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

3

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

5

Instrumentation

• Signal generator

• Amplifier

• Accelerometer and conditioning circuitry

• Data acquisition computer

6

Cantilevered Beam Schematic

7

SDOF Oscillator

( )Mx Cx Kx f t+ + =&& &EOM:

2 ( )2 n nf tx x xM

ζω ω+ + =&& &Canonical form:

8

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−= − + −

9

Example Plot of Decaying Motion

21 10d nω ω ζ= − =

0.2nζω =

0B =(Sine term set tozero)

10

Harmonic Forcing: Effect of Damping Near Resonance

0( ) sinf t F tω=

11

Half-Power Method to Determine Damping

nfff

212 −≈ζ

maxmax 707.0

2max accaccrms U

MUacc ≈=

12

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

13

Perovskite Structure

14

Poling Geometry

Detailed View

15

16

Poling Schedule

17

Field Induced Strain

18

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

20

Relevant Geometry

21

Applied Voltage Distribution

22

Effective Moment Arm of PZT Elements

23

System Wiring Schematic

24

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

25

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.

31

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

32

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:

33

( )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

37

( ) ( )[ ]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