Control of vibration suppression of an smart beam ... - Semantic Scholar
Control of vibration suppression of an smart beam by pizoelectric elements Aydin Azizi, Laaleh Durali, Farid Parvari Rad, Shahin Zareie Department of mechanical engineering, Sharif University of technology, Kish Island, Iran [email protected], [email protected], [email protected], [email protected]
Abstract— Vibration control is an essential problem in different structure. Smart material can make a structure smart, adaptive and self-controlling so they are effective in active vibration control. Piezoelectric elements can be used as sensors and actuators in flexible structures for sensing and actuating purposes. In this paper we use PZT elements as sensors and actuator to control the vibration of a cantilever beam. Also we study the effect of different types of controller on vibration. Keywords: Piezoelectric; PZT sensor; PZT actuator; Vibration control; cantilever beam
I.
INTRODUCTION
A smart structure is basically a distributed parameter system that employs sensors and actuators at different finite element locations on the beam and makes use of one or more microprocessors that analyze the response obtained from the sensor and use different control logics to command the actuators to apply localized strains to plant to response in a desired fashion and bring the system to equilibrium. Smart structure are used to alert system characteristics (such as stiffness and damping) as well as of the system response (such as strain and shape) in a controlled manner [1]. Te main objective of active vibration control is to reduce the vibration of a system by automatic modification of the system's structural response and this process is shown in Fig.2 [1].
Active beam structure composed of conventional beam integrated with smart materials to serves as actuator. Today, active beam have found many application in modern industries, as positioning tool, damping systems, active control, proportional valves, active wing, etc. II.
FURMULATION
We use finite element method for solving this project. Each element has 2 degree of freedom at each nodal point which are, a transverse deflection, an angle of rotation or slop and an electrical degree of freedom as the voltage. When voltage (control input) is applied to the actuator, counteracting moment will be induced by the piezoelectric actuators at each nodal point. The elemental mass and stiffness matrixes of the piezoelectric element Mp and Kp are shown as: (1)
And: (2)
The sensor voltage is: (3)
Figure 1. Block diagrammatic view of a smart structure
This sensor voltage is given as input to the controller and the output of the controller is the controller gain multiplied by s the sensor voltage V (t ) . Thus the input voltage to the actuatorV
a
(t ) , i.e., the control input u is given by
V a (t ) =u= Controller gain * V s (t ) (4) Figure 2. concept of reduction of vibrations using AVC
Aydin Azizi Tel: +98 914 141 6821 E-mail: [email protected]
A. Dynamic equation of smart structure The mass and stiffness matrixes M and K of the dynamic equation of the smart structure include the sensor / actuator mass and stiffness. The equation of motion of the smart structure and sensor output is ⎪⎧ M q + K q = f e x t + f c tr l = f (5 ) ⎨ s T ⎪⎩ y ( t ) = V ( t ) = p q
t
t
when q , q , f ext , f ctrl , f , p are the vector of displacements and slopes, the acceleration vector, the external force vector, the controlling forc3e vector, the total force vector and a constant vector of the beam. If we use transformation q = T g we will have:
hT = E p d31bz[−1 0 1 0 0 0 0 0](8×1) (6) III.
PROBLEM
We suppose a cantilever with 4 element which the sensor and actuator are located at the same place in second element as shown in Fig 3. z actuator
⎧⎪T T MTg + T T KTg = T T f ext + T T f ctrl = T T f t (6) ⎨ ⎪⎩ M * g + K * g = f *ext + f *ctrl T T Which M * = T MT , K * = T KT are called the
generalized mass and stiffness matrixes. The structural modal damping matrixes C * is
C * = α M * +β K *
(7)
Where α and β are the frictional damping constant and the structural damping constant used in C * .The dynamic equation and the sensor output of the smart structure is
⎧ M * g + C * g + K * g = f *ext + f *ctrl (8) ⎨ s T T ⎩ y (t ) = V (t ) = p q = p Tg The SISO state space model of the smart flexible cantilever beam for the first two vibratory modes is
⎧ x = Ax ( t ) + Bu ( t ) + Er ( t ) (9) ⎨ T ⎩ y (t ) = C x ( y ) + Du ( t ) With
sensor Figure 3. Finite element model of the beam
In our project we want to find free response of the beam to an initial condition. Also we design a suitable controller to control vibration of the beam. B.Properties of the beam Properties of the flexible cantilever and the piezoelectric element when beam is divided to 4 finite elements are shown in table1. Table1. Physical parameter of problem
Physical parameters
Cantilever beam
Piezoelectric(PZT): sensor/ actuator
Length(m)
lb=0.3(m)
lp =0.075(m)
Width(m)
b=0.03(m)
b=0.03(m)
0 I ⎡ ⎤ A =⎢ (10) ⎥ −1 −1 ⎣ −M * K * −M * C *⎦ (4×4)
Thickness(m)
tb =0.5(mm)
ts = ta=0.35(mm)
Density(kg/m3)
ρb=8030(Kg/m3)
ρp=7700(Kg/m3)
0 ⎡ ⎤ B =⎢ ⎥ −1 T ⎣ M * T h ⎦ (4×1)
(11)
Young's modulus(GPa)
Eb=193.06(Gpa)
Ep=68(Gpa)
C T = ⎡⎣0 p T T ⎤⎦ (1×4)
(12)
α=0.001
D =0
(13)
Damping constants used in C*
(14)
PZT strain constant
d31=125×10-12 (V/m)
PZT stress constant
g31=10.5×10-3 VmN-1
0 ⎡ ⎤ E =⎢ ⎥ −1 T ⎣M * T f ⎦ And
pT = Gce31zb[0 −1 0 1 0 0 0 0](8×1) (15)
β=0.0001
IV.
RESULTS
In the following sections results of our program will be discussed. 1.Mode shapes of vibration In the following figures first and second modes of vibration are shown. If we want to have smoother shape we should increase number of elements
0 1 0 ⎡ 0 ⎤ ⎢ 0 ⎥ (16) 0 0 1 ⎥ A= ⎢ ⎢−886.27 −1.0994e −009 −0.089627 −1.0994e − 013⎥ ⎢ ⎥ − 29703 −1.0869e − 013 − 2.9713 ⎦ ⎣−1.0869e − 009
0 ⎡ ⎤ ⎢ ⎥ 0 ⎥ B=⎢ ⎢ 0.00026746 ⎥ ⎢ ⎥ ⎣ −0.00024886 ⎦
C = [0
0
(17)
0.020777 0.18826] D= [0]
Figure 4. First mode of vibration of cantilever
Figure 5. Second mode of vibration
2.Free response of vibration In this section we find free response of the beam to an initial condition x 0 = 0.01( m ) .
(19)
3.Controlled vibration Now we design suitable controller for our system and compare response of the beam with controller with free vibration.
Figure 7. Free and controlled vibration of cantilever with P-controller
The state space model of system with P-controller is 0 ⎡ 0 ⎢ 0 0 A= ⎢ ⎢−886.3 −1.099e −009 ⎢ 1.087e 009 − − − 2.97e + 004 ⎣
1 0 − 0.2841 0.181
0 ⎡ ⎤ ⎢ ⎥ 0 ⎥ B=⎢ ⎢ 0.00026746 ⎥ ⎢ ⎥ ⎣ −0.00024886 ⎦ Figure 6. Free response of the cantilever
The state space model of the system is:
(18)
C = [0
0
0.020777 0.18826]
D= [0] Also if we design a PI-controller we have
0 ⎤ 1 ⎥⎥ (20) −1.762 ⎥ ⎥ −1.332⎦
(21)
(22) (23)
Figure 10. Forced response with sinusoidal input Figure 8. Vibration of cantilever with PI-controller
In the following figures we find the P-controller response of the beam to random and sinusoidal input.
Its state space is 0 ⎡ ⎢ 0 ⎢ A=⎢ −886.3 ⎢ ⎢−1.087e − 009 ⎢⎣ 0
0
1
0 − 1.099e − 009
0 − 0.2008
− 2.97e + 004
0.1034
0
0.01039
⎤ ⎥ 0 ⎥ − 0.0002675⎥ (24) ⎥ − 2.034 0.0002489⎥ ⎥⎦ 0.09413 0 0
0
1 − 1.007
0 ⎡ ⎤ ⎢ ⎥ 0 ⎢ ⎥ B = ⎢ 0.00026746 ⎥ ⎢ ⎥ ⎢ −0.00024886 ⎥ ⎢⎣ ⎥⎦ 0 C = [ 0 0 0.020777 0.18826
(25)
0] (26)
D= [0]
(27)
Figure 11. Vibration of the cantilever to random input for P-controller
4.Force response of vibration Here we find the response of the beam to a random and sinusoidal input.
Figure 12. Vibration of the cantilever to sinusoidal input for P-controller Figure 9. Forced response with random input
5.Velocity We can plot the velocity of beam which is shown in fig.12.
Figure 15. Free and controlled vibration of the cantilever with P-controller
VI.
REFRENCES
[1]. Bandyopadhyay, B., Manjunath, T.C. & Umapathy, M. 2007, Modeling, control and implementation of smart structures, Springer, Germany. [2]. Reddy, J.N.1993, An introduction to the finite element method, McGraw-Hill international editions, Singapore..
Figure 13. Velocity of the cantilever with P-controller
6.Sensor voltage In this section we show the sensor voltage for free vibration and P-controller.
Figure 14. Sensor voltage for P-controller
V.
CONCLUSION AND DISCUSSION
In this paper we study how smart materials can be used to control the vibration of a beam. PZT is used as sensor and actuator at the same place; however they can attach to different places. As we see in fig.7 damping time of cantilever for free response is quite large. If we change damping constant used in C* ( α , β ), the structure will be damped faster. This claim is
shown in fig.15 for α = 0.01 ,
β = 0.001 .
Abstract— Vibration control is an essential problem in different structure. Smart material can make a structure smart, adaptive and self-controlling so they are effective in active vibration control. Piezoelectric elements can be used as sensors and actuators in flexible structures for sensing and actuating purposes. In this paper we use PZT elements as sensors and actuator to control the vibration of a cantilever beam. Also we study the effect of different types of controller on vibration. Keywords: Piezoelectric; PZT sensor; PZT actuator; Vibration control; cantilever beam
I.
INTRODUCTION
A smart structure is basically a distributed parameter system that employs sensors and actuators at different finite element locations on the beam and makes use of one or more microprocessors that analyze the response obtained from the sensor and use different control logics to command the actuators to apply localized strains to plant to response in a desired fashion and bring the system to equilibrium. Smart structure are used to alert system characteristics (such as stiffness and damping) as well as of the system response (such as strain and shape) in a controlled manner [1]. Te main objective of active vibration control is to reduce the vibration of a system by automatic modification of the system's structural response and this process is shown in Fig.2 [1].
Active beam structure composed of conventional beam integrated with smart materials to serves as actuator. Today, active beam have found many application in modern industries, as positioning tool, damping systems, active control, proportional valves, active wing, etc. II.
FURMULATION
We use finite element method for solving this project. Each element has 2 degree of freedom at each nodal point which are, a transverse deflection, an angle of rotation or slop and an electrical degree of freedom as the voltage. When voltage (control input) is applied to the actuator, counteracting moment will be induced by the piezoelectric actuators at each nodal point. The elemental mass and stiffness matrixes of the piezoelectric element Mp and Kp are shown as: (1)
And: (2)
The sensor voltage is: (3)
Figure 1. Block diagrammatic view of a smart structure
This sensor voltage is given as input to the controller and the output of the controller is the controller gain multiplied by s the sensor voltage V (t ) . Thus the input voltage to the actuatorV
a
(t ) , i.e., the control input u is given by
V a (t ) =u= Controller gain * V s (t ) (4) Figure 2. concept of reduction of vibrations using AVC
Aydin Azizi Tel: +98 914 141 6821 E-mail: [email protected]
A. Dynamic equation of smart structure The mass and stiffness matrixes M and K of the dynamic equation of the smart structure include the sensor / actuator mass and stiffness. The equation of motion of the smart structure and sensor output is ⎪⎧ M q + K q = f e x t + f c tr l = f (5 ) ⎨ s T ⎪⎩ y ( t ) = V ( t ) = p q
t
t
when q , q , f ext , f ctrl , f , p are the vector of displacements and slopes, the acceleration vector, the external force vector, the controlling forc3e vector, the total force vector and a constant vector of the beam. If we use transformation q = T g we will have:
hT = E p d31bz[−1 0 1 0 0 0 0 0](8×1) (6) III.
PROBLEM
We suppose a cantilever with 4 element which the sensor and actuator are located at the same place in second element as shown in Fig 3. z actuator
⎧⎪T T MTg + T T KTg = T T f ext + T T f ctrl = T T f t (6) ⎨ ⎪⎩ M * g + K * g = f *ext + f *ctrl T T Which M * = T MT , K * = T KT are called the
generalized mass and stiffness matrixes. The structural modal damping matrixes C * is
C * = α M * +β K *
(7)
Where α and β are the frictional damping constant and the structural damping constant used in C * .The dynamic equation and the sensor output of the smart structure is
⎧ M * g + C * g + K * g = f *ext + f *ctrl (8) ⎨ s T T ⎩ y (t ) = V (t ) = p q = p Tg The SISO state space model of the smart flexible cantilever beam for the first two vibratory modes is
⎧ x = Ax ( t ) + Bu ( t ) + Er ( t ) (9) ⎨ T ⎩ y (t ) = C x ( y ) + Du ( t ) With
sensor Figure 3. Finite element model of the beam
In our project we want to find free response of the beam to an initial condition. Also we design a suitable controller to control vibration of the beam. B.Properties of the beam Properties of the flexible cantilever and the piezoelectric element when beam is divided to 4 finite elements are shown in table1. Table1. Physical parameter of problem
Physical parameters
Cantilever beam
Piezoelectric(PZT): sensor/ actuator
Length(m)
lb=0.3(m)
lp =0.075(m)
Width(m)
b=0.03(m)
b=0.03(m)
0 I ⎡ ⎤ A =⎢ (10) ⎥ −1 −1 ⎣ −M * K * −M * C *⎦ (4×4)
Thickness(m)
tb =0.5(mm)
ts = ta=0.35(mm)
Density(kg/m3)
ρb=8030(Kg/m3)
ρp=7700(Kg/m3)
0 ⎡ ⎤ B =⎢ ⎥ −1 T ⎣ M * T h ⎦ (4×1)
(11)
Young's modulus(GPa)
Eb=193.06(Gpa)
Ep=68(Gpa)
C T = ⎡⎣0 p T T ⎤⎦ (1×4)
(12)
α=0.001
D =0
(13)
Damping constants used in C*
(14)
PZT strain constant
d31=125×10-12 (V/m)
PZT stress constant
g31=10.5×10-3 VmN-1
0 ⎡ ⎤ E =⎢ ⎥ −1 T ⎣M * T f ⎦ And
pT = Gce31zb[0 −1 0 1 0 0 0 0](8×1) (15)
β=0.0001
IV.
RESULTS
In the following sections results of our program will be discussed. 1.Mode shapes of vibration In the following figures first and second modes of vibration are shown. If we want to have smoother shape we should increase number of elements
0 1 0 ⎡ 0 ⎤ ⎢ 0 ⎥ (16) 0 0 1 ⎥ A= ⎢ ⎢−886.27 −1.0994e −009 −0.089627 −1.0994e − 013⎥ ⎢ ⎥ − 29703 −1.0869e − 013 − 2.9713 ⎦ ⎣−1.0869e − 009
0 ⎡ ⎤ ⎢ ⎥ 0 ⎥ B=⎢ ⎢ 0.00026746 ⎥ ⎢ ⎥ ⎣ −0.00024886 ⎦
C = [0
0
(17)
0.020777 0.18826] D= [0]
Figure 4. First mode of vibration of cantilever
Figure 5. Second mode of vibration
2.Free response of vibration In this section we find free response of the beam to an initial condition x 0 = 0.01( m ) .
(19)
3.Controlled vibration Now we design suitable controller for our system and compare response of the beam with controller with free vibration.
Figure 7. Free and controlled vibration of cantilever with P-controller
The state space model of system with P-controller is 0 ⎡ 0 ⎢ 0 0 A= ⎢ ⎢−886.3 −1.099e −009 ⎢ 1.087e 009 − − − 2.97e + 004 ⎣
1 0 − 0.2841 0.181
0 ⎡ ⎤ ⎢ ⎥ 0 ⎥ B=⎢ ⎢ 0.00026746 ⎥ ⎢ ⎥ ⎣ −0.00024886 ⎦ Figure 6. Free response of the cantilever
The state space model of the system is:
(18)
C = [0
0
0.020777 0.18826]
D= [0] Also if we design a PI-controller we have
0 ⎤ 1 ⎥⎥ (20) −1.762 ⎥ ⎥ −1.332⎦
(21)
(22) (23)
Figure 10. Forced response with sinusoidal input Figure 8. Vibration of cantilever with PI-controller
In the following figures we find the P-controller response of the beam to random and sinusoidal input.
Its state space is 0 ⎡ ⎢ 0 ⎢ A=⎢ −886.3 ⎢ ⎢−1.087e − 009 ⎢⎣ 0
0
1
0 − 1.099e − 009
0 − 0.2008
− 2.97e + 004
0.1034
0
0.01039
⎤ ⎥ 0 ⎥ − 0.0002675⎥ (24) ⎥ − 2.034 0.0002489⎥ ⎥⎦ 0.09413 0 0
0
1 − 1.007
0 ⎡ ⎤ ⎢ ⎥ 0 ⎢ ⎥ B = ⎢ 0.00026746 ⎥ ⎢ ⎥ ⎢ −0.00024886 ⎥ ⎢⎣ ⎥⎦ 0 C = [ 0 0 0.020777 0.18826
(25)
0] (26)
D= [0]
(27)
Figure 11. Vibration of the cantilever to random input for P-controller
4.Force response of vibration Here we find the response of the beam to a random and sinusoidal input.
Figure 12. Vibration of the cantilever to sinusoidal input for P-controller Figure 9. Forced response with random input
5.Velocity We can plot the velocity of beam which is shown in fig.12.
Figure 15. Free and controlled vibration of the cantilever with P-controller
VI.
REFRENCES
[1]. Bandyopadhyay, B., Manjunath, T.C. & Umapathy, M. 2007, Modeling, control and implementation of smart structures, Springer, Germany. [2]. Reddy, J.N.1993, An introduction to the finite element method, McGraw-Hill international editions, Singapore..
Figure 13. Velocity of the cantilever with P-controller
6.Sensor voltage In this section we show the sensor voltage for free vibration and P-controller.
Figure 14. Sensor voltage for P-controller
V.
CONCLUSION AND DISCUSSION
In this paper we study how smart materials can be used to control the vibration of a beam. PZT is used as sensor and actuator at the same place; however they can attach to different places. As we see in fig.7 damping time of cantilever for free response is quite large. If we change damping constant used in C* ( α , β ), the structure will be damped faster. This claim is
shown in fig.15 for α = 0.01 ,
β = 0.001 .
