Designing piezoelectric modal sensors/actuators J.C. Bellido PICOF - PowerPoint PPT Presentation

Designing piezoelectric modal sensors/actuators J.C. Bellido PICOF 2012, ´ Ecole Polytechnique, April 2012 Universidad de Castilla-La Mancha Departamento de Matem´ aticas (ETSII-Ciudad Real) PICOF 2012, ´ Ecole Polytechnique, April 2012 1 / J.C. Bellido () Designing piezoelectric modal sensors/actuators 19

Designing piezoelectric modal sensors/actuators J.C. Bellido PICOF 2012, ´ Ecole Polytechnique, April 2012 Universidad de Castilla-La Mancha Departamento de Matem´ aticas (ETSII-Ciudad Real) Joint work with A. Donoso PICOF 2012, ´ Ecole Polytechnique, April 2012 1 / J.C. Bellido () Designing piezoelectric modal sensors/actuators 19

Introduction to piezoelectricity Piezoelectricity ability of some materials (notably crystals and certain ceramics) to generate a voltage in response to applied mechanical stress and vice versa. PICOF 2012, ´ Ecole Polytechnique, April 2012 2 / J.C. Bellido () Designing piezoelectric modal sensors/actuators 19

Introduction to piezoelectricity Piezoelectricity ability of some materials (notably crystals and certain ceramics) to generate a voltage in response to applied mechanical stress and vice versa. Sensors : they produce an electric signal proportional to their deformation. PICOF 2012, ´ Ecole Polytechnique, April 2012 2 / J.C. Bellido () Designing piezoelectric modal sensors/actuators 19

Introduction to piezoelectricity Piezoelectricity ability of some materials (notably crystals and certain ceramics) to generate a voltage in response to applied mechanical stress and vice versa. Sensors : they produce an electric signal proportional to their deformation. Actuators : they strain under an applied voltage. PICOF 2012, ´ Ecole Polytechnique, April 2012 2 / J.C. Bellido () Designing piezoelectric modal sensors/actuators 19

Introduction to piezoelectricity Piezoelectricity ability of some materials (notably crystals and certain ceramics) to generate a voltage in response to applied mechanical stress and vice versa. Sensors : they produce an electric signal proportional to their deformation. Actuators : they strain under an applied voltage. These transducers can appear surface bonded to structures or embedded in laminated composites, PICOF 2012, ´ Ecole Polytechnique, April 2012 2 / J.C. Bellido () Designing piezoelectric modal sensors/actuators 19

Introduction to piezoelectricity Piezoelectricity ability of some materials (notably crystals and certain ceramics) to generate a voltage in response to applied mechanical stress and vice versa. Sensors : they produce an electric signal proportional to their deformation. Actuators : they strain under an applied voltage. These transducers can appear surface bonded to structures or embedded in laminated composites, uniformly distributed or like patches. PICOF 2012, ´ Ecole Polytechnique, April 2012 2 / J.C. Bellido () Designing piezoelectric modal sensors/actuators 19

Introduction to piezoelectricity Piezoelectricity ability of some materials (notably crystals and certain ceramics) to generate a voltage in response to applied mechanical stress and vice versa. Sensors : they produce an electric signal proportional to their deformation. Actuators : they strain under an applied voltage. These transducers can appear surface bonded to structures or embedded in laminated composites, uniformly distributed or like patches. Applications lighters, quartz clocks, ultrasonic transducers, bio-sensors, modal control, etc. PICOF 2012, ´ Ecole Polytechnique, April 2012 2 / J.C. Bellido () Designing piezoelectric modal sensors/actuators 19

What can piezoelectric actuators do? + + - + - - + + + + poling - - V PZT V voltage + - - + + - + - - - compression tension - - - + bending up + v v - - + PZT PZT + - BEAM BEAM - - + PZT + PZT v v - + - + bending down PICOF 2012, ´ Ecole Polytechnique, April 2012 3 / J.C. Bellido () Designing piezoelectric modal sensors/actuators 19

Modal sensors/actuators (MSAs) MSAs Those which measure/excite a particular mode of a structure and remain insensitive to the rest (= ⇒ behave as ideal filters). PICOF 2012, ´ Ecole Polytechnique, April 2012 4 / J.C. Bellido () Designing piezoelectric modal sensors/actuators 19

Modal sensors/actuators (MSAs) MSAs Those which measure/excite a particular mode of a structure and remain insensitive to the rest (= ⇒ behave as ideal filters). The reciprocal property of piezoelectric materials continues to be valid in MSAs (i.e. both of them present the same pattern). PICOF 2012, ´ Ecole Polytechnique, April 2012 4 / J.C. Bellido () Designing piezoelectric modal sensors/actuators 19

Modal sensors/actuators (MSAs) MSAs Those which measure/excite a particular mode of a structure and remain insensitive to the rest (= ⇒ behave as ideal filters). The reciprocal property of piezoelectric materials continues to be valid in MSAs (i.e. both of them present the same pattern). Owing to the orthogonality principle, the design problem in 1-D can be reduced to computing the surface electrode width F j ( x ) ∝ φ ′′ j ( x ). Mode 1: 1−2 Mode 2: 1−3 Normalized Surface Electrode Width 1 1(+) 2(+) 0 3(−) −1 0 0.2 0.4 0.6 0.8 1 Axial Position, x [m] PICOF 2012, ´ Ecole Polytechnique, April 2012 4 / J.C. Bellido () Designing piezoelectric modal sensors/actuators 19

Modal sensors/actuators (MSAs) MSAs Those which measure/excite a particular mode of a structure and remain insensitive to the rest (= ⇒ behave as ideal filters). The reciprocal property of piezoelectric materials continues to be valid in MSAs (i.e. both of them present the same pattern). Owing to the orthogonality principle, the design problem in 1-D can be reduced to computing the surface electrode width F j ( x ) ∝ φ ′′ j ( x ). −1 Mode 1: 1−2 Initial response Mode 2: 1−3 −2 Mode 1 response Mode 2 response Normalized Surface Electrode Width −3 Tip−Response, [dB] 1 −4 1(+) 2(+) 0 −5 3(−) −6 −1 −7 0 0.2 0.4 0.6 0.8 1 Axial Position, x [m] −8 −9 0 50 100 150 Frequency, f [Hz] PICOF 2012, ´ Ecole Polytechnique, April 2012 4 / J.C. Bellido () Designing piezoelectric modal sensors/actuators 19

Modelling SIDE VIEW ACTUATOR ELECTRONICS PLATE SENSOR material variable χ m = {0, 1} TOP VIEW poling variable χ p = {-1, 1} SURFACE ELECTRODE x y PIEZOELECTRIC MATERIAL PICOF 2012, ´ Ecole Polytechnique, April 2012 5 / J.C. Bellido () Designing piezoelectric modal sensors/actuators 19

Modelling SIDE VIEW ACTUATOR ELECTRONICS PLATE SENSOR material variable χ m = {0, 1} TOP VIEW poling variable χ p = {-1, 1} SURFACE ELECTRODE x y PIEZOELECTRIC MATERIAL Aim : systematic design of distributed piezoelectric MSAs for plates. PICOF 2012, ´ Ecole Polytechnique, April 2012 5 / J.C. Bellido () Designing piezoelectric modal sensors/actuators 19

Modelling The signal response (electrical charge) of the piezoelectric sensor layer (Lee-Moon 1990, J. Appl. Mech.): � L x � L y ∂ 2 w ∂ 2 w ∂ 2 w q ( t ) = − ( h p + h s ) � � χ m χ p e 31 ∂ x 2 + e 32 ∂ y 2 + 2 e 36 dy dx , 2 ∂ x ∂ y 0 0 where: h p , h s thickness of the plate and sensor layer e 31 = e 32 = e (piezo’s charge the same in both directions), e 36 = 0 (piezo’s axes the same as the plate) piezo stress/charge constants w out-of-plane displacement of the plate piezolectric layers negligable stiffness and mass compared to the plate PICOF 2012, ´ Ecole Polytechnique, April 2012 6 / J.C. Bellido () Designing piezoelectric modal sensors/actuators 19

Modelling Modal-Fourier expansion of w : ∞ � w ( x , y , t ) = φ j ( x , y ) η j ( t ) , j =1 φ j mode shape, η j modal coordinate. PICOF 2012, ´ Ecole Polytechnique, April 2012 7 / J.C. Bellido () Designing piezoelectric modal sensors/actuators 19

Modelling Modal-Fourier expansion of w : ∞ � w ( x , y , t ) = φ j ( x , y ) η j ( t ) , j =1 φ j mode shape, η j modal coordinate. Inserting the expansion of w into the expresion of q we get into ∞ q ( t ) = − e ( h p + h s ) � B j η j ( t ) , 2 j =1 with � L x � L y B j = χ m ( x , y ) χ p ( x , y ) △ φ j ( x , y ) dy dx 0 0 PICOF 2012, ´ Ecole Polytechnique, April 2012 7 / J.C. Bellido () Designing piezoelectric modal sensors/actuators 19

Modelling Taking χ ( x , y ) = χ m ( x , y ) χ p ( x , y ), the optimization problem is given by Maximize χ ( x , y ) ∈{− 1 , 0 , 1 } : B k ( χ ) subject to: B j ( χ ) = 0 , for j = 1 , · · · , M , and j � = k , where � L x � L y B j ( χ ) = χ ( x , y )∆ φ j ( x , y ) dy dx . 0 0 PICOF 2012, ´ Ecole Polytechnique, April 2012 8 / J.C. Bellido () Designing piezoelectric modal sensors/actuators 19

Modelling Taking χ ( x , y ) = χ m ( x , y ) χ p ( x , y ), the optimization problem is given by Maximize χ ( x , y ) ∈{− 1 , 0 , 1 } : B k ( χ ) subject to: B j ( χ ) = 0 , for j = 1 , · · · , M , and j � = k , where � L x � L y B j ( χ ) = χ ( x , y )∆ φ j ( x , y ) dy dx . 0 0 Key point We are looking for an ideal sensor that best observes the k -th mode and filters the rest of the first M modes PICOF 2012, ´ Ecole Polytechnique, April 2012 8 / J.C. Bellido () Designing piezoelectric modal sensors/actuators 19