Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

55, 1, pp. 69-86, Warsaw 2017
DOI: 10.15632/jtam-pl.55.1.69

THE CONSTITUENT EQUATIONS OF PIEZOELECTRIC CANTILEVERED

THREE-LAYER ACTUATORS WITH VARIOUS EXTERNAL LOADS

AND GEOMETRY

Grzegorz Mieczkowski

Bialystok University of Technology, Białystok, Poland

e-mail: g.mieczkowski@pb.edu.pl

This paper presents test results for deformation conditions of three-layer, piezoelectric can-
tilever converters subjected to various electrical and mechanical boundary conditions. A
general solution has been developed based on implementation of piezoelectric triple seg-
ments (PTS) to the beam. A working mechanism and conditions for strain of the PTS
segment have been determined. Basing on the general solution, for the cantilever actuator
subjected to an external force (of single and dual PTS segments) and a uniform load (of
singlePTS segment), particular solutionshave alsobeendeveloped.Moreover, dimensionless
frequencies of the oscillating motion for the analyzed converters have been determined. In
the next step, the influence of such factors as length, quantity andposition ofPTS segments,
their relative stiffness and unit weight on values of the obtained frequencies of vibration ha-
ve been defined. The resulting analytical solutions have been compared with the developed
FEM solution.

Keywords: piezoelectric bender, constitutive equations, deflection, analytical solutions

1. Introduction

Piezoelectric transducers have been used over the years in many devices. These are exploited
as sensors (Ştefănescu, 2011), actuators (Tzou, 1999), energy harvesters (Liu et al., 2014) or
dynamic eliminators (Przybyłowicz, 1999). Their working principle is based on the conversion of
electric energy tomechanical or other way around (Bush-Vishniac, 1999). The relation between
strain and electric field is defined by constitutive equations ((Curie andCurie, 1880; Berlincourt
et al., 1964).
A significant aspect influencing the functionality and durability of converters is selection of

a proper piezoelectric material. The properties of typical piezoelectric materials are presented
in papers by Kawai (1969), Rajabi et al. (2015).
Another important factor is a static characteristic of the converter.Whendesigningpiezoelec-

tric converters for specific application, it is necessary to define and solve adequate simultaneous
equations. These equations bound together geometrical properties,material properties and phy-
sical parameters such as force, deflection and electric field. Solving such simultaneous equations
is very difficult. Materials and geometrical inhomogeneity of the converter global structure and
anisotropy of piezoelectric materials forces the use of some reductions. Smits et al. (1991), by
using energetic methods, formed and solved constitutive equations for a converter made of two
layers of even length (piezoelectric bimorph). In the paper by Wang and Cross (1999) there is
an issue of a three-layer converter extended and solved, whereas in (Xiang and Shi, 2008) – a
multi-layer one. The static characteristics of two-layer converters with different length of layers
are shown in (Park andMoon, 2005; Raeisifard et al., 2014; Mieczkowski, 2016).
Piezoelectric converters as well as other elastic bodies having mass are prone to vibrations.

Therefore, it is necessary to define the nature of their oscillating motion. Many authors have



70 G.Mieczkowski

dealt with this issue. Chen et al.(1998), Askari Farsangi et al. (2013) studied free vibrations of
piezoelectric laminated plates presented,whereasClare et al. (1991)analysed a simply supported
beamwith piezoelectric patches. Also, analyses of dynamic characteristics of transducers where
forced vibration occurred are shown in papers by Rouzegar and Abad (2015), Bleustein and
Tiersten (1968), Djojodihardjo et al. (2015). Dynamical aspects of converters with the piezo-
electric patches, including control strategy, were considered byTylikowski (1993), Pietrzakowski
(2000, 2001), Buchacz and Płaczek (2009).

It is a very rare case that in literature one can find results of tests describing the influence of
geometrical andmaterial characteristics, number and location of piezoelectric patches on static
deflection and free vibration.

In such cases, in order to determine electromechanical behaviour of the converter, usually
the FEM-based analyses are carried out, see (Rahmoune and Osmont, 2010; Mieszczak et al.,
2006). However, carrying out this type of analyses is verywork-consuming and the solutionmay
be subjected to high error.

Therefore, themain purpose of the present paper is to develop a simple analyticalmethod for
determining deflection in function of mechanical and electric loads. By design, the piezoelectric
converters have individual components (layers) of different length and the piezoelectric layer can
be divided,which is an extension to the researches shown in papers by Smits et al. (1991),Wang
and Cross (1999), Xiang and Shi (2008), where lengths of the beam element and piezoelectric
layers were equal, see Fig. 1.

Fig. 1. Three-layer piezoelectric converter, 1 – beam, 2 – piezoelectric elements, 3 – piezoelectric triple
segmentPTS

The proposedmethod involves the implementation ofmodules to a homogeneous beam, fur-
ther referred to as the piezoelectric triple segment (PTS). This allows including a local change in
stiffness and strain caused by the transverse piezoelectric effect within the analytical description
of the beam deflection.

In view of material and geometric discontinuity in the analyzed transducers, natural fre-
quencies of oscillating motion turn out to be different than those in the homogeneous beam.
Therefore, the next aim of the paper is analysis of dynamical behavior of such converters. In
the present work, examination similar to that carried out by Clare et al. (1991) is conducted.
It is extended by analysis of the influence of the number, length and location of piezoelectric
segments and their material properties on the natural frequency values.

In order to verify the correctness of the solutions, it is required to compare the obtained
resultswith experimental data or results obtained usingothermethods.Therefore, for converters
with the diversified material and geometric structure, FEM simulations have been made and
compared with the obtained analytical results.



The constituent equations of piezoelectric cantilevered three-layer actuators... 71

2. Analytical results

2.1. Basic assumptions

The converters analysed in this work are treated as a homogeneous (one-layer) beam with
locally implemented piezoelectric triple segments PTS (Fig. 1). The PTS is made up of three
components – two piezoelectric and one non-piezoelectric element. The non-piezoelectric layer
thickness is the same as the beam thickness. The beam and thePTS both have the samewidth.
In order to simplify the mathematical model, the following assumptions are made:

• bending of the element takes place according to Euler’s hypothesis, and radii of curvatures
of the deflected components are identical,

• in the connection plane between components there is no intermediate layer and no sliding
occurs,

• in the piezoelectric layer, the transverse piezoelectric effect 1-3 takes place causing clear
bending.

2.2. General solution to the piezoelectric converter with implemented PTS segment

Thetask is to consider a section of thepiezoelectric converter (Fig. 2) subjected tomechanical
bending moment M(x) and electric moment Me (following the occurrence of the piezoelectric
effect).

Fig. 2. Section of the piezoelectric converter

In the analysed element subjected to bending, it is possible to determine three characteristic
ranges related to a change in load and stiffness. Within the x1 < x < x2 range, there is a
piezoelectric triple segment PTS (generating Me) with flexural stiffness EpJo. The other two
ranges are a homogeneous beamwith stiffness EbJb. Since there are several characteristic ranges
on the beam, it is convenient to make use of Heaviside’s function. Thus, including the PTS
presence in the beam, the deflection line can be described using the following dependence

∂2y

∂x2
=
M(x)

EbJb
+Meγ(H[x−x1]−H[x−x2]) (2.1)

where: H[x−xi] – Heaviside’s function, Ep, Eb – Young’s moduli of the piezoelectric and non-
piezoelectric element, Jb, Jo –moments of inertia (described in Section 2.3), γ – factor including
the change in stiffness with applied formal notation of Heaviside’s function

γ =
EbJb[Me+M(x)]−EpJoM(x)

EbJbEpJoMe

As determining themechanical momentM(x) in general does not pose any problems, deter-
mining the electrical load Me generated by the PTS is very burdensome and requires solving
the two-dimension problem of thePTS bending.



72 G.Mieczkowski

2.3. Piezoelectric triple segment PTS

The task is to consider the segment PTS (Fig. 3) with constant width b consisting of non-
-piezoelectric (2) and piezoelectric layers (1) and (3).

Fig. 3. Distribution of forces and conditions for strain of the piezoelectric triple segmentPTS

The structure is not subjected to anymechanical load. The longitudinal forces Ni and ben-
dingmoments Mi occurring in individual layers are a result of the applied voltage v. Basing on
the equilibrium equation of forces. the following can be rritten

N1+N2+N3 =0 (2.2)

The sum of moments in relation to the upper interface must be zero, therefore

M1+M2+M3−
N2tb
2
−N3

(

tb+
tp
2

)

+
N1tp
2
=0 (2.3)

According to the adopted Euler hypothesis, bendingmoments can be described as follows

M1 =
Ep
Jp
ρ M2 =

Eb
Jb
ρ M3 =

Ep
Jp
ρ (2.4)

Substituting dependences (2.4) to (2.3) andmaking simple transformations results in the follo-
wing

1

ρ
=
(N2+2N3)tb+(N3−N1)tp

2EbJb+4EpJp
(2.5)

Including the relation between the radius of curvature ρ and deflection w(x)

1

ρ
=
∂2w

∂x2
(2.6)

the differential equation for the converter bending can be found as follows

∂2w

∂x2
=
(N2+2N3)tb+(N3−N1)tp

2EbJb+4EpJp
(2.7)



The constituent equations of piezoelectric cantilevered three-layer actuators... 73

The constitutive equations for all converter layers, including the piezoelectric effect in layer 1
and 3 give the following

∂ux1
∂x
=

N1
EpAp

−d31
(−v

tp

) ∂ux2
∂x
=

N2
EbAb

∂ux3
∂x
=

N3
EpAp

+d31
(−v

tp

)

(2.8)

where Ab = tbb, Ap = tpb are layers cross sectional areas, d31 – piezoelectric constant.
Following the relocation continuity condition (Fig. 3), it is found that

ux1−ux2−
∂w

∂x

(tp
2
+
tb
2

)

=0 ux1−ux3−
∂w

∂x

(tp
2
+
tb
2
+ tb
)

=0 (2.9)

Solving differential equations (2.7) and (2.8) with the following boundary conditions

∂w

∂x
(0)= 0 w(0)= 0

ux1(0)= 0 ux2(0)= 0 ux3(0)= 0
(2.10)

and applying dependence (2.9) and (2.2), the longitudinal force Ni can be determined

N1 =
−bEpvd31(Ebt

3
b +2Ept

3
p)

α
N2 =0 N3 =

bEpvd31(Ebt
3
b +2Ept

3
p)

α
(2.11)

where: α = Ebt
3
b
−2Eptp(3t

2
b
+6tbtp+2t

2
p).

The differential equation for bending PTS in the Me moment function can be written as
follows

∂2w

∂x2
=
−2Me
EpJo

(2.12)

On the basis of comparing equations (2.7) and (2.12), it is possible to determine the bending
moment Me which results from the piezoelectric effect

Me =
EpJo[−(N2+2N3)tb+(N1−N3)tp]

4EbJb+8EpJp
(2.13)

where moments of inertia for the individual layers are, respectively

Jb =
bt3b
12

Jp =
bt3p
12

(2.14)

The averaging value of the moment of inertia Jo can be calculated using the method of
transformation of the cross sectional area (Fertis, 1996). Three materials of different stiffness
moduli and the same width b (Fig. 4a) are replaced with one material of the section composed
of three parts of different widths (Fig. 4b).
The sought moment of inertia, calculated in relation to the neutral layer, is

Jo =
bβ

12Ep
(2.15)

where: β = Ebt
3
b
+2Eptp(3t

2
b
+6tbtp+4t

2
p).

Substituting formula (2.13) with (2.11), (2.14) and (2.15) results in the electric bending
moment value in function of the applied voltage v

Me =−
bEpvβd31(tb+ tp)

2α
(2.16)



74 G.Mieczkowski

Fig. 4. Original (a) and transformed (b) section in the piezoelectric triple segmentPTS

2.4. Particular solutions

This part of the work is concerned with the application of the proposed method based
on implementingPTS segments into the single-layer beam to determine analytical dependences
describingbendingof the converters of fixedgeometry andknownboundaryconditions. Solutions
for the converters of different external loads and thePTS number shall be presented.

2.4.1. Cantilever converter subjected to concentrated force F of a single PTS segment

In the converter, as shown in Fig. 5, the left end is fixed and the right end can move freely.
The load results from the external force F and the electric moment Me is generated by the
applied voltage v. Based on the conditions for equilibrium of forces andmoments, the reactions
in the mounting are established: Ry = F, Rx =0, MF = FL.

Fig. 5. Cantilever converter of the singlePTS segment

Themechanical moment M(x) takes the following form

M(x)=−MF +Ryx =−FL+Fx (2.17)

Substituting expressions (2.16) and (2.17) to the general solution described with formula (2.1),
upon double integration gives a dependence describing the function of bending of the analysed
converter

y(x)= A1v+B1F (2.18)

where



The constituent equations of piezoelectric cantilevered three-layer actuators... 75

A1 =
−3d31Ep(tb+ tp)

α

(

H[x−x1](x−x1)
2−H[x−x2](x−x2)

2
)

B1 =
2

bβEbt
3
b

{

(β −Ebt
3
b)
(

H[x−x1](3L−x−2x1)(x−x1)
2

−H[x−x2](3L−x−2x2)(x−x2)
2
)

−βx2(3L−x)
}

The integration constants are determined on the basis of the following boundary conditions

∂y

∂x
(0)= 0 y(0)= 0 (2.19)

2.4.2. Cantilever converter subjected to concentrated force F of two PTS segments

For theconverter showninFig. 6 theconditionsofmountingandmechanical loadare identical
as in the case described in Section 2.4.1. The electrical load is generated by two PTS segments
powered by voltage v1 and v2.

Fig. 6. Cantilever converter of twoPTS segments

The differential equation for deflection is as follows

∂2y

∂x2
=
M(x)

EbJb
+Me1γ(H[x−x1]−H[x−x2])+Me2γ(H[x−x3]−H[x−x4]) (2.20)

In the formula above, the mechanical moment M(x) is described by equation (2.17), while the
electrical moments are

Me =−
bEpviβd31(tb+ tp)

2α
i =1,2 (2.21)

Solving differential equation (2.20) and assuming boundary conditions (2.19) gives the function
describing bending of the analysed converter

y(x)= A1v1+A2v2+B2F (2.22)

where

A2 =
−3d31Ep(tb+ tp)

α

(

H[x−x3](x−x3)
2−H[x−x4](x−x4)

2
)

B2 =
2

bβEbt
3
b

{

(β −Ebt
3
b)
(

H[x−x1](3L−x−2x1)(x−x1)
2

−H[x−x2](3L−x−2x2)(x−x2)
2+H[x−x3](3L−x−2x3)(x−x3)

2

−H[x−x4](3L−x−2x4)(x−x4)
2
)

−βx2(3L−x)
}



76 G.Mieczkowski

2.4.3. Cantilever converter subjected to uniform external load p of the single PTS segment

In the actuator shown in Fig. 7, the conditions of mounting are identical as those described
in Sections 2.4.1 and 2.4.2. The converter is acted on by a uniform load p. The electrical moment
is generated by a singlePTS located at the left end.

Fig. 7. Cantilever converter of a singlePTS segment

Based on the conditions for equilibriumof forces andmoments, the reactions in themounting
are determined: Ry = p(L−x2), MF = 0.5p(L−x2)(L+x2). The mechanical moment M(x)
takes the following form

M(x)= Ryx−MF −
p(x−x2)

2

2
H[x−x2] (2.23)

Solving differential equation (2.1), assumingmechanical (2.23) and electric (2.16)moments gives
the equation describing deflection of the analysed converter

y(x)= A3v+B3p (2.24)

where

A3 =
−3d31Ep(tb+ tp)

α

(

H[x]x2−H[x−x1](x−x1)
2
)

B3 =
1

2bβEbt
3
b

{

(β −Ebt
3
b)

·
(

H[x−x1](H[x−x2](x−x2)
4+2(x−x1)

2(L−x2)(3L−2x+3x2)
)

−H[x]
(

H[x−x2](x−x2)
4+2x2(L−x2)(3L−2x+3x2)

))

+β
(

H[x−x2](x−x2)
4+2x2(L−x2)(3L−2x+3x2)

)}

Integration constants are determined on the basis of the following boundary conditions

∂y

∂x
(0)= 0 y(0)= 0 (2.25)

2.5. Dynamical behavior of piezoelectric converters

Asmentioned before, piezoelectric transducers similarly to other elastic bodies havingmass,
are prone to vibrations. In this respect, it is advised to include its oscillating nature of motion
in the process of designing and exploitation of piezoelectric structures.Generally, vibrations can
be divided into two groups – free and forced. Free vibrations occur when external forces do not
influence the body and the system vibrates due to action of inherent forces. In that case, the



The constituent equations of piezoelectric cantilevered three-layer actuators... 77

system is going to vibrate with one or more natural frequencies. In vibrating systems natural
damping occurs caused by forces of the internal friction. Damping is usually slight, thus does
not affect the natural frequencies.

In the case when vibrations are caused by external forces, there appear the so called forced
vibrations, and the system is going to vibrate with the excitation frequency.

The phenomenon of resonance is greatly dangerous for a structure. It occurs when the exci-
tation frequency coincides with one of the natural frequencies which causes perilously high
oscillations that may lead to damaging of the structure. Therefore, it is necessary to determine
the natural frequencies and geometrical andmaterial factors affecting their distribution.

Upon elementary theory of bending beams, equation (1) can be presented in the following
way

∂2

∂x2

(

E(x)J(x)
∂2y

∂x2

)

+ρ(x)A(x)
∂2y

∂t2
=
∂2M(x)

∂x2
+Me(δ

′[x−x1]− δ
′[x−x2]) (2.26)

where δ[x−xi], E(x), J(x), ρ(x), A(x) is the derivative of Dirac’s function, Young’s modulus,
moment of inertia, density and cross sectional area of the converter. For free vibration, the
exactly same differential equation can be written in the following way

∂2

∂x2

(

E(x)J(x)
∂2y

∂x2

)

+ρ(x)A(x)
∂2y

∂t2
=0 (2.27)

The solution to equation (2.27) using the method of separation of variables can be written in
the following way

y(x,t)= W(x)T(t) (2.28)

whereW(x) is a functionof space, andT(t) dependsonlyon time. Substituting (2.28)with (2.27)
and performing simple mathematical modifications, the commonly known differential equation
describing the beam boundary problem is obtained

∂2

∂x2

(

E(x)J(x)
∂2W

∂x2

)

−ρ(x)A(x)ω2 =0 (2.29)

where ω is the natural frequency of vibration.

Let us consider the piezoelectric transducer shown in Fig. 8.

Fig. 8. Cantilever converter of n PTS segment

There can be n fragments distributed on the transducer whose total length is equal to
∑n
i=1Lmi = L. Each and every fragment consists of three elements – a PTS segment and



78 G.Mieczkowski

two beam elements. Applying local frames of reference, using the dimensionless coordinates,
differential equation (2.29) is equivalent to

∂4W1,i(ζ1,i)

∂ζ41,i
−ψ4W1,i(ζ1,i)= 0 for ζ1,i ∈ 〈0,κi〉

∂4W2,i(ζ2,i)

∂ζ42,i
−Λ4ψ4W2,i(ζ2,i)= 0 for ζ2,i ∈ 〈0,χi〉

∂4W3,i(ζ3,i)

∂ζ43,i
−ψ4W3,i(ζ3,i)= 0 for ζ3,i ∈ 〈0,

Lmi
L
−κi−χi〉

(2.30)

where

ψ4 = L4ω2
ρbAb
EbJb

Λ4 =
EbJb
ρbAb

ρbAb+2ρpAp
EpJo

ζ1,i =
x

L
ζ2,i =

x−κiL

L
ζ3,i =

x−κiL−χiL

L

ζ1,i+1 =
x−Lmi

L
i =1, . . . ,n

ω is the natural frequency, ρb, ρp is density of beam and piezoelectric material, respectively.

Solutions to differential equations (2.30) can be obtained as (Mahmoud and Nassar, 2000)

W1,i(ζ1,i)= A1,i sin(ψζ1,i)+B1,icos(ψζ1,i)+C1,icosh(ψζ1,i)+D1,i sinh(ψζ1,i)

W2,i(ζ2,i)= A2,i sin(Λψζ2,i)+B2,icos(Λψζ2,i)+C2,icosh(Λψζ2,i)+D2,i sinh(Λψζ2,i)

W3,i(ζ3,i)= A3,i sin(ψζ3,i)+B3,icos(ψζ3,i)+C3,icosh(ψζ3,i)+D3,i sinh(ψζ3,i)

(2.31)

where Aj,i, Bj,i, Cj,i, Dj,i, j =1,2,3 are constants.

The boundary conditions of mounting of the converter (left end is fix-mounted and the right
one freelymove) together with continuity conditions at the intermediate ends lead to a set n×12
linear homogeneous equations.

The continuity conditions adopt the following form

W1,i(ζ1,i)
∣

∣

∣

ζ1,i=κi
= W2,i(ζ2,i)

∣

∣

∣

ζ2,i=0

∂W1,i(ζ1,i)

∂ζ1,i

∣

∣

∣

∣

∣

ζ1,i=κi

=
∂W2,i(ζ2,i)

∂ζ1,i

∣

∣

∣

∣

∣

ζ2,i=0

∂2W1,i(ζ1,i)

∂ζ21,i

∣

∣

∣

∣

∣

ζ1,i=κi

= η
∂2W2,i(ζ2,i)

∂ζ22,i

∣

∣

∣

∣

∣

ζ2,i=0

∂3W1,i(ζ1,i)

∂ζ31,i

∣

∣

∣

∣

∣

ζ1,i=κi

= η
∂3W2,i(ζ2,i)

∂ζ32,i

∣

∣

∣

∣

∣

ζ2,i=0

W2,i(ζ2,i)
∣

∣

∣

ζ2,i=χi
= W3,i(ζ3,i)

∣

∣

∣

ζ3,i=0

∂W2,i(ζ2,i)

∂ζ2,i

∣

∣

∣

∣

∣

ζ2,i=χi

=
∂W3,i(ζ3,i)

∂ζ3,i

∣

∣

∣

∣

∣

ζ3,i=0

η
∂2W2,i(ζ2,i)

∂ζ22,i

∣

∣

∣

∣

∣

ζ2,i=χi

=
∂2W3,i(ζ3,i)

∂ζ23,i

∣

∣

∣

∣

∣

ζ3,i=0

η
∂3W2,i(ζ2,i)

∂ζ32,i

∣

∣

∣

∣

∣

ζ2,i=χi

=
∂3W3,i(ζ3,i)

∂ζ33,i

∣

∣

∣

∣

∣

ζ3,i=0

(2.32)



The constituent equations of piezoelectric cantilevered three-layer actuators... 79

W3,i(ζ3,i)
∣

∣

∣

ζ3,i=φi
= W1,i+1(ζ1,i+1)

∣

∣

∣

ζ1,i+1=0

∂W3,i(ζ3,i)

∂ζ3,i

∣

∣

∣

∣

∣

ζ3,i=φi

=
∂W1,i+1(ζ1,i+1)

∂ζ1,i+1

∣

∣

∣

∣

∣

ζ1,i+1=0

∂2W3,i(ζ3,i)

∂ζ23,i

∣

∣

∣

∣

∣

ζ3,i=φi

=
∂2W1,i+1(ζ1,i+1)

∂ζ21,i+1

∣

∣

∣

∣

∣

ζ1,i+1=0

∂3W3,i(ζ3,i)

∂ζ33,i

∣

∣

∣

∣

∣

ζ3,i=φi

=
∂3W1,i+1(ζ1,i+1)

∂ζ31,i+1

∣

∣

∣

∣

∣

ζ1,i+1=0

where

η =
EpJo
EbJb

φi =
Lmi
L
−κi−χi

The boundary conditions can be written as follows:
— fixed

W1,1(ζ1,1)= 0
∣

∣

∣

ζ1,1=0

∂W1,1(ζ1,1)

∂ζ1,1
=0

∣

∣

∣

∣

∣

ζ1,1=0

(2.33)

— free

EbJb
∂2W3,n(ζ3,n)

∂ζ23,n
=0

∣

∣

∣

∣

∣

ζ3,n=
Lmn
L
−κn−χn

EbJb
∂3W3,n(ζ3,n)

∂ζ33,n
=0

∣

∣

∣

∣

∣

ζ3,n=
Lmn
L
−κn−χn

(2.34)

Using dependences (2.31)-(2.34), as mentioned before, n×12 linear homogenous equation can
be achieved.
The values of dimensionless frequencies ψ are determined from the characteristic equation

representing the zero determinant of the matrix of boundary conditions M12n×12n

M
12n×12n =













M
12×12
i B

12×12 · · · 012×12

C
12×12
i M

12×12
i+1 · · · 0

12×12

...
...

... B12×12

012×12 012×12 C
12×12
n−1 M

12×12
n













(2.35)

where 012×12 is the zero matrix, M12×12i =
[

M
12×4
1,i M

12×4
2,i M

12×4
2,i

]

, the remaining matrices

are shown in Appendix. For the transducer with one segment, matrix (2.35) simplifies to the
following form

M
12×12 =

[

M
12×4
1,1 M

12×4
2,1 M

12×4
2,1

]

(2.36)

An analytical form of the characteristic equation (|M12nx12n| = 0) in special cases where
∑n
i=1χi = 0 (homogenous beam) and

∑n
i=1χi = 1 (PTS segment all along) is described from

subsequent equations

1+cosψcoshψ =0 1+cos(Λψ)cosh(Λψ) = 0 (2.37)

In other cases, in order to determine ψ, the roots of the characteristic equation can be obtained
using numerical methods. In Figs. 9 and 10, there are dimensionless frequencies ψ presented in
function of length of the piezoelectric layer for different locations and amount ofPTS segments.
The results achieved for the transducer with onePTS segment (Lm1 = L) are shown in Fig. 9,
whereas for the transducer with two PTS segments (Lm1+Lm2 = L, Lm1 = Lm2) are shown
in Fig. 10. Furthermore, three variants of piezoelectric segment locations in the transducer have
been examined. Namely:



80 G.Mieczkowski

• PTS segment located on the left end (κi =0),

• transducer withPTS segment located in the middle (κi+0.5χi =0.25),

• PTS segment located on the right end (κi+χi =1).

Moreover, in the transducer with two piezoelectric segments there are identical geometric and
material features adopted for bothPTS segments.

Fig. 9. The dimensionless frequencies ψ for the converter eith a singlePTS segment, ψ1 – first
frequency, ψ2 – second frequency

Fig. 10. The dimensionless frequencies ψ for the converter with a doublePTS segment, ψ1 – first
frequency, ψ2 – second frequency



The constituent equations of piezoelectric cantilevered three-layer actuators... 81

Analyzing the obtained results, it canbe stated that the highest values of thefirst dimension-
less frequency, independent fromPTS length, is acquired for the piezoelectric segment placed on
the left end (Figs. 9 and 10). Similarly is with the second frequency for the converter with two
segments. For the converter with one segment, the second frequency, depending onPTS length,
takes the highest values for the converter with the segment placed either on the left end, or in
the middle.

The dimensionless frequencies ψ depend on the relative stiffness η and the unit weight
µ =(ρbAb+2ρpAp)/(ρbAb) of thebeamand thePTS segment. InFigs. 11a and11b, the influence
of η and µ on the ψ value. Upon the received results, it can be stated that the dimensionless
frequencies decrease with an increase in the parameters η (Fig. 11a) and µ (Fig. 11b).

Fig. 11. The impact of relative stiffness (a) and relative unit (b) on the dimensionless frequency ψ for
the converter with a singlePTS segment, ψ1 – first frequency, ψ2 – second frequency

The circular frequency ω [Hz] of the transducer can be calculated from formula

ω =
ψ2

2π

√

EbJb
ρbAbL

4
(2.38)

3. Numerical calculations

To confirm the correctness of the obtained analytical solutions (static deflection of the trans-
ducer and circular frequencies), it is necessary to perform numerical analyses. FEM simulations
have been prepared and compared with the obtained analytical results. Numerical tests aimed
at determining the bending line and circular frequencies of the actuators for arbitrarily assu-
med material constants and geometry. The tested converters, shown in Figs. 5-7, have been
modelled using the FEM with the help of ANSYS (Mieszczak et al., 2006; Documentation for
ANSYS, 2010). Plane components hve been described using a grid of quadrangular, eight-node
finite elements, with increased concentration at critical points such as sharp corners, mounting
points and places at which the mechanical load was applied. For the piezoelectric component,
PLANE223 type elements have been applied, and non-piezoelectric material has been meshed
with PLANE183 elements with steel material properties. In view of the fact that the actuators
are usuallymade of piezoelectric ceramics, PZ26 has been used as amaterial of the piezoelectric
component. The size of finite elementswas tp/4. The plane issue has been solved for plane strain
conditions.

In the calculations, the following geometrical andmaterial data has been assumed: Young’s
modulus Ep =7.7 ·10

10N/m2, Eb =2.0 ·10
11N/m2; Poisson’s ratio νp =0.3, νb =0.33; density

ρp = 7700kg/m
3, ρb = 7860kg/m

3; piezoelectric strain coefficients d31 = −1.28 · 10
−10m/V,



82 G.Mieczkowski

d33 = 3.28 · 10
−10m/V, d15 = 3.27 · 10

−10m/V; beam length L = 60mm; layers thickness
tp = 0.5mm, tb = 1mm. The values of applied load are: electrode voltage v = 100V; force
F =100N; uniform external load p =100N/m.Coordinates xi, χi (Figs. 5-8), defining thePTS
and external loads application positions are given in Section 4.

4. Results of tests

In this part of thework, the deflection line of converters for which the resulting special solutions
are given in Section 2.4 are graphically presented. The results obtained from the analytical
solutions have been compared with FEM solutions. In the analytical equations, the material
and geometrical data are identical as the data given in Section 3 was applied.

A comparison of analytical solution (2.18)withFEMfor the cantilever converter of the single
PTS segment (Fig. 5) is shown in Fig. 12.

Fig. 12. Deflection of the cantilever converter with a singlePTS segment for x1 =1/12L, x2 =5/12L:
(a) subjected only to electrical voltage, v =100V, F =0; (b) subjected only to force, v =0V,

F =100N

Figure 13 shows the deflection of the cantilever converter of doublePTS segments (Fig. 6),
for which the analytical solution is described by formula (2.22).

Fig. 13. Deflection of the cantilever converter with doublePTS segments for x1 =1/12L, x2 =5/12L,
x3 =7/12L, x4 =11/12L: (a) subjected only to electrical voltage, v1 =100V, v2 =150V, F =0;

(b) subjected only to force, v1 = v2 =0, F =100N

Figure 14 shows the strain of the cantilever converter subjected to a uniform external load p
of the single PTS segment (Fig. 7) for which the analytical solution is described by formula
(2.24).

Basing on the obtained results of static deflections of the converters, the qualitative and
quantitative compliance of analytical and numerical solutions can be stated. Generally, the
difference between the analytical and numerical solutions is approx. 1% for the electrical load,
and 2-3% for the mechanical load.



The constituent equations of piezoelectric cantilevered three-layer actuators... 83

Fig. 14. Deflection of the cantilever converter subjected to a uniform external load p for x1 =1/3L,
x2 =2/3L: (a) subjected only to electrical voltage, v =−100V, p =0; (b) subjected only to uniform

external load, v =0V, p =100N/m

By means of FEM, it has been helpful to determine circular frequencies of the transducers
with one (Fig. 7) and two (Fig. 6) PTS segments. The material and geometric data has been
adopted identically as in the above-described static analysis. The obtained results (Table 1) have
been compared to the analytical solution (formula (2.38) and Figs. 9 and 10).

Table 1.Comparison of the first two frequencies between the analytical and FEM results

Mode
sequence ω∗ [Hz] ω∗∗ [Hz] Error [%] ω∗ [Hz] ω∗∗ [Hz] Error [%]

1 368.5 379.4 2.9 252.1 256.4 1.7

2 1799.8 1837.7 2.1 1575.4 1543.8 2
∗ – analytical solution, ∗∗ – FEM solution

In the dynamical analysis as well as in the static analysis, a satisfactory compatibility of
both obtained solutions has been indicated. The disparity in these results ranges less than 3%.

5. Summary and conclusions

The paper deals with the issue of bending of three-layer piezoelectric actuators subjected to
electric field and mechanical load. A general solution has been developed, based on the im-
plementation of piezoelectric segments PTS to a homogeneous (one-layer) beam. The working
mechanism and conditions for strain of thePTS segment have been determined. Basing on the
general solution, for arbitrarily selected three different types of converters, special solutions ha-
ve been developed (for the cantilever actuator of single and double PTS segments subjected
to external force and the converter with a single PTS acted on by a uniform external load).
Moreover, dynamical analysis of transducers has been performed. Also, a matrix whose deter-
minant enables determination of the characteristic equation for the transducerwith any amount
of piezoelectric segments has been formulated. On the basis of characteristic equations, for the
converter with one and two PTS, the natural frequencies and the influence of relative stiffness,
size and placing of a segment on their value have been determined. The obtained analytical
solutions have been compared with the developed FEM solution.
On the basis of the performed analytical and numerical tests, it is found that:

• the developedmethod involving the implementation ofPTS segments into a homogeneous
beam allows obtaining solutions for piezoelectric converters:

– of any either type of the external load,



84 G.Mieczkowski

– of diverse lengths and heights of piezoelectric and non-piezoelectric layers,

– with any number of piezoelectric components;

• the obtained particular solutions allows determination of the deflection at any point of the
converter;

• themaximum values of the first dimensionless frequency, independently of the length and
number of PTS, are to be obtained for segments with mountings located closer;

• for a transducer with two PTS segments, the distribution of the second frequency is the
same as for the first frequency;

• for a transducer with one piezoelectric segment, the second frequency depending on PTS
length holds the highest value for either the converter with the segment located on the left
end or for the one with the segment located in the middle;

• the dimensionless frequencies decrease with an increase in the relative stiffness and unit
mass of the piezoelectric segment;

• the particular solutions of static behaviour confirm with the results obtained from FEM
(for the electrical load themaximumdifference is approx. 1%, and for themechanical load
– approx. 3%);

• a similar discrepancy between the analytical solution and FEM (less than 3%) has been
obtained while calculating circular frequencies.

Appendix

M
12×4
1,i =



































0 1 1 0
ψ 0 0 ψ

sin(ψκi) cos(ψκi) cosh(ψκi) sinh(ψκi)
ψcos(ψκi) −ψsin(ψκi) ψsinh(ψκi) ψcosh(ψκi)
−ψ2 sin(ψκi) −ψ

2cos(ψκi) ψ
2cosh(ψκi) ψ

2 sinh(ψκi)
−ψ3cos(ψκi) ψ

3 sin(ψκi) ψ
3 sinh(ψκi) ψ

3cosh(ψκi)
0 0 0 0
...

...
...

...
0 0 0 0



































M
12×4
2,i =















































0 0 0 0
0 0 0 0
0 −1 −1 0
−Λψ 0 0 −Λψ
0 ηΛ2ψ2 −ηΛ2ψ2 0

ηΛ3ψ3 0 0 −ηΛ3ψ3

−sin(Λψχi) −cos(Λψχi) −cosh(Λψχi) −sinh(Λψχi)
−Λψcos(Λψχi) Λψ sin(Λψχi) −Λψsinh(Λψχi) −Λψcosh(Λψχi)
ηΛ2ψ2 sin(Λψχi) ηΛ

2ψ2cos(Λψχi) −ηΛ
2ψ2cosh(Λψχi) −ηΛ

2ψ2 sinh(Λψχi)
ηΛ3ψ3cos(Λψχi) −ηΛ

3ψ3 sin(Λψχi) −ηΛ
3ψ3 sinh(Λψχi) −ηΛ

3ψ3cosh(Λψχi)
0 0 0 0
0 0 0 0

















































The constituent equations of piezoelectric cantilevered three-layer actuators... 85

M
12×4
3,i =































0 0 0 0
...

...
...

...
0 1 1 0
ψ 0 0 ψ
0 −ψ2 ψ2 0
−ψ3 0 0 ψ3

−ψ2 sin(ψφi) −ψ
2cos(ψφi) ψ

2cosh(ψφi) ψ
2 sinh(ψφi)

−ψ3cos(ψφi) ψ
3 sin(ψφi) ψ

3 sinh(ψφi) ψ
3cosh(ψφi)































B
12×12=













0 0 0 0 0 · · · · · · 0
...
...

...
...
... · · · · · ·

...
0 ψ2 −ψ2 0 0 . . 0
ψ3 0 0 −ψ3 0 . . 0













C
12×12
i =













0 · · · −sin(ψφi) −cos(ψφi) −cosh(ψφi) −sinh(ψφi)
0 · · · −ψcos(ψφi) ψ sin(ψφi) −ψ sinh(ψφi) −ψcosh(ψφi)
...
...

...
...

...
...

0 · · · 0 0 0 0













References

1. Askari Farsangi M.A., Saidi A.R., Batra R.C., 2013, Analytical solution for free vibrations
of moderately thick hybrid piezoelectric laminated plates, Journal of Sound and Vibration, 332,
5981-5998

2. Berlincourt D.A., Curran D.R., Jaffe H., Mason W.P., 1964,Physical Acoustics, Princi-
ples and Methods, Academic Press, NewYork

3. Bleustein J.L., Tiersten H.F., 1968, Forced thickness-shear vibrations of discontinuously pla-
ted piezoelectric plates, Journal of the Acoustical Society of America, 43, 6, 1311

4. Buchacz A., Płaczek M., 2009, Damping of mechanical vibrations using piezoelements, inclu-
ding influence of connection layer’s properties, on the dynamic characteristic, Solid State Pheno-
mena, 147/149, 869-875

5. Busch-Vishniac I.J., 1999,Electromechanical Sensors and Actuators, Springer-Verlag

6. ChenW.Q.,XuR.Q.,DingH.J., 1998,On freevibrationof apiezoelectric composite rectangular
plate, Journal of Sound and Vibration, 218, 4, 741-748

7. ClarcR.L.,FullerCh.R.,WicksA., 1991,Characterizationofmultiple piezoelectricactuators
for structural excitation, Journal of Acoustical Society of America, 90, 346-357

8. Curie P.J., Curie J., 1880, Crystal physics-development by pressure 0/ polar electricity in he-
mihedral crystals with inclined faces (in French), Comptes rendus hebdomadaires des sances de
l’Acadmie des sciences, 294, pp. 91

9. Djojodihardjo H., Jafari M., Wiriadidjaja S., Ahmad K.A., 2015, Active vibration sup-
pression of an elastic piezoelectric sensor and actuator fitted cantilevered beam configurations as
a generic smart composite structure,Composite Structures, 132, 848-863

10. Documentation for ANSYS, 2010, Coupled-Field Analysis Guide

11. Fertis D.G., 1996,Advanced Mechanics of Structures, Marcel Dekker, NewYork

12. Kawai H., 1969, The piezoelectricity of poly (vinylidene fluoride), Japanese Journal of Applied
Physics, 8, 975

13. Liu X., Wang X., Zhao H., Du Y., 2014, Myocardial cell pattern on piezoelectric nanofiber
mats for energy harvesting, Journal of Physics: Conference Series, 557, 012057



86 G.Mieczkowski

14. Mahmoud A.A., Nassar M.A., Free vibration of a stepped beam with two uniform and/or
tapered parts,Mechanics and Mechanical Engineering, 4, 2, 165-181

15. Mieczkowski G., 2016, Electromechanical characteristics of piezoelectric converters with freely
defined boundary conditions and geometry,Mechanika, 22, 4, 265-272

16. Mieszczak Z., Krawczuk M., Ostachowicz W., 2006, Interaction of point defects in piezo-
electric materials – numerical simulation in the context of electric fatigue, Journal of Theoretical
and Applied Mechanics, 44, 4, 650-665

17. Park J.K., Moon W.K., 2005, Constitutive relations for piezoelectric benders under various
boundary conditions, Sensors and Actuators A, 117, 159-167

18. PietrzakowskiM., 2000,Multiple piezoceramic segments in structural vibration control,Journal
of Theoretical and Applied Mechanics, 38, 378-393

19. Pietrzakowski M., 2001, Active damping of beams by piezoelectric system: effects of bonding
layer properties, International Journal of Solids and Structures, 38, 7885-7897

20. Przybyłowicz P.M., 1999, Application of piezoelectric elements to semi-adaptive dynamic ele-
minator of torsional vibration, Journal of Theoretical and Applied Mechanics, 37, 2, 319-334

21. Raeisifard H., Bahrami M.N., Yousefi-Koma A., Fard H.R., 2014, Static characteriza-
tion and pull-in voltage of a micro-switch under both electrostatic and piezoelectric excitations,
European Journal of Mechanics A/Solids, 44, 116-124

22. Rahmoune M., Osmont D., 2010, Classic finite elements for simulation of piezoelectric smart
structures,Mechanika, 86, 6, 50-57

23. Rajabi A.H., Jaffe M., Arinzeh T.J., 2015, Piezoelectric materials for tissue regeneration: A
review, Review Article,Acta Biomaterialia, 24, 15, 12-23

24. Rouzegar J., Abad F., 2015, Free vibration analysis of FG plate with piezoelectric layers using
four-variable refined plate theory,Thin-Walled Structures, 89, 76-83

25. Smits J.G., Dalke S.I., Cooney T.K., 1991, The constituent equations of piezoelectric bimor-
phs, Sensors and Actuators A, 28, 41-61

26. ŞtefănescuD.M. 2011,PiezoelectricForceTransducers (PZFTs), [In:]Handbook of Force Trans-
ducers, 109-130

27. Tylikowski A., 1993, Stabilization of beam parametric vibrations, Jurnal of Theoretical and
Applied Mechanics, 31, 3, 657-670

28. Tzou H.S., 1999,Piezoelectric Shells: Distributed Sensing and Control of Continua, Kluwer Aca-
demic Publishers, Dordrecht

29. Wang Q., Cross L.E., 1999, Constitutive equations of symmetrical triple-layer piezoelec-
tric benders, IEEE Transactions on Ultrasonics, Ferroelectronics, and Frequency Control, 46,
1343-1351

30. Xiang H.J., Shi Z.F., 2008, Static analysis formulti-layered piezoelectric cantilevers, Internatio-
nal Journal of Solids and Structures, 45, 1, 113-128

Manuscript received October 6, 2015; accepted for print May 6, 2016