Post on 30-Apr-2020
ME242 Vibrations-Mechatronics Experiment
Daniel. S. StuttsAssociate Professor of
Mechanical Engineering and
Engineering Mechanics
Wednesday, September 16, 2009
2
Purpose of Experiment• Learn some basic concepts in vibrations and
mechatronics.• Gain hands-on experience with common
instrumentation used in the study of vibrations
• Gain experience in taking and reporting experimental results in written and verbal form
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Basic Concepts in Vibrations• Free vibration of a Single DOF system• Damping measurement via the logarithmic
decrement method and half-power method.• Natural frequencies and modes of a beam in
bending • Harmonic forcing via piezoceramic elements and
the steady-state response
4
Basic Concepts in Mechatronics
• Material properties and behavior of a piezoceramic, PZT (Lead Zirconate Titanate)
• Electro-mechanical coupling: Actuation and Sensing
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Instrumentation
• Signal generator
• Amplifier
• Accelerometer and conditioning circuitry
• Data acquisition computer
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Cantilevered Beam Schematic
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SDOF Oscillator
( )Mx Cx Kx f t+ + =&& &EOM:
2 ( )2 n nf tx x xM
ζω ω+ + =&& &Canonical form:
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Solution to free-vibration problem
22 0n nx x xζω ω+ + =&& &
( )2 2( ) cos 1 sin 1nt n nx t e A t B tζω ω ζ ω ζ−= − + −
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Example Plot of Decaying Motion
21 10d nω ω ζ= − =
0.2nζω =
0B =(Sine term set tozero)
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Harmonic Forcing: Effect of Damping Near Resonance
0( ) sinf t F tω=
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Half-Power Method to Determine Damping
nfff
212 −≈ζ
maxmax 707.0
2max accaccrms U
MUacc ≈=
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Piezoelectric Effect
• Direct effect: the charge produced when a piezoelectric substance is subjected to a stress or strain
• Converse effect: the stress or strain produced when an electric field is applied to a piezoelectric substance in its poled direction
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Perovskite Structure
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Poling Geometry
Detailed View
15
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Poling Schedule
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Field Induced Strain
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Piezoelectric Constitutive Relations
EεeSDEeScT
S
tE
+=−=
where
Etcde =
T = resultant stress vectorD = electric displacement vectorS = mechanical strain vectorE = electric field vectore = piezoelectric stress tensoret = piezoelectric stress tensor transposed = piezoelectric strain tensorcE = elastic stiffness tensor at constant fieldεS = dielectric tensor at constant strain
and where
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1-D Constitutive Equations
ESYdDEYdYSTε+=
−=
31
31
Y = Young’s modulus
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Relevant Geometry
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Applied Voltage Distribution
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Effective Moment Arm of PZT Elements
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System Wiring Schematic
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Interconnection Diagram
CHAN 1 CHAN 2 CHAN 3
X1 X1 X1 X1 X1 X1
X100X100X100X100X100X100
X10 X10
Data Acquisition Input
Piezo Inputconnections
GND P1 P2 P3 P4 P5 P6
Attenuator Outputs
Isolated Attenuators
GND P IN GND P IN
PowerAmp SignalGenerator
AttenuatorCables
MUST be useddata Acq Card
Interface
DO NOTc onnec t
anyting to thisbox!!
ch0
ch1
ch2
ch3
ch3
ch1
ch2
AccelerometerIntegrator
Accelerometer
ch1
ch1
ch2
ch2
PZT Ground
PZT #1 drive line
PZT #2 drive line
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Mathematical model of an Ultrasonic Piezoelectric Toy
The following is an example of the use of vibrations and mechatronics theory to model (or design) a simple piezoelectric toy.
All of the theory presented in this example directly applies to modeling the piezoelectriclly driven cantilevered beam used in the ME242 lab, and explained in the vibrations mechatronics manual --http://web.mst.edu/~stutts/ME242/LABMANUAL/Piezo-Beam_F09.pdf.
http://web.umr.edu/~stutts/ME242/LABMANUAL/MechVibLab.pdf
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•PZT – Lead Zirconate Titanate (PbZrTiO3)
• Applied voltage –> strain (converse effect)
• Alternating strain in PZT “buckles” beam into first mode
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• Crawler “gallops” due to beam flexingin its first natural mode – U(x)
• First natural or “resonant” modecorresponds to first resonant frequencyat approximately 26k Hz – inaudibleto most humans – hence, “ultrasonic”
• Beam is supported at nodes whereU(x) is zero so little vibratory energy islost.
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Euler-Bernoulli Beam with Moment Forcing Equation of Motion
ρ∂2u∂t2
+ c∂u∂t
+ YI∂ 4u∂x4
= b∂ 2 Me(x,t)
∂x2
Where, M(x, t) = rPZTd31YPZTV (x,t),and,
[ ] txxHxxHVtxV ωsin)()(),( 210 −−−=
⎩⎨⎧ ≥=−
otherwise ,0for ,1
)(ax
axHand,
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Free Vibration SolutionThe general form of the spatial solution for the Euler-Bernoulli Beam is
)sinh()cosh()sin()cos()( 43213 xAxAxAxAxU λλλλ +++=
And the free-free boundary conditions are:
0)()0( 23
2
23
2
=∂
∂=∂
∂x
lUx
U
0)()0( 33
3
33
3
=∂
∂=∂
∂x
lUx
Uand
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( ) ( ) ( )( ) ( ) ( )( )⎥⎦
⎤⎢⎣
⎡+⎟
⎠
⎞⎜⎝
⎛++= xxAAxxAxU nn
n
nnnn λλλλ sinhsincoshcos1
21
The general eigen-solution for discrete eigenvaluesIs given in terms of the unknown constants:
The leading constant is arbitrary, and may be set to unity.
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Forced Free-Free Beam Solution
),(3 txbfxVubh +∂∂−=&&ρ
Equation of Motion:
Where u3 is the transverse deflection, V is the shear, b and h are the beam width and height respectively,and f(x,t) is an applied pressure in the 3-direction.
xtxMtxV
∂∂= ),(),(
For the Euler-Bernoulli beam, we have
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32
2
3 FxMuh =
∂∂+&&ρ
32
2
3 bFxMbuA =
∂∂+&&ρ
),(121
3123
23 txVYdr
xu
YhM pztpztss −∂∂
=
bhA =
( ) ( ) ( )txMtxMtxM em ,,, +=
Hence:
The moment, ignoring the stiffness of the PZT layer, is given by:
where
So, the total moment may be divided into mechanical and electrical components:
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( )23
23
121,
xu
YhtxM ssm
∂∂
=
( ) ),(, 31 txVYdrtxM pztpzte −=
( ) ( )[ ] ( )txxHxxHVtxV o ωsin),( 21 −−−=
( ) ( ) ( )[ ] ( )txxxxVYdbrtxbFxu
YIuu opztpzt ωδδγρ sin, 2131343
4
33 −′−−′+=∂∂
++ &&&
[ ]2
damping ddistributelength
timeforce ⋅=≡γ ρρ bh=≡ lengthmassand
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( ) ( ) ( )∑∞
==
13 ,
nnn txUtxu η
[ ] ( )( ) ( ) ( )[ ] ( )txxxxVYdbrtxF
xUYI
opztpzt
nnnnnn
ωδδ
ηληγηρ
sin, 213131
4
−′−−′+
=++∑∞
=
&&&
( ) ( )tFtF mnnnnnn ˆˆ2 32 +=++ ηωηωξη &&&
Seeking a solution in terms of the natural modes viathe modal expansion process, we have
Canonical form, we have:
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( ) ( ) ( )
( ) ( ) ( ) ( )[ ]
( )
ρξωρ
λω
ρωρ
c
YI
dxxUN
NxUxUtVYdbr
tF
NdxxUtxF
tF
n
nn
l
nn
n
nnopztpztm
n
l
n
=
=
=
′−′=
=
∫
∫
2
sinˆ
,ˆ
2
02
1231
0 33
where
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( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( )( )xxllllxxxU nn
nn
nnnnn λλλλ
λλλλ sinhsinsinhsin
coscoshcoshcos +⎟⎟⎠
⎞⎜⎜⎝
⎛−−++=
( ) ( ) ( ) ( ) ( )( )[ ]xxAxxxU nnnnnnn λλλλλ coshcossinsinh 2 ++−=′
( ) ( )nnn tt φωη −Λ= sin
( ) 22222*
41 nnnnn
nrr
Fξω +−
=Λ
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( ) ( )[ ]n
nnopztpztn N
xUxUVYdbrF
ρ1231* ′−′=
nnr ω
ω=and where
where we have ignored the contribution of any externaltransverse forcing (F3).
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Crawler Steady State Simulation
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Crawler displacement magnitude.
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Ultrasonic Motor Example
ME242 Vibrations-Mechatronics ExperimentPurpose of ExperimentBasic Concepts in VibrationsBasic Concepts in MechatronicsInstrumentationCantilevered Beam SchematicSDOF OscillatorSolution to free-vibration problemExample Plot of Decaying MotionHarmonic Forcing: Effect of Damping Near ResonanceHalf-Power Method to Determine DampingPiezoelectric EffectPerovskite StructurePoling GeometryDetailed ViewPoling ScheduleField Induced StrainSlide Number 181-D Constitutive EquationsRelevant GeometryApplied Voltage DistributionEffective Moment Arm of PZT ElementsSystem Wiring SchematicInterconnection DiagramSlide Number 25Slide Number 26Slide Number 27Euler-Bernoulli Beam with Moment Forcing Equation of MotionFree Vibration SolutionSlide Number 30Forced Free-Free Beam SolutionSlide Number 32Slide Number 33Slide Number 34Slide Number 35Slide Number 36Slide Number 37Crawler Steady State SimulationSlide Number 39Ultrasonic Motor Example