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International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:19 No:06 131
192706-5353-IJMME-IJENS © December 2019 IJENS I J E N S
Finite Element Modeling of a Cantilever for
Counteracting Vibration Induced Deflection Mohammed W. Al-Hazmi1,
Mohammad S. Alsoufi1, Ahmed H. Backar1,2
1- Mechanical Engineering Department, College of Engineering and Islamic Architecture, Umm Al-Qura University,
Makkah, Saudi Arabia
mwhazmi@uqu.edu.sa 2- Production Engineering Department, Faculty of Engineering, Alexandria University, Egypt
ahmedbackar@gmail.com
Abstract-- Excessive vibration can shorten the life of high-cost
engine components. Moreover, given that vibration is
unavoidable, procedures to control and reduce its effect are
proposed in this paper.
In this study, the vibration characteristics of a cantilever plate
structure excited with piezoelectric actuator elements attached
to the top and bottom surfaces of the plate were investigated.
The piezoelectric actuators were attached to the plate in two
different configurations: straight (parallel to the axis of the plate)
and angled (at an angle of 45o to the axis of the plate).
In this study, a finite element modal analysis of the cantilever
plate structures was conducted using ANSYS to determine the
mode shapes of the plate under investigation. The effect of
exciting the piezoelectric actuators to counteract the deflections
detected by the modal analysis of the plate was then evaluated.
Based on the analysis results, it was concluded that piezoelectric
actuators can be used to overcome the effects of different modes
of cantilever plate vibration.
Index Term-- FEM, modal Analysis, piezoelectric, dynamics
1. INTRODUCTION
Vibration control is critical for high-precision machine
operations. The techniques of vibration control can be
generally divided into active and passive techniques. The
active technique is adopted by utilizing electromechanical, pneumatic, fluidic, electromagnetic, and/or piezoelectric
forces to actively counteract or eliminate the undesirable
oscillation. Active control techniques typically require a
transducer or sensor for the measurement of the real-time
dynamic condition whereas passive techniques involve the
use of absorbers with springs for the dissipation of vibration
energy.
The active vibration reduction technique is more reliable with
respect to vibration suppression over a wide range of
operating conditions, with a lower weight penalty than
passive techniques, i.e., aerospace structures may prevent several mechanical complexities and weight penalties
associated with the use of structure control schemes by
utilizing active (smart) material actuators. Such actuators can
be employed in discrete actuation mechanisms or distributed
along the structure. Moreover, they are advantageous in that
they only require electrical power operation.
In this study, the vibration of a cantilever plate was
investigated using ANSYS finite element analysis software,
and the first six mode shapes and frequencies were detected.
Thereafter, piezoelectric actuators were set on the cantilever
plate in different configurations, and their roles in
counteracting deformations to eliminate the effect of
vibration were then investigated.
2.1 Review
Active vibration control has attracted significant research
attention, and different analysis models have been employed.
Itsuro Kajiwara et al. [1] evaluated the fundamental
characteristics of smart structures that are composed of
dielectric elastomer actuators (DEAs) for the suppression of
vibrations. Experiments were carried out on an acrylic plate,
and piezoelectric actuators were then used to apply a
sinusoidal signal onto the plate. Thereafter, dielectric
elastomer actuators (DEAs) were used to suppress vibrations.
Tuan A. Z. Rahman et al. [2] employed active vibration control to suppress the unwanted vibrations of a flexible
beam structure. The research can be considered as a
theoretical study in which a chaotic fractal search (CFS)
algorithm in the MATLAB programming environment was
employed. In particular, a flexible beam system was
considered as an optimization problem in which a CFS
algorithm was employed to determine the optimal parameters.
Moreover, an algorithm capable of developing an adequate
and stable model was proposed.
Herold et al. [3] introduced an analytical model for designing a piezoelectric stack actuator to act as an adaptive vibration
absorber. The introduced system is a resonance frequency
tuneable one by static pre-stress forces capable of tuning the absorber’s damping. The researchers tested it in laboratory. They concluded that the design presented in their work possesses the potential for vibration control applications, however having design issues limiting its performance. They stated that further investigations and improvements of the absorber are required to enable for practical applications.
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Shunqi et al. [4] investigated the vibration characteristics of
thin-walled structures, and then designed a controller for
vibration suppression in such structures using an electro-
mechanically coupled finite element (FE) model, which was
developed based on first-order shear deformation (FOSD)
hypothesis. A proportional–integral–derivative (PID) controller was used to damp the free and forced vibrations.
Moreover, an optimal linear–quadratic (LQR) controller was
employed for comparison. The implemented control methods
were validated by a piezo-laminated composite plate under
various excitation conditions.
It was concluded that the proposed solution has a negative
influence on free vibration suppression.
Kamel et al. [5] conducted an FE analysis for the
development of a dynamic model for vibration control by
comparing the effects of different controllers on the system
performance. Moreover, the ANSYS software was used to
derive the flexible beam model by conducting a frequency
analysis (modal and harmonic) in four different cases. In each
of the cases, MATLAB was used to extract the state space
model of the beam based on the ANSYS analysis results. The
researchers designed a PID controller based on the model obtained from the first case in MATLAB and subsequently
validated the other three cases. To improve the system
performance, the researchers considered three more
intelligent controller systems: PID auto-tuning (PID-AT),
proportional–derivative (PD) such as a fuzzy controller, and
a self-tuning fuzzy controller (STFC). By comparing the
different controllers, the system subject to the STFC
exhibited the lowest degree of overshoot, shortest rise time,
and shortest settling time. The three abovementioned
controllers based on fuzzy logic (PID-AT, PD, and STFC)
resulted in an improved system performance; whereas, the STFC demonstrated the best system performance.
Fig. 2. Beam and attached piezoelectric transducer (PZT) element
configuration [5]
The researchers glued one side of piezoelectric material to the
upper surface of the beam. A modal analysis was carried out
to determine the structural performance. During the analysis,
the PZT beam was subjected to tip loadings of 2.5 N and 5 N,
which were the two cases of forced vibration. It was
concluded that the ANSYS finite element analysis is a
reliable and effective method for the modeling of dynamic
structures. Moreover, it was stated that the scope of future of
work should include more advanced options in ANSYS, with
different types of beam materials and dimensions. In addition,
experiments should be conducted based on the derived model.
Zuzana Lasovaa et al. [6] employed the FE method to
simulate the piezoelectric effect in thin ceramic layers. First,
a piezoelectric bi-morph beam model was investigated. The
beam was composed of two actuators loaded by opposite
electric potentials, thus causing the beam to bend. Thereafter,
the approximation of the electric potential based on the
thickness of a simple piezoelectric cantilever beam loaded by
an external force at its tip was investigated. The numerical
results obtained were compared using mutually deferent
finite element types, with a simplified analytical solution. A
formula was obtained for the bi-morph beam deflection u at position x using the Euler–Bernoulli beam theory and the
linear variation of electric potential through the thickness of
the beam, as follows:
𝑢 = 3𝑒31𝑈
𝐸ℎ2𝑥2
Biswal et al. [7] developed an FE model of a non-prismatic
piezo-laminated cantilever for voltage generation. The model is based on the Euler–Bernoulli beam formulation. The
Hamilton principle was used to derive the governing equation
of motion. Moreover, a two-node beam element with a two
degree-of-freedom (DoF) per node was used to solve the
governing equation, and the effect of structural damping in
the model was considered. The effects of the taper width and
height on the output voltage were also evaluated.
Fig. 1. Piezoelectric composite plate [4]
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Fig. 3. Piezo-laminated tapered beam [7]
It was concluded that a non-prismatic beam produces a higher voltage for a given length of the PZT patch in comparison
with a prismatic beam, due to the uniform strain distribution.
In addition, a higher increase in voltage was observed in the
case wherein the taper height was varied, and the width was
kept constant than that in which the taper width was varied,
and the height was kept constant.
H. Karagulle et al. [8] modeled smart structures with
piezoelectric materials using ANSYS/Multiphysics. Active
vibration control was first applied to the model with a two
DoF system. The results were then obtained analytically
using the Laplace transform method, and then were compared with those obtained using ANSYS. Thereafter, smart
structures were investigated using ANSYS. The authors
studied a cantilever model with a piezoelectric batch on its
top and a sensor at the other side, and then applied a force at
the free end of the cantilever, as shown in the following figure.
Fig. 4. Configuration of the beam type structure [8]
2.2 Current work
In this study, the analysis was conducted on a plate structure
of a thin rectangular aluminum plate with a thickness of 1 mm,
width of 200 mm, and length of 300 mm. The piezoelectric
material used in the analysis was lead zirconate titanate (PZT-
4) with a thickness of 0.7 mm in different dimensions. The
ANSYS FE software was employed in the study.
2.2.1 Piezoelectric materials In general, piezoelectric materials are solids that generate a
charge in response to mechanical deformation; or
alternatively, they undergo mechanical deformation when
subjected to an electrical field. Consequently, they are
employed as actuators or sensors in the development of smart
structures. However, the use of piezoelectric devices is
dependent on the shape of the element, the polarization
direction of the element, and whether the element is
electrically or mechanically excited.
Piezoelectric materials generally exhibit two
electromechanical phenomena:
The direct piezoelectric effect.
The converse piezoelectric effect.
The piezoelectric element was attached to the cantilevered
beam, as shown in Figure 5. Thin sheet piezo-element
actuators were bonded to the upper and lower faces of the
beam, as shown in Figures 5 (a), (b), and (c), to create
extensional forces, bending moments or coupled with
extensional, bending, and torsional actuation of the structure;
thus allowing for the vibration response to be modified to
achieve specified performance characteristics. For pure
bending, an equal but opposite potential was applied to top and bottom piezoelectric elements, as shown in Figure 5a;
whereas, for pure extension, the same potential was applied
to both piezoelectric elements, as shown in Figure 5b. The
correct positioning of the piezoelectric elements on the upper
and lower faces of the beam allows for the maximum bending
moment to be exerted on the structure in accordance with the
properties of the element. If the actuators are oriented at
opposite angles at the top and bottom surfaces of the beam,
as shown in Figure 5c, and similar electrical voltage of
identical potential is applied to both actuators, a pure twisting
moment is applied to the beam structure.
Fig. 5a. Pure bending excitation Fig. 5b. Pure extensional excitation Fig. 5c. Pure torsional excitation
Fig. 5. Piezoelectric cantilevered beam configurations upon application of an external voltage
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As an analogy and to evaluate the effect of piezoelectric actuators in overcoming the deflections due to the vibration of an
airplane wing, a simplified aluminum cantilever beam model was used to check the results of applying piezoelectric actuators
with different strain distribution cases.
2.2.2 Governing equations
To apply FE modelling, the following parameters and equations were employed.
The piezoelectric permittivity matrix [ε] is a measure of
the charge stored on an electrode material at a given
voltage. The piezoelectric matrix [e] is related to the
electric field vector {E} in order X, Y, Z, to the stress
vector {T} in the order X, Y, Z, XY, XZ, YZ. In the
analysis, the piezoelectric element was assumed to be of
an isotropic material.
The details of the smart plate, in addition to the details
of the PZT patches and epoxy considered in this study, are presented in the following table.
Table I
The dimensions and properties of PZT and aluminum plate and epoxy
parameter Value parameter Value
PZT length 36 mm Aluminum length 300 mm
width 10 mm width 200 mm
thickness 0.7 mm thickness 1 mm
Young’s modulus 99 GPa Young’s modulus 70 GPa
Density ρ 7600
kg/m3
Density ρ 2700 kg/m3
Poisson's ratio v 0.31 Poisson's ratio v 0.33
In this study, an analysis was conducted for the vibration response with 9 and 12 actuator configurations at different positions,
as shown in Figures 6 and 7.
Fig. 6. Plate1 and the piezoelectric actuator co-ordinated systems for bending excitation
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Fig. 7. Piezoelectric actuator co-ordinated and configurations systems for torsional excitation
In this study, two types of analyses were conducted on the cantilever plate structure:
1) Modal analysis to determine the natural frequencies and mode shape of the cantilever plate structure.
2) Harmonic analysis to determine the cantilever plate structural responses and excitation using attached piezoelectric
actuators with different configurations and selective resonance frequencies; for the prediction of the plate dynamic
behavior, and to enable the verification of the optimal actuator position and configuration. A finite element analysis using ANSYS was conducted with a piezoelectric actuator bonded to the structure, SOLID elements
187 and 5 were employed for the cantilever plate and piezoelectric actuators, respectively.
3. MATERIALS AND METHODS
3.1 Modal analysis for cantilever plate structure
Figure 8 and Table 2 present the first five natural frequencies and mode shapes for the cantilever plate structure, as obtained
by the modal analysis using ANSYS.
Fig. 8a Mode shape 1 Fig. 8b Mode shape 2 Fig. 8c Mode shape 3
Fig. 8d. Mode shape 4 Fig. 8g. Mode shape 5
Fig. 8. Mode shapes of cantilever plate structure
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Table II
Modal analysis results for the cantilever plate structure using FEM
Mode shapes Frequency (Hz) Mode shape type
Mode 1 9.46 First Flapping
Mode 2 42.521 First Twisting
Mode 3 62.697 Second Flapping
Mode 4 132.653 Second Twisting
Mode 5 203.256 Fifth Flapping
3.2 Harmonic analysis of the cantilever plate structure
excited by patches of PZT actuators: The harmonic response analysis was used to determine the
steady response of the linear structure under the harmonic
loads.
Under normal conditions, the PZT patches were actuated
using sinusoidal-wave power from the power supply. This
type of PZT structure coupled analysis accorded with the
conditions of the harmonic response analysis.
The harmonic analysis of the plate was conducted under the
assumption that the piezoelectric patch was perfectly bonded
at a different position and double-bonded, as shown in
Figures 6 and 7. The response of the harmonic analysis of aluminum is shown
in Figure 8. As can be seen from the figure, the peak occurred
at the frequencies that correspond to the frequencies
determined by the modal analysis.
For the lateral vibration (bending moment) excitation of the
cantilever plate structure, five different configurations were selected. First, the cantilever plate structure excited with the
first row (1-3x) (Configuration 1) of the piezoelectric
actuators within a frequency range of 0–120 Hz and 120 sub-
steps was investigated with an applied voltage of 100 V.
Second, the cantilever plate structure excited with the second
row (2-3x) (Configuration 2) of piezoelectric actuators on the
plate was evaluated at the same frequency and voltage; in
addition to the plate excited with the third row (3-3x)
(Configuration 3) of piezoelectric actuators. Moreover, the
cantilever plate excited with the first and second rows (1-3x;
2-3x) (Configuration 4), in addition, all the cantilever plate excited with the whole attached piezoelectric actuators rows
(1, 2, and 3) (Configuration 5), as shown in Figure 6.
Table III
Piezoelectric actuators configuration for lateral vibration
Configuration No. Excitation by Active actuators
Configuration 1 First row of attached actuators Row (1) - 3 actuators
Configuration 2 Second row of attached actuators Row (2) - 3 actuators
Configuration 3 Third row of attached actuators Row (3) - 3 actuators
Configuration 4 First and Second rows of attached actuators Row (1 and 2) - 6 actuators
Configuration 5 First, Second and third rows of attached actuators Row (1, 2, and 3) - 9 actuators
For the torsional excitation of the cantilever plate structure,
six different configurations were selected. First, the
cantilever plate structure excited with the first row (1-3x)
(Configuration 45-1) of piezoelectric actuators on the plate within a frequency range of 0–300 Hz and 300 sub-steps was
investigated with an applied voltage of 100 V.
Second, the cantilever plate structure excited with the second
row (2-3x) (Configuration 45-2) of piezoelectric actuators on
the plate at the same frequencies and voltage, in addition to
the plate excited with the third row (3-3x) (Configuration 45-
3) of actuators and that excited with the fourth row (4-3x)
(Configuration 45-4) of actuators, were investigated. Moreover, the cantilever plate excited with the first and
second rows (1-3x; 2-3x) (Configuration 45-5) and that
excited with all the piezoelectric actuators rows (1, 2, 3 & 4)
(Configuration 45-6) are shown in Figure 7.
Table IV
Piezoelectric actuator configuration for torsional vibration
Configuration No. Excitation by Active actuators
Configuration 45-1 First row of attached actuators Row (1) - 3 actuators
Configuration 45-2 Second row of attached actuators Row (2) - 3 actuators
Configuration 45-3 Third row of attached actuators Row (3) - 3 actuators
Configuration 45-4 Third row of attached actuators Row (4) - 3 actuators
Configuration 45-5 First and second rows of attached actuators Row (1 and 2) - 6
actuators
Configuration 45-6 First, second, third, and fourth rows of attached
actuators
Row (1, 2, 3, and 4) - 12
actuators
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4. RESULTS AND DISCUSSIONS
This section presents the vibration characteristics of
the cantilever plate structure sub-structure with piezoelectric
actuator elements attached to the top and bottom surfaces of
the plate. A total of 11 different actuator configurations were
used in this investigation: five lateral actuator configurations (labelled 1, 2, 3, 4, and 5), and six angled actuator
configurations for torsional vibration (labelled 45-1, 45-2,
45-3, 45-4, 45-5, and 45-6) with different frequencies of the
voltage applied to the piezoelectric actuators.
The FE results below present the excitation of the actuators
according to the configurations shown in Figures 6 and 7,
over a range of excitation frequencies:
1) Lateral vibration analysis for the plate structure with
different configurations:
a) Configuration 1:
Figure 9 presents the vibration profiles of the cantilever plate
structure obtained by the FE analysis, presented with
excitation frequencies close to resonant Mode shape 1 and 3 applied to Configuration 1. The results clearly indicate that
the profile of the plate deflection with Configuration 1 was
identical to Mode shapes 1 and 3 when the actuator was
excited at a frequency close to the natural mode frequency.
However, the plate deflection with actuator Configuration 1
and an excitation frequency close to resonance 1 (0.047 mm)
was more significant than the excitation frequency close to
Resonance 3 (0.024 mm). Therefore, it was concluded that
Configuration 1 is very effective for Excitation mode 1.
Fig. 9. Cantilever plate excited by Configuration 1 for Mode shapes 1 and 3
b) Configuration 2
Figure 10 presents the vibration profiles of the cantilever
plate structure obtained by FE analysis, subjected to
excitation frequencies close to that of the resonant Mode
shapes 1 and 3, as applied to Configuration 3. The results
clearly indicate that the profile of the plate deflection with
Configuration 3 was identical to Mode shapes 1 and 3 when
the actuator was excited at a frequency close to the natural
mode frequency. However, the plate deflection Configuration
3 and an excitation frequency close to Resonance 3 (0.052
mm) was more significant that with an applied excitation
frequency close to Resonance 1 (0.023 mm). Therefore, it
was concluded that Configuration 2 is the very effective for
Excitation mode 3.
Fig. 10. Cantilever plate excited by Configuration 2 for Mode shapes 2 and 3
c) Configuration 4
Figure 11 presents the vibration profiles of the cantilever
plate structure obtained by the FE analysis, under the
application of excitation frequencies close to the resonant
Mode shapes 1 and 3 to Configuration 4. The results clearly
indicate that the profile of the plate deflection with
Configuration 4 was identical to Mode shapes 1 and 3 when
the actuator was excited at a frequency close to the natural
mode frequency. However, the plate deflection with
Configuration 4 and an excitation frequency close to
0.E+00
1.E-05
2.E-05
3.E-05
4.E-05
5.E-05
0 50 100 150
AMPLITUDE conf 1
0.00E+00
2.00E-05
4.00E-05
6.00E-05
0 50 100 150
AMPLITUDE conf 2
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Resonance 1 (0.065 mm) was similar to that with an
excitation frequency close to Resonance 3 (0.064 mm).
Therefore, it was concluded that Configuration 4 is very
effective for Excitation modes 1 and 3.
Fig. 11. Cantilever plate excited by Configuration 4 for Mode shapes 1 and 3
d) Configuration 5:
Figure 12 presents the vibration profiles of the cantilever
plate structure obtained by FE analysis, under the application
of excitation frequencies close to the resonant Mode shapes
1 and 3 to Configuration 5. The results clearly indicate that
the profile of the plate deflection with Configuration 5 was
identical to Mode shapes 1 and 3 when the actuator was
excited at a frequency close to the natural mode frequency.
However, the plate deflection with Configuration 5 under the
application of an excitation frequency close to Resonance 1
(0.065 mm) was similar to that under the application of an
excitation frequency close to Resonance 3 (0.086 mm).
Therefore, it was concluded that Configuration 5 is very
effective for Excitation mode 3.
-2.00E-05
0.00E+00
2.00E-05
4.00E-05
6.00E-05
8.00E-05
0 50 100 150
AMPLITUDE conf 1&2
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Fig. 12. Cantilever plates excited by Configuration 5 for Mode shapes 1 and 3
2) Torsional vibration analysis for the plate structure
with different configurations:
The FE results for the excitation of the actuators with respect to the angle of the actuator configuration is shown in Figure
6.
a) Configuration 45-1:
Figure 13 presents the vibration profiles of the cantilever
plate structure obtained by FE analysis, under the application
of excitation frequencies close to the resonant Mode shapes
2 and 4 to Configuration 45-1. The results clearly indicate
that the profile of the plate twisting deflection with
Configuration 45-1 was identical to Mode shapes 2 and 4 when the actuator was excited at a frequency close to the
natural mode frequency. However, the plate deflection with
Configuration 45-1 under the application of an excitation
frequency close to Resonance 2 (0.045 mm) was more
significant than the excitation frequency close to Resonance
4 (0.007 mm). Therefore, it was concluded that Configuration
45-1 is very effective for Excitation mode 2.
Fig. 13. Cantilever plate excited by Configuration 45-1 for Mode shapes 2 and 4
b) Configuration 45-2
Figure 14 presents the vibration profiles of the cantilever
plate structure obtained by FE analysis, under the application
of excitation frequencies close to resonant Mode shapes 2 and
4 to Configuration 45-2. The results clearly indicate that the
profile of the plate twisting deflection with Configuration 45-
2 was identical to Mode shapes 2 and 4 when the actuator was
excited at a frequency close to the natural mode frequency.
0.00E+00
2.00E-05
4.00E-05
6.00E-05
8.00E-05
1.00E-04
0 50 100 150
AMPLITUDE conf 1,2&3
0.00E+00
1.00E-04
2.00E-04
3.00E-04
4.00E-04
5.00E-04
0 50 100 150 200
AMPLIT…
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However, the plate deflection with Configuration 45-2 under
the application of an excitation frequency close to Resonance
2 (0.046 mm) was more significant than that under the
application of excitation frequency closed to Resonance 4
(0.003 mm). Therefore, it was concluded that Configuration
45-2 is very effective for Excitation mode 2.
Fig. 14. Cantilever plate excited by Configuration 45-2 for Mode shapes 2 and 4
c) Configuration 45-3
Figure 15 presents the vibration profiles of the cantilever
plate structure obtained by FE analysis, under the application
of excitation frequencies close to resonant Mode shapes 2 and
4 to Configuration 45-3. The results clearly indicate that the
profile of the plate twisting deflection with Configuration 45-
3 was identical to Mode shapes 2 and 4 when the actuator was
excited at a frequency close to the natural mode frequency.
However, the plate deflection with Configuration 45-3 under
the application of an excitation frequency close to Resonance
2 (0.072 mm) was more significant than that under the
application of an excitation frequency close to Resonance 4
(0.0845 mm). Therefore, it was concluded that Configuration
45-3 is very effective for Excitation mode 4.
0.00E+00
1.00E-04
2.00E-04
3.00E-04
4.00E-04
5.00E-04
0 50 100 150 200
AMPLIT…
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Fig. 15. Cantilever plate excited by Configuration 45-3 for Mode shapes 2 and 4
d) Configuration 45-4
Figure 16 presents the vibration profiles of the cantilever
plate structure obtained by FE analysis, under the application
of excitation frequencies close to resonant Mode shapes 2 and
4 to Configuration 45-4. The results clearly indicate that the
profile of the plate twisting deflection with Configuration 45-4 was identical to Mode shapes 2 and 4 when the actuator was
excited at a frequency close to the natural mode frequency.
However, the plate deflection with Configuration 45-4 under
the application of an excitation frequency close to Resonance
2 (0.072 mm) was more significant than that under an
excitation frequency close to Resonance 4 (0.0845 mm).
Therefore, it was concluded that Configuration 45-4 is very effective for Excitation mode 4.
Fig. 16. Cantilever plate excited by Configuration 45-4 for Mode shapes 2 and 4
e) Configuration 45-5:
Figure 17 presents the vibration profiles of the cantilever
plate structure obtained by the FE analysis, under the
application of excitation frequencies close to resonant Mode shapes 2 and 4 to Configuration 45-5. The results clearly
indicate that the profile of the plate twisting deflection with
Configuration 45-5 was identical to Mode shapes 2 and 4
when the actuator was excited at a frequency close to the
natural mode frequency. However, the plate deflection with
Configuration 45-5 under the application of an excitation
frequency close to Resonance 2 (0.169 mm) was similar to that under the application of an excitation frequency close to
Resonance 4 (0.153 mm). Therefore, it was concluded that
0.00E+00
2.00E-05
4.00E-05
6.00E-05
8.00E-05
1.00E-04
0 50 100 150 200
AMPLIT…
0.00E+00
2.00E-05
4.00E-05
6.00E-05
8.00E-05
1.00E-04
0 50 100 150 200
AMPLIT…
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Configuration 45-5 is very effective for Excitation modes 2
and 4.
Fig. 17. Cantilever plate excited by Configuration 45-5 for Mode shapes 2 and 4
f) Configuration 45-6:
Figure 18 presents the vibration profiles of the cantilever
plate structure obtained by FE analysis, under the application
of excitation frequencies close to resonant Mode shapes 2 and
4 to Configuration 45-6. The results clearly indicate that the
profile of the plate twisting deflection with Configuration 45-
6 was identical to Mode shapes 2 and 4 when the actuator was excited at a frequency close to the natural mode frequency
However, the plate deflection with Configuration 45-6 under
the application of an excitation frequency close to Resonance
2 (0.3 mm) was more significant than that under the
application of an excitation frequency close to Resonance 4
(0.042 mm), thus yielding the maximum twisting deflection
for the plate structure. Therefore, it was concluded that
Configuration 45-6 is very effective for Excitation mode 2.
0.00E+00
5.00E-05
1.00E-04
1.50E-04
2.00E-04
0 50 100 150 200
AMPLITUDEconf 45 1&2…
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Fig. 18. Cantilever plate excited by Configuration 45-6 for Mode shapes 2 and 4
Fig. 19. Cantilever plate excited by Configurations 1, 2, 3, 4, and 5
Fig. 20. Cantilever plate excited by Configurations "45-1"; "45-2"; "45-3"; "45-4"; "45-1" and " 45-2"; and "45-1", " 45-2", " 45-3", and "45-4"
CONCLUSIONS The results indicate the following:
1) The straight piezoelectric actuator configuration was
found to be the most effective for exciting the plate in
the bending (flapping) lateral vibration:
a. Configuration 1 was found to be the most
effective in the Excitation mode 1 vibration of
the cantilever plate structure.
b. Configuration 2 was found to be the most
effective in the Excitation mode 3 vibration of
the cantilever plate structure.
0.00E+00
1.00E-04
2.00E-04
3.00E-04
4.00E-04
0 50 100 150 200
AMPLITUDE…
-2.00E-05
0.00E+00
2.00E-05
4.00E-05
6.00E-05
8.00E-05
1.00E-04
0 20 40 60 80 100 120 140
AMPLITUDE conf 1 AMPLITUDE conf 2
AMPLITUDE conf 3 AMPLITUDE conf 1&2
AMPLITUDE conf 1,2&3
0.00E+00
1.00E-04
2.00E-04
3.00E-04
4.00E-04
5.00E-04
0 50 100 150 200 250 300 350
AMPLITUDE (m) conf 45 1
AMPLITUDE conf 45 2 (m)
AMPLITUDE conf 45 3 (m)
AMPLITUDE conf 45 1&2 (m)
AMPLITUDE conf 45-4 (m)
AMPLITUDE conf 45- 123&4 (m)
International Journal of Mechanical & Mechatronics Engineering IJMME-IJENS Vol:19 No:06 144
192706-5353-IJMME-IJENS © December 2019 IJENS I J E N S
c. Configuration 4 was found to be most efficient
and effective in the Excitation modes 1 and 3
vibration of the cantilever plate structure.
2) The angled piezoelectric actuator configuration was
found to be the most effective for the excitation of the
plate in twisting vibration:
a. Configuration 45-2 was found to be most
effective in the Excitation mode 2 vibration of
the cantilever plate structure.
b. Configuration 45-4 was found to be the most
effective in the Excitation mode 4 vibration of
the cantilever plate structure.
c. Configuration 45-5 was found to be the most
efficient and effective in the Excitation modes
2 and 4 vibration of the cantilever plate
structure.
d. Configuration 45-6 was found to be the most efficient and effective in the Excitation mode
2 vibration of the cantilever plate structure,
thus yielding the maximum twisting
deflection for the cantilever plate structure.
Finally, the input excitation frequency was found to have a
significant impact on the modal responses of the plate
structure upon which the actuators were correctly positioned.
Hence, it is necessary to appropriately select the actuator
position to dampen the vibrations of the plate at the respective
resonant frequencies.
ACKNOWLEDGEMENT
The authors would like to acknowledge the Deanship of
scientific research and the Center of Engineering and
Architecture researchers at Umm al-Qura University Saudi
Arabia for the financial support (Research Project No.
43108003) and for the logistic support throughout the project.
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