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Engineering, Technology & Applied Science Research Vol. 10, No. 6, 2020, 6549-6556 6549 
 

www.etasr.com Aktas & Esen: State-Space Modeling and Active Vibration Control of Smart Flexible Cantilever Beam … 

 

State-Space Modeling and Active Vibration Control 

of Smart Flexible Cantilever Beam with the Use of 

Finite Element Method  
 

Kerim Gokhan Aktas 

Department of Mechanical Engineering 
Karabuk University 
Karabuk, Turkey 

kerimgokhanaktas@karabuk.edu.tr 

Ismail Esen 

Department of Mechanical Engineering 
Karabuk University 
Karabuk, Turkey 

iesen@karabuk.edu.tr 
 

 

Abstract-The aim of this study is to design a Linear Quadratic 

Regulator (LQR) controller for the active vibration control of a 

smart flexible cantilever beam. The mathematical model of the 

smart beam was created on the basis of the Euler-Bernoulli beam 

theory and the piezoelectric theory. State-space and finite 
element models used in the LQR controller design were 

developed. In the finite element model of the smart beam 

containing piezoelectric sensors and actuators, the beam was 

divided into ten finite elements. Each element had two nodes and 

two degrees of freedom were defined for each node, transverse 

displacement, and rotation. Two Piezoelectric ceramic lead 

Zirconate Titanate (PZT) patches were affixed to the upper and 

lower surfaces of the beam element as pairs of sensors and 

actuators. The location of the piezoelectric sensor and actuator 

pair changed and they were consecutively placed on the fixed 

part, the middle part, and the free end of the beam. In each case, 

the design of the LQR controller was made considering the first 

three dominant vibratory modes of the beam. The effect of the 

position of the sensor-actuator pair on the beam on the vibration 
damping capability of the controller was investigated. The best 

damping performance was found when the sensor-actuator pair 
was placed at the fixed end. 

Keywords-vibration control; piezoelectric; LQR control; finite 

element analysis; smart beam   

I. INTRODUCTION  

Because of the low resistance of lightweight and flexible 
structures, vibration problems occur at low frequencies and 
their control is a major problem. Active vibration control 
systems are preferred because passive methods are considered 
inadequate in dealing with this problem. One of the most 
effective solutions of the vibration problem in light and flexible 
structures is to make them smart using smart materials [1]. 
Piezoelectric materials come to the forefront in smart materials 
due to advantages such as high sensitivity in sensing and drive, 
being applicable in a wide frequency range and being supplied 
in different shapes and sizes. Piezoelectric materials have been 
widely used recently in various areas especially in aviation, 
flexible robot arms, energy harvesting, and noise and shape 
control due to their ability to convert mechanical into electrical 
and electrical into mechanical energy [2-5]. Flexible beam 

elements are distributed parameter systems with infinite 
Degrees of Freedom (DoF). The active vibration control 
generally requires a low-order mathematical model of the 
system containing a limited number of DoF [6]. In particular, 
care must be taken so that the control of low-frequency modes 
does not affect the stabilization of high-grade modes, since 
sensors and actuators have limited bandwidth [7, 8]. Many 
techniques have been used to solve this problem, but the most 
effective is the use of collocated sensors and actuators that 
work in harmony with each other [9]. Vibration can be 
controlled by different control techniques. The most used 
control techniques are classical control algorithms such as 
Positive Position Feedback (PPF), velocity feedback and PID 
control [10-14]. It is very important to specify the sensor and 
actuator position optimally to keep the control capability at 
maximum. Otherwise, control spillover and observation 
spillover may occur, and this will affect the efficiency of the 
control system [6]. 

Authors in [15] proposed a controller combining PPF and 
PD control methods to suppress the first three vibration modes 
of the cantilever plate. Authors in [16] used two types of 
actuators (an inertial and a distributed strain actuator) and two 
types of controllers (PVF and H∞ control) to suppress the 
vibration of a fully clamped beam, and found that the best 

result was obtained with the H∞ controller and a distributed 
strain actuator. Authors in [17] designed a LQR controller 
based on the independent mode space control techniques. LQR 
control is one of the methods frequently used in active 
vibration control [18]. In LQR control, the performance index 
is minimized with optimum weight matrices. The LQR control 
provides the optimal response between the response speed and 
energy required to suppress vibration. [Q] and [R] are the most 
important parameters in keeping the permissible actuator 
voltage within certain limits while keeping the control 
performance at maximum level. Therefore, the important thing 
in LQR control is to determine the optimum state weighting 
matrix [Q] and the input weighting matrix [R]. Significant 
research has been done on the determination of weight matrices 
[19-21].  

Corresponding author: Kerim Gokhan Aktas



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In this study, the method of constructing a comprehensive 
mathematical model including piezoelectric mass and stiffness 
sensor-actuator and controlling the first three dominant 
vibration modes of a cantilever beam using the LQR controller 
are presented. The collocated piezoelectric sensor and actuator 
pair were placed on various finite elements. This allowed us to 
examine the effect of the sensor-actuator pair position on the 
performance of the controller. Since piezoelectric actuators 
have a certain operating voltage range, it is important to keep 
the actuator voltage within the allowable limits for a realistic 
control performance. For this reason, unlike most studies in the 
literature, the necessary actuator voltage was determined while 
ensuring optimum controller performance. As a result of the 
simulation study, system response and required actuator 
voltage were determined. 

II. SYSTEM MODELING 

In this section, the mathematical model of a smart 
cantilever beam with a piezoelectric collocated sensor and 
actuator pair is presented. The mathematical model of the smart 
beam was created on the basis of the Euler-Bernoulli beam 
theory. As shown in Figures 1 and 2, the beam is divided into 
10 finite elements, each element with two nodes and two DoF 
were defined for each node: transverse displacement and 
rotation ( �� , �� , �� , �� ). The physical and mechanical 
properties of the beam and PZT patches are given in Table I. 

TABLE I.  MECHANICAL AND PHYSICAL PROPERTIES 

Parameters Beam PZT Patches 

Length Lb=0.25 m lp=0.025 m 

Width w=0.03 m w=0.03 m 

Thickness hb=1.0 mm ha=hs=0.55 mm 

Density ρb=8030 kg/m
3
 ρp=7700 kg/m

3
 

Young’s modulus Eb=193 GPa Ep=68 GPa 

PZT strain constant  d31=125x10
-12 
m/V 

 

 
Fig. 1.  Finite element discretization of the smart cantilever beam. 

 
Fig. 2.  Beam element and piezoelectric beam element. 

As shown in Figure 2, the section of the beam without the 
piezoelectric patch is expressed as a beam element, and the 
section with the piezoelectric patch is expressed as a 

piezoelectric beam element. In Figures 1-2, Fext, Lb, lb, lp (ls, la), 
and P represent the external disruptive force applying at the 
free end of the beam, the length of the entire beam, the length 
of the beam element, the length of the piezoelectric sensor and 
of the actuator patch, and the polarization direction. 

A. Modeling of the Beam Element 

In the modeling, beam and piezoelectric patches were 
considered as an Euler-Bernoulli beam. The governing 
equations of motion of the smart beam for forced motion can 
be written as a fourth order partial differential equation [22] : 

�� ��	
�,
��
� � �� ��	
�,
���� � ���
    (1) 
where � is the displacement of the beam, � is the mass density, � is the cross-section area, � is the Young’s Modulus, � is the 
moment of inertia, and ���
  is the external force. In order to 
obtain the shape functions, the beam displacement function is 
taken as a cubic displacement function as in (2): �
�, �� � ���� � ���� � ��� � ��    (2) 
where ��  indicates the total degree of freedom, including 
displacement and rotation in each element (��, �� , ��, ��). 
Constants ��, ��, ��and �� are obtained by using cantilever 
beam boundary conditions. 

Substituting the constants ( �� , �� , �� , �� ) in (2), the 
displacement function can be obtained as: 

�
�, �� � ���
�� … … ��
���  
��������! � ����"�    (3) 

where ��� represents the shape function and �"� represents the 
nodal displacement vector. The strain energy in bending and 
bending moment can be found as [23] : 

# � $ %��&'() *�, + � �� ��	���    (4) 
Using (4), the potential energy is obtained as in (5): 

# � &,',� $ -��	���.� *� � &,',� $ �"�/�0��/�0���"�*�1,)1,)   � �� �"�/ -�2�2 $ �0��/�0��*�1,) .    (5) 
The kinetic energy can be found as: 

3 � 4,5,� $ -�	�
.� *�1,) � 4,5,� $ �"6�/�0�/1,) �0��"6�*�  � �� �"6�/ -�2�2 $ �0�/1,) �0�*�.�"6�    (6) 
where 0� and 0 are the second spatial derivative and the time 
derivative of the shape function, respectively. Total kinetic and 
potential energy can be found as: 

# � �� �"�/�72��"�    (7) 3 � �� �"6�/�82��"6�    (8) 
Using the Lagrange equations, the stiffness and mass matrix 

of each beam element can be written as in (9) and (10): 



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�72� � &,',1,9 :;;
;< 12 6@2 −12 6@26@2 4@2� −6@2 2@2�−12 −6@2 12 −6@26@2 2@2� −6@2 4@2� CDD

DE
    (9) 

�82� � 4,5,1,��) :;;
;< 156 22@2 54 −13@222@2 4@2� 13@2 −3@2�54 13@2 156 −22@2−13@2 −3@2� −22@2 4@2� CDD

DE
    (10) 

B. Modeling of the Piezoelectric Beam Element and of the 
Smart Beam Element 

Similarly, when the piezoelectric patches are considered as 
Euler-Bernoulli beam elements, the stiffness and mass matrix 
of the piezoelectric patches can be obtained as: 

H7IJ � &K 'K1K9 :;
;;
< 12 6@I −12 6@I6@I 4@I� −6@I 2@I�−12 −6@I 12 −6@I6@I 2@I� −6@I 4@I� CD

DD
E
    (11) 

H8IJ � 4K 5K1K��) :;;
;< 156 22@I 54 −13@I22@I 4@I� 13@I −3@I�54 13@I 156 −22@I−13@I −3@I� −22@I 4@I� CD

DD
E
    (12) 

As shown in Figure 2, two piezoelectric patches are bonded 
to the upper and lower surface of the beam element as a sensor 
and actuator. This state of the beam is called a smart beam 
element. The stiffness of a smart beam element can be obtained 
by the equivalent flexural stiffness method [24] . 
����L � 	�2�2 � 2�N�N    (13) 
With the help of the parallel axis theorem, the moment of 

inertia of the smart beam element can be obtained as: 

�N � OPK9�� � QℎN SPKTP,U��     (14) 
The mass per unit length of the beam element can be found 

as: �� � Q
�2ℎ2 � 2�IℎI�    (15) 
where ℎI and ℎ2 represent the thickness of the PZT patch and 
core beam respectively. The elemental stiffness and mass 
matrix of the smart beam can be written as: 

�7V� � 
&'�WX1Y9 :;;
;< 12 6@V −12 6@V6@V 4@V� −6@V 2@V�−12 −6@V 12 −6@V6@V 2@V� −6@V 4@V� CDD

DE
    (16) 

�8V� � 451Y��) :;;
;< 156 22@V 54 −13@V22@V 4@V� 13@V −3@V�54 13@V 156 −22@V−13@V −3@V� −22@V 4@V� CDD

DE
    (17) 

III. PIEZOELECTRIC CONSTITUTIVE EQUATIONS 

Piezoelectric materials can be used as both sensors and 
actuators. If used as an actuator, the piezoelectric patch 
undergoes mechanical strain when voltage is applied as input 
(inverse piezoelectric effect). If used as a sensor, the patch 
generates voltage as an output when mechanical stress is 
applied to it (direct piezoelectric effect) [25]. When a 
disruptive external force is applied to the beam, the beam is 
exposed to vibration. In the vibration damping process, 
mechanical stress occurs in the beam exposed to vibration, and 
the patch used as a sensor produces a voltage against this stress. 
The generated voltage is applied to the controller after the 
signal conditioning process. After the signal is evaluated, it is 
applied to the actuator as a control signal. The patch used as 
actuator undergoes a mechanical strain and suppresses 
vibration by producing an equal amplitude and an opposite 
response to vibration. The IEEE piezoelectricity standard was 
taken into account in defining the electromechanical properties 
of the piezoelectric material and the piezoelectric material is 
considered to exhibit linear behavior [26]. The 
electromechanical equations of linear piezoelectric materials 
can be written as: Z� � [��&\� � *���]    (18) 

]̂ � *��\� � _��`�]    (19) 
where Z� , [�� , \� , *�� , �] , ]̂ and _�� represent the strain 
(m/m), compliance (m

2
/N), stress (N/m

2
), piezoelectric strain 

constant (m/V), electric field (V/m), electric displacement 
(C/m

2
), and permittivity (F/m). Also, the superscripts E and σ 

represent the constant electric field and the constant voltage. 

A. Sensor Equations 

Equation (19) expresses the direct piezoelectric effect and is 
used to determine the total charge obtained from the PZT 
sensor. Since no extra electric field is given to the sensor, (19) 
can be written as: 

]̂ � *��\� � *��a��Z� � b��Z�    (20) 
where b�� is the piezoelectric stress/charge constant and a�� is 
the inverse of compliance (1/[��). The total charge obtained 
from the sensor layer can be calculated as: c
�� � $ ]̂5 *�    (21) 
The charge can be converted to current as: 

d
�� � ef

�e
 � ee
 $ ^]5 *� � ee
 $ b��Z�5 *�   (22) 
As we know, strain at any point can be found as [23]: 

Z� � − ]g ,			�g = e�	e��,			Z� = h e�	e��    (23) 
where h is the distance to the neutral axis and i is the radius of 
curvature. Equation (22) can be written as: 

d(�) = ee
 $ b��h0�"5 *� = hb��Q $ 0�"61K) *�    (24) h = (P,� + ℎj)    (25) 



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The current obtained from the sensor is converted into 
voltage with the signal conditioning device. The voltage value 
obtained is given to the actuator with a suitable control gain: 

k V(�) = lhb��Q $ 0�"61K) *� = ["6    (26) 
where l is the signal conditioning device gain. This sensor 
voltage is given as input to the controller and the controller 
output is multiplied by the controller gain and is given to the 
actuator. The actuator input voltage can be found as: kj(�) = m = n	�	kV(�)    (27) 
where n is the control gain. 
B. Actuator Equations 

Equation (18) expresses the inverse piezoelectric effect and 
is used to calculate the total strain. Since no extra stress is 
given to the actuator, (18) can be written as: Z� = *���]    (28) 
When an input voltage is applied to the actuator in the 

polarization direction, the actuator strains in one direction. The 
stress and electrical field generated in the actuator can be 
obtained as: 

�] = op(
)Pp , \� = �IZ� = �I*�� op(
)Pp     (29) 
where �I and ℎj are the Young’s Modulus and the thickness of 
the actuator respectively. As a result of the stress, the bending 
moment occurs in the beam element. The bending moment in a 
small cross section can be written as: 

*+jq
 = �I�I e�	e��    (30) 
The sum of the moments of the stresses on the x axis 

relative to the y axis can be written as: 

+jq
 = $ h̿\�*h,   h̿ = sPpTP,� t    (31) 
By using (29) and (30), the bending moment can be found 

as: +jq
 = �I*��h̿kj(�)    (32) 
The control force applied by the actuator to the beam 

element can be obtained using (27) and (32): ujq
 = �I*��Qh̅ $ 0�/1K *�kj(�) = wkj(�)    (33) 
where w is a constant vector of size (4×1).  

IV. EQUATIONS OF MOTION, STATE SPACE 
REPRESENTATION, AND MODEL VERIFICATION 

A. Equations of Motion for the Smart Beam 

The dynamic equation of a smart beam can be written as: �+�x"yz + �n�x"z = ���
 + �q
{1    (34) 
Rayleigh proportional damping method is used to include 

the damping effect in the model. Structural damping matrix can 
be obtained as [27]: a = |+ + }n    (35) 

where a  is the damping matrix, |  is the mass proportional 
damping coefficient, and }  is the stiffness proportional 
damping coefficient. In obtaining the damping matrix of the 
smart beam, |  and }  coefficients were taken as 0.001 and 
0.0001 respectively. +, aand n represent the global mass and 
the damping and stiffness matrices of each finite element. The 
mass and stiffness matrices of the entire beam are obtained by 
combining the local mass and stiffness matrices using the finite 
element technique. The equation of motion of cantilever beam 
and sensor output equation can be written as: �+�x"yz + �a�x"6z + �n�x"z = ���
 + �jq
    (36) ~(�) = kV(�) = ["6    (37) 
In order to examine the dominant first three vibration 

modes of the system, " = Φ* conversion was made and the 
transition from generalized coordinates to principal coordinates 
was made [28]. Here, Φ denotes the modal matrix. Using the 
transformation " = Φ*, (36) and (37) can be written as: +Φ*y + aΦ*6 + nΦ* = ���
 + �jq
    (38) ~(�) = kV(�) = [Φ*6    (39) 
By pre-multiplying (38) with Φ/ , generalized mass, 

stiffness and damping matrices can be obtained as: Φ/+Φ*y + Φ/aΦ*6 + Φ/nΦ* = Φ/���
 + Φ/�jq
    (40) +�*y + a̅*6 + n�* = ����
 + ��jq
    (41) 
The +� , a̅ and n� matrices are generalized mass, damping 

matrix, and stiffness matrix respectively, each (3×3) in size. 

The generalized external force vector ����
 is obtained as: ����
 = Φ/���
 = Φ/u�(�)    (42) 
where �(�) refers to the external force input to the smart beam. 
The generalized control force vector is expressed by: ��jq
 = Φ/�jq
 = Φ/wkj(�)    (43) 
B. State Space Repsentation of the Smart Beam 

The state and output equation of a dynamic system can be 
written as: �6 = �� + �m    (44) ~ = a� + ^m    (45) 
where � is the system matrix, � is the input matrix, a is the 
output matrix, ^ is the direct transmission matrix. �, m and ~ 
represent the state vector, input vector and output vector 
respectively. x is used as the state variable to obtain the state 
equations. The generalized coordinates in (44) can be written in 
terms of the state variable as follows: 

�*� = �*�*�*�� = �
������� , H*6J = �

*�6*�6*�6 � = �
������� , H*yJ = �

�6��6��6��	    (46) 
The state space form of the (36) can be written as: 



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+� ��6��6��6�� + a̅ �
������� + n� �

������� = ����
 + ��jq
    (47) 
The state equation form of (47) can be written as: 

:;
;;
;<�6��6��6��6��6��6�CD

DD
DE = - 0 �−+���n� −+���a̅.

:;;
;;<
������������CD

DDD
E
+ - 0+� ��Φ/w. m(�)  

+ � 0+���Φ/u� �(�)    (48) 
The sensor voltage can be taken as output and is obtained 

as: ~(�) = a/�(�) + ^m(�)    (49) 
~(�) = �0 [Φ� ���⋮������    (50) 

C. Model Verification 

Before starting the simulation study, the created 
mathematical model was compared with two previous 
numerical studies and the results are shown in Tables II and III. 
In [28], considering a cantilever beam divided into four finite 
elements, four different models were created by placing 
piezoelectric elements in different positions and the dominant 
first two vibration mode natural frequency values were 
obtained for each model. These values were compared with the 
values obtained from the current study in Table II. The 
mechanical and physical properties of the considered beam are 
given in [28]. In [29], a bimorph cantilever beam consisting of 
two piezoelectric PVDF layers divided into five finite 
elements, was considered and tip displacements were obtained 
by applying various voltage values in the range of 0-200 V. 
The dimensions of the PVDF beam are L=100 mm, w=5 mm 
and h=1 mm. The tip displacement values were compared with 
the values obtained from the current study in Table III. The 
mechanical properties of the PVDF layers are given in [29]. 

TABLE II.  NATURAL FREQUENCY COMPARISON 

Model 

Natural frequency (Hz) 

[28] Present 

Mode 1 Mode 2 Mode 1 Mode 2 

Model 1: Fixed end 6.66 36.89 6.6581 36.8735 

Model 2 4.73 27.43 4.7246 27.4143 

Model 3 3.85 30.52 3.8212 30.2934 

Model 4: Free end 3.16 25.34 3.0433 25.0924 

TABLE III.  TIP DISPLACEMENT COMPARISON 

Voltage (V) 
Tip displacements (×10

-4
m) 

[29] Present 

50 0.1755 0.1736 

100 0.3409 0.3377 

150 0.5067 0.4991 

200 0.6819 0.6706 

V. LQR CONTROL DESIGN   

LQR control theory is used in active vibration control of 
smart cantilever beams. The cost function can be written as 
[30]: 

� = �� $ (�/c� + m/im)�) *�    (51) 
where [c] and [i] are the positive semi-definite symmetric and 
positive definite symmetric weighting matrices respectively. 
Larger [c] means that a better control performance is requested 
from the controller. On the other hand, the meaning of the 
larger [i] is the reduction of the permissible control effort. 
Using the linear control law, the control input can be found in: m = −n�    (52) 
where n is the feedback control gain which can be obtained as: n = i���/�    (53) 
where � is a algebraic Ricatti equation solution of (54): �� + �/� − ��i���/� = 0    (54) 
Once �  is calculated, the actuator input voltage can be 

found with the help of (27). [Q] and [R] matrices can be written 
as: 

�c��� =  ��� 0 0 00 ⋱ 0 000 00 ⋱ 00 ���!    (55a) 
�i��� =  ��� 0 0 00 ⋱ 0 000 00 ⋱ 00 ���!     (55b) 

The [c ] and [i ] weighting matrices are obtained using 
Bryson’s method [31]: 

��� = s �|��|�p�� t , ��� = s �|��|�p�� t 			d = 1,2,3, . , 0    (56) 
In the Bryson’s method, in a system with n state variables, 

the state weight coefficient of each state variable is determined 
according to the maximum acceptable error amount of that 
variable (|��|�j�). The determination of the R control weight 
coefficient depends on the maximum acceptable value of the 
control signal (|m�|�j�).  

VI. RESULTS AND DISCUSSION 

As shown in Figure 3, the beam was divided into ten finite 
elements and the actuator-sensor pair was placed in three 
different positions to examine the effect of the pair's position 
on the beam. As a result, three different models were created, 
and a state space model was obtained for each model. Finite 
element model and LQR controller were programmed in 
MATLAB. The maximum acceptable error was 2% and the 
maximum voltage applied to the actuator was 200V. The 
feedback control gain n obtained for each model can be written 
as: �n�� = �−25.3 146.7 −129.2 355.2 95.1 28.6�    (57) �n�� = �52.2 434.8 326.6 −426.3 145.9 37.7�    (58) 



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�n�� = �53.8 410.4 364.5 A432.1 166.6 42.1�    (59) 
The natural frequencies for the first three vibration modes 

of the three models are given in Table IV. The step responses 
for the first three vibration modes of the system are given in 
Figures 4-6. 

 

     
            (a)                                 (b) 

 
(c) 

Fig. 3.  Position of the PZT sensors. (a) model 1, (b) model 2, (c) model 3. 

TABLE IV.  NATURAL FREQUENCIES   

Model 
Natural frequency (rad/s) 

Mode 1 Mode 2 Mode 3 

Model 1: Fixed end 92.51 569.42 1568.50 

Model 2 77.92 504.73 1405.32 

Model 3: Free end 68.15 458.34 1326.90 
 

 
Fig. 4.  Closed loop displacement responses of model 1. 

 
Fig. 5.  Closed loop displacement responses of model 2. 

When Table IV and Figures 4-6 are examined, it is seen 
that as the actuator-sensor pair moves from the fixed end to the 
free end, the stiffness of the system increases depending on the 
natural frequency and thus the vibration amplitudes and settling 
time decrease. The required actuator voltage for the vibration 
control of the three models is shown in Figures 7-9. 

 
Fig. 6.  Closed loop displacement responses of model 3. 

 

Fig. 7.  Maximum actuator voltage requirement of model 1. 

 

Fig. 8.  Maximum actuator voltage requirement of model 2. 

 
Fig. 9.  Maximum actuator voltage requirement of model 3. 

The obtained settling time and actuator voltage values, 
according to the simulation results are shown in Table V. 

1 2 3 4 5 6 7 8 910 1 2 3 4 5 6 7 8 910

1 2 3 4 5 6 7 8 910



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TABLE V.  SETTLING TIME AND ACTUATOR VOLTAGE 

Model Settling time (s) Maximum Voltage (V) 

Model 1: Fixed end 2.33 28.05 

Model 2 3.39 23.12 

Model 3: Free end 4.56 19.88 

 

As seen in Table V, the best settling time values were 

obtained in model 1. Although the required actuator voltage in 

model 1 is higher than that of other models, the difference is 
not significant when settling time values are considered. 

VII. CONCLUSION 

This paper presents the active vibration control of a smart 
flexible cantilever beam bonded with a collocated PZT actuator 
and sensor pair. An LQR controller was designed for the 
control of first three vibration modes of the beam. The natural 
frequency and tip displacement values obtained to verify the 
finite element model and equations of motion created within 
the scope of the study were compared with existing studies in 
the literature. It has been determined that the natural frequency 
values are similar with a maximum error rate of 0.98% and the 
tip displacement values are similar to the 1.66% error rate. The 
position of the PZT sensor-actuator pair on the beam was 
changed to optimally determine the sensor-actuator position. 
As a result of the simulation study, system response, settling 
time, and required actuator voltage were determined for the 
first three vibration modes. The simulation results demonstrate 
that the best vibration damping results were obtained when the 
PZT actuator and sensor were affixed to the fixed end. 

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