Finite Element Modeling of a Cantilever for Counteracting ...ijens.org/Vol_19_I_06/192706-5353-IJMME-IJENS.pdf · formula was obtained for the bi-morph beam deflection u at position

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

[email protected] 2- Production Engineering Department, Faculty of Engineering, Alexandria University, Egypt

[email protected]

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.



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]



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



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



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



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



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.


c) Configuration 4


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



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.


d) Configuration 5:


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


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


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



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:



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



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


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…



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



c) Configuration 45-3






3 was identical to Mode shapes 2 and 4 when the actuator was






(0.0845 mm). Therefore, it was concluded that Configuration


0.00E+00

1.00E-04

2.00E-04

3.00E-04

4.00E-04

5.00E-04

0 50 100 150 200

AMPLIT…




d) Configuration 45-4





profile of the plate twisting deflection with Configuration 45-4 was identical to Mode shapes 2 and 4 when the actuator was




2 (0.072 mm) was more significant than that under an


Therefore, it was concluded that Configuration 45-4 is very effective for Excitation mode 4.


e) Configuration 45-5:


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…



Configuration 45-5 is very effective for Excitation modes 2

and 4.


f) Configuration 45-6:






6 was identical to Mode shapes 2 and 4 when the actuator was excited at a frequency close to the natural mode frequency





(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…




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



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)



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



b. Configuration 45-4 was found to be the most



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.

REFERENCES [1] I. Kajiwara, S. Kitabatake, N. Hosoya, S. Maeda, Design of

dielectric elastomer actuators for vibration control at high

frequencies, International Journal of Mechanical Sciences 157–

158 (2019) 849–857.

[2] A. Tuan, Z. Rahman, A. As’arry, N. Aswan, A. Jalil, Active

Vibration Control of a Flexible Beam Structure using Chaotic

Fractal Search algorithm, Procedia Engineering 170 (2017) 299–

306.

[3] Sven Herold * and Dirk Mayer, “Adaptive Piezoelectric

Absorber for Active Vibration Control”, Actuators 2016, 5, 7.

[4] Z. Shunqi, R. Schmidt, Q. Xiansheng, Active vibration control of

piezoelectric bonded smart structures using PID algorithm,

Chinese Journal of Aeronautics (2014).

[5] M A. Kamel, K. Ibrahim, A. El-Makarem Ahmed, Vibration

control of smart cantilever beam using finite element method,

Alexandria Engineering Journal (2019).

[6] Z. Lasovaa, R. Zemik, Comparison of finite element models for

piezoelectric materials, SciVerse ScienceDirect (2012).

[7] A. R. Biswal, T. Roy, R. K. Beheraa, P. K. Parida, S. K. Pradhan,

Finite Element Based Modeling of a Piezolaminated Tapered

Beam for Voltage Generation, 12th International Conference on

Vibration Problems, ICOVP (2015).

[8] H. Karagulle, L. Malgaca, H. F. Oktem, Analysis of active

vibration control in smart structures by ANSYS, Smart Materials

and Structures (2004).

[9] M. Al-Hazmi, H. Ghulman, Theoretical Investigation of Simply

Supported Plate Structures Excited by Patches of Piezoelectric

Actuators, International Journal of Factory Automation, Robotics

and Soft Computing 2 (2007) 57–66.

[10] H. Ghulman, M. Al-Hazmi, Finite Element Analysis of simply

Supported Plate Structure Excited by Patches of Piezoelectric

Actuators, The Second International Conference on Modeling

Simulation, and Applied Optimization ICMSA2007, Abu Dhabi

(2007).

[11] A. Chattopadhyay, H. Gu, D. Dragomir-Daescu, Dynamics of

Delaminated Composite Plates with Piezoelectric Actuators,

AIAA Journals 37(2) (1999) 248–254.

[12] E. F. Crawley, E. H. Anderson, Detailed models of the

piezoceramic actuation for beams J. Intell. Mater. Syst. Struct. 1

(1990).

[13] K. R. Kumar, S. Narayanan, Active vibration control of beams

with optimal placement of piezoelectric sensor/actuator pairs,

Smart Mater. Struct. 17 (2008) 055008.

[14] D. Sun, D. Wang, Z. L. Xu, H. Wu, Distributed Piezoelectric

Element Method for Vibration Control of Smart Plates, AIAA

Journal 37(11) (1999) 1459–1463.

[15] J. K. Hwang, C. H. Choi, C. K. Song, J. M. Lee, Identification of

a Thin Plate with Piezoelectric Actuators and Sensors,

Transaction of the ASME Journal of Vibration and Acoustics 120

(1998) 826–828.

[16] I. Chopra, Review of State of Art of Smart Structures and

Integrated Systems, AIAA Journal 40 (2002) 2145–2187.

[17] V. Giurgiutiu, Review of Smart-Materials Actuation Solutions

for Aero elastic and Vibration Control.” Journal of Intelligent

Material Systems and Structures 11 (2000) 525–544.

[18] H. Benaroya, Mechanical Vibration: Analysis, Uncertainties, and

Control, Prentice Hall, New Jersey, USA (1998).

[19] I. Chopra, Status of Application of Smart Structures Technology

to Rotorcraft Systems, Journal of American Helicopter Society

(2000) 228–252.

https://iopscience.iop.org/journal/0964-1726

https://iopscience.iop.org/journal/0964-1726

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