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ADVANCED VIBRATIONS AND ADVANCED VIBRATIONS AND NOISE ENGINEERING 1
ADVANCED VIBRATIONS ANDADVANCED VIBRATIONS AND NOISE ENGINEERING
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Vibration Engineeringb at o g ee g
T iTopics:• Introduction to Vibration• Practical Examples• Fundamentals• Mathematical System Modeling• Analysis of Vibrating System• Fourier Analysis• Application
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Introduction to Vibrations
Vibrations are present everywhere in life ------ Atomic vibrations (temperature)
Human body (heart)
Machines (large and small)
E th k tEarth quake tremors
Musical instruments …….
So we live in a world of vibrationsSo we live in a world of vibrations….
Let us , from now onwards look at the world from this view point
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Practical Examples of Vibrations
Everyday examples where vibrations are involved :
Cars BikesCars Bikes
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Practical Examples of Vibrations
5Heavy machinery ( turbines and airplane engines )
Galileo Galilei Italian mathematician his great work in dynamics
Practical Examples of Vibrations
Galileo Galilei ----- Italian mathematician ---- his great work in dynamics
SIMPLE PENDULUMSIMPLE PENDULUM
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Vibration Engineeringb at o g ee g
T iTopics:• Introduction to Vibration• Practical Examples• Fundamentals• Mathematical System Modeling• Analysis of Vibrating System• Fourier Analysis• Application
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Fundamentals of Vibrations
Vibration Analysis:
PHYSICALDYNAMICSYSTEM
MATHEMATICALMODEL
GOVERNINGEQUATIONS
SOLVE GOVERNINGEQUATIONS
INTERPRETERESULTS
Example:
8FBD
Fundamentals of Vibrations
Single degree of freedom (DOF)
Two DOFModeling of the systems
Two DOF
Multi DOF
Continuous system
Each system can be under
Free Damped Forced
9Or a combination of these modes
DOF:
Fundamentals of Vibrations
DOF:In vibrations DOF refers to the number of independent co-ordinates with which we can completely define the system at any point of timewhich we can completely define the system at any point of time
Select DOF for any given system considering:Select DOF for any given system considering:
•Simplicity of analysis
•Validation with real life approximations•Validation with real life approximations
Let us see how this works ----
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Vibration Engineeringb at o g ee g
T iTopics:• Introduction to Vibration• Practical Examples• Fundamentals• Mathematical System Modeling• Analysis of Vibrating System• Fourier Analysis• Application
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SDOF reduction of bike rider system
Mathematical System Modeling
SDOF reduction of bike rider system
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Two DOF modeling of bike rider system
Mathematical System ModelingTwo DOF modeling of bike rider system
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Three (multi) DOF modeling of bike rider system
Mathematical System ModelingThree (multi) DOF modeling of bike rider system
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M d li f bik id
Mathematical System Modeling
More accurate modeling of bike rider system
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Other Examples of vibration systems
Single cylinder Internal Combustion Engine
16Equivalent model representation ( Torsional )
Vibration Engineeringb at o g ee g
T iTopics:• Introduction to Vibration• Practical Examples• Fundamentals• Mathematical System Modeling• Analysis of Vibrating System• Fourier Analysis• Application
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Analysis of vibrating systems
F ib iFree vibrations
When a body is set into vibrations and left to itself , it vibrates about a mean positionposition.
This is called free vibration . And the frequency of vibration is called Natural q yFrequency
Natural Frequency
18Best example
Analysis of vibrating systems
Forced vibrationForced vibration
A body acted upon by an external force of particular frequency responds at that frequency and hence the body under consideration is said to be inat that frequency and hence the body under consideration is said to be in forced vibration
FF
Forced vibrations are very common in mechanical systemssyste s
Eg:
C
Single degree of freedom system with forced vibration
Car
Turbine , turbine blades
Machine shafts19
vibrationMachine shafts
Even earthquake excitation
Analysis of vibrating systems
Vibrations are both useful as well as harmful…… just like coefficient of friction
We prefer to travel in a train than by an old bus…….. (vibrations)
Machines are not supposed to vibrate at more than particular level to prod ce eno gh acc racproduce enough accuracy
Life of the systems depends on the DYNAMICS of the machineLife of the systems depends on the DYNAMICS of the machine
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Analysis of vibrating systems
Simple harmonic motion (SHM)Simple harmonic motion (SHM)
Best example : Simple pendulum
Displacement equation : x Xcos(pt) p frequencyDisplacement equation : x = Xcos(pt) p – frequency
X – amplitude
The variation of displacement velocity and acceleration of a SHM system with time are given in the figure
2
cos( )sin( )
x Xp ptx Xp pt
′ = −
′′ = − sin( )x Xp pt=
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Analysis of vibrating systems
Combining two SHM ‘sCombining two SHM s
When we combine two SHM of almost same frequency are combined , a phenomenon called beats is observed p
1 1 2 2cos sin( )x X p t X p t ϕ= + +
P1=(2 πm)/T P2=(2 πn)/T 1p mp n
=2p n
The resultant signal oscillates with two frequencies
2π 2πand
222p
1pand
Analysis of vibrating systems
Beat Phenomenon
( )1 2cos cosx X p t p t= + p1 and p2 are nearly equal
1 2 1 22 cos cos2 2
p p p px X t t− +=
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Vibration Engineeringb at o g ee g
T iTopics:• Introduction to Vibration• Practical Examples• Fundamentals• Mathematical System Modeling• Analysis of Vibrating System• Fourier Analysis• Application
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Fourier Analysis
Most Practical Excitations are Periodic in NatureMost Practical Excitations are Periodic in Nature
• Unbalance excitation in reciprocating mechanism )2cos(cos2 trtmeFP ωωω +=engine
)(uP
• Gas torque in IC engine
• Displacement excitation in Cam- Follower system
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Fourier Analysis
Periodic Seismic Excitation
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Fourier Analysis
2
1.2
1
2
plitu
de =+
mpl
itude
0.6
0.8
1.0
0 40-1
0Am A
m
0.2
0.4
0.05
0.1
0.15
0.20
10
20
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Frequency
TimeFrequency (Hz)
0 10 20 30 40 50 600.0
Decomposition of time domain periodic signal in frequency domain
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Fourier Analysis
6
8
T
2
4
litud
e
0
2
Am
pl
0 0 05 0 1 0 15 0 2 0 25 0 3 0 35-4
-2
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35Time(sec)
1. To find different frequency components
282. Amplitudes of different components
0.5
1
1.5
e
RESPONSE ESTIMATION
1 10 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
-1
-0.5
0
Time (sec)
Am
plitu
de
Excitation function to systemf(t)=cos(2πft)+0.33 cos(2π(2f)t)
f=20Hzω=2πf
f2=2f=40Hz0 8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
Am
plitu
de
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
Am
plitu
de
f1=f=20Hzω1=2πf1
Harmonic components of excitation f ti f2 2f 40Hz
ω2=2πf20 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1-1
-0.8
Time (sec) 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1-1
-0.8
Time (sec)
SYSTEM SYSTEM
function
-0.5
0
0.5
1
1.5
Am
plitu
de
-0.2
0
0.2
0.4
0.6
0.8
1
Am
plitu
de
Response of the system to these harmonic
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1-1.5
-1
Time (sec) 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1-1
-0.8
-0.6
-0.4
Time (sec)+
components of excitation function
2
2 .5
C l t
-0 .5
0
0 .5
1
1 .5
Am
plitu
de
Complete response of the system
290 0 .0 1 0 .0 2 0 .0 3 0 .0 4 0 .0 5 0 .0 6 0 .0 7 0 .0 8 0 .0 9 0 .1
-1 .5
-1
T im e (s e c )
The break up of excitation frequency and response of the system to the excitation
Fourier Analysis
Frequency analysis to help detect possible resonance condition
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Fourier Analysis
F i S i
Any periodic function can be expressed as a summation of
Fourier Series
)sin()cos()( tbttf ∑∑∞∞
+
y p pinfinite sum of a series of pure sinusoids
)sin()cos()(10
tnbtnatfn
nn
n ωω ∑∑==
+=
where, the coefficients of the sinusoids are given by
ωπ /2
/2
∫=ωπ
πω /2
00 )(
2dttfa
31∫=ωπ
ωπω /2
0
)cos()( dttntfan and ∫=ωπ
ωπω /2
0
)sin()( dttntfbn
Fourier Analysis
)2i ()i ()( AAA
Amplitude variation of Fourier coefficients with number of terms (n)
.....)2sin(2
)sin(2
)( −−−= tAtAAtx ωπ
ωπ
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Vibration Engineeringb at o g ee g
T iTopics:• Introduction to Vibration• Practical Examples• Fundamentals• Mathematical System Modeling• Analysis of Vibrating System• Fourier Analysis• Application
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Applications
Modeling of such systems should be done as MDOF34
Modeling of such systems should be done as MDOF
Applications
Big and heavy penstock
B tt b d l dBetter be modeled a MDOF or a continuous system
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Development of Tuned Mass Damper
Applications
Development of Tuned Mass Damper
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Applications
Another important application of vibration analysis is signature analysiswhere
Di ti d h lth it iDiagnostics and health monitoring
Life estimation
Life extensionLife extension
Initial the structures are made robust and vibration data is recorded . After some time or after some effects( like earthquake on bridges) the new vibration data tells us about the changes in the structure
The changes are due to
cracks developed
t t t37
support movement etc
Applications
Health equipment (vibrator38
Health equipment (vibrator for massaging)
Forging HammerApplications
Forging HammerVehicle Dynamic Simulatory
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Assignment1
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Assignment
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Assignment
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