velocity sees peak shock (bogey) deflection.

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2.4 inches

Vel change 120

130 g's

Rel Vel SS and PVSS for Reed

VelSSComparison 12

2.4 inches

130 gs

Blue rel vel does not hit asymptotesAgrees in center severe section

Reed SS’s, 5% Damping

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HS54 Time History

VelSSComparison 14

130 g's

270 ips

I should have drawn 54 inch line.

Rel Vel SS and PVSS for HS54

VelSSComparison 15

Blue is rel vel, doesn’t hitaccel asymptote.Magenta is PV

HS54 SS’s, 5% Damping

VelSSComparison 16

900 g's

300 ips Vel change

8 inches

HW4 Time History

VelSSComparison 17

8 inches

300 ips

Blue is rel velMagenta is PVAt low frequencies, rel vel ismax velocity, PV is max z

Rel Vel SS and PVSS for HW4

VelSSComparison 18

900 g's

Low frequency SDOF see peak rel vel

PV shows peak spring deflection

HW4 SS’s, 5% Damping

VelSSComparison 19

Meaningless

HW4 SS’s, 10% Damping

VelSSComparison 20

Peak g: 0.35

Peak vel is 15, vel change is about 20 ips

Max z about 70 in.

El Centro EQ Time History

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Rel Vel SS and PVSS for El Centro

VelSSComparison 22

Velocity change 20 ips

70"0.35 g

Peak velocity 15 ips

Rel Vel SS and PVSS for El Centro EQ

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CONCLUSION: PVSS Gives More Useful Information than Relative Velocity SS

• PVSS has HF peak acceleration asymptote vice rel vel SS not.

• Both have about equal levels in plateau• PVSS has LF max deflection asymptote

vice rel vel SS not• Rel Vel SS has LF max velocity asymptote

vice PVSS not. Not that helpful.

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The Magnitude of the Fourier Transform of a Shock is its Undamped Residual PVSS

VelSSComparison 25

Background

• Proved by many and Rubin in [Shock and Vibration Handbook, 2002, pp 23.24,23.25],

• Hopefully more clear here.• Undamped shock spectrum equation: (Set 1, Sli 24)

0

1 ( )sin ( )t

z y t dτ ω τ τω

= − −∫

VelSSComparison 26

Complex vector ideas

ie α

• is a complex vector at angle . ie α

α

( )

0

1 Im ( )t

i tz y e dω ττ τω

−⎡ ⎤= − ⎢ ⎥

⎣ ⎦∫

0

1 Im ( )t

i t iz e y e dω ωττ τω

−⎡ ⎤= − ⎢ ⎥

⎣ ⎦∫

The integral on the right is the FT. Call it Y.

( )

cos sin:

Im sin

i

i

e iso

e

α

α

α α

α

= +

=

0

1 ( )sin ( )t

z y t dτ ω τ τω

= − −∫

VelSSComparison 27

Continue vector ideas

1 Im i tz e Yω

ω⎡ ⎤= − ⎣ ⎦

iY Y e θ=

0

( ) −= ∫T

iY y e dωττ τ

Im i t iz e Y eω θω ⎡ ⎤= − ⎣ ⎦

Put this in here.

VelSSComparison 28

Combine the exponents and use the sine

( )Im i tz Y e ω θω +⎡ ⎤= − ⎣ ⎦

sin( )z Y tω ω θ= − +

( )maxz Yω =

z at every frequency is vibrating continuously at that frequency with the magnitude or value of Y. Y is the residual PV.

VelSSComparison 29

How do we calculate? Make it discrete.

ττ=ω ωτ−∫ de)(yz i

t

0

sf1nf2iN

1ns

e)n(yf1)f(Y

−π−

=∑=

A row times a column is a sum of the products. Actually used a row of N complex numbers or exponentials times a column of N accelerations times.

( )

21

1, 1:

s

s

i t ft

df

nt from n N

f

ω π

τ

=

=

−= =

VelSSComparison 30

Code crux

• for kk=1:nfreqs• Y(kk)=1/fs*abs(exp(-i*tpi*f(kk)*t)*yy);• end• loglog(f,Y);

sf1nf2iN

1ns

e)n(yf1)f(Y

−π−

=∑=

VelSSComparison 31

Example applied to 2000 g 0.4 ms half sine shock

• Fourier Transform Calculation In Matlab• Compare with Undamped PVSS

VelSSComparison 32

Plot Undamped PVSS and Fourier Transform of 2000g, 0.4 ms half sine

Fourier transform is residual undamped PVSS so it must lie under the overall undamped PVSS, which it does.

VelSSComparison 33

FT and DFT

sf1nf2iN

1ns

e)n(yf1)f(Y

−π−

=∑=

∑−=

=

π−

=1Nn

0n

Nkn2i

nk exN1X

Very similar; FT evaluates for a freq list, and divides by fs. Fewer values. Matlab’s DFT has N differently; confusing.

VelSSComparison 34

Matlab's DFT has 1/N in DSP position

21

0

1( 1) ( 1)i knN

N

k

x k X k eN

π−

=

+ = +∑

21

1 10

i knn NN

k nn

X x eπ−= −

+ +=

= ∑

This needs more explanation Matlab’s DFT has N differently; confusing.

VelSSComparison 35

HOW TO LOOK AT THE DFT

• Process involves these calculations.

• First Eq analyzes x’s into X’s which are complex sine wave amplitudes.

• Second Eq inverse transforms X's back to x's. Builds x’s from X’s. Amazing! But I have the 1/N in best place. Usually placed in the second Eq.

• This way, second Eq says add up for each n, content from each spiral at each frequency, k.

∑−=

=

π−

=1Nn

0n

Nkn2i

nk exN1X

∑−

=

π

=1N

0k

Nnk2i

kn eXx

VelSSComparison 36

TEST ANALYSIS METHODS ON EQUALLY SEVERE SHOCKS

• Develop several equally severe shocks; all able to just fail same equipment.

• Find analysis methods that show them all somewhat equally severe.

• Such an analysis is needed to describe shock severity.

• Needed to describe equipment hardness or fragility.

VelSSComparison 37

Equipment Fragility-Hardness

• The most severe shock equipment survives• Shock severity concept needed• Pseudo-velocity shock spectrum on four

coordinate paper: Recommended by Eubanks and Juskie, 1963 SVB

• Somewhat adopted by the seismic and nuclear defense community

• Chalmers and I advocated PVSS

VelSSComparison 38

Equipment Lowest Modal Frequency

• Postulate equipment can only accept energy and be damaged at its modal frequencies

• To damage equipment shock must have high PV at equipment modal frequencies.

• Shock isolation reduces the higher frequency, high PV severe range

• An item may be damaged by dropping on a concrete floor but survive a fall on a carpet because the carpet cushion reduced the severe high PV range to frequencies below the equipment's lowest frequency.

VelSSComparison 39

Equipment to test

• $50. Squirrel cage blower selected• Failure: blower can’t blow (pressure drop

through orifice)• Failure Equipment becomes hazardous• Purchased about 15

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Develop equally severe shocks

• Test same equipment on progressively increasing different shock tests

• Drop tests: terminal peak, half sine, hard phenolic block

• Mil-s-901: Light weight, medium weight, heavy weight

• TP60, HS54, PB24, LW72, MW36, HW4

VelSSComparison 41

Abbreviations

• TP60, 60 inch drop to a terminal peak saw tooth• HS54, 54 inch drop on rubber pad to get half sine• PB24, 24 inch drop on hard phenolic block• LW72, 72 inch hammer drop to shelf bracket on light

weight shock machine• MW36, 36 in hammer drop on the medium weight shock

machine• HW4, 4th shot on the floating shock platform at Hunters

Point

VelSSComparison 42

Failures consisted of

• Spider members broken or deformed: belt flies off

• Motor knocked out of its supports: belt flies off or motor can’t turn

• Self tapping bolts deformed, loosened, pulled out

• Feet bolt holes deformed

VelSSComparison 43

f1

Test blower as originallyconfigured. Arrows point to thin spider members supportingbearings, and to sheet metalscrews. Notice sheet metal legs.

Weak sheet metalmotor mount beam.;

Blowers as purchased; trivial failure.

VelSSComparison 44

Original configuration, trivialfailure.

Trivial failure

VelSSComparison 45

Original bent sheet metalmotor support. Trivial failure.

Trivial failure, another view.

VelSSComparison 46

Motor mounted tobase plate.

Motor Mounted on Base Plate

VelSSComparison 47

f3a

Rubber pad forhalf sine test.

Drop table shockmachine.

Tagami’s China Lake MRC Shock

Machine

VelSSComparison 48

f3b

Mylar covered openingwith orifice to simulaterealistic delta p.

Manometer to checkfor adequate delta p

Terminal peaksawtooth programmer

Shock Machine with Terminal

Peak Programmer

VelSSComparison 49

f3c

Phenolic blocktaped to shockmachine anvil forshort duration halfsine test, pb24

Phenolic Block High Impact Set

Up

VelSSComparison 50

f4

Mounted todeck fixtureon light-weight shock mach.

Blower on Navy

Lightweight Shock

Machine

VelSSComparison 51

f4a5

Hammer

Blower on lightweightshock machine

Perspective View of Lightweight Shock Machine

VelSSComparison 52

f2

Mounted onmedium wgtshock machine.Extra plates added to increaseweight.

Blower Mounted on

Navy Medium Weight Shock

Machine

VelSSComparison 53

f5

Two blowers installedin FSP. One foundationstiff, and the other lessstiff

Two Blowers Mounted in FSP

VelSSComparison 54

f6

Spot weld fail

HS54 failure.Spot weld pulledout allowing beltto loosen.

HS54 Failures

VelSSComparison 55

f7

TP60 failure. Pulley side bearing supportspot weld failedallowing bearingsupport to deflectand loosen belt

TP60 Failures

VelSSComparison 56

f8

PB24 failure. Spidermember pulled outof housing, loosingbelt.

PB24 Failures

VelSSComparison 57

f9

MW 36 failure

Leg deformed

Leg pulled away from blower body. Sheet metalscrews gone.

Motor mount bentrubbing pulley

Bearing support bent

Accelerometer

MW36 Failures

VelSSComparison 58f10

Bent spider members

Deformed legs

Loosened self tappingscrews

MW36 failure

More MW36 Failures

VelSSComparison 59f11

MW36 damage. Motorbearing housing poppedout of end bracket

Accels

MW36 Failures Detail

VelSSComparison 60

f12

HW4 failure.

Sheet metal leg pulledaway from blower body

Bent spider members

Motor clamp bent andrubbing against pulley.Belt gone.

Loose self tappingscrews

HW4 Failures

VelSSComparison 61f13

Screw gone

BendingHW4 failure

VelSSComparison 62

f14

Non pulley end of motor supportbracket deformed. Motor bearingsupport popped out of bracket.

HW4 Motor Damage

VelSSComparison 63

f15

HW4 Failure. Bentmotor clamp. Motor

pivoted about clampand lost belt.

HW4 Motor Collar Bent

VelSSComparison 64

Failures consisted of

• Spider members broken or deformed: belt flies off

• Motor knocked out of its supports: belt flies off or motor can’t turn

• Self tapping bolts deformed, loosened, pulled out

• Feet bolt holes deformed

VelSSComparison 65

HS54 Time History

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HW4 Time History

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LW72 Time History

VelSSComparison 68

MW36 Time History

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PB24 Time History

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TP60 Time History

VelSSComparison 71

Here's all 6 shocks plotted to same scale. Only one of these did not fail the blower.

Time histories, same scale

VelSSComparison 72

HS54=r,HW4=g,LW72=bMW36=blk,PB24=m,TP60=c

5 of these 6 shock spectra failed the blower; which 5?

Composite Undamped SRS

VelSSComparison 73

HS54=r,HW4=g,LW72=bMW36=blk,PB24=m,TP60=c

5 of these 6 shock spectra failed the blower; which 5?

Composite 5% Damped SRS

VelSSComparison 74

HS54=r,HW4=g,LW72=bMW36=blk,PB24=m,TP60=c

Composite 10% Damped SRS

VelSSComparison 75

HS54=r,HW4=g,LW72=bMW36=blk,PB24=m,TP60=c

5 of these 6 shock spectra failed the blower; which 5?

Composite 20% Damped SRS

VelSSComparison 76

HS54=r,HW4=g,LW72=bMW36=blk,PB24=m,TP60=c

FT magnitude of a shock is it’s residual PVSS.5 of these 6 FT magnitudes failed the blower; which 5?

Composite FT Magnitudes

VelSSComparison 77

HS54=r,HW4=g,LW72=bMW36=blk, PB24=m,TP60=c

5 of these 6 shock spectra failed the blower; which 5?

Composite Undamped

PVSS on 4CP

VelSSComparison 78

HS54=r,HW4=g,LW72=bMW36=blk,PB24=m,TP60=c

5 of these 6 shock spectra failed the blower; which 5? (Any light?)

Composite 5% Damped PVSS

on 4CP

VelSSComparison 79

HS54=r,HW4=g,LW72=bMW36=blk,PB24=m,TP60=c

5 of these 6 shock spectra failed the blower; which 5?

Composite 10% Damped PVSS on 4CP

VelSSComparison 80

Here's all 6 shocks plotted to same scale. Only one of these did not fail the blower.

Time histories, same scale

VelSSComparison 81

HS54=r,HW$=g,LW72=bMW36=blk,PB24=m,TP60=c

5 of these 6 shock spectra failed the blower; which 5?

Composite 20% Damped PVSS on 4CP

VelSSComparison 82

Conclusions Presented evidence that:

• Damped Pseudo-velocity shock spectrum is the best severity indicator

• Peak g’s and time histories are useless• Shock spectrum calculation must become

widely available• Collect digitized time histories and make

available so others can test

VelSSComparison 83

This completes Velocity Shock Spectrum and Analyses Evaluations

Next is Shock Severity Estimation• Editing and Integration • Filtering Effects on the PVSS• Damping and Polarity• Fragility Concepts• Final Comments