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Lecture 1: Single Degree of
Freedom Systems (III)
Prof. Kuo-Shen Chen
Department of Mechanical EngineeringNational Cheng-Kung University
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Outline
Rotordynamics, an introduction
Impulse Responses
Arbitrary excitation
Transfer function and Laplace Domain
Shock Isolation
Simple problems
Youtube Demos
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Part I: Rotordynamics: an Introduction
Rotor phenomenon
Unbalance
Campbell diagrams
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Rotordynamics
A specialized branch concerned with the behavior
and diagnosis of rotating structures.
commonly used to analyze the behavior of structures
ranging fromjet engines and steam turbines to auto
engines and computer disk storage.
Vibration, noise, bearing damages
Key issues to be introduced
Critical speed
Whirling
Campbell diagram
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Jeffcott Rotors
A single disk mounted on a flexible and
massless shaft with rigid bearings
Serves as the fundamental model for
studying rotor phenomena
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Whirling
Rotation of bent shaft
: rotating speed, : whirling speed
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Critical Speeds
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Whirling: Mathematics
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Whirling Behavior
Synchronous whirling:
As rotating speed passes the critical speed, the imbalance
location would actually move toward the center
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Campbell Diagram
known as "Whirl Speed
Map" or a "Frequency
Interference Diagram
Basic concept
Natural frequencies
depends on rotating
speed
Resonance occurs as
natural frequencies hit
rotating speed
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Turbine Example
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Dynamic Balancing
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Responses Under a General Force (I)
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Responses Under a General Force (II)
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Response Under a Periodic Force with
Irregular Form
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Part III: Vibration Subject to Arbitrary
Excitations
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Introduction
It is important to evaluate the response to a
general, non-periodic input
No, exact analytical solutions available
However, the task can be performed by either
Convolution approach
Fourier transform or Laplace transform approach
Transfer function
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Impulse Response
The response of a vibration system subjected
to a unit impulse input
Impulse function
0
0
0
0 t t
( ) t=tt t
0( ) 1
t t dt
2
F
- +
is a small positive number
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Effect on spring-mass-damper ?
Ft mv
impulse=momentum change
m[v( t0) v(t0
)]
?F mv0 v0?F
m
Ft
m
Just after
impulse
Just before
impulse
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For an underdamped system
x(t)?Fent
md
sindt (response with zero I.C.)
x(t) ?F h(t), where h(t)e
nt
md
sindt
impulse response function
0 10 20 30 40-1
-0.5
0
0.5
1
Time
h(t)
m
k
x(t)
c
F
h(t )1
mn
sinn(t)
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h(t )
0 te
n ( t)
md
sind(t ) t
for the case that the impulse occurs at
note that the effects of non - zero initial conditions
and other forcing terms must be super imposed on
this solution
0 10 20 30 40-1
0
1
h1
0 10 20 30 40-1
0
1
h2
0 10 20 30 40-1
0
1
Time
h1+h2
=0
=10
For example: If two
pulses occur at two
different times then
their impulseresponses will
superimpose
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Impulse Response of a
Vibration System (I)
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Impulse Response
of a Vibration
System (II)
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Convolution Integrals: Introduction
Decompose a general input to the
combination of a series of impulse train
Superimpose the impulse response to form
the final response The method itself can be treated as an
analytical relation
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Convolution Integral (I)
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Convolution Integral (II)
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Linear Superposition Method
A complicate input can be decomposed into a
few simple input with
scaling, multiplexing, and time shift operations
The response are then becomes the linear
superposition of the simple output after these
linear operations
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Example: Pulse Input
0
2
1( ) 1 cos( )
1
nt
d
Fx t e t
k
0( )00
2
1( ) 1 cos( ( ) )
1
n t t
d
Fx t e t t
k
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Time Shift Operation
E.g., for the pulse input
The total response = x1 + x2
Where
X1
x2
0
2
1( ) 1 cos( )
1
nt
d
Fx t e t
k
0( )00
2
1( ) 1 cos( ( ) )
1
n t t
d
Fx t e t t
k
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Part IV: Transfer Function and
Laplace Transform
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Transfer Functions
A black box approach to correlate the input-
output relation
Based on linear superposition principle
Usually performed by Laplace Transform
approach
Or Fourier Transform
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Laplace Transform
Changes ODE into algebraic equation
Solve algebraic equation then compute the
inverse transform
Rule and table based in many cases
Is used extensively in control analysis to
examine the response
Related to the frequency response function
dtetxtxsX st
0
)())(()( L
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02
0
s
sFsXkcsms
tFkxxcxm
)()(
cos
Take the transform of the equation of motion:
Now solve algebraic equation in s for X(s)
))((
)(222
0
skcsms
sFsX
To get the time response this must be inverse
transformed
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Laplace Transform: Fundamentals (I)
Consider a SDOF vibration system
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Laplace Transform: Fundamentals (II)
Transfer Functions: Z(s) Impedance; Y(s): Admittance
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Typical
Laplace
Transforms
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Propertiesof Laplace
Transform
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Example: Rao Ex. 4.16
Find the response
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where
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Part V: Response Spectrum
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Introduction
Shock
A sudden application of a force input to a SDOF
system to result a transient response
The maximum value of the response can be used
to measure the shcok sensitivity
Response spectrum is a plot of the maximum
peak response of the SDOF oscillator as a
function of natural frequency
Different shock inputs result in different response
spectra
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Typical Response Spectra
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General Response Spectra (1)
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Part VI: Simple Problems
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Problem 1. Periodic Vibration of a
Hydraulic Valve (Rao 4.1)
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Problem 2. Response of a Structure under
Impact (Rao 4.4, 4.5)
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Problem 3. Step Force of a
Compacting Machine (Rao 4.6)
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Problem 4. Blast Load on a Building
Frame (Rao 4.10)
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Problem 5: Water Tank Subjected to
Base Acceleration (Rao 4.12)
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Problem 6. Response of a Building
Frame to an Earthquake (Rao. 4.13)
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Part VII: Youtube Demonstrations
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