Dynamic Analysis Of Piezoelectric Transducers Using Finite Element Method

Dynamic Analysis Of Piezoelectric Transducers Using Finite Element Method

Dr. V. CHITTI BABU, Associate Professor,Department of Mechanical Engineering

GMRIT, RAJAM, Srikakulam,AP

Mr. L.V.SATEESH, M.Tech. (CAD/CAM), GMRIT, RAJAM

Sri P. Govinda Rao Associate Professor,Department of Mechanical Engineering

GMRIT, RAJAM, Srikakulam,AP

Abstract

Piezoelectric transducers have been extensively applied over the last decade to diverse areas such as band-pass filters and highenergy ultrasonic devices. They are typically utilized for generating an acoustic pulse proportional to the input external loading and detecting the same and converting it in to electric signal. Other application of piezoelectric are in loudspeaker, microphone, phonograph, wrist watches, oscillator, resonator, transducer etc. Piezoelectric crystals are widely used in aerospace industry, automobile industry and geological instruments as sensors and actuators. In the current engineering world automation is an integral part of the machines, processes, automobiles etc. In automation we need sensors and actuators. Piezoelectric material is widely used for the same purpose. Various type of analysis can be done by Finite Element Modelling of Piezoelectric Transducer e.g. Calculation of Eigen values, mode shapes, electrical output on application of strain and vice versa.

1. Introduction

   The word piezoelectricity literally means

   pressure electricity; the prefix piezo is derived from the Greek word piezein which means to press. Piezoelectricity is the property of a crystal by which electric polarization is produced by mechanical strain in crystal and conversely production of strain on application of voltage. The electric polarization produced is proportional to the applied strain and changing direction with it. Piezoelectricity is different from electrostriction, another effect which causes a solid dielectric to change shape on application of a voltage. In piezoelectricity, a reversal of voltage reverses the sign of the resulting strain whereas for

   electrostriction the strain is an even function of the applied voltage.

   Piezoelectric transducers have been extensively applied over the last decade to diverse area such as band-pass filters and highenergy ultrasonic devices. They are typically utilized for generating an acoustic pulse proportional to the input external loading and detecting the same and converting it in to electric signal. Different other application of piezoelectric are in loudspeaker, microphone, phonograph, wrist watches, oscillator, producing very selective filter circuit, resonator, transducer etc. Piezoelectric crystals are widely used in aerospace industry, automobile industry and geological instruments as sensors and actuators.

   A lot of work has been done in the field of finite element modelling of piezoelectric transducer on different type of uses. A.V. Belokon et al [1] proposed new scheme which develops the technique for the finite element analysis to take account of attenuation in piezoelectric media. K.Y.Sze, Y.S.pan [4] employed hybrid variational principles for finite element formulation of piezoelectric transducer. F.Cote et al [6] validates both theoretically and experimentally the implementation of a multilayered three- dimensional model based on the analogy between thermal strains and piezoelectric strains.

   Application of piezoelectric crystals can be divided in to two categories e.g. sensors and actuators. In the following work, main emphasis is given to the analysis of piezoelectric crystals as sensors. An analysis of transducers have also been given for dynamic loading, which includes the analysis of simple cantilever beam made of piezoelectric crystal, analysis of shell made of piezoelectric material with ends of brass, analysis of cantilever beam with piezoelectric strip as

   transducer, and analysis of proving ring with piezoelectric strips as sensors. The above analysis

   where i,j,k and l take the values 1,2 and 3, and p,q

   take the values 1,2,3,4,5 and 6.

   has been done for dynamic loading considering the sinusoidal load as dynamic load. Other than the

   So cijkl

   c pq

   above work, a separate work is presented to understand the dependency of piezoelectric response on working temperature.

2. Theoretical concept about Piezoelectric Properties

   Piezoelectricity is electric polarization produced by mechanical strain in crystals belonging to certain classes, the polarization being proportional to the strain and changing direction with it. The production of an electric polarization by mechanically inducing a strain in a crystal is called the direct piezoelectric effect. The converse effect, whereby a mechanical strain is produced in a crystal by a polarizing electric field, also exists.

   All the crystalline materials are anisotropic and do not have the same properties in all the directions as do the isotropic material. Crystals can be divided into 32 classes on the basis of symmetry they possess and, of these 32 classes , only 20 possess the property of piezoelectricity and 12 do not because these 12 classes of crystals have the centre of symmetry where as a crystal possessing the centre of symmetry can not be piezoelectric because no combination of uniform stresses will produce a separation of the centres of gravity of

   +ve and ve charges and produce an induced dipole moment which is necessary for the polarization by stresses. Most unsymmetrical type of system is the triclinic system, all three of whose crystallographic axis makes oblique angles, and the length of the unit cells on the three-axis is unequal. Such a material will have 21 elastic constants, 18 piezoelectric constants, 6 dielectric constants. A most symmetric crystal (cubic crystal) has 3 elastic constants, 1 piezoelectric and 1 dielectric constants. Thus on the basis of symmetry the piezoelectric crystals are classified in different classes.

   According to the linear theory of piezoelectricity, the constitutive equations that couples the deformation and electric field in the piezoelectric plate are given by-

   ij cijkl Skl ekij Ek

   eikl eip

   ij Tij Tp p

   .

3. Piezoelectric Analysis

   The piezoelectric effect is the coupling of stress and electrical field in a material: an electrical field causes the material to strain, and vice versa. ANSYS/Standard has the capability to perform fully coupled piezoelectric analysis. The elements that are used in this case contain both displacement degrees of freedom and the electric potential as nodal variables.

   1. Equilibrium and Flux Conservation

      The piezoelectric effect is governed by coupled mechanical equilibrium and electric flux conservation equations.

      The mechanical equilibrium equation is

      V :dV S t.udS V f .udV

      where

      is the true (Cauchy) stress at a point currently at x;

      t is the traction across a point of the surface of the body;

      f is the body force per unit volume in the body

      u is an arbitrary, continuous vector field (the virtual velocity field).

      The electrical flux conservation equation is

      V q.EdV S qS .dS V qV dV

      where is the electric flux vector;

      qs is the electric flux per unit area entering the body at a point on its surface;

      qV is the electric flux entering the body per unit volume;

      is an arbitrary, continuous, scalar field (the virtual potential).

      The electric flux vector q is known as the electrical displacement, and the potential gradient E isknown as the electrical field.

      Constitutive Behavior: Material Coupling

      Di eikl

      Skl

      ik Ek

      Currently the assumption of linear materials is utilized. The basic equations for a piezoelectric

      Where- cijkl = stiffness coefficient

      ekij = piezoelectric constants ik = dielectric constants

      In order to determine the solution of piezoelectric vibration problem, we will have to know array of material coefficients for the

      linear medium are defined in the following. Mainly three alternative forms of the constitutive equations are presented as e-form, d-form, and g-form.

      In ANSYS the constitutive equations in the e- form are used:

      ij DE l kl e Em

      particular symmetry of the material. Here we are considering the compressed matrix notation that

      and

      ijk

      mkl

      turns out to be more useful then the extended tensor notation. When discussing symmetry. The matrix notation consist of replacing ij and kl by p or q

      qi eijl jk Dij E j

      These are expressed in terms of the

      K MN

      BM : D : BN dV

      is the dielectric

      V

      piezoelectric stress coefficient matrix

      emkl .

      stiffness matrix,

      However, ANSYS also allows the input of

      K MN

      BM .e : BN dV

      is the piezoelectric

      u V u

      piezoelectric constants in terms of the piezoelectric

      coupling matrix,

      strain coefficient matrix d .

      PM

      NM .P dV

      NM .P dS PM

      is the

      mkl V v S s c

      The constitutive equations in the -form can also be expressed as

      mechanical force vector, and

      Dq

      • Dq g q

        QM NM .QvdV NM .Qs dS QM

        is the

        ij ijkl kl

        and

        q D g

        ijkl mkl m

        D E

        V S c

        electrical charge vector.

        In these expressions the constitutive properties are

        i im

        mjk jk ij j

        specified in a matrix form where is the

        These equations can be convenient in interpreting and verifying the results of piezoelectric analyses.

   2. Kinematics

      For the piezoelectric elements both displacements and electric potentials exist at the nodal locations. The displacements and electrical potentials are approximated within the element as

      u N Nu N

      and

      =NNN

      where NN is the array of interpolating functions and uN and N are nodal quantities. The body forces and charges as well as the surface forces and charges are interpolated in a similar manner.

      mechanical relationship, is the electrical relationship, and e is the piezoelectrical relationship.

      3.4 Specifying Piezoelectric Material Properties

      ijk

      ijk

      m

      m

      A piezoelectric material responds to an electric potential gradient by straining, while stress causes an electric potential gradient in the material. This coupling between electric potential gradient and strain is the material's piezoelectric property. The material will also have a dielectric property so that an electrical charge exists when the material has a potential gradient. Piezoelectric material behavior is discussed in the mechanical properties of the material must be modeled by linear elasticity. The mechanical behavior can be defined by

      The strains and electrical potential gradients are given as

      ij DE l kl

      • eij Em

      in terms of the

      u

      u

      BNu N

      and

      piezoelectric stress coefficient matrix, e ,

      mij

      mij

      E

      E

      and,

      N N

      N N

      E B

      BN N

      ij Dijkl kl

      • d

      • d

      mkl

      Em

      in terms of the

      where

      u and

      B are the spatial derivatives of

      piezoelectric strain coefficient matrix, d

      . The

      NN .In geometrically nonlinear analyses these spatial derivatives are defined in the current configuration.

      electrical behavior is defined by

      qi e jk D E j

      mkl

      3.3 System Equations

      With these approximate fields and the constitutive properties given above, in conjunction with the equilibrium and conservation equations,

      ijk ij

      Where

      ij is the mechanical stress tensor;

      ij is the strain tensor;

      qi is the electric displacement vector;

      the following system of equations is derived in

      E

      D

      D

      ijkl

      is the material's elastic stiffness matrix

      terms of nodal quantities:

      M MN u N K MN u N K MN N PM

      defined at zero electrical potential gradient (short circuit condition);

      and

      uu u

      e is the material's piezoelectric stress

      K MN u N K MN N QM

      ijk

      u

      coefficient matrix,

      where

      M MN

      NM .N N dV

      d

      d

      mkl

      is the material's piezoelectric strain

      V

      is the mass matrix (no inertia terms exist for the

      coefficient matrix.

      D

      D

      is the electrical potential;

      electrical flux conservation equation),

      is the mass density,

      ij

      is the material's dielectric property,

      K MN

      BM : D

      : BN dV

      is the displacement

      Ei is the electrical potential gradient vector,

      uu V u m u

      / xi .

      stiffness matrix,

      1. Electrical Output of Piezoelectric

         In the analysis we are more interested in the

         You specify the single value

         D for the

         electrical output. The electrical response of a piezoelectric material is assumed to be made up of piezoelectric and dielectric effects

         ijl ij

         ijl ij

         a qi e jk D E j a

         where

         – the electrical potential,

         qi – the component of the electric flux vector (also known as the electric displacement) in the ith material direction,

         dielectric constant. D must be determined for

         a constrained material. Isotropic behavior is the default.

         For orthotropic behavior you must specify three

         11 22 33

         11 22 33

         values in the dielectric matrix( e , e and e ). For fully anisotropic behavior you must specify six

         values in the dielectric matrix

         e – the piezoelectric stress coupling,

         ijk

         ij – a small-strain component,

         D – the material's dielectric matrix for a fully

         ( e12 , e22 , e22 , e13 ,

4. Modelling and Analysis

e23

and

e33 ).

ij

constrained material, and

Ei – the gradient of the electrical potential along the

ith material direction, / xi .

1. Specifying Piezoelectric Material Properties

The piezoelectric material properties can be

mij

mij

defined by giving the stress coefficients, e (this is the default), or by giving the strain

mkl

mkl

coefficients, d . In either case, 18 components

must be given in the following order (substitute d

for e for strain coefficients):

4.1 Finite element modelling of cantilever beam made of piezoelectric material PZT- 4 and determination of eigenvalues and corresponding mode shapes.

To start with the piezoelectric transducer, first take the simplest case i.e. a cantilever beam having piezoelectric material. To understand the nature of piezoelectric as a single material this condition has been taken. To start the analysis we divide the real structures in three parts , beam, plate and shell.

e

e

111

e

e

211

e

e

311

e

e

122

e

e

222

e

e

322

e

e

133

e

e

p>233

e

e

333

e

e

112

e

e

212

e

e

312

e

e

113

e

e

213

e

e

313

e

e

123

e

e

223

e

e

323

Model of the first analysis has been prepared as shown in Fig 4.1.

The first index on these coefficients refers to the component of electric displacement (sometimes called the electric flux), while the last pair of indices refers to the component of mechanical stress or strain.

Thus, the piezoelectric components causing electrical displacement in the 1-direction are all given first, then those causing electrical displacement in the 2-direction, and then those causing electrical displacement in the 3-direction. (Some references list these coupling terms in a different order.)

ij

ij

The dielectric matrix can be isotropic, orthotropic, or fully anisotropic. For non-isotropic dielectric materials a local orientation for the material directions must be specified

Fig 4.1. Model of simple beam made of piezoelectric (PZT-4)

In the above problem the material used is PZT-4. The main purpose of the analysis is to learn about the proper modeling of the piezoelectric material structures. Generally for a simple material we are

having only 6 degree of freedoms but for the

The dielectric matrix isotropic, so that

D D

D can be fully

material like piezoelectric we have one more

degree of freedom which is directly related to the electrical properties of the piezoelectric material. In

ij ij

stead of the eigen values and corresponding mode

shapes , analysis of electric potential at points 1,2,3,4( fig4.1.1 ) is also done.

Element Type

CPE4E A 4 node bilinear plain strain piezoelectric quadrilateral.

.Material

PZT -4

Boundary condition

Surface 1-4 is rigidly fixed i.e. U1= U2=UR3=0

Total no of elements- 306

Total no of nodes- 419

Fig 4.2 Meshed Cantilever made of Piezoelectric PZT-4

Results and discussion

Other then the eigen values and the mode shapes, the voltage output of the beam is also calculated at different points. This output is shown in table 4.1

Electrical potential output

Frequency (cycle/time)	EPOT -1	EPOT -2	EPOT -3	EPOT -4
1031.9	-8.838	-8.846	-8.846	-8.854
6259	5.056	5.003	5.003	4.949
16515	-0.405	-0.561	-0.561	-0.716
20726	35.7	38.68	38.68	35.7
29990	-4.829	-4.522	-4.522	-4.215
45594	-22.48	-22.98	-22.98	-23.48
62059	33.65	36.59	36.59	3.65

Table 4.1 Electrical potential at point 1,2,3,4 with frequencies.

Frequency vs Electric Potential

1. Finite element modeling of cylindrical shell made of PZT-4 and having end cover of Brass and determination of eigenvalues and mode shapes

   The idea behind the solving such situation was to have a good understanding about the assembly of piezoelectric with the other materials. Also to understand the effect of the piezoelectric material while it is in the combination of other material.

   The modeling of this type of shell is done by taking 2-D asymmetric element. The upper half of the diagram as shown in following figure is constructed. And then analysis is done

   Fig- 4.4 Model of piezoelectric shell with Brass end cap

   Considering the following;-

   Element type

   Brass

   CAX4R: A 4 node bilinear axisymmetric quadrilateral

   Total no of elements- 600 Total no of nodes- 656

   PZT-4

   CAX8RE: An 8 node biquadratic axisymmetric piezoelectric quadrilateral

   Total no of elements- 600

   50

   40

   30 EPOT-1

   EPOT

   EPOT

   20

   Total no of nodes- 1941

   10

   0

   -10 0 20000 40000 60000 80000

   -20

   -30

   EPOT-2

   EPOT-3

   EPOT-4

   Frequency

   Fig 4.3 Graphical representation of frequency vs. electric potential on the basis of data given inTable4.1

   Fig: 4.5 The mesh representing the shell element in axisymmetric Form

   Boundary Condition

   At axis of symmetry no displacement in radial direction

   At plane of symmetry no displacement in vertical direction

   Result and discussion

   Mode shapes

   1. Mode-1(19347cycle/sec)

   2. Mode -2 (42564 cycles/sec)

   3. Mode -3 (60009 cycles/sec)

   4. Mode -4 (66174 cycles/sec)

   5. Mode -5 (90844 cycles/sec)

   6. Mode -6 (97955 cycles/sec)

   Fig 4.6

   Mode shapes of the model for first 6 eigen values

   Electric potential output

   In this problem the electrical potential is calculated at the plane of symmetry of the shell.

   Sr. N0.	Frequency (cycles/sec)	Electric potential
   1	19347	2.958
   2	45564	-7.174
   3	60009	4.531
   4	66174	-2.606
   5	90844	-4.034
   6	97955	-9.35
   7	118651	-6.565

   Table 4.2 Electrical potential at plane of symmetry with frequencies.

2. Finite element modelling of cantilever beam with piezoelectric transducer PZT-4 and determination of its response under sinusoidal loading.

   To solve this problem we consider a 2-D planer cantilever beam fixed at one end. In this cantilever

   beam we attach a piezoelectric element (PZT-4) as shown in figure with the TIE option The cantilever beam is kept at zero potential difference. A periodic load is applied on the right most corner with different frequencies.

   Fig 4.7 Model of beam with piezoelectric element.

   Material

   Piezoelectric Element – PZT-4

   Element

   Beam

   CPS4R A 4 node bilinear plane stress quadrilateral.

   Total no of elements- 322 Total no of nodes 504 Transducer

   CPS4E – A 4 node bilinear plane stress piezoelectric quadrilateral.

   Total no of elements- 80 Total no of nodes-105 Boundary condition

   Rigidly fixed i.e. U1= U2=UR3=0

   Fig: 4.8 Meshed structures of Beam and Transducer

   Result and Discussion

   Following is the table of the electrical output with respect to the natural frequency of the system .

   Frequency	EPOT	Frequency	EPOT
   671.1	19.01	6040	22.36
   1342	69.18	6711	7.918
   2013	17.22	7383	9.147
   2685	18.74	8054	1.076
   3356	20.73	8725	0.432
   4698	76.94	9396	10.22
   5369/p>	3.262

   Electric Potential

   Electric Potential

   Table 4.3. Frequency Vs Electric Potential in Piezoelectric Element.

   Frequency vs Electric potential

   Frequency vs Electric potential

   100

   80

   60

   40

   20

   0

   100

   80

   60

   40

   20

   0

   0

   2000

   4000

   6000

   8000

   10000

   0

   2000

   4000

   6000

   8000

   10000

   Frequency

   Frequency

   Fig: 4.9 Frequency Vs output voltage

3. Finite element modelling of proving ring with piezoelectric layers as a sensor and dynamic analysis under sinusoidal loading with the variation in frequency with constant load and with the variation of load under constant frequency.

   Proving ring is a device which is used to calibrate the instruments which apply the static loading eg. Universal Testing Machine etc. The following model is presented with an expectation that the same ring can be used to calibrate the instruments involving the dynamic loading if we use the piezoelectric strips as a sensor.

   The modeling of proning ring has been done as follows. It contains five parts, one is the ring made of stainless steel and rest of the four parts are made of piezoelectric material PZT-4. The steel ring is the main body and the rest of the four parts are made as integral part of the main ring with TIE option. The construction is as thefollowing diagram.

   Fig 4.10 Model of Proving Ring with Piezoelectric Strip

   Now each part will be described individually. The first part is the ring made of stainless steel as shown in the following figure. The steel ring is modeled as followed-

   Inner Diameter 175mm Outer Diameter 185mm No. of elements 70

   No. of nodes – 350 Element type CPS8R

   Fig 4.11 Ring made of Stainless steel

   Other four part are made of piezoelectric material PZT-4.All the four strips are having thickness of 1mm. All the strips are modeled as follows

   No. of elements 40 No. of nodes – 203 Element type CPS8RE

   Fig 4.12 Piezoelectric Strips

   1. Static Analysis

      After the modeling of the transducer , the first part of the analysis is static analysis. In the part of static analysis , the analysis is done by applying the static load on the top point in the downward direction. The range of load applied is 20N to 500 N. The result of the static analysis is shown in the Table 4.5

      S. No.	Load	Max. Strain	Plastic Strain
      1	20	6.937 x 10-12	0
      2	40	1.137 x 10-11	0
      3	60	2.081 x 10-11	0
      4	80	2.775 x 10-11	0
      5	100	3.469 x 10-11	0
      6	150	5.203 x 10-11	0
      7	200	6.937 x 10-11	0
      8	250	8.672 x 10-11	0
      9	300	1.041 x 10-10	0
      10	350	1.214 x 10-10	0
      11	400	1.387 x 10-10	0
      12	450	1.561 x 10-10	0
      13	500	1.734 x 10-10	0

      Table 4.5 Result of Static Analysis

      After going through the result of the above table, we see that the proving ring is safe for a load of 500N . Though the dynamic analysis is done for a small loads but static analysis shows that the ring is safe for the load of 500N. It may be safe for the higher load also but as per our current requirement the testing up to 500N is enough. Also we see that there is no plastic strain developed in the ring so our working limit is the elastic region only. So the conclusion of the static analysis is that we can use this ring for dynamic analysis to see the effect of various type of loading.

   2. Dynamic Analysis

      Before going to the dynamic analysis of the proving ring with piezoelectric layers , the natural

      frequencies of the model has been calculated by using the step frequency and then model dynamics. After the calculation of natural frequencies of the system , the analysis has been divided in two parts.

      1. The analysis under the variation of load keeping the frequency constant.

      2. The analysis under the constant load with the variation of frequency.

After the analysis the result are collected in graphical and numerical form and presented. Accordingly this analysis gives us the idea that a proving ring can also be used in calibration of equipments which apply dynamic loading.

1. The analysis under the variation of load keeping the frequency constant.

   In this part of the analysis the analysis is done by keeping the frequency of loading constant at 100 cycles /second. The analysis time is 0.1 second. The output was calculated in the form of Electric flux at the outermost and inner most surfaces of the piezoelectric strips. The load is varied from 5N to 65 N in various steps. Following are the graphical results of the output with respect to the time.

   Load 5N Frequency 100 cycles/sec
   Inner Layer	Outer Layer

   Fig 4.13 Electric Flux Generated with time for ( 5N, 100 cycles/sec)

   Load 10N Frequency 100 cycles/sec
   Inner Layer	Outer Layer

   Fig:4.14 Electric Flux Generated with time for ( 10N, 100 cycles/sec)

   Load 15N Frequency 100 cycles/sec
   Inner Layer	Outer Layer

   Fig 4.15 Electric Flux Generated with time for ( 15N, 100 cycles/sec)

   Load 20N Frequency 100 cycles/sec
   Inner Layer	Outer Layer

   Fig 4.16 Electric Flux Generated with time for ( 20N, 100 cycles/sec)

   Load 25N Frequency 100 cycles/sec
   Inner Layer	Outer Layer

   Fig 4.17 Electric Flux Generated with time for ( 25N, 100 cycles/sec)

   Load 30N Frequency 100 cycles/sec
   Inner Layer	Outer Layer

   Fig 4.18 Electric Flux Generated with time for ( 30N, 100 cycles/sec)

   Load 55N Frequency 100 cycles/sec
   Inner Layer	Outer Layer

   Load 55N Frequency 100 cycles/sec
   Inner Layer	Outer Layer

   Load 35N Frequency 100 cycles/sec
   Inner Layer	Outer Layer

   Load 35N Frequency 100 cycles/sec
   Inner Layer	Outer Layer

   Fig 4.19 Electric Flux Generated with time for ( 35N, 100 cycles/sec)

   Load 40N Frequency 100 cycles/sec
   Inner Layer	Outer Layer

   Fig 4.20 Electric Flux Generated with time for ( 40N, 100 cycles/sec)

   Load 45N Frequency 100 cycles/sec
   Inner Layer		Outer Layer
   Load 50N Frequency 100 cycles/sec
   Inner Layer	Outer Layer

   Fig 4.21 Electric Flux Generated with time for 45N, 100 cycles/sec, 50N, 100cycles/sec.

   Fig 4.22 Electric Flux Generated with time for ( 55N, 100 cycles/sec)

   Load 60N Frequency 100 cycles/sec
   Inner Layer	Outer Layer

   Fig 4.23 Electric Flux Generated with time for ( 60N, 100 cycles/sec)

   Load 65N Frequency 100 cycles/sec
   Inner Layer	Outer Layer

   Fig 4.24 Electric Flux Generated with time for ( 65N, 100 cycles/sec)

   After seeing the above result , and analyzing minutely, we find that the output we are getting is in the form of sinusoidal curves, we also find that on increasing the load the output flux value is increasing but the response is almost the same wave form . It implies that on increasing the load the out put flux value is affected not the type of out put. The numerical data related to electric flux range with the variation in loading is given in the Table 4.6

   S. No .	Load (N)	Frequency (cycles/sec ond)	Range of Electric Flux
   			Inner Layer	Outer Layer
   1	5	100	-15 x 10-12 — 15 x 10- 12	-15 x 10-12 — 20 x 10-12
   2	10	100	-0.04x 10-9 — 0.02 x10-9	-0.02x 10-9 — 0.04 x10-9
   3	15	100	-0.05x 10-9 — 0.04 x10-9	-0.04x 10-9 — 0.06 x10-9
   4	20	100	-0.06x 10-9 — 0.05 x10-9	-0.05x 10-9 — 0.07 x10-9
   5	25	100	-0.10x 10-9 — 0.06 x10-9	-0.06x 10-9 — 0.10 x10-9
   6	30	100	-0.12x 10-9 — 0.07 x10-9	-0.28x 10-9 — 0.16 x10-9
   7	35	100	-0.13x 10-9 — 0.08 x10-9	-0.30x 10-9 — 0.20 x10-9
   8	40	100	-0.15x 10-9 — 0.10 x10-9	-0.35x 10-9 — 0.25 x10-9
   9	45	100	-0.16x 10-9 — 0.11 x10-9	-0.40x 10-9 — 0.25 x10-9
   10	50	100	-0.18x 10-9 — 0.12 x10-9	-0.42x 10-9 — 0.27 x10-9
   11	55	100	-0.20x 10-9 — 0.15 x10-9	-0.50x 10-9 — 0.30 x10-9
   12	60	100	-0.23x 10-9 — 0.16 x10-9	-0.55x 10-9 — 0.35 x10-9
   13	65	100	-0.25x 10-9 — 0.20 x10-9	-0.60x 10-9 — 0.40 x10-9

   S. No .	Load (N)	Frequency (cycles/sec ond)	Range of Electric Flux
   			Inner Layer	Outer Layer
   1	5	100	-15 x 10-12 — 15 x 10- 12	-15 x 10-12 — 20 x 10-12
   2	10	100	-0.04x 10-9 — 0.02 x10-9	-0.02x 10-9 — 0.04 x10-9
   3	15	100	-0.05x 10-9 — 0.04 x10-9	-0.04x 10-9 — 0.06 x10-9
   4	20	100	-0.06x 10-9 — 0.05 x10-9	-0.05x 10-9 — 0.07 x10-9
   5	25	100	-0.10x 10-9 — 0.06 x10-9	-0.06x 10-9 — 0.10 x10-9
   6	30	100	-0.12x 10-9 — 0.07 x10-9	-0.28x 10-9 — 0.16 x10-9
   7	35	100	-0.13x 10-9 — 0.08 x10-9	-0.30x 10-9 — 0.20 x10-9
   8	40	100	-0.15x 10-9 — 0.10 x10-9	-0.35x 10-9 — 0.25 x10-9
   9	45	100	-0.16x 10-9 — 0.11 x10-9	-0.40x 10-9 — 0.25 x10-9
   10	50	100	-0.18x 10-9 — 0.12 x10-9	-0.42x 10-9 — 0.27 x10-9
   11	55	100	-0.20x 10-9 — 0.15 x10-9	-0.50x 10-9 — 0.30 x10-9
   12	60	100	-0.23x 10-9 — 0.16 x10-9	-0.55x 10-9 — 0.35 x10-9
   13	65	100	-0.25x 10-9 — 0.20 x10-9	-0.60x 10-9 — 0.40 x10-9

   Table 4.6 : Electric Flux Range for Load Variation at Constant Frequency

   EFLX on Inner Layer with Load Variation

   0	10	20	30	40	50	60	7

   0	10	20	30	40	50	60	7

   3.00E-10

   2.00E-10

   Electric Flux

   Electric Flux

   1.00E-10

   0.00E+00

   0

   -1.00E-10

   -2.00E-10

   -3.00E-10

   Load,N

   Fig 4.25 Flux on Inner layer with load variation

   EFLX On Outer Layer with Load Variation

   0	10	20	30	40	50	60	7

   0	10	20	30	40	50	60	7

   6.00E-10

   4.00E-10

   Electric Flux

   Electric Flux

   2.00E-10

   0.00E+00

   electrical response on the application of mechanical loading. On the basis of the result of the following- In the section 4.1, the analysis of cantilever beam made of only piezoelectric material is done. The study shows that such a beam can be used as a transducer for the dynamic loading. Natural frequency of the system is also an important part so that we can avoid the loading frequency with the

   natural frequency.

   In section 4.2, analysis of a shell made of PZT- 4 with the brass cap is done. In which mode shape has been drawn and also the electrical response has been calculated. The same model has been tested for dynamic loading for a frequency of 250 cycles/sec and for a loading of 20N to 200 N. In which we find that electrical flux output is continuously increasing with increment of load. Thus we can say that such a transducer can be used as dynamic loading.

   In section 4.3 analysis of a cantilever beam with a piezoelectric strip as a transducer is done. It is analyzed at a frequency of 250 cycles/ second and a load from 10N to 120N. We observe that with the increase in load the variation in electrical flux output is linear as shown in figure 4.22. Also we can understand that in stead of a complete beam of piezoelectric material, we can use a small strip of piezoelectric as transducer in big structures to analyze dynamic loading.

   In section 4.4 the analysis of proving ring with piezoelectric strip by varying load and frequency is done. When we vary the load, we find that there is regular increment in electrical flux output while the frequency remains constant. and when the load is constant at 5N and frequency varied from 1000 cycles/sec to 15000cycles/sec then we find there is no change in the electrical flux output value even though the frequency changes. Hence by using such transducers we can get the magnitude as well as frequency of the dynamic load with proper calibration. The effect of temperature has been observed by which we can say that temperature has a good impact on the response of piezoelectric transducer. In the studies as mentioned above, the main emphasis is given to the sensor. Piezoelectric behave as sensors when it gives electrical response on the application of mechanical loading. After completing the above studies, we can now conclude that a piezoelectric sensor can be modeled

   -2.00E-10

   -4.00E-10

   -6.00E-10

   -8.00E-10

   0

   Load,N

   successfully by finite element method even without any experimentation. After the detailed study of the work is done, we find that the electrical response of any sensor made of piezoelectric is dependent on

   Fig 4.26 Flux on Outer layer with load

   variation

   In the studies as described in earlier sections, the main emphasis is given to the sensor. Piezoelectric behave as sensors when it gives

   the following factors

   • Loading condition i.e. what type of loading is there, loading frequency and magnitude of load.

   • Temperature of the work environment.

   • Natural frequency of the system.

• Curie temperature of the piezoelectric material.

Other then the above mentioned factors the electrical response also depend on the material properties, type of boundary condition and also the type of application.

Future Prospect

An important significance of the work presented in the previous sections is to analyze the piezoelectric transducers by finite element analysis and thus actual experiments can be avoided which is a time consuming and costly process. Piezoelectric transducers are widely used for the system where dynamic loading is present such as miniature accelerometer, high frequency accelerometer , load cell etc. all such systems can be modeled and analyzed by finite element analysis. Other then sensors, the reverse of it i.e. actuators can also be modeled analyzed by the finite element method.

We can also develop some systems on FEA packages and can understand the effect of various type of geometries , shape , size etc on the response of transducer. Also the temperature effect can be analyzed more elaborately .

REFERENCES

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[3]K.Y. Szea, X. M. Yanga, L.-Q Yao. Stabilized plane and axisymmetric piezoelectric finite element models. Finite Elements in Analysis and Design 40 (2004) 11051122

[4]K. Y. SZE Y. S. PAN. Hybrid finite element models for piezoelectric materials Journal of Sound and vibration (1999) 226(3), 519-547 Article No. jsvi.1999.2308

[5]Jin H. Huang, Wen-Long Dai. Static and dynamic electromechanical responses of piezoelectric transducers. Materials Letters 50(2001).209218

[6]F. Cote, P. Masson, N. Mrad , V. Cotoni. Dynamic and static modelling of piezoelectric composite structures using a thermal analogy with MSC/NASTRAN. Composite Structures 65 (2004)

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[7]R. Garcia Lage, C.M. Mota Soares, C.A. Mota Soares, J.N. Reddy. Modelling of piezolaminated plates using layer wise mixed finite elements. Computers and Structures 82 (2004) 18491863.

[8]Benjeddou, J.-F. Deu¨, S. Letombe. Free vibrations of simply-supported piezoelectric adaptive plates: an exact sandwich formulation. Thin-Walled Structures 40 (2002) 573593 [9]Ayech Benjeddou, Jean-Francois Deu. A two- dimensional closed-form solution for the free- vibrations analysis of piezoelectric sandwich plates. International Journal of Solids and Structures 39 (2002) 14631486

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2. Christophe Poizat, Matthias Sester. Effective properties of composites with embedded piezoelectric fibers. Computational material science 16(1999) 89-97

3. Amancio Fernandes, Joel pouget. Two- dimensional modelling of laminated piezoelectric composites: analysis and numerical results. Thin walled structures 39(2001) 3-22

4. Tirupati R. Chandrupatala, Ashok D. Belegundu (2004), Introduction to Finite Elements in Engineering Pearson Education Pte. Ltd [14]H.F. Tiersten (1969), Linear Piezoelectric Plate Vibration Plenum Press, New York [15]Warren P. Mason (1960), Piezoelectric Crystal and Their Application to Ultrasonic (1960)

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