applications of bode plots - university of minnesota · applications of bode plots ... transfer...
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Rajesh Rajamani
ME 3281 Department of Mechanical Engineering
University Of Minnesota
Applications of Bode Plots
FREQUENCY RESPONSE
Assume the transfer function G(s) is
asymptotically stable
)(sG)sin()( ttx
)sin()( tAty
|)(| jGA
)( jG
Answer
?A
?
2
BODE PLOTS
ssG
1)( ssG )(
1|)(| jG
ojG 90)0/(tan)( 1
|)(| jGojG 90)(
BODE PLOTS
Relative degree of transfer function
= order of denominator - order of numerator
ssG
1)( Relative degree = 1
Transfer function rolls off at - 20 dB/dec
1
1)(
TssG Relative degree = 1
Transfer function rolls off at - 20 dB/dec
Relative degree = 2
Transfer function rolls off at - 40 dB/dec
22
2
2)(
nn
n
sssG
3
BODE PLOTS
1
1
|)(|
22
T
jG
)(tan
)(
1
T
jG
1
1)(
TssG
BODE PLOTS
1)( TssG
1|)(| 22 TjG
)(tan)( 1 TjG
4
BODE PLOTS
22
2
2)(
nn
n
sssG
2222
2
2
|)(|
nn
njG
)2
(tan)(22
1
n
njG
APPLICATIONS OF BODE PLOTS
Experimentally determining the dynamic model for a system
Design of vibration isolation mounts
Design of sensors
Design of actuators
Signal processing filters
Control system design
Countless other applications …….
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SYSTEM IDENTIFICATION
Measuring the dynamic model of a system experimentally is called system identification
Such a measurement can be done in the frequency domain
Transfer function relation between input and output
If is a sinusoid at any one frequency
Find the Bode magnitude plot of by using the above relation and having sinusoidal inputs, one frequency at a time
)()()( sXsGsY
x
yjG
of amplitude
of amplitude)(
)(tx
)( jG
SYSTEM IDENTIFICATION
|)(| jG
)(sG)(sV )(s
Example: The following data shows experimentally measured data on the response of a DC servo motor
What is the dynamic model of the servo motor?
x x
x x
x
x
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SYSTEM IDENTIFICATION
m
K
1
|)(| jG
10
-20 dB/dec
-40
dB/dec
1sTs
K
m
)(sV )(s
First, determine the form of the transfer function that matches the experimental data
Next, find gain and corner frequency
VIBRATION ISOLATION
Designing vibration isolation mounts for a machine
k c
x
m
F
Objective
• Motion of the machine
should be less than 1 mm
from equilibrium
• At 25 Hz, less than 250 N
of force should be
transmitted to the structure Structure
Machine
F - mean 5000 N and sinusoidal 1000 N at 25Hz
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VIBRATION ISOLATION
Designing vibration isolation mounts for a machine
k c
x
m
F
Fkxxcxm
kcsmssF
sXsG
2
1
)(
)()(
kxxcFt
kcsms
kcs
sF
sFsH t
2)(
)()(
Motion of machine
Force transmitted to structure
VIBRATION ISOLATION
k c
x
m
F
Fkxxcxm
22
2
22
11
)(
)()(
nn
n
sskkcsmssF
sXsG
Motion of machine
10-1
100
101
102
10-5
10-4
10-3
10-2
10-1
100
101
de
flecti
on a
mp
litu
de (
m)
frequency (Hz)
Frequency response of deflection for a 5000 N force
k=16000
k=100,000k=5,000,000
|)(|5000 jG
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VIBRATION ISOLATION
22
2
2
2
1
)(
)()(
nn
n
t
ss
sk
c
kcsms
kcs
sF
sFsH
Force Transmitted
10-1
100
101
102
100
101
102
103
104
105
tra
ns
mit
ted
fo
rce
a
mp
litu
de
(N
)
frequency (Hz)
Frequency response of transmitted force for a 1000 N excitation
k=16000
k=100,000k=5,000,000
|)(|1000 jH
VIBRATION ISOLATION
Designing vibration isolation mounts for a machine
k c
x
m
F
Conclusions • A very hard spring can
restrain motion to be
less than 1 mm
• But a hard spring
increases the high
frequency forces
transmitted to the
structure Structure
Machine
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SENSOR DESIGN
Design an accelerometer to measure inertial acceleration
k c
x
m
Objective
• Estimate acceleration of
the casing:
• Use the physically
measured variable:
– Relative position
Proof mass
y
yA
yxz
k c
x
m Proof mass
y
22
2
2
2
2
2)(
)(
)(
)()(
nnss
s
sY
sZ
kcsms
ms
1- kcsms
kcs
sY
sZsH
10-2
100
102
104
10-10
10-9
10-8
10-7
10-6
frequency (Hz)
magnitude
22 2
1
)(
)(
nnss
sA
sZ
SENSOR DESIGN
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SENSOR DESIGN Fabricated two-axes MEMS device
IMPORTANCE OF BODE PLOTS
Experimentally determining the dynamic model for a system
Design of vibration isolation mounts
Design of sensors
Design of actuators
Signal processing filters
Control system design
Countless other applications …….