Transcript Active Vibration Control for a Cantilevered Beam

Embedded Control of
Smart Structures
Mid-Summer Presentation By:
Alicia Vaden (Tennessee Technological University)
Graduate Advisor:
Tao Tao
Faculty Advisor:
Ken Frampton, PhD
Objectives
 Model and analyze the
vibration response of a
cantilevered beam.
 Design a smart materialbased controller that will
reduce the vibrations of
the beam.
Piezoelectric Patch
Implementation
 Theory
of project

Building a smart structure

Matlab/Simulink with dSPACE

Initial Displacement Disturbance

White Noise Disturbance
Smart Structures
 Ordinary structure with sensory network
 Sensors
 Actuators
 Smart Materials
Piezoelectri
Material
 Aluminum Alloy 2024-T4
 Cost efficient
 Modulus of elasticity
PL3
v ( L)  
3EI
E = 73.1 GPa
 Exaggerated movements
Vibration
 A rapid linear motion of a particle about an
equilibrium position
 Undesirable
 Stress
 Energy loss
 Reduce vibration!
Underdamped System
 Given initial displacement or force,
oscillates until transition state is reached
 Damping ratio, ζ , represents the amount
of damping in a system
0<ζ<1
Modeling Beam
 Many different equations and strategies
 Transfer Function
 Natural Frequency, ωn
 Damping ratio, ζ
n
G 2
2
s  2 n s  n
2
Modeling Beam from Graph
n
G 2
2
s  2 n s  n
2
• From the graph of the
sinusoid, the period,
T,
can be found.
T = 0.14 seconds
With period now known,
the natural frequency
1
f 
can be calculated T
Exponential Decay
c(t )  e
nt
sin(nt   )
Exponential Decay and ζ
 Equation of vibration ignoring sine
a(t )  c1e
nt
 Natural frequency, ωn , never changes
 To increase exponential decay, increase ζ
Modeled Beam
 With ωn and ζ known, the
equation can just be plugged in.
2 x 103
G( s)  2
s  1.5s  (2 x 103 )
Implementation

Theory of project
 Building
a smart structure

Matlab/Simulink with dSPACE

Initial Displacement Disturbance

White Noise Disturbance
Building Smart Structure
1.
2.
3.
4.
5.
Cut beam to desired length (24”)
Clean attaching area with ethyl alcohol
Use epoxy for insulating area, let cure
Attach copper tape to negative side
Use epoxy to attach piezoelectric, keeping
edges down
6. Remove excess tape and epoxy
7. Solder wires to respective side
Implementation

Theory of project

Building a smart structure
 Matlab/Simulink
with dSPACE

Initial Displacement Disturbance

White Noise Disturbance
dSPACE with Matlab
 Using a block diagram designed in
Simulink, code is generated
 Real-Time Workshop in Matlab works in
conjunction with the Real-Time Interface in
dSPACE
 C code is generated that will run outside of
dSPACE through the physical model
 RTI works as interface between Matlab and
dSPACE
Implementation

Theory of project

Building a smart structure

Matlab/Simulink with dSPACE
 Initial

Displacement Disturbance
White Noise Disturbance
System Flow Diagram
reference
Feedback
+
-
DAC
Low
Pass
Filter
ADC
Power
Amp
Low
Pass
Filter
acceleration
Piezo
Beam
Sensor
1. Sensor - Accelerometer
 Detects acceleration of beam
 Outputs voltage
 More sensitive than piezoelectric patch
1
2. Signal Conditioner
 Provide power to accelerometer
 Pre-amplify signal from accelerometer
1
2
3. Butterworth Low-Pass Filter
 Filters out outside noise
 Cleaner graph
 Easier to read and more accurate results
 Cut-off frequency of 100 Hz
C1
R1
R2
(dc coupled)
+
C2
1
-
(K-1)R
R
2
3
4. AD17 and Oscilloscope Input 1
 With noise filtered out, sample sent into
dSPACE
 Runs through gain
 Splits to oscilloscope
 View input voltage going into dSPACE
 Voltage cannot exceed ±10 Volts
1
2
3
4
5. Feedback Gain and Oscilloscope
Input 2
 Multiplies sensor signal by gain
 During loops, vibrations signal begins to decay
 Splits to oscilloscope
 Monitor output voltage
 Voltage cannot exceed ±10 Volts
1
2
3
4
5
6. Low Pass Filter/Power Amplifier
 Filter dSPACE output through low-pass filter
 Amplifies voltage from low-pass filter
 Since dSPACE voltage cannot exceed ±10 Volts
then signal must be amplified to increase voltage
to piezoelectric
 Amplifier calibrated to 15 Volts
1
2
3
4
6
5
7. Actuator – Piezoelectric Patch
 Voltage is applied to piezoelectric
 Piezoelectric will bend or contract to apply
control force to beam
 Reduces beam vibrations
1
2
3
7
4
6
5
Closed Loop Feedback
 The loop has reached its beginning, thus the
process begins again until the beam has
stopped vibrating
1
2
Closed Loop
3
7 Feedback
Control
System
4
6
5
Results
 Tested beam with gains
 Positive feedback gains made the beam
unstable
 Negative feedback gains proved to be the
best control thus far
 Higher gains proved to be most effective
Results
 Beam vibration without controller
ζ = 0.0168
Results
 Beam vibration with -3 feedback gain applied,
followed by 15 Volt amplification
ζ = 0.05
Results
 Beam vibration with - 3 feedback gain applied,
followed by 15 Volt amplification
Aqua sine wave
decays considerably
faster
With gain applied,
beam vibration is
controlled
Implementation

Theory of project

Building a smart structure

Matlab/Simulink with dSPACE

Initial Displacement Disturbance
 White
Noise Disturbance
System Flow Diagram
White Noise
(disturbance)
reference
Feedback
+
-
DAC
Low
Pass
Filter
ADC
Power
Amp
Low
Pass
Filter
acceleration
Piezo
Beam
Sensor
White Noise
 Before disturbance depended on displacement
 White noise is produced by a stimulus containing all
frequencies of vibration.
 frequency range limited from 0 to 100 Hertz
 magnitude will be ± 10 Volts.
Beam
 Extra piece of piezoelectric added to side
completely opposite of actuator.
 All previous connections remain the same
 New connections
 Extra output from dSPACE (white noise)
 dSPACE low-pass filter power amplifier beam
Simulink Diagram
??? Voltage-Time domain ???
 -3 gain in Simulink
 White noise emits random frequencies
 Very difficult to determine damping when
viewing from time domain
Frequency Domain
 Power Spectral Density Function
 describes how the power of a time series is
distributed with frequency
 Mathematically, it is defined as the Fourier
Transform of a successive correlation of the time
series
 psd command in Matlab

1 

f ( ) 
  0  2  (k ) cos(k ) 
2 
k 1

Frequency Domain -3 Gain
 Multiple peaks??
 Different modes
 More than one ωn in beam
 2nd order modeling is not most accurate
approach
Power Spectral Density Estimate via Welch
Without Controller
With Controller
-50
Power/frequency (dB/Hz)
-60
-70
-80
-90
-100
40
50
60
70
80
90
Frequency (Hz)
100
110
120
Frequency Domain -3 Gain
 First Peak
 Very little damping
 Controller too weak
Power Spectral Density Estimate via Welch
Without Controller
With Controller
-50
Power/frequency (dB/Hz)
-60
-70
-80
-90
-100
40
50
60
70
80
90
Frequency (Hz)
100
110
120
Increased Power Amplifier to 15V
 Dampened white noise by 20 dB
 Voltage emitted to piezo approximately 100 V
 Piezo is designed to accept the voltage
 Repeated high voltage can cause damage
Power Spectral Density Estimate via Welch
-50
Without Controller
With Controller
-55
-60
Power/frequency (dB/Hz)
-65
-70
-75
-80
-85
-90
-95
20
30
40
50
60
70
80
Frequency (Hz)
90
100
110
120
Change Conditions of Beam
 Shortening beam by
2 inches changes
results drastically
 Placement and
boundary conditions
 Proportional
feedback gain
strategy not good
enough for white
noise
-45
-50
-55
Power/frequency (dB/Hz)
 Dampens first peak,
but excites second
Power Spectral Density Estimate via Welch
-60
-65
-70
-75
-80
Without Controller
-85
With Controller
30
40
50
60
70
Frequency (Hz)
80
90
100
Future Work
 System ID of beam
 Design more complicated controller
 Bode Plots
 Root-Locus Method
 Design new circuit




Cut down on wires involved in experiment
Use BNC cables instead of Alligator clips
Less of a hassle
Neater workspace
Questions???
Thank you!!
Embedded Control of
Smart Structures
Mid-Summer Presentation By:
Alicia Vaden (Tennessee Technological University)
Graduate Advisor:
Tao Tao
Faculty Advisor:
Ken Frampton, PhD