ME242 Vibrations-Mechatronics Experiment

Daniel. S. Stutts Associate Professor of

Mechanical Engineering and Engineering Mechanics

Wednesday, January 27, 2010

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

EOM:

Canonical form:

8

Solution to free-vibration problem

9

Example Plot of Decaying Motion

(Sine term set to zero)

10

Harmonic Forcing: Effect of Damping Near Resonance

11

Half-Power Method to Determine Damping

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

where T = resultant stress vector D = electric displacement vector S = mechanical strain vector E = electric field vector e = piezoelectric stress tensor et = piezoelectric stress tensor transpose d = piezoelectric strain tensor

cE = elastic stiffness tensor at constant field εS = dielectric tensor at constant strain

and where

19

1-D Constitutive Equations

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

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://www.mst.edu/~stutts/ME242/LABMANUAL/PiezoBeam_F08.pdf.

26

• PZT – Lead Zirconate Titanate (PbZrTiO3)

•  Applied voltage –> strain (converse effect)

•  Alternating strain in PZT “buckles” beam into first mode

27

•  Crawler “gallops” due to beam flexing in its first natural mode – U(x) •  First natural or “resonant” mode corresponds to first resonant frequency at approximately 26k Hz – inaudible to most humans – hence, “ultrasonic” •  Beam is supported at nodes where U(x) is zero so little vibratory energy is lost.

28

Euler-Bernoulli Beam with Moment Forcing Equation of Motion

Where,

and,

and,

29

Free Vibration Solution The general form of the spatial solution for the Euler- Bernoulli Beam is

And the free-free boundary conditions are:

and

30

The general eigen-solution for discrete eigenvalues Is given in terms of the unknown constants:

The leading constant is arbitrary, and may be set to unity.

31

Forced Free-Free Beam Solution

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.

For the Euler-Bernoulli beam, we have

32

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

and

34

Seeking a solution in terms of the natural modes via the modal expansion process, we have

Canonical form, we have:

35

where

36

37

and where

where we have ignored the contribution of any external transverse forcing (F3).

38

Crawler Steady State Simulation

39

Crawler displacement magnitude.

40

Ultrasonic Motor Example