Fuzzy-PID Controller Design for Random Vibration Attenuated Smart Cantilever Timoshenko Beam based on MOGA

3rd
.International Conference on Researches in Science & Engineering
31 Aug. 2017, Kasem Bundit University, Bangkok, Thailand
Fuzzy-PID Controller Design for Random Vibration Attenuated
Smart Cantilever Timoshenko Beam based on MOGA
M.Hasanlu*1
, M.Siavashi2
, M.Soltanshah3
, M.Zamanian4
1,2
Department of Mechanical Engineering, Faculty of Engineering, University of Guilan, P.O. Box 3756, Rasht,
Iran
3
Faculty of Mechanical Engineering, Isfahan University of Technology, Isfahan, Iran
4
Department of Mechanical Engineering, Faculty of Engineering, Kharazmi University, Tehran, Iran.
*
Corresponding Author: mhasanlu@webmail.guilan.ac.ir
Abstract
Today, vibration control of continuous systems that is also referred as structure has been
remarkably developed. Due to varied application of structures such as beam, plate, shell,
panel, truss, etc., engineers have faced with new and more complex challenges in order to
control these structures for their own purposes. The life of an engineering structure in each
industry is a major factor for maintenance of equipment. The life of engineering structures
reduces due to vibration amplitude or the design for mechanical system can be far from the
goals of designers. In this study, a thin cantilever Timoshenko beam has been considered. A
smart structure model has been designed using PID controller combined with fuzzy logic for
vibration attenuation. Here, PID controller has been used in order to control the structure. In
this controller there are three unknown coefficients in Kp, Ki and Kd matrices using multi-
objective genetic algorithm for or MOGA, and the unknown coefficients are determined as
result of searching this algorithm during the beam in order to determine an optimal point or
points of beam to place sensor and piezoelectric actuator in the top and down for the structural
suppression at the least possible time. Vibration beam model is based on finite element
numerical method. It has been embedded on the beam regarding the piezoelectric direct and
inverse property as the sensor and actuator in order to be a factor of active control for
attenuation of transverse displacement.
Keywords: vibration attenuation, cantilever Timoshenko beam, Fuzzy-PID controller, piezoelectric, Genetic
Algorithm
Symbols
A Cross-Section Transverse Beam Mass Matrix
System Matrix Transverse Piezo Mass Matrix
b width Acceleration Shape Function
Input Matrix Angular Shape Function
Output Matrix Transverse Shape Function
Piezo Strain Constant q Local Coordinate
Smart Structure Damping Matrix ̇ Local Velocity of Node
Young’s Modulus of Beam t Time
Control Effort Matrix Beam Thickness

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Young’s Modulus of Piezo Piezo-Patch Thickness
f Force Local T Kinetic Energy
F Force Global u(t) Control Input
Total Force U Potential Energy
External Force Actuator Voltage
Piezoelectric Force Poisson’s Ratio
Electrical Resistance Constant w(t) External Input
I Inertia Moment w Transverse Displacement
[I] Identity Matrix w(x,t) Transverse Dynamic Displacement
i(t) Electrical Current Virtual Work
Beam Stiffness Matrix x Horizontal Coordinate
Piezo – Patch Stiffness Matrix Output System
Smart Structure Stiffness Matrix z Transverse Coordinate
K Shearing Parameter Angular Displacement
Controller Gain Angular Dynamic Displacement
Length Element Shearing Constant
Length Piezo-Patch Density
m Local Torque Axial Stress
Beam Mass Matrix Shearing Stress
Piezo-Patch Mass Matrix Damping Constants
Smart Structure Mass Matrix [ ] Transpose Matrix
Angular Beam Mass Matrix
Angular Piezo Mass Matrix
1. Introduction
According to the requirement of various industries to control their structures due to dynamic
stimulations including disturbance and noise, designers have considered using accurate,
accessible, and simple controllers with respect to the control of these unwanted inputs.
However, developing fuzzy logic and deploying it for equipment construction, robust control
of systems has been possible. This study attempted both to make a simple and reliable
controller available and to stabilize robustly the system using PID-controller combined with
fuzzy logic. M.Ebrahimnejad et al. used piezoelectric for vibration attenuating of a truss
structure, pole placement, and LQR methods to recognize the best place to embed the
piezoelectric sensor and actuator [1]. J.Arushankar et al. using sliding mode controller
attempted to control a smart beam (beam along with piezoelectric) due to external
stimulations and disturbances. The beam was a model of a part of aircraft structure frequently
used in aviation industry [2]. N.Ghareeb et al. embedded PVDF, which acted as sensor and
actuator, on a cantilever beam. Then, using Luapunov linear stability theory, they extracted
the first vibration materials of a beam and found out remarkable results [3]. N.M.Sridevi et al.
using H∞ controller controlled a beam along with piezoelectric in several vibrating modes.
The results of this controller also showed more optimized performance compared with LQR
controller [4]. T.C.Manjunath et al. designed a kind of robust controller named RDPOF. They
conducted Euler-Bernoulli beam model with four elements in six vibration attenuated modes
[5]. In an empirical research, L.R.Karlmax et al. used feedback Intel 8061 output feedback
microcontroller in a smart structure, and they designed and controlled its vibrating control [6].
J.L.Fanson et al. experimentally designed a beam with piezoelectric using positive feedback
location [7]. W.Hwang et al. presented finite element model of piezoelectric actuator [8].
T.C.Manjunath et al. designed the best location for piezoelectric actuator and sensor to
achieve optimal attenuation in Euler-Bernoulli beam [9]. J.M.S.Moita et al. studied

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comparative and active control of multilayer structures with piezoelectric [10]. M.R. Safizade
et al. studied the optimal place for a plate with all edges clumped by using controllability
gramian performance index and genetic algorithm. Structural equations of plate were
extracted analytically and were incorporated with analytical equations of piezoelectric
actuator and the equation of an intelligent structure was obtained. Then by using a controlling
method, the optimal placement of their system was conducted. In this method, the main
responsibility is system's controllability and expressing an optimum control input so that by
applying forces on this optimal place of structure, system, can be damped [11]. J. Yang et al.
in two researchers studied the optimal placement of piezoelectric sensor and actuator on a
plate. Their theory was that in order to increase the controlling performance of system or in
other words controlling the system by piezoelectric results in vibration suppression,
piezoelectric actuator should affect a specific direction on plate. Now there are coordinates on
the plate that show their potential effect by the placement of piezoelectric actuator and system
is controlled more efficiently. They used two types of Simulated Annealing for the TSP
(SATSP) algorithm and another algorithm called Hopfield-Tank for the TSP (HTTSP) to
optimize the place of plate. The results of SATSP optimization algorithm were better than
HTTSP. In this article, by using SATSP algorithm alongside Genetic Algorithm for TSP
(GATSP) algorithm, better results from GATSP were provided compared to SATTSP [12,
13]. V. Gupta et al. in a practical and helpful research used 6 objective functions for
optimizing the placement of piezoelectric as sensor and actuator on the structure of beam and
plate. These functions include [14]:
1) The maximization of force or torque released from piezoelectric actuator
2) The maximization of deflection in a structure
3) The minimization of controlling effect or the maximization of wasted energy
4) The maximization of controllability degree
5) The maximization of observability degree
6) The minimization of spillover phenomenon
M. Trajkov et al. analyzed the placement based on controllability and observability criteria.
The optimization was conducted based on H2 and H00 soft and controllability and
observability gramian function which is dependent on vibrational modes. The structure model
was designed using finite element and after the reduction of order process, optimization
operations were done on the reduced model and the optimal place was suggested for the plate
and cantilever beam [15]. F. Bachman et al. conducted a research on optimal placement of
two piezoelectric pieces on a turbo machinery blade which is a carbon / epoxy composite and
by using the criterion of increasing potential energy for piezoelectric and structure and
increasing electromechanical coupling damping coefficient, they tried to increase the energy
saved in piezoelectric in the best place of a composite blade [16]. J. Zhang et al. analyzed the
vibration suppression of a cantilever beam by finding the best place in that beam for the
placement of piezoelectric actuator and sensor in order to from an intelligent system for

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automatically controlling of the vibration and reaching a logical stability. For system's
steering, they used Linear Quadratic Gauss (LQG) controller as an optimal controller method
and analyzed 4 vibration modes of beam [17]. A. Molter et al. studied the optimal placement
of piezoelectric on a flexible Manipulator as a cantilever beam with its mass concentrated on
top of the beam. They studied and controlled the beam based on Euler-Bernoulli theory
analytically and numerically. In addition to optimal placement, they optimized the size of
piezoelectric sensor and actuator in this model and their controlling method was done by
using Lyapunov function [18]. L. Nowak et al. proposed a method for calculating the optimal
place of piezoelectric sensor and actuator. Thin beam, thin plate and thin panel were tested as
structures with different boundary conditions under acoustic vibration. The dynamic analysis
of structure was conducted analytically [19]. G. Rosi et al. controlled the inactive released
sound on an aluminum plate with non-standard boundary conditions and by optimal
placement on it; they considered coordinates for locating piezoelectric pieces. Generally the
objective was reaching the best efficiency of this plate in reducing the released sound from it
[20]. A.H. Daraji et al. used an isotropic cantilever plate through finite element method in
ANSYS software by using 2 elements; element Solid45 for 3D meshing and element Shell63
for 2D meshing of plate. They also used element Solid45 for meshing piezoelectric sensor and
actuator pieces and prepared the model for controlling and by using Linear Quadratic
performance index (minimization) and genetic algorithm in MATLAB software, they were
able to relate these two software simultaneously in order to accelerate the vibration
suppression through the optimal placement of piezoelectric pieces and showed their results
[21]. J.M. Hale studied their placement by 10 piezoelectric pieces as sensors and actuators on
a cantilever plate. They used genetic algorithm for optimization of objective function.
Modeling method was in accord with first order shear theory and was conducted numerically
by finite element and Hamilton relationship. Their objective for optimal placement was
reaching the damping of first 6 vibrational modes of plate which were implemented
analytically by ANSYS software by using element Solid45 for 3D meshing and element
Shell63 for 2D meshing of plate and compared their results with finite element method.
Optimized objective function was modified by H2 soft in genetic algorithm to find the
optimal place [22]. Nemanja D. Zoric et al. tried to optimized the size and location of
piezoelectric on a thin composite beam by modeling finite elements based on the third order
shear deformation theory, using fuzzy logic combination and PSO optimization algorithm

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with the swarm of one objective particles[23]. Sergio L. Schulz et al. used piezoelectric pieces
for the optimal location on a cantilever beam and a plate with simply support. Their
optimization method was genetic binary algorithm and Linear Quadratic Gauss (LQG)
method. In this study, their optimization objective function was designed based on Lyapunov
function. During this research, they reached to this goal that optimal location in many first
modes that are considered as energetic modes can control the system and cover other
vibration modes. State variables, kinetic energy, strain energy and input voltage of actuator
are considered and considerable results were provided [24]. D. Chhabra et al. in their article
optimally located 10 piezoelectric actuators on a square plate with simply support by using
Modified Control Matrix and Singular Value Decomposition (MCSVD) method and genetic
algorithm and finally for vibration control, they used LQR controlling rule in order to
investigate the performance and efficiency of system and this way, they optimized and
controlled first 6 vibration modes in square plate [25]. F. Botta et al. suggested a new
objective function for optimal locating of piezoelectric. Optimization of the objective function
results in simultaneously damping of several vibration modes. Heir studied structure was a
cantilever beam with using Euler-Bernoulli theory. Their modeling of beam was done in two
ways; analytically and numerically and showed the conformity of these two methods. Also in
another article, they obtained experimental results from optimal placement of piezoelectric as
sensor and actuator on turbo machinery blades. They considered a turbo machinery blade as a
cantilever beam and then analytically solved the equation of beam coupled with piezoelectric
sensor and actuator and presented their theoretical and experimental results. Their objective
was to reduce vibration and fatigue of the beam [26].
A.L. Araujo et al. conducted the optimal placement of piezoelectric on a Sandwich plate with
viscoelastic core and multi-layered procedure by using DMM optimization method. Because
of this research, they were able to analyze the vibration suppression of sandwich plate by
finding the best place and by analyzing 6 vibration modes. They found considerable results
[27]. S. Wrona et al. in two different articles used local directional controllability theory to
find the optimal place for piezoelectric sensor and actuator. Their studied structure was
isotropic square plate with four sides clamped. They used Memetic algorithm to optimize the
controllability index. Memetic algorithm has a proper performance due to faster convergence
and better statistical solution [28-29]. A.Takezawa et al. proposed an optimization methodology for
piezoelectric sensor and actuator for vibration control truss structure based on numerical method

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optimization [30]. W.Dafang et al. experimental studied about beam structure by using patch
piezoelectric and for comparison experimental and analytical method was used the independent modal
space method[31]. F.M.Li and X.X.Lv investigated vibration control of lattice sandwich beam using
the sensor and actuator piezo-patch and designing controller with the velocity feedback control
method and linear quadratic regulator for suppressing lattice sandwich beam displacements [32].
M.K.Kwak and D.H.Yang investigated fluid structure interaction (FSI) in rectangular plate structure
submerged in a fluid for analyzing active vibration control piezo-structure and for controlling system.
They used MIMO (Multi- Input Multi-Output)positive position feedback controller[33]. K.Khorshidi
et al. in their studies which is controlled circular plates coupled with piezoelectric layers excited by
plane sound wave based on classical plate theory and using LQR and Fuzzy-Logic controllers [34].
M.A.Trindade et al. designed semi-model active vibration control of plate structure and using modal
filter with distribution piezoelectric patches on structure [35]. S.Q.Zhang et al. using various
nonlinearities method for designing vibration suppression composite thin-walled structure with 2 types
laminated such as cross-ply and angle-ply. They used negative proportional velocity feedback control
and piezoelectric material for reducing vibration structure [36]. K.Yildirim and I.Kucuk designed
optional controller for vibration autention simply-supported Timoshenko beam by using
piezoelectric[37]. B.A.Selim et al. in their study that could investigated to active vibration control of
sunctionally graded material(FGM) plate and using Reddy’s higher-order shear deformation
theory(HSDT) and implementd piezo-Layers for reducing vibration stricture[38]. O.Abdeljaber et al
discussed neurocontroller for active vibration control cantilever plate for designing optimal voltage
piezoelectric actuator by using neural network [38]. J.Plattenburg et al. investigated vibration control
cylindrical shell with piezo-patches and disturbuted cardboard liner[39]. B.A.Selim et al. designed
CNT-reinforced composite plates and cylindrical shells with Reddy’s higher shear deformation and
using piezoelectric layers [40]. In this study, a cantilever beam is modeled using Timoshenko
Theory and finite element numerical method. Then, cantilever beam was divided into 20
elements and piezoelectric sensor and actuator patches are embedded at the bottom and the
top of the element. Then, control process for attenuation and transverse vibration control of
the beam against Random, White Noise, Random Integer, Random Binary inputs was done.
For structure control against these movements, Fuzzy-PID controller and Multi-Objective
Genetic Algorithm (MOGA) were used to complete the control loop.
2. Constitutive Equations

3rd
As shown in Figure (1), the studied structure in this model is a thin cantilever beam under external effects
including transverse force. Timoshenko beam is closer to reality than Euler-Bernoulli beam model because of
considering the effect of shear stress.
Fig.1. Cantilever Beam
As shown in Figure (2), after bending Timoshenko beam the cross section does not remain perpendicular to the
neutral axis.
Fig.2. Shearing Stress Effect on Beam Element[46].
Therefore, a gradient is formed in the neutral axis, which is considered as two parameters of and β according
to the following figure.
Fig.3. Deformation angles [46].
b
L
𝑡𝑏
x
y
z
F(t)

3rd
In finite element method, there are different methods to describe each element by using shape functions. For a
hypothetical element such as Figure (5), we used a 2-node element in its beginning and end, each of which has
degrees of freedom including angular and transverse.
Shape function for an element is defined as follows:
Where q is defined as displacement vector for nodes of one element according to Figure (4). Each coefficients of
q are function matrices that are completely in accordance with boundary conditions of the cantilever beam [33].
Fig.4. Degree of freedom beam element
Transverse Displacement
Shape Function
[ ]
[
{ ( ) ( ) ( ) }
{( ) ( ) ( ) ( ) ( )}
{ ( ) ( ) ( )}
{( ) ( ) ( ) ( )}
]
(
1
2
)
(8)
[ ][ ]
(9)
[ ][ ]
(10)
[ ][ ]
(11)
̇ [ ][ ̇]
[ ]
𝑤
𝜃
𝑤
𝜃
Element Node 2
Node 1

3rd
The derivative of the equation (12) due to x is:
Angular Displacement Shape Function
[ ]
[
{( ) ( )}
{ ( ) ( ) }
{( ) ( )}
{ ( ) ( )}
]
(13)
The derivative of the equation (12) due to time is:
Acceleration Shape Function
[ ]
[
{ }
{ }
{ }
{ ( )}
]
(14)
As noted before, the only difference between Timoshenko beam and Euler-Bernoulli beam is the following
parameter:
( )
(15)
Depending on the type of the cross section, the structure can be different and K parameters can be considered for
rectangular and circular cross-sections.
Rectangular
Circular
From (12) and (13), we can extract stiffness and mass matrices:
[ ] ∫ [
[ ]
[ ]
] [ ][
[ ]
[ ]
]
(16)

3rd
As each node in one element has two degrees of freedom including angular and transverse motions, mass matrix
has two parts. is related to transverse motion, and is related to angular motion.
[ ] [ ] [ ] (17)
[ ] * + * + (18)
Exactly the same as before in order to find piezoelectric stiffness and mass matrices, each element of a
piezoelectric patch is considered with 2 nodes, each of which has two degrees of freedom (Figure 4). Therefore:
Equation (4) is related to strain energy, from which the system stiffness is calculated[46].
[ ] ∫ [
[ ]
[ ] [ ]
] * +[
[ ]
[ ] [ ]
] (19)
Finally, from the virtual work equation (6), external force matrix of system can be extracted:
[ ] ∫ [
[ ]
[ ]
] * +
(20)
Now, in order to describe a smart structure, piezoelectric must be combined with the structure. Therefore,
assembled beam stiffness and mass matrices are combined with piezoelectric stiffness and mass matrices.
Therefore:
[ ] [ ] [ ] (21)
[ ] [ ] [ ] (22)
[ ] [ ] [ ] (23)
Where α and β can be achieved through Rayleigh method.
and matrices are mass and stiffness matrices for piezoelectric patch in equations 21-22,
respectively, which are achievable using equations 16 and 19. The only difference is that in equations 16 and 19,
mechanical properties of piezoelectric should be replaced according to Table 4.
3. Piezoelectric Electromechanical Model
As shown in Figure 2, we divide a cantilever beam into nodes or grids through finite elements. The greater the
number of nodes is, the more accurate the computations and design analysis will be. Now assume beam of
Figure 2 is similar to Figure 5. Then, piezoelectric actuator and sensor are respectively supposed to be placed on
the top and bottom beam on one of the divided elements. The mechanism of piezoelectric patch is in a way that it
would sense each element on which the piezoelectric sensor is embedded by vibration and angular and transverse
displacement, and convert it into current using the equation (24). In this paper, electromechanical model is
designed as Figure 5, i.e. piezoelectric sensor and actuator element is placed in the second element and then
Fuzzy-PID controller is designed based on this model [46].
z Digital to Analog
Fuzzy-PID
Actuator
x

3rd
Fig.5. Electromechanical beam model
There
̃
Then, it would convert current into voltage by multiplying Gc, which indicates electrical resistance along the
signal path from the sensor to the controller.
The most important part of this study is the design of the best controller gain that is optimized by using Fuzzy-
PID controller and GA. Fuzzy controller has applied the best order from voltage to piezoelectric actuator on the
beam through equation (26) to end with system's vibration attenuation. Controlling force that piezoelectric
actuator must actually apply on the beam is defined as follows [10]:
(26)
̅ ∫
(27)
̅ is the distance between the natural beam axis and natural piezo-patch.
4. Fuzzy-PID Controller Design
In this section, comparative PID controller design is done with variable coefficients. Controller coefficients are
set up by fuzzy logic. Fuzzy regulator considers error signal and its derivatives as the inputs in order to regulate
PID controller and calculates Kp, Kd, and Ki. The designed Fuzzy controller calculates PID controller coefficient
values, each of which has three membership functions, by assigning three membership functions to error signal,
̃ ∫ ̇
)24)
(25)

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three membership functions to the first derivative of the error, and nine controlling rules. In the following,
membership functions and control surfaces are shown for each inputs and outputs.
Table 1. Fuzzy rules
Rule Base Derivative Error
Error
N Z P
N N N Z
Z N Z P
P Z P P
5. State-space model
In dynamic description of continuous systems from Newton's second law, we face equation (28). It is noteworthy
that each coefficient in this equation indicates square matrix with identical order. In fact, they are the same
stiffness, mass, and damping matrices combined with piezoelectric.
̈ ̇ (28)
However, total stimulating force ft is made up of two parts. A force that is applied by external force and a force
that is applied by actuator on the beam or structure, preventing from unwanted displacement. Therefore:
(29)
One method to analyze and design system control is description based on state space. Therefore:
̇ (30)
Where x is state and y is system output. State space method converts nth
dynamic equation to n first order
equation. All system simulation is done based on state space.
[
[ ]
] [ ] [ ]
0 [ ] (31)
6. Simulation Modeling
In this paper, a cantilever beam with a uniform rectangular cross-section based on Timoshenko theory was
divided into 20 elements, as shown in Figure 1. As shown in Figure 4, each node has two degrees of
freedom. Now, consider a pair of piezoelectric sensor and actuator as shown in Figure 5. That means the
length of piezoelectric patch is equal to the length of each element of the beam. On the other hand,

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piezoelectric sensor and actuator are the only producers of torque at each node to reduce volatility of the
structure, as shown in Figure 6. Now, we displace the piezoelectric sensor and actuator patches so that they
would be embedded on the bottom and top of the beam, respectively. The best element of beam along with
the best coefficients of PID controller and 36 parameters of membership functions of fuzzy trigonometric
are detected by multi-objective genetic algorithms in a way that logic conditions of the controller are not
violated and structure vibration control is adopted. As can be seen in Figures 16, 19, 22, and 25,
piezoelectric optimum location for different inputs is at the end of almost the last element of the beam. This
indicates that piezoelectric optimum location for the inputs and incoming forces is the part of the beam with
the most displacement.
Fig. 1. Dynamic Model of Piezo-Patch [47].
Mechanical properties of cantilever beam and piezoelectric material and geometry information as
showed in Table 2 and 3.
Table 2. Beam information
Table 3. Piezo-patch information [34].
It was validated numerical method for modeling cantilever beam with analytical method and modeling
beam in ANSYS software as Table 4 is showed.
Table 4. Natural Frequencies for 9 Modes in Timoshenko Beam
Parameters Values
L Length 0.5 m
b Width 0.024 m
Young’s Modulus 193.096 GPa
Density 8030 kg/m3
Damping Constants 0.001,0.0001
Thickness 1 mm
Parameters Values
Length 0.125m
Width 0.024m
Thickness 0.5mm
Young’s Modulus 68GPa
Density 7700kg/m3
Strain Constant m/V
Stress Constant Vm/N
Analytical Numerical ANSYS

3rd
For comparison 3 methods based on Table 4 it is considered in Figure 2:
Fig. 2. Camparison Naturel Frequencies of Timoshenko Beam
6.1. Inputs
6.1.1. Membership Functions
Fuzzy-PID Controller design is based on table rules 1 with membership functions for PID controller
coefficients, error coefficients and derivative coefficients are showed in figure 3-7.
Fig. 3. Error Membership Function Fuzzy-PID Controller
-1 -0.5 0 0.5 1
0
0.2
0.4
0.6
0.8
1
e
Degree
of
membership
N Z P
16.706 16.696 16.698
104.699 104.572 105.01
292.751 292.528 296.26
572.186 572.467 587.65
942.512 944.736 987.59
1402.112 1408.498 1506.4
1949.703 1962.947 2157.4
2584.256 2607.317 2957.5
3304.567 3340.965 3926.8

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Fig. 4. Derivative Error Membership Function Fuzzy-PID Controller
Fig. 5. Kp Parameter of Controller
Fig. 6. Ki Parameter of Controller
Fig. 7. Kd Parameter of Controller
6.1.2. Surface Control
In Figure 8-10 are showed surface control of PID controller coefficients.
-1 -0.5 0 0.5 1
0
0.2
0.4
0.6
0.8
1
de
Degree
of
membership
N Z P
100 150 200 250 300 350 400 450 500
0
0.2
0.4
0.6
0.8
1
Kp
Degree
of
membership
N Z P
10 12 14 16 18 20
0
0.2
0.4
0.6
0.8
1
Ki
Degree
of
membership
N Z P
0 5 10 15 20
0
0.2
0.4
0.6
0.8
1
Kd
Degree
of
membership
N Z P

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Fig. 8. Kp Surface Control
Fig. 9. Ki Surface Control
Fig. 10. Kd Surface Control
6.1.3. Random Inputs
Active vibration simulation is investigated in this section. Firstly Random external distribution is
assumed in figures 11 and vibration suppression tip cantilever beam based on Random input is
showed as figure 12 and 13.
-1
-0.5
0
0.5
1
-1
-0.5
0
0.5
1
0
200
400
600
e
de
K
p
Surface K
p
-1
-0.5
0
0.5
1
-1
-0.5
0
0.5
1
10
15
20
e
de
K
i
Surface K
i
-1
-0.5
0
0.5
1
-1
-0.5
0
0.5
1
0
10
20
e
de
K
d
Surface Kd

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Fig. 11. Finite Interval Time Random Signal
Fig. 12. Control-Uncontrolled Flexural Displacement Tip Beam
Fig. 13. Control Flexural Displacement Tip Beam based on Random Input
6.1.4. White Noise Input
White Noise is kind of noise input used for distribution force at tip beam is showed at figure 14.
0 0.02 0.04 0.06 0.08 0.1
-300
-200
-100
0
100
200
300
Time (sec)
Amplitude
(N)
Signal Input
0 5 10 15
-0.03
-0.02
-0.01
0
0.01
0.02
0.03
Time (sec)
Dispalacement
(m)
Flexural Vibration Tip Beam
Uncontrolled
Fuzzy-PID
0 5 10 15
-0.01
-0.005
0
0.005
0.01
Time (sec)
Dispalacement
(m)
Active Vibration Control Tip Beam

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Fig. 14. External White Noise Signal Input on Tip Beam
Figure 15is without control displacement tip beam and with Fuzzy-PID Controller design and for more
resoultion , active vibration displacement amplitude at tip beam is showed figure 16.
Fig. 16. Control Flexural Displacement Tip Beam based on White Noise Input
6.1.5. Random Integer
Random Integer distribution input at tip beam is second input as using in this simulation as figure
17 and figure 18 and 19 are determined uncontrolled and controlled vibration respectively.
0 5 10 15
-3
-2
-1
0
1
2
3
x 10
4
Time (sec)
Amplitude
(N)
Signal Input
0 5 10 15
-0.02
-0.01
0
0.01
0.02
Time (sec)
Dispalacement
(m)
Uncontrolled
Fuzzy-PID
0 5 10 15
-2
-1
0
1
2
x 10
-3
Time (sec)
Dispalacement
(m)

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Fig. 17. External Random Integer Signal Input on Tip Beam
Fig. 19. Control Flexural Displacement Tip Beam based on Random Integer Input
6.1.6. Random Binary
Third external input as shown in figure 20 is Random Binary at tip cantilever beam and with
investigation vibration control by using Fuzzy-PID controller as shown in figure 21 and 22.
0 5 10 15
0
10
20
30
40
50
Time (sec)
Amplitude
(N)
Signal Input
0 5 10 15
-0.01
-0.005
0
0.005
0.01
0.015
0.02
Time (sec)
Dispalacement
(m)
Uncontrolled
Fuzzy-PID
0 5 10 15
-5
0
5
10
15
x 10
-3
Time (sec)
Dispalacement
(m)

3rd
Fig. 20. External Random Binary Signal Input on Tip Beam
Fig. 22. Control Flexural Displacement Tip Beam based on Random Binary Input
Conclusion:
This study attempted to automatically control a cantilever beam model from external stimulations by
piezoelectric and conduct vibration attenuation in system. Due to wide use of industrial PID controller and using
fuzzy calculations, smart control of the beam occurred. Using finite element method, system equations were
designed in MATLAB Software. Then, by embedding fuzzy PID controller attenuated the system. Here, it has
been attempted to stimulate the system with variety of inputs so that fuzzy PID controller design verification
would be reliable.
0 5 10 15
-10
0
10
20
30
40
50
60
Time (sec)
Amplitude
(N)
Signal Input
0 5 10 15
-0.02
-0.01
0
0.01
0.02
0.03
Time (sec)
Dispalacement
(m)
Uncontrolled
Fuzzy-PID
0 5 10 15
-5
0
5
10
15
x 10
-3
Time (sec)
Dispalacement
(m)

3rd
Table 5. Optimal Location Piezo-Patch
Input PZT Element Location
Random 18
White Noise 19
Random Integer 19
Random Binary 18
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