2017 5th Iranian Joint Congress on Fuzzy and Intelligent Systems (CFIS) 7-9 March, Qazvin Islamic Azad University, Tehran, Iran
Fuzzy Active Vibration Control of an Orthotropic Plate Using Piezoelectric Actuators Seyed Rashid Alavi
Mehdi Rahmati
Dep. of Mechanical Eng. Isfahan University of Technology Isfahan, Iran
[email protected]
Dep. of Mechanical Eng. Isfahan University of Technology Isfahan, Iran
[email protected]
Dep. of Industrial Eng. Najafabad Branch, Islamic Azad University
Isfahan, Iran
[email protected]
adaptive fuzzy sliding mode controller for active control of a smart beam with mass uncertainty through experimental studies. Li et al. [6] used fuzzy logic-based controllers for active vibration control of a smart flexible beam with mass uncertainty. They developed an adaptive fuzzy sliding mode (AFSM) controller, which can deal with model uncertainties. Li et al. [7] proposed a novel piezoelectric multimode control strategy for an all clamped stiffened plate. They employed an active disturbance rejection control method to ensure performance of the vibration suppression of the closed-loop system. An active control based on the Positive Position Feedback technique was proposed and tested by Zippo et al. [8] for vibration suspension of a free-edge rectangular sandwich plate. On the other hand, in addition to the studies related to small-size and light-weight structures [9-11], there are a substantial number of research efforts carried out for the purpose of protecting and improving other engineering systems such as data storage systems, micro-accelerometers, bio-related systems, mass and liquid level sensors, redundant pumping systems, and cooling towers [12-20]. In the present study, an orthotropic square plate is studied in order to compensate vibrations due to initial conditions. The square plate is bonded at the top and bottom with four piezoelectric actuators. Rectangular piezoelectric patches are polarized along the Z-axis and a voltage across their thickness is applied to them in order to control vibrations. The firstorder shear deformation plate theory is utilized to model the orthotropic plate. After that, the generalized differential quadrature method is employed in order to convert the continuous equations of plate motion to discrete equations. Finally, the LQR controller and fuzzy logic controller are used to compensate vibrations of plate.
Abstract— Common light-weight structures are often subjected to extensive vibrations contributing to mechanical damages and failures; thus, active vibration control of them is of crucial importance. In this article, a fully clamped orthotropic square plate is investigated so as to compensate transversal vibrations. The square plate is bonded at the top and bottom with four piezoelectric actuators. Rectangular piezoelectric patches are polarized along the Z-axis and a voltage across their thickness is applied to them in order to control vibrations. The first-order shear deformation plate theory is employed to model the orthotropic plate. After that, the generalized differential quadrature method is utilized to convert the continuous equations of plate motion to discrete ones. Finally, both the LQR and fuzzy logic controller are used in order to compensate plate vibrations. Keywords— Orthotropic plate; First-order shear deformation theory; Generalized differential quadrature method; Smart structure; Fuzzy logic controller; Piezoelectric actuator
I. INTRODUCTION Today’s common light-weight structures are often subjected to extensive vibrations which can reduce the structural life and contribute to mechanical damages and failures; as a result, active vibration control of them is very important, and a remarkable number of research efforts have been conducted into it over the past years. Takawa et al. [1] employed a fuzzy logic controller (FLC) for a hybrid smart composite beam actuated by electro-rheological fluids and piezoceramics actuators through the simulation and experiment. A fuzzy-controlled genetic-based optimization technique for optimal vibration control of cylindrical shell structures incorporating piezoelectric sensor/actuators (S/As) was presented by Jin et al. [2]. They proposed the geometric design variables of the piezoelectric patches, including the placement and sizing of the piezoelectric S/As by using fuzzy set theory. Sharma et al. [3,4], in two separate articles, established a control law for controlling vibrations of a cantilevered beam and plate by using fuzzy logic based independent modal space control and fuzzy logic based modified independent modal space control. Singla et al. [5] developed a positive position feedback controller and an
978-1-5090-4008-7/17/$31.00 ©2017 IEEE
Seyed Mohammad Mortazavi
II. MATHEMATICAL MODELING As shown in Fig. 1, the square plate of length a and thickness h is bonded at the top and bottom with four piezoelectric actuators. Rectangular piezoelectric patches are polarized along the z-axis and a voltage across their thickness is applied to them in order to control vibrations.
207
K s A 55 ( = I0
∂ 2w ∂φx ∂ 2w ∂φ y + + + ) ( ) K A s 44 ∂x 2 ∂x ∂y 2 ∂y
∂ 2w ∂t 2
(2.3)
[[[[[[[
∂ φy
∂ 2φ y ∂ 2φx ∂ φx ) + D12 + D 66 ( 2 + ∂x ∂y ∂x ∂x ∂y
2
D 22
2
∂y 2
(2.4)
∂ φy ∂w + φy ) = I 2 ∂y ∂t 2 2
− K s A 44 (
2 ∂ 2φ y ∂ 2φx ∂ 2φx ∂ φ y + D12 + D 66 ( 2 + D11 ) ∂x 2 ∂x ∂y ∂y ∂x ∂y
Fig.1. The orthotropic plate model with distributed actuators A. first-order shear deformation plate theory The first-order shear deformation plate theory (FSDT) [21] is employed in this article to model the orthotropic plate, which is clamped from all sides. The displacement field of the FSDT is expressed as follows:
u x (x , y , z , t ) = u (x , y , t ) + z φx (x , y , t ) u y ( x , y , z , t ) = v (x , y , t ) + z φ y ( x , y , t )
∂w ∂ 2φx − K s A 55 ( + φx ) = I 2 ∂x ∂t 2
In the equations above, the factor Ks is the shear correction coefficient, Aij and Dij are the plate stiffnesses, and I0 and I2 are the mass moments of inertia, which are given by:
(1)
A11 =
u z (x , y , z , t ) = w (x , y , t )
A11 (
3
2
∂ 2v ∂ 3w ∂ 2u ∂ 3w A 22 ( 2 + 3 ) + A12 ( + ) ∂y ∂y ∂y ∂x ∂x 2 ∂y + A 66 (
2
3
2
∂v ∂u ∂w ∂v + + 2 2 ) = I0 2 2 ∂x ∂x ∂y ∂x ∂y ∂t
A12 =
υ12 E 2 h 1 − υ12υ 21
E 1h 3 D11 = 12(1 − υ12υ 21 )
E 2h 3 υ12 E 2 h 3 D 22 = 12(1 − υ12υ21 ) 12(1 − υ12υ21 )
D 66 =
G 12 h 3 12
A 66 = G 12 h
(2.1)
∂ 2u ∂ 2v ∂ 3w ∂ 2u + A 66 ( 2 + +2 ) = I0 2 ∂y ∂x ∂y ∂x ∂y 2 ∂t
2
D12 =
3
∂ u ∂w ∂v ∂w + 3 ) + A12 ( + 2 ) 2 ∂x ∂x ∂y ∂x ∂y ∂x
E 1h 1 − υ12υ21
E 2h A 22 = 1 − υ12υ21
where ux, uy and uz are displacements in the x, y and z directions, respectively. Also, u, v and w denote displacements of a point on the plane z=0, and ௫ and ௬ are rotations of a transverse normal about the y- and x-axes, respectively. In continuation, equations of motion could be written in terms of displacements: 2
(2.5)
A 44 = G 23 h I 0 = ρh
(3)
A 55 = G13 h I2 = ρ
h3 12
where h and ߩ are plate thickness and plate density, respectively. E1, E2, G12, G13, G23, ࢜ͳʹ ࢜ʹͳ denote Young’s modulus, shear modulus, and Poisson’s ratio of orthotropic plate in different axial directions. (2.2)
B. Generalized differential quadrature method Generalized differential quadrature (GDQ) method [22] is employed in this investigation to convert the continuous equations of plate motion to discrete ones, which are easier to be solved. The main concept of GDQ is to approximate the partial derivative of a function with respect to a space variable
2 0 8
ሼܵሽ ൌ ሾܿ ா ሿሼܶሽ െ ሾ݀ሿሼܧሽ
at a grid point by an equivalent weighted linear sum of the function values of all grid points in the whole domain: N d nF = ¦C ijn F (x j ) dx n x = x i j =1
where
C ij(1) =
π (x i ) ; (x i − x j )π (x j )
Here, {T}, {S}, {E}, [c], and [d] indicate stress, strain, electric field intensity, elastic compliance, and piezoelectric constant matrices, respectively. The load (moment) generated from each pair of piezoelectric actuators can be expressed as:
(4)
where d31 is piezoelectric strain constant, and V is the voltage applied to the top and bottom piezo-layers in the opposite directions. Ep and tp are the Young’s modulus and thickness of piezo-layers. After assuming the synchronic motion and discretization of plate governing equations by GDQ, actuator loads are inserted and then, new discrete and coupled equations of motion are obtained, which can be expressed in the standard form:
where the definition of ߨሺݔ ሻ is: N
j =1
(6)
and for i=j N
C ij(1) = C ii(1) = −¦ C ik(1) , k =1
[ M ]{r} + [ K ]{r } = {Fext } + {Fp }
(7)
i = 1, 2,..., N ; i ≠ k ; i = j where
In which N is number of grid points along the x direction. In addition, the higher-order derivatives are determined by using weighting coefficients through matrix multiplication. N
k =1
N
k =1
k =1
N
N
k =1
k =1
C ij(3) = ¦C ik(1)C kj(2) = ¦ C ik(2)C kj(1)
G wT
G
φx T
G
T
φ y T º¼
{r (t )} ≈ ¦ φi q i (t ) = [ Φ ]{q (t )}
(8)
(13)
i =1
Here, ሼݍሺݐሻሽ is the modal coordinate vector, m is number of the retained vibration modes, which are used in the control process, and [߶] is the truncated modal matrix, which can be expressed as:
i , j = 1, 2,..., N
The Chebyshev–Gauss–Lobatto distribution is used as a grid pattern, which could be stated as:
aª πi º 1 − cos( ) » , « N ¼ 2¬ bª πj º y j = «1 − cos( ) » i , j = 0,1,..., N N ¼ 2¬
G vT
m
C ij(4) = ¦ C ik(1)C kj(3) = ¦C ik(3)C kj(1) ;
xi =
G
{r } = ª¬u T
(12)
D. Reduced-order modal equations The mode superposition method is used to reach an approximate reduced-order model of the system; thus, the GDQ nodal displacement vector is transformed to the modal coordinate vector by using the system modal matrix.
C ij(2) = ¦ C ik(1)C kj(1) , N
(11)
ܯ ൌ ܾ ܧ ݀ଷଵ ൫ݐ ݄൯ ൈ ܸ (5)
i , j = 1, 2,..., N ; i ≠ j
π (x i ) = ∏ (x i − x j ); i ≠ j
(10)
[ Φ ] = [Φ1 ,..., Φ m ]
m ≤n
(14)
After reducing the order of equations, the linear decoupled modal equation of the feedback control system is:
(9)
ª¬ M º¼ {q} + ª¬ K º¼ {q } = {Fp } + {Fext }
C. Piezoelectric-mechanical constitutive equations The linear coupled piezoelectric-mechanical [23] constitutive relation should be considered in order to enter the actuator forces into the basic equation. This relation is written as:
(15)
where
ª¬ M º¼ = [ Φ ] [ M
][Φ ] T ª¬ K º¼ = [ Φ ] [ K ][ Φ ] T
209
(16)
{F } = [Φ ] {F } {F } = [Φ ] {F } T
p
state variables are controlled to be zero. In addition, a Mamdani type fuzzy inference system is used in this control problem, and center of gravity (COG) is selected for the type of defuzzification. The rule base is combination of 75 IF-Then rules for each of FLCs.
T
p
ext
ext
The parameters given in (16) denote the modal mass, modal stiffness matrix, modal actuator force, and modal external force, respectively. In addition, ª¬ K a º¼ {V a } =
{F } , p
where ª¬ K a º¼ is the modal actuator stiffness matrix. III. COMPENSATOR DESIGN Fig. 2. Membership functions of inputs and outputs
In order to prepare equations for the control process, the second-order governing system equation should be converted to the first-order state-space form. The state space conversion of (15) is employed as follows:
{x } = [ A ]{x } + [ B ]{u }
IV. RESULTS AND DISCUSSION In this study, a specific control problem is solved by MATLAB as a numerical example to compensate vibrations of plate. Table 1 shows the numerical properties of the considered orthotropic plate and piezoelectric patches. After solving the eigen-value problem, related natural frequencies and mode shapes are determined. Fig. 3 illustrates the first four mode shapes of the specific orthotropic plate considered in this article. Finally, the results of control methods are shown in Fig. 4, and it is obvious that vibrations (caused by initial condition) are compensated by using each of controllers. According to Fig. 4, it can be observed that both the controllers could compensate vibrations under 0.1 (s).
(17)
where the state vector is:
{x } = [q1
q 2 q 3 q 4 q1 q2 q3 q4 ]
T
(18)
And the actuator input is ሼݑሽ ൌ ሾܸଵ ܸଶ ሿ் . In continuation, the full state feedback control law is applied to the control system. Then, the LQR controller and FLC are used in this article to compensate vibrations of plate.
Table 1. Numerical properties of the considered orthotropic plate and piezoelectric patches.
A. The LQR controller The LQR controller [24] is designed to minimize a cost function (performance index), which is written as:
Plate
Piezoelectric actuator
a
25 cm
bp
3 cm
b
25 cm
hp
2.5 cm
h
2.5 mm
tp
0.15 mm
where cost matrices are chosen to be ܳ ൌ ܴ ൌ ܫ. Then, an auxiliary matrix (P) is found from the solution of the algebraic Riccati equation (ARE):
E1
10 E2
2 GPa
E2
5 GPa
Ep ߩp
1780 Kg/m3
(20)
G12
0.5 E2
d31
23e-12 mV-1
G13
0.5 E2
G23
0.2 E2
ȣ12
0.25
ȣ21
0.025
Ks
0.8601
ߩ
1000 Kg/m3
∞
J = ³ ({q } Q {q } + {u } R {u })dt T
T
(19)
0
A T P + PA − PBR −1B T P + Q = 0 Finally, the control gain is calculated by:
G = R −1B T P
(21)
B. The fuzzy logic controller In this paper, two sets of FLCs [25] are used to compensate vibrations of the plate. The designed FLCs use five membership functions for both inputs and outputs, which are: negative large (NL), negative small (NS), zero (Z), positive small (PS) and positive large (PL). Membership functions of the inputs and outputs are defined in [-1, 1], and the Gaussian functions are selected for them as shown in Fig. 2. All of the
210
Fig. 4. The controlled displacement of middle point using LQR and FLC.
V. CONCLUSIONS In this paper, an orthotropic square plate was studied in order to compensate vibrations due to initial condition. The square plate was bonded at the top and bottom with four piezoelectric actuators. Rectangular piezoelectric patches were polarized along the Z-axis, and a voltage across their thickness was applied to them in order to control vibrations. Then, the first-order shear deformation plate theory was employed to model the orthotropic plate, which was clamped from all sides. After that, the generalized differential quadrature method was employed to convert the continuous equations of plate motion to discrete equations, which can easily be solved. Then, the linear coupled piezoelectric-mechanical constitutive relation was considered in order to enter the actuator forces into the basic equation, and the mode superposition method was used to reach an approximate reduced-order model of the system. Finally, the LQR controller and fuzzy logic controller were used by the full state feedback control to compensate vibrations of plate, and good vibration compensation was observed. REFERENCES [1]
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[4]
[5]
Fig .3. The first four mode shapes of the fully clamped orthotropic plate.
[6]
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