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How to perform analysis on structures with piezoelectric components?

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- 1. Piezoelectric Materials Dr. Mohammad Tawfik
- 2. What is Piezoelectric Material? • Piezoelectric Material is one that possesses the property of converting mechanical energy into electrical energy and vice versa.
- 3. Piezoelectric Materials • Mechanical Stresses Electrical Potential Field : Sensor (Direct Effect) • Electric Field Mechanical Strain : Actuator (Converse Effect) Clark, Sounders, Gibbs, 1998
- 4. Conventional Setting Conductive Pole
- 5. Piezoelectric Sensor • When mechanical stresses are applied on the surface, electric charges are generated (sensor, direct effect). • If those charges are collected on a conductor that is connected to a circuit, current is generated
- 6. Piezoelectric Actuator • When electric potential (voltage) is applied to the surface of the piezoelectric material, mechanical strain is generated (actuator). • If the piezoelectric material is bonded to a surface of a structure, it forces the structure to move with it.
- 7. Applications of Piezoelectric Materials in Vibration Control
- 8. Collocated Sensor/Actuator
- 9. Self-Sensing Actuator
- 10. Hybrid Control
- 11. Passive Damping / Shunted Piezoelectric Patches
- 12. Passively Shunted Networks Resistive Capacitive Resonant Switched
- 13. Modeling of Piezoelectric Structures
- 14. Constitutive Relations • The piezoelectric effect appears in the stress strain relations of the piezoelectric material in the form of an extra electric term • Similarly, the mechanical effect appears in the electric relations s11 1 d 31 E D d 31 1 33 E
- 15. Constitutive Relations • • • • ‘S’ (capital s) is the strain ‘T’ is the stress (N/m2) ‘E’ is the electric field (Volt/m) ‘s’ (small s) is the compliance; 1/stiffness (m2/N) • ‘D’ is the electric displacement, charge per unit area (Coulomb/m2)
- 16. The Electromechanical Coupling • Electric permittivity (Farade/m) or (Coulomb/mV) • d31 is called the electromechanical coupling factor (m/Volt)
- 17. Manipulating the Equations • The electric displacement is the charge per unit area: • The rate of change of the charge is the current: • The electric field is the electric potential per unit length: Q D A 1 I D Idt A As V E t
- 18. Using those relations: • Using the relations: • Introducing the capacitance: • Or the electrical admittance: d 31 1 s11 1 V t A 33 s I Ad 31s 1 V t I Ad 31s 1 CsV I Ad 31s 1 YV
- 19. For open circuit (I=0) • We get: • Using that into the strain relation: • Using the expression for the electric admittance: Ad 31s V 1 Y 2 31 Asd 1 s11 1 1 tY 2 d 31 1 1 s11 1 s 33 11
- 20. The electromechanical coupling factor • Introducing the factor ‘k’: 1 s11 1 k 1 2 31 • ‘k’ is called the electromechanical coupling factor (coefficient) • ‘k’ presents the ratio between the mechanical energy and the electrical energy stored in the piezoelectric material. • For the k13, the best conditions will give a value of 0.4
- 21. Different Conditions • With open circuit conditions, the stiffness of the piezoelectric material appears to be higher (less compliance) 1 s11 1 k 1 s 1 2 31 D • While for short circuit conditions, the stiffness appears to be lower (more compliance) s11 s E
- 22. Different Conditions • Similar results could be obtained for the electric properties; electric properties are affected by the mechanical boundary conditions.
- 23. Zero-strain conditions (S=0) • Using the relations: • Introducing the capacitance: • Or the electrical admittance: d 31 0 s11 1 V t 2 As 33 d 31 1 V I t 33 s11 I Y 1 k V 2 31
- 24. Other types of Piezo!
- 25. 1-3 Piezocomposites 3 c E e33 E 3 33 3 S D3 e33 3 33 E3
- 26. Active Fiber Composites (AFC) c eff 11 c E11 e eff 31 v C 2 v p e31 v p S 33 33 33e31 v C 33 v p S 33 33 S 33 eff 33 C v 33 v p S 33
- 27. Actuation Action • PZT and structure are assumed to be in perfect bonding
- 28. Axial Motion of Rods • In this case, we will consider the case when the PZT and the structure are deforming axially only
- 29. Zero Voltage case • If the structure is subject to axial force only, we get: a Ea a s Es s • And for the equilibrium: F Aa a As s Aa Ea a As Es s F Aa a As s Aa Ea As Es x
- 30. Zero Voltage case • From that, we may write the force strain relation to be: F F b x Aa Ea As Es 2ta Ea t s Es
- 31. Zero Force case • In this case, the strain of the of the PZT will be less than that induced by the electric field only! E E E E d V a a s a p a s a 31 t s Es s • For equilibrium, F=0: V F Aa a As s Aa Ea s Aa Ea d31 As Es s 0 t V Aa Ea d 31 t s Aa Ea As Es
- 32. Homework #2 • Solve problems 1,2,&3 from textbook • Due 27/11/2013 (11:59PM)
- 33. Beams with Piezoelectric Material
- 34. Review of Thin-Beam Theory • The Euler-Benoulli beam theory assumes that the strain varies linearly through the thickness of the beam and inversely proportional to the radius of curvature. d 2v y 2 dx d 2v E Ey 2 dx
- 35. Equilibrium • The externally applied moment has to be in equilibrium with the internally generated h/2 h/2 moment. d 2v M bydy h / 2 Ey dx 2 h/2 • For homogeneous materials: 2 h/2 d v d 2v 2 M E 2 y bdy EI 2 dx h / 2 dx bydy
- 36. Equilibrium • Rearranging the terms: M d 2v 2 EI dx My I
- 37. With piezoelectric materials • Introducing change in the material property: h/2 M ydy b h / 2 t s / 2 V Ea a d 31 ydy Es s ydy ta h / 2 t s / 2 ts / 2 V Ea a d 31 ydy ta ts / 2 h/2
- 38. With piezoelectric materials • Expanding the integral 2 M d v Ea 2 b dx ts / 2 V / 2y dy Ea d31 ta h ts / 2 ydy 2 2 ts / 2 h / 2 2 h/2 d v d v V 2 2 Es 2 y dy Ea 2 y dy Ea d 31 dx t s / 2 dx t s / 2 ta h/2 ydy ts / 2
- 39. With piezoelectric materials • Rearranging ts / 2 h/2 ts / 2 2 M d v 2 Ea y dy Es y 2 dy Ea y 2 dy b dx h / 2 t s / 2 ts / 2 2 V Ea d 31 ta t s / 2 V / 2ydy Ea d31 ta h h/2 ydy ts / 2
- 40. With piezoelectric materials • Integrating M 1 d 2v 3 3 3 Ea h 3 t s 2 E s t s Ea h 3 t s b 24 dx 2 V 2 V 2 2 2 Ea d 31 t s h Ea d 31 h ts 8t a 8t a Ea d 31V 2 M 1 d 2v 3 3 2 3 Ea h t s E s t s h ts 2 b 12 dx 4ta
- 41. Remember: • For homogeneous structures: Eh3 d 2v M 2 12 dx b • Thus, in the absence of the voltage: Ea h t s Es t s EI Equivalent b 12 • OR: M EI Equivalent 3 3 3 d 2 v Ea bd31V 2 2 h ts 2 dx 4t a
- 42. In the absence of load 2 Ea bd31V d v 2 2 h ts 2 dx 4ta EI Equivalent • Thus, the structure will feel a moment: Es I s Eabd31V 2 d 2v 2 M s Es I s 2 h ts dx 4ta EI Equivalent
- 43. Piezoelectric forces • The above is equivalent of having a force applied by the piezoelectric material that is equal to: Ms Es I s Ea bd 31V 2 2 Fa ts 4t s t a EI Equivalent h ts
- 44. Homework #3 • Solve problems 4,5,&6 from textbook • Due 30/11/2013 (11:59PM)

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