ME242 Vibrations- Mechatronics Experiment

Transcription

1 ME4 Vibratios- Mechatroics Experimet Daiel. S. Stutts Associate Professor of Mechaical Egieerig ad Egieerig Mechaics Wedesday, September 16, 009

2 Purpose of Experimet Lear some basic cocepts i vibratios ad mechatroics. Gai hads-o experiece with commo istrumetatio used i the study of vibratios Gai experiece i takig ad reportig experimetal results i writte ad verbal form

3 Basic Cocepts i Vibratios Free vibratio of a Sigle DOF system Dampig measuremet via the logarithmic decremet method ad half-power method. Natural frequecies ad modes of a beam i bedig Harmoic forcig via piezoceramic elemets ad the steady-state respose

4 Basic Cocepts i Mechatroics Material properties ad behavior of a piezoceramic, PZT (Lead Zircoate Titaate) Electro-mechaical couplig: Actuatio ad Sesig 4

5 Istrumetatio Sigal geerator Amplifier Accelerometer ad coditioig circuitry Data acquisitio computer 5

6 Catilevered Beam Schematic 6

7 SDOF Oscillator EOM: Caoical form: Mx && + Cx& + Kx f() t f () t && x+ ζωx& + ωx M 7

8 Solutio to free-vibratio problem && x+ x& + x ζω ω 0 ( ) ω ζ ω ζ ζω xt () e Acos 1 t+ Bsi 1 t t 8

9 Example Plot of Decayig Motio ω ω ζ d 1 10 ζω 0. B 0 (Sie term set to zero) 9

10 Harmoic Forcig: Effect of Dampig Near Resoace f () t F siωt 0 10

11 Half-Power Method to Determie Dampig ζ f f f 1 Uacc max rms M accmax U accmax 11

12 Piezoelectric Effect Direct effect: the charge produced whe a piezoelectric substace is subjected to a stress or strai Coverse effect: the stress or strai produced whe a electric field is applied to a piezoelectric substace i its poled directio 1

13 Perovskite Structure 1

14 Polig Geometry 14

15 Detailed View 15

16 Polig Schedule 16

17 Field Iduced Strai 17

18 Piezoelectric Costitutive Relatios T S e S D es + ε E where T resultat stress vector D electric displacemet vector S mechaical strai vector E electric field vector e piezoelectric stress tesor e t piezoelectric stress tesor traspose d piezoelectric strai tesor c E elastic stiffess tesor at costat field ε S dielectric tesor at costat strai ad where e d t c E c E t E 18

19 1-D Costitutive Equatios T YS Yd 1 E D Yd 1 S + εe Y Youg s modulus 19

20 Relevat Geometry 0

21 Applied Voltage Distributio 1

22 Effective Momet Arm of PZT Elemets

23 System Wirig Schematic

24 Itercoectio Diagram data Acq Card Iterface ch0 Atteuator Cables MUST be used DO NOT coect aytig to this box!! ch1 ch ch ch1 Power Amp ch Sigal Geerator ch1 ch ch ch ch1 X100 Atteuator Outputs X100 X100 X100 X100 X100 Data Acquisitio Iput CHAN 1 CHAN CHAN Accelerometer Itegrator X1 X1 X1 X1 X1 X1 Isolated Atteuators X10 X10 GND P1 P P P4 P5 P6 Piezo Iput coectios GND P IN GND P IN Acceleromete r PZT # drive lie PZT #1 drive lie PZT Groud 4

25 Mathematical model of a Ultrasoic Piezoelectric Toy The followig is a example of the use of vibratios ad mechatroics theory to model (or desig) a simple piezoelectric toy. All of the theory preseted i this example directly applies to modelig the piezoelectriclly drive catilevered beam used i the ME4 lab, ad explaied i the vibratios mechatroics maual

26 PZT Lead Zircoate Titaate (PbZrTiO ) Applied voltage > strai (coverse effect) Alteratig strai i PZT buckles beam ito first mode 6

27 Crawler gallops due to beam flexig i its first atural mode U(x) First atural or resoat mode correspods to first resoat frequecy at approximately 6k Hz iaudible to most humas hece, ultrasoic Beam is supported at odes where U(x) is zero so little vibratory eergy is lost. 7

28 Euler-Beroulli Beam with Momet Forcig Equatio of Motio ρ u t + c u t + YI 4 u x 4 b M e (x,t) x Where, ad, M(x,t) r PZT d 1 Y PZT V (x,t), V [ H ( x x ) H ( x x )] si t ( x, t) V0 1 ω ad, H ( x a) 1, for x a 0, otherwise 8

29 9 Free Vibratio Solutio The geeral form of the spatial solutio for the Euler- Beroulli Beam is ) sih( ) cosh( ) si( ) cos( ) ( 4 1 x A x A x A x A x U λ λ λ λ Ad the free-free boudary coditios are: 0 ) ( (0) x l U x U 0 ) ( (0) x l U x U ad

30 U The geeral eige-solutio for discrete eigevalues Is give i terms of the ukow costats: A ( x) A ( cos( λ x) + cosh( λ x) ) + ( si( λ x) + sih( λ x) ) 1 The leadig costat is arbitrary, ad may be set to uity. A 1 0

31 Forced Free-Free Beam Solutio Equatio of Motio: V ρbh u&& + bf ( x, t) x Where u is the trasverse deflectio, V is the shear, b ad h are the beam width ad height respectively, ad f(x,t) is a applied pressure i the -directio. For the Euler-Beroulli beam, we have V ( x, t) M ( x, t) x 1

32 Hece: M ρh u&& x + F M ρa u&& + b bf x where A bh The momet, igorig the stiffess of the PZT layer, is give by: 1 u M h Y s s r d 1Y V ( x, t) pzt pzt 1 x So, the total momet may be divided ito mechaical ad electrical compoets: M m e ( x, t) M ( x, t) + M ( x, t)

33 M m 1 1 ( x, t) hs Ys u x M e ( x, t) r d 1Y V ( x, t) pzt pzt [ H ( x x ) H ( x x )] si( t) V x, t) Vo ω ( 1 4 u ρu&& + γu YI bf & + 4 pzt 1 pzt o 1 x ( x t) + br d Y V [ δ ( x x ) δ ( x x )] si( ωt), γ force time legth distributed dampig [ ] ad ρ mass legth bhρ

34 Seekig a solutio i terms of the atural modes via the modal expasio process, we have 1 F u ( x t) U ( x) η ( t), 1 [ 4 ρη&& + γη& + YIλ η ] U ( x) ( x t) + br d Y V [ δ ( x x ) δ ( x x )] si( ωt), pzt 1 pzt o 1 Caoical form, we have: Fˆ ξ ω & η + ω && η + η () t + Fˆ () t m 4

35 where Fˆ m () t br Fˆ pzt () t d 1 Y N pzt l V F 0 o ( x, t) U ( x) si l 0 U ρn ( ωt) [ U ( x ) U ( x )] ρn ( x) YI ω λ ρ ξω c ρ dx dx 1 5

36 U ( x) ( λ x) + cosh( λ x) U ( λl) cos( λl) ( λ l) sih( λ l) cosh cos + + si ( si( λ x) sih( λ x) ) ( x) λ [ sih( λ x) si( λ x) + A ( cos( λ x) cosh( λ x) )] + η () t Λ si( ωt φ ) Λ F * ( r ) 4ξ r ω + 1 6

37 ad where r ω ω F [ U ( x ) U ( x )] brpztd YpztV * 1 o ρn where we have igored the cotributio of ay exteral trasverse forcig (F ). 1 7

38 Crawler Steady State Simulatio 8

39 Crawler displacemet magitude. 9

40 Ultrasoic Motor Example 40