Shock & Vibration:

Case Study

For Northrop Grumman Information Systems

LX Course: 3rd Quarter 2011

The presentation material is a proprietary property of Luxea & Dunamis Inc.

Contact the company for appropriate distribution.

Luxea Inc. / Dunamis Tech Inc.

2 The presentation material is a proprietary property of Luxea & Dunamis Inc. Contact the company for appropriate distribution.

Week 2 - Shock & Vibration I Review

CLASS NOTES & SOFTWARE

• Class notes will be available for download from Luxea.com next week.

• LuxCalc Tools v1.2.3 will be downloadable from NG ESL server by tomorrow.

• Expect a revised Week1 class notes this week.

HOMEWORK

• Homework description is included in the class notes.

• Answers will be posted every Thursday pm.

• Discussions and comments are encouraged through Luxea HW blog – more details to

come. www.luxea.com/Blog/Categories

Blog

Announcements

3

SCHEDULE

Week Topic/Case Study HW

1 Overview and Introduction

2 Review of Shock & Vibration for Electroincs I

3 Case I: Transportation shock and vibration

4 Case II: Rack on isolators – pulse shock

5 Case II: Rack on isolators – drop shock

6 Case II: Rack on isolators – random vibration

7 Case II: Rack on isolators – multi-DOF and nonlinear effect

8 Case III: Chassis/PCB – shock

9 Case III: Chassis/PCB – random vibration

10 Case IV: Transit Case Analysis – MIL-HDBK-304

11 Case V: Transit Case Analysis – Nonlinearity

12 Summary and Closing

4

Luxea Inc. / Dunamis Tech Inc.

5 The presentation material is a proprietary property of Luxea & Dunamis Inc. Contact the company for appropriate distribution.

Shock & Vibration: Case Study

Stiffness k in Shock and Vibration

Think in terms of stiffness k

m

kfn

2

1

m

k

Spring vs. Solid Body

F

k

x

F = kx

Force-displacement relation in spring:

Stress-strain relation in deformable body:

= E

F/A= E x/L

F F E

L

F= AE/L x

k =AE/L

F = kx and = E are equivalent

x

Bar in Axial Load

l

AEk

AE

Flx

Axial deflection

Axial stiffness

F F E

L

k = F/x

Beam in Bending

3

3

3

3

l

EIk

EI

Fly

Bending deflection (Cantilever beam)

Bending stiffness

l

k = F/y

F

y

Kinematics

Displacement, velocity and acceleration, may be expressed

in sinusoidal functions as:

• Y = Yo sint

• v = Yo cost

• a = -Yo 2 sint

= 2f

m

k

vmax= Yo

amax = Yo 2

Use to relate the displacement and

acceleration in sinusoidal vibration

SDOF Natural Frequency by Energy Method

Vibration = periodic motion of energy transfer

between SE and KE

Conservation of energy

• SE max = ½ kx2 = ½ kYo2

• KE max = ½ mv2 = ½ m(Yo )2

• KE = SE

m

k

m

k

m

kf

2

1

Natural Frequency of a Single-mass System

Natural frequency in terms of spring deflection

Static deflection due to the weight = st

st

mgk

m

k

st

m

kf

2

1

st

gf

2

1

A

L

m

st

Sample Exercise Problem – Natural Frequency

13

A

A"10 AAt.sec

Natural frequency=?

• Use beam deflection equation • E = 10e6 psi

• The weight of beam is small compared to the weight of end mass.

EI

WLst

3

3

10

lbs

0.3

0.3

5.0

5.0

Key Features

Multiple Platforms

PC & Mobile Devices

iPhone Shock Mobile

iPhone Vibration Mobile

iPhone MOI Mobile

Comprehensive

9 modules and expanding

Thermal

Dynamics

Structural

For real engineering problems

Based on 100 + years experience

14

LuxCalc © Tools System

LuxCalc MOI and Beam Modules

15 Input table

Case selection area

Results table

Sample Exercise Problem - Answer

16

inEI

WLst

5

7

33

1065.9)453.3)(101(3

)10)(10(

3

Hzg

fst

47.3181065.9

4.386

2

1

2

15

I = 3.453 in4

st = 9.65 X 10-5 in

Multiple Spring System

Springs in Parallel

K1 K2

21 KKKeq

Springs in Series

21

111

KKKeq

K1

K2

Springs in series with fixed ends

21 KKKeq

K1

K2

m1

Frequency, Acceleration, Displacement

Displacement, velocity and acceleration, may be expressed

in sinusoidal functions as:

• Y = Yo sint

• v = Yo cost

• a = -Yo 2 sint

= 2f

m

k

vmax= Yo

amax = Yo 2

386

4 0

22

max Yf

g

aG

(Yo in inch, G in number of g’s)

8.9

0

2YfG

20

8.9

f

GY Dynamic displacement

Acceleration response

SDOF Forced Vibration with Damping

k c

x

tPo cos

tPkxxcxm o cos

2/1

222

2

)2()1(

)2(1

RR

RQ

2/1

2

2

)2(

)2(1

Q

2

1Q

n

R

cc

c

At resonance Light damping

m

Random vs. Sinusoidal Vibrations

Characteristics of random vibration

• Non-periodic

• Predicts probability of occurrence of various acceleration &

displacement magnitudes

Difference between sinusoidal and random vibrations

• Random vibration:

All of the frequencies within a given bandwidth are present all of

the time.

• Sinusoidal vibration

Each frequency excited individually.

No coupling of modes

1m

2m

vibration

Random Vibration Input Curves

Log-Log plot:

• Power spectral density P (acceleration squared per hertz) plotted

along the vertical axis

• The frequency plotted along the horizontal axis

Input RMS acceleration levels by integrating under random vibration

curve

RMSGGHzHz

Garea 2

2

f

GP

f

2

0lim

G : RMS of the acceleration expressed in gravity units

f : narrow bandwidth of the frequency range expressed in hertz

1f 2

f

P

Response of a SDOF System

k c

mdfPG

f

f outout 2

1

2inout

PQP 2

222

2

//2/1

1

ncnffccff

Q

)/(8

22

c

inn

outcc

PfG

Qcc c

2

1/ For lightly damped system

QfPGninout

2

Response of a SDOF system to a white noise is derived as:

For 1, Miner’s equation

1f 2

f

P

Q for forced vibration

Multiple DOF System

Multi-degree-of-freedom systems

2

iiiout QfPG

f

f

Q

For 1 Input PSD

Response PSD

:area under the curve

Exercise – SDOF Random Vibration

24

Random vibration input spectra: 0.1 g2/Hz from 10 to 2000 Hz.

a. Determine the chassis stiffness K.

b. Determine the input grms (Gin) and response grms (Gout) of chassis.

c. Determine Q of Chassis response.

1f 2

f

P30 lbs Chassis

Chassis resonance was found to be 100 Hz at 5% damping.

Exercise - LuxCalc Tools – Input View

25

inlbk /71.30682)1002(386

30 2 Spring constant for Chassis

Damping

K and m

Integration points,

30000

PSD input Random vibration

26

Exercise – grms View

Gout = 12.556 grms

rms

in

g

areaG

071.14

)102000(1.0

rms

ninout

g

QfPG

533.12

)10)(100)(1.0(2

2

Gin = 14.107 grms

27

Exercise – PSD View

Q = (10.125/0.1)1/2

= 10.06

0.10

)05.0(2

1

2

1

Q

Homework 2.1 Vibration

28

Random vibration input spectra: 0.1 g2/Hz from 10 to 2000 Hz.

a. Determine the PCB and chassis stiffness K1 and K2.

b. Determine the input grms (Gin) and response grms (Gout) of PCB and chassis.

c. Determine Q of PCB and Chassis.

1f 2

f

P

1 lb

30 lbs

PCB

Chassis

PCB resonance at 200 Hz and Chassis resonance at 100 Hz at 5% damping.

Luxea Inc. / Dunamis Tech Inc.

29 The presentation material is a proprietary property of Luxea & Dunamis Inc. Contact the company for appropriate distribution.

Shock & Vibration: Case Study

Shock Environment Specifications

Shock types:

• Pulse shock

• Velocity shock (drop shock)

• Shock response spectrum

Pulse shock: MIL-E-5400, MIL-STD-810, MIL-T-5422

Velocity shock (drop shock)

• Drop shock (falling package),

• Hammer shock (sudden velocity to the specimen)

Shock response spectrum

• Shock specifications based on the structure’s expected response

to shock input as a function of frequency.

Pulse Shock

Pulse shocks do not represent the real environment.

Effective in revealing the weak area

½ sine pulse is the most common form.

Gin

p

p

pf2

1

p = 11 ms

fp = 45.5 Hz

SDOF Response to Half Sine Pulse

Response of a single DOF system to a half sine pulse input

Closed-form solution

c

m

k

Shock attenuation area

pulsen ff /

Am

plif

ication

A=

Go

ut/G

in

Shock isolator selection criteria depends on

fp

p

n

ff

Shock amplification area

m

kfn

2

1

fn/fp ~0.6

Half Sine and Saw-tooth Pulses

Comparison of responses to half sine pulse and the saw-tooth pulse

• 30g ½ sine vs. 40g saw-tooth pulse

• For the same Gin, the sine pulse causes higher response.

0

10

20

30

40

50

60

0 50 100 150 200 250

G r

esp

on

se

natural frequency (Hz)

Comparison of shock pulses

30 g sine

40g sawtooth

30 g input sine

40 g input saw-tooth

Pulse Response Comparison (SDOF)

0 1 2 3 4 5 6 7 8 9 10

Frequency ratio, f/fp

Am

pli

fica

tio

n, G

ou

t/G

in

Approximate max amplification

~1.8

~2.0

~1.4

~2.0

Exercise – SDOF Pulse Shock

35

Exposed to 11 ms, 20 g, ½ sine shock.

a. Calculate the pulse frequency

b. Determine the maximum acceleration transmitted to the chassis.

1f 2

f

P30 lbs Chassis

Chassis resonance was found to be 100 Hz at 5% damping.

Exercise - LuxCalc Tools Shock Input View

36

inlbk /71.30682)1002(386

30 2 Spring constant for Chassis

Damping

K and m

Shock pulse data

Pulse shock

Observation time

37

Exercise – Acceleration View

Peak acceleration = 31.83 g

Frequency ratio, f/fp = 2.2

Amplification, Gout/Gin ~1.6

Gout ~ 32 g

0 1 2 3 4 5 6 7 8 9 10

Homework 2.2 Shock

38

1f 2

f

P

1 lb

30 lbs

PCB

Chassis

Exposed to 11 ms, 20 g, ½ sine shock.

Determine the maximum acceleration responses of the PCB and chassis.

PCB resonance at 200 Hz and Chassis resonance at 100 Hz at 5% damping

Luxea Inc. / Dunamis Tech Inc.

39 The presentation material is a proprietary property of Luxea & Dunamis Inc. Contact the company for appropriate distribution.

Shock & Vibration: Case Study

Case Study I

Transportation Random Vibrations

NG package responses to various MIL-STD-801F transportation

random vibrations are compared.

Random vibration per MIL-STD-

810F transportations

Package had been qualified for

truck transportation

Alternative transportation via C5

& C130 aircraft

Use of LuxCalc Vibration for

comparisons

Various input spectra result in

surprising results

41