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SHOCK ANALYSIS USING THE PSEUDO VELOCITY SHOCK

SPECTRUMPART 1

- Howard A. Gaberson, Ph.D., P.E.
- Consultant
- 234 Corsicana Drive
- Oxnard, CA 93036-1300
- (805)485-5307
- hagaberson_at_att.net

Dick Chalmers1931 - 1998

Joel Leifer

Henry Pusey

Points to Learn

- Shock spectrum definitions
- Equations Conceptual and Practical
- Calculate Do loop, Filter, Residual
- Plotting SS's with PV as ordinate on (4CP)

log-log four coordinate paper. - Proofs that stress is proportional to velocity

for rods, beams, plates. - Fundamental Maximum modal velocities
- Shock data needs least range in velocities
- PV use Civil/Structural, nuclear defense, Army

Conventional Weapons Effect, Earthquake. Hall's

chapter in Shock and Vibration Hdbk - (PV)2 ,energy, severe frequencies.
- PVSS on 4CP asymptotes peak displ, severity or

vel change, and peak accel. - PV vs Rel Vel, low freq. problem.
- Integrated shock presentation

More Important Points

- Half sine example. Compare to explosive and EQ
- Tests of which motion analysis is best for

damage. PV on 4CP wins. - Shaker shock wimp. Both synthesized or pre and

post pulses. - Applies to MDOF
- Susceptible Frequencies
- Shock Isolation
- Damping. Precludes swept sines.
- Heavy damping shows polarity pos neg spectra.
- Mean removal and detrending. Show El Centro Calc

ss of a trend. Reasonable fudge Add trend to

adjust net displacement.

Shock Spectrum Definition

- SS is a plot of an analysis of a transient motion

that calculates the maximum response of many

different frequency damped SDOFs exposed to the

motion. (explosions, earthquakes, package drops,

railroad car bumping, vehicle collisions, etc.) - Can be positive, negative, or maximum of the

two. Maximax maximum of the maximum (refers to

residual and during) - Can be for during or after the motion or both.

Residual means after. SDOF can be damped or

undamped. And could use relative or absolute

acceleration, velocity, or displacement, and

pseudo velocity. - REPEAT FOR US ITS A PLOT OF AN ANALYSIS OF A

MOTION. - .

ABSOLUTE ACCELERATION SHOCK SPECTRUM, SRS

- LOG LOG PLOT OF ABSOLUTE ACCELERATION VS

FREQUENCY MOST POPULAR SHOCK SPECTRUM FORM

CALLED SRS. NOT GOOD FOR DAMAGE. ALLOWS WEAK

SHOCKS TO APPEAR SEVERE.

PVSS on 4CP

- Pseudo velocity is best for severity or capacity

to cause damage. - PSEUDO VELOCITY EXACTLY MEANS PEAK RELATIVE

DISPLACEMENT, Z, MULTIPLIED BY THE NATURAL

FREQUENCY IN RADIANS, . - PVSS4CP (PSEUDO VELOCITY SHOCK SPECTRUM PLOTTED

ON FOUR COORDINATE PAPER) IS A SPECIFIC

PRESENTATION OF THE RELATIVE DISPLACEMENT SHOCK

SPECTRUM THAT IS EXTREMELY HELPFUL FOR

UNDERSTANDING SHOCK.

FOUR COORDINATE PAPER

The best way to plot PVSS is on 4CP

(tripartite.) 4CP is a logarithmic graph that has

four sets of lines relating frequency,

displacement, velocity, and acceleration of

sinusoidal motion. For a sine wave, the

displacement, velocity, and acceleration are

Equating maximum values yields

The equations mean

are related. Knowing two, can calculate other.

two. On loglog paper, lines of constant

versus

are all straight lines. Look at a 4CP plot.

4CP example. Lines slanting down to the right

are constant gs. Slanting down to left are

constant displacement. Horizontal pseudo

velocity. Vertical frequency.

On 4CP, if you know any two, you can get the

other two from

SHOCK SPECTRUM EQUATION DWGS THIS SYSTEM A

BOGY GETS THE SHOCK RATTLES THE MASS

Mental Model

Free Body Diagram

Shock Spectrum Equation

Duhamels Integral Solution

Use on digitized shock values and apply eqs. from

one data point to the next. Over and over for

whole, say, 10,000 point shock.

Aside Undamped

Calculating Details

Equations become

After integrating from 0 to h, we get O'Hara's

and EQ Eqs.

Manipulating you can get

These get during values. Pick off max and final

values and then calc residual

Residual Calculation

- One way is to append at least one period of zeros

to the shock and consider all of the file as

during and pick off the max and min values. - Second procedure takes the final during

displacement and velocity as initial conditions

and evaluate the free damped vibration equations

for the maximum and minimum values. - Both procedures have merit. I happen to use the

second.

PLOTTING PVSS ON 4CPDISPLACEMENT, VELOCITY,

ACCELERATION

ACCELERATION MEANING ON 4CP

Response of a 20 Hz 15 damped SDOF

toExplosion.. Notice max acceleration does not

occur at same time as max displacement.

Arguments for shock spectra with PV as ordinate

on log-log 4CP

- Velocity Theoretically stress proportional to

modal velocity - Velocity Experimentally spectral and time plots

require least range. - PVSS on 4CP relate experimentally severe shocks
- Related to stored elastic energy Indicates

capacity to deliver energy - Shows Three Regions
- 1. Peak deflection, Rattle space reqts.
- 2. High PV mid plateau, severe energy

delivery frequencies. - 3. Max acceleration
- Unmasks weak shocks with damping
- Reveals polarity with damping

Why Pseudo Velocity and not Absolute or Relative

Velocities are Best For Shock Spectra

Precedent for PVSS on 4CP

- Eubanks and Juskie "Shock Hardening of

Equipment" SVB 1963 - Crede in ASME, "Shock and Structural Response"

1960 - Fung in same ASME publication
- Civil Seismic Community Hall's Chapt in recent

SV Hdbk - Roberts in "Explosive Shock in our 1969 SVB

Session - Vigness in 1964 SVB "Elementary Considerations of

SS - Currently UERD, EQ, Structural, Nuclear

MECHANICAL SHOCK

- Explosion, a shock wave in air or water,

instantaneous intense pressure spike, 1000's of

psi. - Impacts metal structure.
- Excites 100's of modes.
- Accelerometer sensing that motion reads the build

up and ring down of modes as a complicated

multi-frequency shock.

Near Miss Explosion, Army Tank

Near Miss Explosion, Army Tank

EXPERIMENTAL DATA REQUIRE THE LEAST DYNAMIC RANGE

WITH VELOCITY

- From Walsh, J.P., A Review of the Report on

the Cameron Trials, SVB, No. 4, p 62, July

1947 Royal Navy "Cameron Trials", experiment. - Conducted an instrumentation study, of shock

damage to specimen with many structure types

(high/low freqs, brittle/ductile), - Specimen designed to provide failure modes like

those of underwater blast. - Many different shocks applied to specimen until

all designed failure modes occurred.

EXPERIMENTAL DATA REQUIRE THE LEAST DYNAMIC RANGE

WITH VELOCITY, II

EXPERIMENTAL DATA REQUIRE THE LEAST DYNAMIC RANGE

WITH VELOCITY, II

- Analysis wide ranges of usual shock parameters

were needed to describe failure levels for all

modes designed into the specimen. - Accelerations associated with failure ranged from

2.45 to 9x103 g's. - Displacements varied from 1.9 to 7.6x10-4 inches.

Extremes in each would cause failure in one mode

but not in another. - Velocity at these accelerations and displacements

ranged from 30 to 240 inches/second.

EXPERIMENTAL DATA REQUIRE THE LEAST DYNAMIC RANGE

WITH VELOCITY, III

- Accelerations range for damage was 3700 to 1.
- Displacement range for damage was 2500 to 1.
- Velocity range for damage was 8 to 1.
- Expressed in dB 71 dB for acceleration 68

dB for displacement 18 dB for velocity. - If their test specimen, representative, velocity

was best parameter for shock. - Velocity was prominent during "Cameron Trials",

and has been important in U. S. Navy ship shock

trials since that time.

Gunfire Shock plotted as acceleration shock

spectrum shows constant velocity character

Data falling along 12 ips line

Railroad humping on acceleration shock spectrum

shows constant velocity

If this is g's it's 61.4 ips,

Existing Knowledge that stress proportional to

modal velocity

- Reinhart's 1954 work cited by Roberts in 69 SVB
- Hunt 1960, JASA
- Ungar 1962, ASME Trans.
- Crandall, late 62, ltr. to JASA
- Ungar 71 used in Damping Chapt. In Beranek's

Noise Vibe Control - Lyon, 75 used in SEA book
- MIL Std 810F acknowledges
- Gaberson Chalmers Modal Velocity 1969 SVB
- Shock and Vibration Handbook Equipment Design

Stress is Proportional to Velocity

ABSOLUTELY ESSENTIALRIGHT HERE AND NOW THAT I

CONVINCE YOU THAT STRESS IS PROPORTIONAL TO

VELOCITY AND NOT ACCELERATION.

Shock Spectrum Equation,Variables and Constants

y is the shock. We calculate max z. Free body

diagram of m. F ma, gives

Divide by m and use

Shock Spectrum Equation

Strain in a rod

F ma

Stress Velocity Rods

More Stress Velocity Rods

SUM UP STRESS VELOCITY IN RODS

BEAMS STRESS VELOCITY EQUATIONS

Develop beam vibration equation. From strength

of materials we have

When the beam is vibrating, (accelerating up and

down) the ma load on the little chunk of beam

is, then uniform load on the

beam, (force per unit length) is

Substituting this value for w, in (1c) gives us

the Bernoulli Euler beam vibration equation

STRESS VELOCITY IN BEAMS

MORE BEAM STRESS VELOCITY

BEAM SHAPE FACTORS

Ted Hunts Analysis(Frederick Vinton Hunt from

Harvard)

K was a beam shape factor, but Hunts careful

scholarly analysis in JASA, 1960, proves that the

relation.holds for plates, tapered rods, and

wedges. He felt that for practical situations the

constant stays under two, but found a value for a

cone of 6.89. He used the phrase half and order

of magnitude. He also argues that it applies to

sums of modes.

SEVERE VELOCITIES

Velocity proportional to stress higher stress

allows higher velocity. Vibrational velocity is

the parameter that proportional to stress, and as

such indicates the severity of the vibration in

structure. Only taught in statistical energy

analysis. Hunt 4 knew this well before 1960

Ungar 5 understands it. Crandall commented on

it 7. Lyon 6 knows this and uses it in his

book. But it still is not in machine design,

materials, or vibration texts. I keep bringing

it up to encourage its use.

HIGH PVSS SHOWS SHOCK CAPACITY TO DELIVER ENERGY

TO AN SDOF SYSTEM

MAXIMUM PSEUDO VELOCITY IS RELATED TO THE ENERGY

STORED IN THE SDOF SPRING The shock spectrum

algorithm finds the peak relative displacement

for a base excited SDOF. That's the spring

stretch which is how the SDOF system stores

energy. Consider the elastic energy stored in

the spring, and remember k/m ?2, This

is our pseudo velocity. Pseudo velocity is the

square root of twice the peak energy per unit

mass that is stored in the oscillator during the

shock. HIGH PV FREQUENCY RANGE IS WHERE IS WHERE

SHOCK HAS GREATEST CAPACITY TO DELIVER ENERGY.

Before I Look at the Extremely Important

Asymptotes

- Consider Half Sine and other Simple Shocks First
- Then well do
- 3 Regions on the PVSS on 4CP
- Important
- Acceleration and Displacement Asymptotes
- Severity

Half Sine and Other Simple Shocks

- Include drop in analyzed signal. Have chuckle re

Hdbk. SRS drop. - You need to learn why Drop needed for low

frequency - No rebound
- Other simple shocks same SS
- Impact Velocity Change
- Peak acceleration
- Max displacement

PVSS-4CP Example 800 g, 1 ms half sine

ZERO MEAN SIMPLE SHOCK

- THAT LAST SHOCK WAS A ZERO MEAN SIMPLE SHOCK.
- ZERO MEAN ACCELERATION MEANS SHOCK BEGINS AND

ENDS WITH ZERO VELOCITY - THE SHOCK INCLUDES THE DROP AND ANY REBOUND.
- THE INTEGRAL OF THE ACCELERATION IS ZERO IF IT

HAS A ZERO MEAN. - BY SIMPLE SHOCK I MEAN ONE OF THE COMMON SHOCKS

HALF SINE, INITIAL PEAK SAW TOOTH, TERMINAL PEAK

SAW TOOTH, TRAPEZOIDAL, HAVERSINE

ZERO MEAN, SIMPLE SHOCK, PVSS-4CP HILL SHAPE

- WHEN A ZERO MEAN SIMPLE SHOCK PVSS IS PLOTTED ON

4CP IT HAS A HILL SHAPE - THE LEFT UPWARD SLOPE IS A PEAK DISPLACEMENT

ASYMPTOTE - THE RIGHT DOWNWARD SLOPE IS THE PEAK

ACCELERATION ASYMPTOTE. - THE TOP IS A PLATEAU AT THE VELOCITY CHANGE

DURING IMPACT.

Collision and Kickoff Shocks

- Collision shock, (a car slamming into a wall)
- Kickoff shock (environment on a ball when it is

kicked) - Do not have a zero mean.
- The collision starts with a high velocity and

ends with zero velocity - The kickoff starts with zero velocity and ends

with a high velocity. - These PVSS's on 4CP have damped max velocity

change low frequency asymptote, vice down to the

left maximum deflection asymptote line.

Half sine equations

HALF SINE SHOCK WITHOUT THE DROPI'm going to

plot this half sine shock along with it's two

integrals. Let's illustrate it with a moderately

severe shock with a 100 ips velocity change and a

peak acceleration of 200 g's. From the above

formula, we find the duration to be 2.035 ms.

Assume a shock machine table could do this, and

consider the integrals.

The shock machine table has a final velocity and

just keeps on going.

BAD SHOCK SPECTRUM OF HALF SINEThis what I want

everyone to learn to expect. Because the velocity

did not end at 0, this is an unrealistic SS.

It's indicating that a 0.1 Hz SDOF would have a

peak deflection of about 140 inches, 12 ft. No

way. However notice that the velocity change of

100 ips shows up as it should and the curve at

high frequency is assympotic to 200 g's, as it

must.

Here's its PVSS on 4CP. Get used to these 4

axes. AND 100 ips ands 200 gs

Correct concept. Half sine time history with

drop, and integrals

Include no rebound drop. Half sine brings to

rest. Realistic. Drop drop it 12.95. Use

half sine programmer.

SS of 100 ips, 200 g half sine with drop

Here's its PVSS

100 ips

13"

200 gs

Note The three regions 12.9" drop, 100 ips

velocity change, and the 200 g peak acceleration.

COSINE RAMP TRAPEZOID SHOCK EQUATIONS

Acceleration during cosine ramp to max in ?td

Integrate to get velocity change during ramp.

Final velocity change equation for cosine ramp

trap shock.

HOW MUCH COSINE RAMP IS REASONABLE

Now let's try and figure out how much of a cosine

ramp would be reasonable within the confines of

the IEC Specification, according to Figure 3., on

p40. I'm did it graphically in Matlab. Now

studying this picture, their nominal trapezoid

has a linear ramp with phi0.1 the blue and

green limits allow one to increase td by 0.4, and

I used a cosine with a phi0.3. Ive drawn 0.3,

and by increasing the max and minimum levels.

Trapezoid, 30 cosine ramp, 100 ips, 200 g with

drop

Simple pulses have similar PVSS's. Here's a 200

g, 100 ips trapezoidal shock, with a 30 cosine

ramp up and down.

Again it's caught by a programmer that brings it

to zero velocity.

Terminal peak sawtooth, 200 g. 100 ips, with 10

cosine fall off.

Here's the same for a 200 g, 100 ips terminal

peak shock.

Initial peak shock time plot

Here's the same for a 200 g, 100 ips initial

peak shock.

We shall see they all have a similar PVSS.

All the simple shocks are equally severe.

The simple pulses have similar shock spectra.

This is not well known, Vigness showed in 1964

paper, Gertel in 1967 Frankford Arsenal report..

If related by velocity change, they only differ

at high frequencies beyond PLATEAU, where

severity drops.

And with 5 damping we get the same result

No one knows this, but now you do. All simple

shocks have equally severe SSs. Gertel tossed

that comment off in a '67 report, and Vigness

plotted in his 64 SVB Paper Shock and Vibe

Hdbk, Rubin and Ayre Chpts go overboard on the

droop.

HIGH AND LOW FREQUENCY ASYMPTOTES FOR PVSS ON

4CP TRADITIONAL RATIONALIZATIONS

- High frequency oscillators have very stiff

springs and light masses. The mass follows the

acceleration of the foundation thus, these SDOFs

record a peak absolute acceleration equal to the

peak acceleration of the shock. - Low frequency oscillators are very flexible. If

their foundations are given a very quick or short

duration wiggle, the mass barely moves until the

foundation motion is over. If the shock being

analyzed is one that begins and ends with zero

velocity, the peak relative deflection will be

the peak shock displacement. - For intermediate values of the frequency, the

peak pseudo velocity is often almost constant.

In this region the pseudo velocity closely

approximates the relative velocity. This region

is the high PV region contains the frequencies

where the shock has the greatest capacity to

deliver energy to the SDOFs.

Low Frequency Asymptote Carefully

- In low frequency region, spring very soft, mass

heavy. Mass just sits there during shock.

Maximum spring stretch or zmax is just maximum y,

ymax We plot ?zmax - Since ymax is constant, log of the pv here is

straight line with positive slope.

HIGH FREQUENCY ASYMPTOTE CAREFULLY

- In the PVSS on 4CP, we are plotting ?z focus on

that. - In high frequency region mass very light and

spring very stiff Mass exactly follows input

motion. Acceleration of mass equal to

foundation. Max z, is spring force over

stiffness, k. - On log log paper, since the maximum acceleration

is a constant, pv is a straight line with a

negative slope.

PLATEAU, MID FREQUENCY

- The mid frequency, plateau region of the spectrum

is the important region. It shows the velocity

change of a simple pulse. - No one explains it well. And this is the really

important part. - Pick up any civil structural textbook and go to

the earthquake section and read the traditional

explanations. - Vigness 1 declared without any explanation that

four regions exist on the shock spectrum, and

broke the center region into two parts this is

typical of our business. - This middle region is the most severe and thereby

most important region of the of the shock's SS.

Here the PV is at its maximum. Vigness's

comments

VIGNESS ON MID FREQUENCY REGION

- Vigness's very important, yet somewhat weakly

justified paper 1964 paper "Elementary

Considerations of Shock Spectra" from the SVB

n34, pp 211-222. - In paragraph 16 which is on page 213, "At some

intermediate frequency, region B, peaks in the

shock spectral curves indicate sustained

frequencies of the shock motions. - At some lower frequency (generally), a section of

the shock spectral curve, region V, remains at a

constant velocity value. This corresponds to a

frequency region over which the shock motion can

be considered to be an impulsive (step) velocity

change." - That's a very interesting few sentences, probably

correct, but not justified. In his intro to the

paper he warns that these are observations, and

suggestions, not necessarily new. So maybe in

1964, many people had agreed on this.

Roberts, W.H., Explosive Shock, Shock and

Vibration, Bulletin 40, Part 2, Dec 1969, pp

1-10.

- This is a difficult paper for me to follow and

has not received attention. - However he reasons, he gets many of the right

answers. For example - It is reasonable to conclude therefore, by

analogy to the discussion on the shock spectrum

that vibration velocity measures internal

stresses, not vibration acceleration. This is

exactly the result that our analyses lead to. - He uses Rinehart and Pearson, Behavior of Metals

Under Impulsive Loads, 1954 Dover, which

establishes 1200-2400 inches per second as

theoretical wave velocity limits in structural

materials. He cites his experience that

structure tolerates 360 ips. He next reasons

that mechanical components velocity limits are

60 120 ips. - His reasoning allows him to conclude that the 3

shock spectrum regions are logical. - Clarifying of how Roberts thinks would help me.

Roberts Drawing

UNDAMPED PLATEAUIMPORTANT CONVINCE YOURSELF

- Instant shock
- Bogey and mass fall h
- Shock over before spring compresses
- Uundamped free vibration equation govern

PLATEAU FINISH

PLATEAU SUMMARY

SUMMARIZE UNDAMPED ZERO MEAN SIMPLE SHOCK SSs

- UNDAMPED SIMPLE DROP TABLE SHOCKS HAVE A FLAT

CONSTANT PV PLATEAU AT THE VELOCITY CHANGE THAT

TOOK PLACE DURING THE SHOCK. - THE HIGH FREQUENCY LIMIT OF THE PLATEAU IS SET BY

THE MAXIMUM ACCELERATION OF THE SHOCK. - THE HIGH FREQUENCY ASYMPTOTE IS THE MAXIMUM

ACCELERATION LINE. - THE LOW FREQUENCY LIMIT OF THE PLATEAU IS SET BY

THE MAXIMUM DEFLECTION OF THE SHOCK. - THE LOW FREQUENCY ASYMPTOTE IS THE MAXIMUM

DISPLACEMENT LINE

ADJUST A SPECTRUM

2g LINE CONCEPT UNDAMPED SIMPLE SHOCK NO

REBOUND DROP HEIGHT IS WHERE THE PLATEAU

INTERSECTS 2 gs

2 g Line for Drop Height of a Simple Pulse as a

Function of Velocity Change

The red line is the 2 g line. The drop height is

the displacement at the intersection of the red

line and the desired velocity change. Accurate

for simple pulses where deflection during the

pulse is small.

Exception Joshs 200ips, 5 g Halfsine

Velocity level when numerical value of

acceleration in gs equals numerical value of

frequency in Hz

If Ng is numerical value of g level, and Nf is

numerical value of frequency, we find

(1)

Explosive shock time history and integrals

Here's a chunk of an explosive test. I

removed the mean to make the velocity end at

zero.

Peak accel 918 g's Peak vel 302, total vel

change abt 400, during first 4-5 ms abt 300

ips Max displacement, 8.6 inches.

Explosive shock PVSS on 4CP

Here's its PVSS.

We see the 8.6 inches. The undamped is hovering

around 300 ips. I interpolated by 3 to get fs

150,000 Hz, to see if undamped would go to 900

gs. The damped is heading for 900 g's.

We could simulate with 9 inch bungee assisted,

13.8 g drop (to get 900 gs in 9 in), 300 ips,

900 g half sine.

Bad Editing Example Pyroshock ExampleHimelblau

Piersol 95-96 SVS Proceedings

Pyroshock Shock SpectrumVery High FrequencyBad

at 200 Hz

NOTE Axes on 4CP have had to be shifted

one decade to the right to allow frequencies to

100,000.

This completes Part I. In Part IIExperimental

and computational proof, additional concepts.

- Pseudo velocity compared to relative velocity

shock spectra indicating pitfalls for relative

velocity. - Relative velocity low frequency problem.
- 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.

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