Lecture 11-2
Frequency Domain Analysis II
Control System Design in Frequency Response
1. large fast time response
2. : function of damping ratio large large
3. good damping characteristics
4. minimum effect of any undesirable noise
m
,
pM M
M
m M
p
m
1 M
m 1.4
1.4 1.0
m
( ) M
Experimental Determination of Transfer Function
( )
r t Unknown y t ( )
System
( ) sin
i; ( ) (
i) sin
ir t t y t M t
some frequency measure steady state
i
M
i
M
( ) ( )
G s G j
M G j
Transfer Functions and Frequency Magnitude Plot
Bode Plot (Logarithmic Plots) i) Log Magnitude
ii) dB : decibel, Logarithm of the magnitude iii) 1 decade : frequency width
Drawing the Bode Plots
The reason of using logarithm - mathematical operation
- typical three types : simple straight line asymtotic
approximations
( ) 20 log ( ) L G j
m G j dB
1 ~ 10 2.5 ~ 25
Hz Hz
Hz Hz
, ,
1 2
2 2
3
1 2
2 3
1 1
( ) 1 1
1 1
2 1 1
) 20 log
1 1
) 20 log 20 log
m m m m
m m m
m
m
K sT sT
G s s sT as bs
L G j L K L j T L j T
L j L j T L aj b j
i L K K dB
ii L j j
Transfer Functions and Frequency Magnitude Plot
0.1 1 10
straight line
straight line
With negative slope -20dB/decade
log
log ( )
M
( ) M
ex)
2 2
1 2 2
1
1
) 1
1 20 log 1 20 log 1
1 1 20 log1 0
1 1 20 log
20 log 20 log ) 1
1
1 20 log 1 20 log 1
1 1 20 log1 0
1 1 20 log
20 log
m
m m
m
m
m
iii j T
L j T j T T
T L j T dB
T L j T T
T iv j T
L j T j T T
T L j T dB
T L j T T
20 log T
log ( )
M
( ) M
1/T
1/T
Transfer Functions and Frequency Magnitude Plot
20dB decade/
20dB decade/
log
2 2
2 2
2 1/ 2 2
2
2 2
2 2
2 2
1 1
)
2 1 2 1
1 1
1 1 2
20 log 20 log 1
2 1 2 1
1 1
1
( ) 20 log1 0 1
( ) 20 log 4
n n n n
m
n n
n n n n
n
m
n
m
n
v
s s j j
L
j j j j
L G j dB
L G j
0 log
n
Transfer Functions and Frequency Magnitude Plot
( ) M
log
( )
M
1
( ) M ( )
M
1
1 20 logK
1 20 logK
20dB decade/
20dB decade/
1 1
1 1 1
K K
j j s s
Transfer Functions and Frequency Magnitude Plot
20dB decade/
40dB decade/
ex)
Vibration Isolation in Rotating Systems
Vibration due to rotating unbalance
2sin mr t
2
2
(1) ( )
sin
( ) 1
( ) ( )
M x bx k x p t
mr t
X s G s
P s Ms bs k
2
2
1 1
2 2 2
2 2 2 2
2 2
2 2
(2)
( ) 1
( ) ( )
( ) sin( ) sin( )
2 /
tan tan
1 /
1
1/
1 / 2 /
n n
n n
x j G j
p j M bj k
x t X t G j t mr
b k M G j
k M b
k
Frequency response
Vibration Isolation in Rotating Systems
small
n
range
1 k ( ) M
Vibration Isolation in Rotating Systems
Vibration isolators
system isolator
system isolator
Machine Suspensions
Reduce the magnitude of force transmitted from a machine to its foundation
Reduce the magnitude of motion transmitted from a vibratory foundation to a system
Isolator i) - load supporting elements, spring.
- energy-dissipating elements, damper.
ii) - Synthetic rubber : both load supporting and energy dissipating
Force Force
Motion
Motion
Transmissibility :
A measure of the reduction of transmitted force or motion afforded by an isolator.
( )
( )
the force amplitude transmitted to the foundation TR machine
the amplitude of the exciting force the vibration amplitude of the system TR suspensions
the vibration amplitude of the foundation
Transmissibility
Transmissibility for Force Excitation
2
0
2
2
2 2
( ) sin sin
( ) sin
( ) 1 ( )
( ) ( ) ( ) ( )
/ /
( )
( ) / /
t
t
t
p t mr t F t
f t bx k x F t
X s P s
Ms bs k
F s bs k X s bs k P s Ms bs k
b M j k M
F j bj k
p j M bj k b M j k M
Frequency response
Transmissibility for Force Excitation
2
2 2
0
2
2 2
2 2
2
2 2
2
1 2 /
( )
/ , / 2
( ) 1 / 2 /
( ) ( )
1 2 /
1 / 2 /
1 2
1 2
, 2 , 1
t n
n n
n n
t t
n
n n
F j j
k M b M
p j j
F F j TR F p j
Note TR
for any
n
2
small
1 TR
small
spring&damper
Simplified model
sin p P t ( )
x t
2 2
2 2 2
2 2 2 2
2
2 2
2
( ) ( ) 0
( ) ( )
( ) , ( )
( ) ( )
1 2
1 2
mx b x p k x p
X s bs k x j bj k
P s ms bs k p j m bj k
x j b k
TR p j k m b
Automobile Suspension Systems
If b is small and
Then resonance - excessive vibration
- extremely large force transmitted
2
2 2 2 2
2 2 2 2
P mr
mr b k
TR
k m b
x m
n
X j
1
/
k m
is very close to , critical speed,
=> A dynamic vibration absorber
nDynamic Vibration Absorbers
x y
( ) sin p t P
t maM
2
2
2
2 2 2
( )
( ) ( ) ( )
( ) ( ) 0
( ) ( )
a
a a
a a
a a a
a a
a a a a
M x kx bx k x y p t m y k y x
Ms bs k k X s k Y s P s m s k Y s k X s
m s k X s
P s Ms bs k k m s k k
k
aLaplace Transform
Frequency response
2
2 2 2
( ) ( )
( )
a a
a a a a
m k
X j for small b
P j M k k m k k
f t kx bx kx
Dynamic Vibration Absorbers
The transmitted force
Dynamic Vibration Absorbers
2
2 2 2
2 2
2 2 2
2
( ) ( )
, 0
( ) 0, ( ) 0
a a
a a a a
a a
a a a a
a a
a a
k m
X j P j
M k k k m k
mr k m
M k k k m k
k m
k m
X j f t
If we choose so that,
then, at this frequency.
If
we use dynamic vibration absorbers.
exc n
n 2
1
X j
Physically, f ta( ) p t( ) Psin
t
( ) ( ), , 0
( )
a a a exc n
a
f t k x y k y p t at x
k y p t
Spring force cancels .
2
2
22
( ) ( )
, 0
a
a a a a
a a a a
k Y j
P j M k k m k k
m k k m
If we choose so that,
then magnitude ratio
( ) 1
( )
aY j
P j k
1 k
a
( ) sin
( ) 1 sin 180 sin 180
a a
p t t
y t P t P t phase
k k
Dynamic Vibration Absorbers
M
Seismograph
x
y
z x y
A device used to measure ground displacement during earthquakes
0
mx k y x b y x z x y
m y z bz kz mz bz kz my
Equation of motion : measurement :
2 2
2 2 2
2
: ( )
( ) 2
, 2
n n
n n
Z s ms s
Transfer Function
Y s ms bs k s s
k b
m m
2 2
2 2 2
( )
( ) 2 1 2
( ) ( ) ( )
1, 1
( )
n
n n
Z j
Y j j j
z t y t Z j Y j
We hope, If,
Z Y 1
n
( ) ( )
nn
n
z t y t for
k m
Choose as small as possible.
(large mass, soft spring)
Seismograph
Accelerometer
z x y
The system configuration is basically the same as the seismograph, butchoice of undamped natural frequency is different.
mz bz kz my
Equation of motion : the same as the seismograph,
2 2 2 2
2
( ) 1
: ( ) 2
( ) 1
( )
n n
n
n
Z s m
Transfer Function
s Y s ms bs k s s
Z s s Y s
Accelerometer
y
n 2
1
n
operating range Frequency response
End of Lecture 11-2
