Ijmet 09 11_017

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International Journal of Mechanical Engineering and Technology (IJMET)
Volume 9, Issue 11, November 2018, pp. 146–160, Article ID: IJMET_09_11_017
Available online at http://www.iaeme.com/ijmet/issues.asp?JType=IJMET&VType=9&IType=11
ISSN Print: 0976-6340 and ISSN Online: 0976-6359
© IAEME Publication Scopus Indexed
CONVERTING FORCED VIBRATIONS INDUCED
IN PIEZOELECTRIC CANTILEVER PLATE INTO
NANO ELECTRICAL POWER
Dr. Hatem Hadi Obeid
Assist. Professor
Department of Mechanical Eng. / University of Babylon
ABSTRACT
In current paper, unimorph cantilever laminated plate was constructed using
piezoelectric material and aluminum metal to study their ability to generate energy under
action of wind load. The Euler Bernoulli method was used to investigate the behavior of
power harvesting structures theoretically. The cantilever plate structure examined under
simple harmonic and random point loads. The obtained results showed that, the harmonic
excitation gives power more than random excitation and the frequency is of contrariwise
effect on the harvested power.
The cantilever plate structure that was analyzed under harmonic action and random.
The obtained results showed that, the structure gives powers (0.00061834 W in Pin-force
method,0.00049586 W in Enhanced pin-force method and 0.0000059969 W in Euler
Bernoulli method) within 3 second.
KEYWORDS: Unimorph Cantilever plate, PZT, Euler Bernoulli, Random Force,
Finite Element Analysi
Cite this Article Dr. Hatem Hadi Obeid, Converting Forced Vibrations Induced in
Piezoelectric Cantilever Plate into Nano Electrical Power, International Journal of
Mechanical Engineering and Technology, 9(11), 2018, pp. 146–160.
http://www.iaeme.com/IJMET/issues.asp?JType=IJMET&VType=9&IType=11
1. INTRODUCTION
Most forms of energy and power can be converted from form to another. Of course, energy exists
in many forms, such as solar energy, water flow, wind, temperature gradients, vibrations, sound
waves, etc. The energy generated from natural resources refers to the term renewable energy. Its
indicated in 2006 that more than 18% of the global consumed power came from different forms
of renewable power which is considered non-negligible ratio [P. Basset,2005]. The production
of renewable energies lies between the enormous of Mega Watts to low of Nano Watts range
according to the method of generation. figure (1) Shows three divisions of renewable energies
based upon the generated power ratings such that they are depending upon their applications i.e.

Figure 1 Different renewable energies sources and applications
• Enormous power generation sources rated in megawatts.
• Low power generation sources rated in micro to miliwatt.
• Nanotechnology power Sources.
Through the rest years it was possible to convert energy associated with vibration induced in
structures to electrical one based on properties of piezoelectric materials. The Brothers Pierre and
Jacques Curie in 1880 were the first to prove and demonstrate the effect of piezoelectric to convert
vibration energy into electrical one. Briefly, in the piezoelectric material a dipole deformation
and charge formation occur when subjected to loads and inducing stresses. On the contrary,
stresses are induced when a voltage is generated through piezoelectric material of a polarized
attitude. The polarity will be lost in piezoelectric material when temperature is raised above
Curing [ Ikeda,1990], Priya,2007] and [ Sodano,2005].
A piezoelectric material is that generates an electric voltage drop when subjected into
squeezed or stretched load and stresses are induced. On other hand, shrink or expand
deformations are induced when an electric charge is applied on the PZT. These effects are
generated in crystal structural that have central un symmetry.
Let us first understand the common dielectric material to illustrate the piezoelectric effect. High
permittivity dielectrics [Jyoti K. Ajitsaria,2008] can be formulated as:
Figure 2 Piezoelectric Transducers [Priya,2007]

Converting Forced Vibrations Induced in Piezoelectric Cantilever Plate into Nano Electrical
Power

€

€ €

€

1
And

€

2
The electric displacement (denoted by D), which is the charge density or charge to area ratio
of the capacitor:

€

3
And the electric field formulated as:
Equations (1) to (4) are valid for isotropic dielectrics. PZT ceramic material is considered
isotropic in the state of unpolarizing, but they may be considered anisotropic in the state of poling.
For anisotropic material, the electric field and electric displacement can be formulated in three
dimensions vector form. That lead to the dependent of the dielectric displacement and electric
field ratio on the direction or capacitor orientation with respect to the crystal axes. Then the
electric displacement is written as an equation of state variables [Jyoti K. Ajitsaria,2008] as:
Electric displacement is acting or generated in parallel with the electric field. That let the
electric displacement vector (Di), is the product of summations of field vector (Ej) and dielectric
constant, ij [Jyoti K. Ajitsaria,2008] as:
6
Most of the constants of dielectric for PZT ceramics are characterized to be opposed to single
crystal PZT material, then they are eliminated to zeros [Jyoti K. Ajitsaria, 2008]. The remaining
non-zero terms are formulated as: .
According to ANSI/IEEE [1988], [ANSI / IEEE Standard,1988], the PZT bender constitutive
equations are formulated as:
The boundary conditions that may be used are assigned as: Constant stresses (T), Constant
strains (S), Constant electrical displacements (D) and Constant field (E).
2. MODELING OF PZT SENSER
PZT sensor model is considered to be a lamination of PZT and metal bended as shown in figure
(3). Euler-Bernoulli propose mathematical model of the PZT – metal lamination plate response.
The bonding is assumed to be not permit of slipping between layers. Neutral axis can be assigned
using modulus weighted algorithm, [Wang,K,2001].

Figure 3 Euler Bernoulli model of PZT - metal [Wang,K,2001]
The location of neutral axis can be specified as: [Eggborn,2003]
The average strain ϵa that induced in PZT is estimated and then used to find the generated voltage drop
as:
Substitution of Equation (9) into (10) lead to obtain the induced average strain as: [Eggborn,2003]
The voltage on the PZT poling surfaces is also can be obtained in relation to the stress as:
[Wang, K,2001]
The substitution of equation (13) into (14) and applying hooks law leads to the voltage as:
[Eggborn,2003]
3. CANTILEVER PTZ PLATE POWER HARVESTING
Figure (4) shows a cantilever rectangular lamination plate proposal model. The model is
considered to be thin plate satisfying the conditions of the geometric ratios between the
dimensions according to [Wang,K,2001]. The laminated plate is consisting of aluminum metal
and PTZ bonded together perfectly as assumed such that he PZT covers all the top surface of the
aluminum plate. The PZT elastic module is (62×109)Pa, while that for the aluminum is
(71×109)Pa. the analysis is required to estimate an equivalent Young’s Modulus for the laminated
plate as: [Wang,K,2001]

Power
Where is the plate’s Young’s Module, and is the plate thickness. The equivalent
modulus Estiff is estimated as (68.943×109)Pa.
For the purpose of estimating natural frequencies of laminated plate, Ritz method was used
as applied by Blevins (1987). For a vibrating elastic plate, equation of maximum potential energy
is given as:
Where w(x,y) is the lateral deformation (displacement) through x and y domains, is the
equivalent Poisson’s ratio for laminated plate with a value of 0.28, Beer [1992].
Where D is the flexural rigidity of aluminum plate, t is thickness of the PZT and aluminum
referred as the plate thickness. The generated voltages are estimated by applying equation (15),
a small difference will be appeared due to the variable , which take into the account a ratio of
laminated plate total thickness to thickness of PZT, which is a part of the total thickness (t).
The plate’s flexural vibration equation is:
Where:
plate’s vibrational displacement w(x,y) can be estimated as, Young [1950]
Where is an indexed coefficient that satisfying boundary conditions of cantilever plate.
Then, the mode shapes Plate can specify for each mode. The first three modes are intended for
that purpose. The first mode shape is represented as shown in Figure (5) which is much similar
to the multiplication two orthogonal beams with proper boundary conditions through x and y axis.
The plate has clamped-free boundary conditions in x axis:

And free-free boundary conditions in y axis:
The clamped-free modes are given by the equation
Table 1 gives coefficients and , where (a) is the length of plate. The free-free modes
through y axis are given by the equation: [Young (1950)]
Also, Table 1gives coefficients and , where b is the width of plate. Equation (23a)
represents rigid-body translation while equation (23b) represents rigid-body rotation, and
equation (23c) specifies free - free boundary conditions. Table 2 shows the correct combination
of both of Xi(x) and Yi(y). [Young [1950]
Table 1: The coefficients , , and Young [1950]
i 1 2 3 4 5 6 7 8 9
1.875104
1
4.6940911 7.8547574
10.995540
7
14.137168
4
11*π/2 13*π/2
15*π/
2
17*π/
2
0.734095
5
1.0184664
4
0.9992245
0
1.0000335
5
0.9999985
5
1 1 1 1
0 0 4.7300408 7.8532046
10.995607
8
14.137165
5
17.278759
6
13*π/
2
15*π/
2
0.9825022
2
1.0007773
1
0.9999664
5
1.0000014
5

Power
Table 2 Mode shapes pairings
Mode,Φ(x,y) pairings
1
2
3
4
5
Next, the following integrals is needed to solve:
The values of indices i, k, m and n are from 1 to 6 that set equations (24a-f) form (6*6) matrix.
Then after some derivation, Blevins reaches to the characteristic equation:
Where the product and for mn = ik, while set to zero for mn ≠ ik.
The matrix C is formed from the equations ( ) by the two equations:
It may be valid for the off-diagonal where mn ≠ ik.
While for diagonal terms mn=ik:
4. FINITE ELEMENT MODELLING
Finite element model was created to the laminated plate using the ANSIS Package. The elements
SOLID5, PLANE13, SOLID98, PLANE223, SOLID226, and SOLID227 were used to create the
finite element model. The element of (SOLID5) have been used as the cantilever beam model
and for the electrical solution the element (CIRCU94) have been used and the model can be
shown in figure 6, The element of (CIRCU94) used to add resistor to the model as shown and the
point of node must wiring to the node of the resistor and then the voltage will be compute.

Figure 6 unimorph Cantilever Plate finite element model and modal analysis
5. HARVESTING ENERGY IN TERM OF DYNAMIC EXCITATION
It’s about to derive mathematical model that describing the energy generated in the laminated
plate when subjected into dynamic excitation. Two types of excitations will be considered. The
first is harmonic while the second is random excitation.
5.1. Harmonic Driving Force:
The dynamic harmonic excitation is given by [Eggborn, 2003] as
Where:
F0: is the amplitude of load function.
: is excitation frequency.

Power
is displacement of the forcing function through beam length, , , is displacement of
the forcing function through width.
After applying orthogonality principle, it can be reduce the force model into:
Where:
Then it’s possible to propose the time response of the plate through x and y domains as:
Researchers had been found when the plate is excited by dynamic load applied at fundamental
natural frequency of the plate, only odd modes are excited. Thus, the modal deflections,
and that corresponding with the even modes can be eliminated and equal to zero. The
number of modes used in the deflection calculation is reduced to three. The flexural profile shape
of the plate through x and y axis can now be detected using curvature equations of plate as:
Then, the total or overall plate curvature can be estimated by the product two separate
curvatures together (similarly as in the plate’s mode shapes) as:
For elimination of the curvature’s dependent on the location through plate domain, average
curvature can estimate as:
Considering the first five mode shapes, only fifth mode has a dependent on width along y
axis. While Second and fourth modes dependent on y but do they aren’t contributing to the plate
curvature since they aren’t excited; therefore, only the first and third modes will be considering
leads to a suitable estimation of the average curvature, thus equation (34) can be reduced to:
The moment that applied on the plate can be estimated as: Timoshenko [1959]

where is the moment applied per unit length which can be multiplied by width of plate to
get units of N.m as:
To get maximum harvesting power, the time response must be maximizing but does not
exceed the allowable safety limit. That may be occurred under resonance when the plate is excited
closer to fundamental natural frequency. In addition to excitation closer to second, third …. Fifth
natural frequencies.
5.2. Random Driving Force:
The dynamic random excitation is given by Inman[2000] as:
The frequency is that excited within the first five natural frequencies range, ( )
is the phase shift within the range between 0 and π. n is number of iterations that generating
random excitation function sufficiently. The range of frequency that considered is 0 to 1000 Hz
such that the first two resonant modes are excited and included. [Inman2000] proposed a
generalized time response using mode summation method as:
given as:
The curvature equation for the plate can be described as:
The equation used to calculate power from an AC voltage signal is:
Where V is the source voltage, is the Resistance load, is the resistance of source
and n is the number of time steps.
6. RESULTS AND DISCUTIONS
6.1. Case study properties:
The dimensions and the properties of the aluminum plate can be shown in table3

Power
Table 3: the dimension and properties of Aluminum plate
Aluminum plate properties
ParameterValueUnit
Length0.010m
Width0.050m
Thickness9*10-4m
density2715Kg/m3
Young's Modulus71×109Pa
The dimensions and the properties of the PZT can be shown in table 4
Table 4 the dimension and properties of PZT plate
PZT properties
ParameterValueUnit
Length0.10m
Width0.05m
Thickness2.667*10-4m
Young's Modulus62×109Pa
Dielectric constant-320×10-12m/v
Voltage constant-9.5×10-3Vm/N
Internal resistance3900Ω
Table 5 shows the five natural frequencies calculated by the Ritz method Young (1950). The
estimated natural frequencies and mode shapes are using for both harmonic and random
excitations.
Table 5 The first five natural frequencies
iω (rad/s)if (Hz)Mode
1274202.61
50388022
660910523
1616025714
2343037295
6.2. Plate with harmonic excitation:
Figure 7 shows the estimated harvesting power in addition to the corresponding load impedance.
It’s obvious the maximum harvested power occurs when the load impedance coinciding with the
internal resistance of the PZT. The harvesting power estimated with each method is indicated in
Table 6. The results show that Euler Bernoulli method gives lower magnitudes of harvesting
power if compared with the other two methods.
Table 6 the power harvest from plate model, harmonic forces.
MethodPower (W )Power (µW )
Pin force0.00061834618.34
Enhanced Pin force0.00049586495.86
Euler Bernoulli0.00000599695.9969

Figure 8 illustrates the power harvest from the analytical plate model and the voltage output
from model.
6.3. Plate with random excitation:
The first five simulation presented in table 7 and the average power estimated will be compute
and figures 9, 10, and 11 shows the voltage of the 1st, 2nd
and 3th simulation respectively, of
three methods and the figures 12, 13, and 14 show the power estimation from first three
simulation with time taking 3 second for three methods.
Figure 8 of the 1st simulation of the plate model, the voltage alternate signal about 0.5V,-
0.5V in Euler-Bernoulli method.5V,-5V in the Pin-force and Enhanced Pin-Force.
In the figure 9 of the 2nd simulation, the shape of the signal output voltage was changed
because the random excitation was not predicted in value and range of frequency.
The same thing was found in the figures 10, of the 3th simulation, the shape of the output
voltage was changed and the value of voltage was changed too.
Table 7 The first five simulation of power estimated from plate
Method
( )
=1500H
z
Power (W )Power (W )
Power (W
)
Power (W )Power (W )Power (W )
1st
simulation
2nd
simulation
3rd
simulation
4th
simulation
5th
simulation
Average
power from
external
random
force
Pin force0.000867620.000452080.000763530.000870050.000379280.000667
Enhance
d Pin
force
0.000695750.000362530.000612290.000697710.000304150.000534
Euler
Bernoull
i
0.000008414
5
0.000004384
4
0.00000740
5
0.000008438
1
0.000003678
4
0.000004640
8
Figure 7 powers of the three methods with load impedance, plate model

Power
Figure 8 power harvest with time from the analytical plate model, harmonic force and output voltage
from model.
Figure 9 PZT voltages calculated from analytical plate model, random forcing, and 1st
simulation
Figure 10 PZT voltages calculated from analytical plate model, random forcing, and 2nd
simulation.
Figure 11 PZT voltages calculated from analytical plate model, random forcing, and 3rd
simulation.

Figure 12 power harvest with time from the analytical plate model, random force and 1st
simulation
Figure 13 power harvest with time from the analytical plate model, random force and 2nd
simulation.
Figure 14 power harvest with time from the analytical plate model, random force and 3rd simulation.
The purpose of this paper is to estimate harvesting power of plate structures, and the best
configuration of vibratory plate illustrate to get maximum power, know from Table (7) we can
taken the average power harvesting within 3 second of Euler Bernoulli method and compute the
power per time. A (0.000667 w) is compute for 3 second and it's be equal to57.6288w per 24
hours.
CONCLUSION
1. The power can be harvesting from any plate structural vibrate by harmonic or random
load according like speed of wind and it is easy to get power along time from the
unimorph plate. A (0.000667 W) per 3 second that mean (57.6288 W) per day is useful
to use as long-life battery.
2. Euler Bernoulli method that used analysis of the behavior the of piezoelectric
elements. Such that deflection, voltage, and power generation from a plate were
estimated under action of excitation of either harmonic or random loads. The force
used to excite the plate is a simulation to the wind load which more practical.
3. You can use this study on bridge structural as a large area of plate and big excitation
random load by moving cars to get a harvesting power that not endless.
Peb
Time
(Second)
Powe
r(W)
Ppin
Time
(Second)
Powe
r(W)
Penh
Time
(Second)
Pow
er(W
)
Peb
Time
(Second)
Power
(W)
Ppin
Time
(Second) *10-
Power(
W)
Penh
Time
(Second) *10-
Pow
er(W
)

Power
REFERENCES
[1] ANSI / IEEE Standard. (1988). IEEE Standard on Piezoelectricity, 176-1987.
[2] Ikeda, T. et al.," Fundamentals of Piezoelectricity". New York: Oxford University Press.
1990.
[3] Blevins, R. D. Formulas for Natural Frequency and Mode Shape. 4th Edition, Robert
E.Krieger Publishing Co., Florida. 1987, p.254.
[4] Beer, F. P. and Johnston, Jr., E. R. Mechanics of Materials, 2nd Ed, McGraw-Hill, Inc.,New
York, 1992.
[5] Young, D. Vibration of Rectangular Plates by the Ritz Method. Journal of Applied Mechanics,
December 1950, pp.448-453.
[6] Stephen P. Timoshenko and S. Woinowsky-Krieger. Theory of Plates and Shells (1959).
[7] Inman, D. J. Engineering Vibration, 2nd edition, Prentice Hall, 2000.
[8] Jyoti K. Ajitsaria, MODELING AND ANALYSIS OF PZT MICROPOWER GENERATOR,
December 19, 2008
[9] P. Basset et al., “Chip-size antennas for implantable sensors and smart dusts”, Proc. of
Transducers’05, Seoul, Korea, 2005
[10] Priya et al., "Advances in energy harvesting using low profile piezoelectric transducers".
Journal of Electro Ceram, 19(3), 165-182. 2007.
[11] Sodano, H., Inman, D. & Park, G. (2005). Comparison of piezoelectric energy harvesting
devices for recharging batteries. Intelligent Material Systems and Structures, 16(10), 799-807.
[12] Eggborn, T., Analytical models to predict power harvesting with piezoelectric materials,
Master’s Thesis Virginia Polytechnic Institute and State University, 2003.
[13] Wang, K. Modeling of Piezoelectric Generator on a Vibrating Beam. For completion of Class
Project in ME 5984 Smart Materials, Virginia Polytechnic Institute and State University,
April 2001.
LIST OF ABBRIVATIONS
R Resistance source Ω
s Elastic Compliance m N⁄
t Thickness Or Plate Separation m
t PZT thickness m
t beam thickness m
V Voltage v
Veb Voltage of Euler Bernoulli method v
Vpin Voltage of Pin-forced method v
Venh Voltage of Enhanced pin-force methods v
V RMS Source Voltage v
ω Frequency rad/s
€ Relative Dielectric Constant Non
€ Dielectric Constant Of Air = 8.85 * 10-12
V m⁄
€ Dielectric Constant V m⁄
ε Mechanical Strain m m⁄
σ Mechanical Stress N m⁄
ρ Density Kg m#⁄
Ζ Damping Ratio Non

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