shock & vibration: case study - luxeashock & vibration: case study for northrop grumman...
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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