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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)

J.C. Bellido () Designing piezoelectric modal sensors/actuators PICOF 2012, ´ Ecole Polytechnique, April 2012 1 / 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

J.C. Bellido () Designing piezoelectric modal sensors/actuators PICOF 2012, ´ Ecole Polytechnique, April 2012 1 / 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.

J.C. Bellido () Designing piezoelectric modal sensors/actuators PICOF 2012, ´ Ecole Polytechnique, April 2012 2 / 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.

J.C. Bellido () Designing piezoelectric modal sensors/actuators PICOF 2012, ´ Ecole Polytechnique, April 2012 2 / 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.

J.C. Bellido () Designing piezoelectric modal sensors/actuators PICOF 2012, ´ Ecole Polytechnique, April 2012 2 / 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,

J.C. Bellido () Designing piezoelectric modal sensors/actuators PICOF 2012, ´ Ecole Polytechnique, April 2012 2 / 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.

J.C. Bellido () Designing piezoelectric modal sensors/actuators PICOF 2012, ´ Ecole Polytechnique, April 2012 2 / 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.

J.C. Bellido () Designing piezoelectric modal sensors/actuators PICOF 2012, ´ Ecole Polytechnique, April 2012 2 / 19

What can piezoelectric actuators do?

poling voltage

+

• +

+

V V

• +

+

PZT BEAM PZT

tension compression

v v v v

bending up bending down

PZT PZT BEAM PZT

+

• +

+ + + +

+

• +

+ +

• +

+ + +

J.C. Bellido () Designing piezoelectric modal sensors/actuators PICOF 2012, ´ Ecole Polytechnique, April 2012 3 / 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).

J.C. Bellido () Designing piezoelectric modal sensors/actuators PICOF 2012, ´ Ecole Polytechnique, April 2012 4 / 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).

J.C. Bellido () Designing piezoelectric modal sensors/actuators PICOF 2012, ´ Ecole Polytechnique, April 2012 4 / 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 Fj(x) ∝ φ′′

j (x).

0.2 0.4 0.6 0.8 1 −1 1 Axial Position, x [m] Normalized Surface Electrode Width Mode 1: 1−2 Mode 2: 1−3

1(+) 3(−) 2(+)

J.C. Bellido () Designing piezoelectric modal sensors/actuators PICOF 2012, ´ Ecole Polytechnique, April 2012 4 / 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 Fj(x) ∝ φ′′

j (x).

0.2 0.4 0.6 0.8 1 −1 1 Axial Position, x [m] Normalized Surface Electrode Width Mode 1: 1−2 Mode 2: 1−3

1(+) 3(−) 2(+)

50 100 150 −9 −8 −7 −6 −5 −4 −3 −2 −1 Frequency, f [Hz] Tip−Response, [dB] Initial response Mode 1 response Mode 2 response J.C. Bellido () Designing piezoelectric modal sensors/actuators PICOF 2012, ´ Ecole Polytechnique, April 2012 4 / 19

Modelling

= {0, 1}

SURFACE ELECTRODE PIEZOELECTRIC MATERIAL PLATE ACTUATOR SENSOR

SIDE VIEW

material variable poling variable

χm

= {-1, 1}

χp

x y TOP VIEW

ELECTRONICS

J.C. Bellido () Designing piezoelectric modal sensors/actuators PICOF 2012, ´ Ecole Polytechnique, April 2012 5 / 19

Modelling

= {0, 1}

SURFACE ELECTRODE PIEZOELECTRIC MATERIAL PLATE ACTUATOR SENSOR

SIDE VIEW

material variable poling variable

χm

= {-1, 1}

χp

x y TOP VIEW

ELECTRONICS

Aim: systematic design of distributed piezoelectric MSAs for plates.

J.C. Bellido () Designing piezoelectric modal sensors/actuators PICOF 2012, ´ Ecole Polytechnique, April 2012 5 / 19

Modelling

The signal response (electrical charge) of the piezoelectric sensor layer (Lee-Moon 1990, J. Appl. Mech.): q(t) = −(hp + hs) 2 Lx Ly χmχp

• e31

∂2w ∂x2 + e32 ∂2w ∂y 2 + 2e36 ∂2w ∂x∂y

• dy dx,

where: hp, hs thickness of the plate and sensor layer e31 = e32 = e (piezo’s charge the same in both directions), e36 = 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

J.C. Bellido () Designing piezoelectric modal sensors/actuators PICOF 2012, ´ Ecole Polytechnique, April 2012 6 / 19

Modelling

Modal-Fourier expansion of w: w(x, y, t) =

∞

• j=1

φj(x, y) ηj(t), φj mode shape, ηj modal coordinate.

J.C. Bellido () Designing piezoelectric modal sensors/actuators PICOF 2012, ´ Ecole Polytechnique, April 2012 7 / 19

Modelling

Modal-Fourier expansion of w: w(x, y, t) =

∞

• j=1

φj(x, y) ηj(t), φj mode shape, ηj modal coordinate. Inserting the expansion of w into the expresion of q we get into q(t) = −e (hp + hs) 2

∞

• j=1

Bj ηj(t), with Bj = Lx Ly χm(x, y) χp(x, y) △φj(x, y) dy dx

J.C. Bellido () Designing piezoelectric modal sensors/actuators PICOF 2012, ´ Ecole Polytechnique, April 2012 7 / 19

Modelling

Taking χ(x, y) = χm(x, y)χp(x, y), the optimization problem is given by Maximizeχ(x,y)∈{−1,0,1}: Bk(χ) subject to: Bj(χ) = 0, for j = 1, · · · , M, and j = k, where Bj(χ) = Lx Ly χ(x, y)∆φj(x, y) dy dx.

J.C. Bellido () Designing piezoelectric modal sensors/actuators PICOF 2012, ´ Ecole Polytechnique, April 2012 8 / 19

Modelling

Taking χ(x, y) = χm(x, y)χp(x, y), the optimization problem is given by Maximizeχ(x,y)∈{−1,0,1}: Bk(χ) subject to: Bj(χ) = 0, for j = 1, · · · , M, and j = k, where Bj(χ) = Lx Ly χ(x, y)∆φj(x, y) dy dx.

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

J.C. Bellido () Designing piezoelectric modal sensors/actuators PICOF 2012, ´ Ecole Polytechnique, April 2012 8 / 19

Modelling

Likewise, such optimal profiles let the actuator layer control both the magnitude and location of the forces induced by the electric filed ǫ(t) to the plate through the actuator equation and therefore to excite the mode at interest (Lee-Moon 1990): Eh3

p

12(1 − ν2) ∂4w ∂x4 + 2 ∂4w ∂x2∂y 2 + ∂4w ∂y 4

• + ρhp

∂2w ∂t2 = −hpha(hp + ha)eǫ(t)∆(χmχp(x, y))

J.C. Bellido () Designing piezoelectric modal sensors/actuators PICOF 2012, ´ Ecole Polytechnique, April 2012 9 / 19

Optimization problem

Taking χ(x, y) = χm(x, y)χp(x, y), the optimization problem is given by Maximizeχ(x,y)∈{−1,0,1}: Bk(χ) subject to: Bj(χ) = 0, for j = 1, · · · , M, and j = k, where Bj(χ) = Lx Ly χ(x, y)∆φj(x, y) dy dx.

J.C. Bellido () Designing piezoelectric modal sensors/actuators PICOF 2012, ´ Ecole Polytechnique, April 2012 10 / 19

Optimization problem

Taking χ(x, y) = χm(x, y)χp(x, y), the optimization problem is given by Maximizeχ(x,y)∈{−1,0,1}: Bk(χ) subject to: Bj(χ) = 0, for j = 1, · · · , M, and j = k, where Bj(χ) = Lx Ly χ(x, y)∆φj(x, y) dy dx. The set of functions where we optimize is not compact, so, in principle, we cannot guarantee the existence of optimal solutions for (P).

J.C. Bellido () Designing piezoelectric modal sensors/actuators PICOF 2012, ´ Ecole Polytechnique, April 2012 10 / 19

Optimization problem

Taking χ(x, y) = χm(x, y)χp(x, y), the optimization problem is given by Maximizeχ(x,y)∈{−1,0,1}: Bk(χ) subject to: Bj(χ) = 0, for j = 1, · · · , M, and j = k, where Bj(χ) = Lx Ly χ(x, y)∆φj(x, y) dy dx. The set of functions where we optimize is not compact, so, in principle, we cannot guarantee the existence of optimal solutions for (P). A relaxed formulation is required (= ⇒ just replace χ(x, y) by ρ(x, y) ∈ [−1, 1]).

J.C. Bellido () Designing piezoelectric modal sensors/actuators PICOF 2012, ´ Ecole Polytechnique, April 2012 10 / 19

Analysis of the relaxed formulation

The relaxed problem is given by Maximizeρ(x,y)∈[−1,1]: Bk(ρ) subject to: Bj(ρ) = 0, for j = 1, · · · , M, and j = k, Both objective function and constraints are linear.

J.C. Bellido () Designing piezoelectric modal sensors/actuators PICOF 2012, ´ Ecole Polytechnique, April 2012 11 / 19

Analysis of the relaxed formulation

The relaxed problem is given by Maximizeρ(x,y)∈[−1,1]: Bk(ρ) subject to: Bj(ρ) = 0, for j = 1, · · · , M, and j = k, Both objective function and constraints are linear. By using the Lemma I in [Artstein 1980], it is analytically proved that

• ptimal solutions for (RP) just take either -1 or 1.

J.C. Bellido () Designing piezoelectric modal sensors/actuators PICOF 2012, ´ Ecole Polytechnique, April 2012 11 / 19

Analysis of the relaxed formulation

The relaxed problem is given by Maximizeρ(x,y)∈[−1,1]: Bk(ρ) subject to: Bj(ρ) = 0, for j = 1, · · · , M, and j = k, Both objective function and constraints are linear. By using the Lemma I in [Artstein 1980], it is analytically proved that

• ptimal solutions for (RP) just take either -1 or 1.

Optimal solutions for (P) correspond to taking: χm ≡ 1 and χp ∈ {−1, 1}.

J.C. Bellido () Designing piezoelectric modal sensors/actuators PICOF 2012, ´ Ecole Polytechnique, April 2012 11 / 19

Analysis of the relaxed formulation

The relaxed problem is given by Maximizeρ(x,y)∈[−1,1]: Bk(ρ) subject to: Bj(ρ) = 0, for j = 1, · · · , M, and j = k, Both objective function and constraints are linear. By using the Lemma I in [Artstein 1980], it is analytically proved that

• ptimal solutions for (RP) just take either -1 or 1.

Optimal solutions for (P) correspond to taking: χm ≡ 1 and χp ∈ {−1, 1}. The discrete problem can be easily solved by the simplex method.

J.C. Bellido () Designing piezoelectric modal sensors/actuators PICOF 2012, ´ Ecole Polytechnique, April 2012 11 / 19

Numerical simulations

Example 1: Isolate 1st flexural mode in a plate simply-supported at all four sides

(i) M = 5 (j) M = 10 (k) M = 15 (l) M = 26

J.C. Bellido () Designing piezoelectric modal sensors/actuators PICOF 2012, ´ Ecole Polytechnique, April 2012 12 / 19

Numerical simulations

Example 2: Isolate 6th extensional mode in a plate cantilevered in its left side.

(m) M = 12

J.C. Bellido () Designing piezoelectric modal sensors/actuators PICOF 2012, ´ Ecole Polytechnique, April 2012 13 / 19

Numerical simulations

Example 3: Isolate 1st flexural mode in a plate cantilevered in its left side.

(n) M = 20 (flexural) (˜ n) M = 20+12 (flexural + extensional)

J.C. Bellido () Designing piezoelectric modal sensors/actuators PICOF 2012, ´ Ecole Polytechnique, April 2012 14 / 19

Numerical simulations

Example 4: Isolate 2nd mode in a half cylindrical shell cantilevered in its left curved side.

X S

L

X Y Z

π

R

J.C. Bellido () Designing piezoelectric modal sensors/actuators PICOF 2012, ´ Ecole Polytechnique, April 2012 15 / 19

Manufacturing and Experimental validation

Joint work with the Microsystems, actuators and sensors Group leaded by J.L. Sanchez-Rojas in ETSII-UCLM.

−3 −2 −1

−5

J.C. Bellido () Designing piezoelectric modal sensors/actuators PICOF 2012, ´ Ecole Polytechnique, April 2012 16 / 19

Manufacturing and Experimental validation

Joint work with the Microsystems, actuators and sensors Group leaded by J.L. Sanchez-Rojas in ETSII-UCLM. Example: Isolate 1st flexural mode in a microbridge clamped in both sides.

50 µm

10 20 30 40 50 60 10 20 −3 −2 −1 x 10

−5

2 4 6 8 10 1 10 100 1000 62 52 71 42 32 61 22 60 51 41 40 31 30 21 20 Displacement (pm) 2 4 6 8 10 1 10 100 1000 42 32 22 51 41 31 40 30 21 20 Displacement (pm) Frequency (MHz)

J.C. Bellido () Designing piezoelectric modal sensors/actuators PICOF 2012, ´ Ecole Polytechnique, April 2012 16 / 19

Sensitivity analysis - gap

20 21 30 31 40 41 50 22 51 60 32 61 42 70 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

modo respuesta norm.

gap = 10 gap = 5 gap = 2 gap = 0

J.C. Bellido () Designing piezoelectric modal sensors/actuators PICOF 2012, ´ Ecole Polytechnique, April 2012 17 / 19

Future work

Simultaneous optimization of both the supporting structure and the polarization profile of the piezoelectric sensor/actuator.

J.C. Bellido () Designing piezoelectric modal sensors/actuators PICOF 2012, ´ Ecole Polytechnique, April 2012 18 / 19

Future work

Simultaneous optimization of both the supporting structure and the polarization profile of the piezoelectric sensor/actuator. Doctoral student David Gracia working on that.

J.C. Bellido () Designing piezoelectric modal sensors/actuators PICOF 2012, ´ Ecole Polytechnique, April 2012 18 / 19

References

• A. Donoso, J.C. Bellido. Systematic design of distributed modal

sensors/actuators for rectangular plates by optimizing the polarization profile.

• Struc. Mult. Opt., 38, 347-356, 2009.
• A. Donoso, J.C. Bellido. Tailoring distributed modal sensors for in-plane

modal filtering. Smart Materials and Struc. 18, 2009.

• A. Donoso, J.C. Bellido. Distributed piezoelectric modal sensors for circular
• plates. Journal of Sound and Vibration. 319, 50-57, 2009.
• A. Donoso, J.C. Bellido, J.M. Chacon. Numerical and analytical method for

the design of modal sensors/actuators for shell-type structures. Int. J. Num. Methods Eng. 81, 1700-1712, 2010. J.L. Sanchez-Rojas, J. Hernando, A. Donoso, J.C. Bellido, T. Manzaneque,

• A. Ababneh, H. Seidel, U. Schmid. Modal optimization and filtering in

piezoelectric microplate resonators. J. Micromechanics and Microengineering. 20, 2010.

J.C. Bellido () Designing piezoelectric modal sensors/actuators PICOF 2012, ´ Ecole Polytechnique, April 2012 19 / 19