COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE
EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China
©2006 Tsinghua University Press & Springer
Active Vibration Control Analysis of Piezoelectric Intelligent Beam
Tao Wang, Rong Qin*, Guirong Li
Institute of Civil and Architectural Engineering, Guangxi University, Nanning, 530004 China
Email: freewangtao@eyou.com, cmssi@sina.com, lgrhl@163.com
Abstract The new theory and new method of the analysis of intelligent structures are studied. Based on intelligent
constitutive relation theory, instantaneous variational principle and spline dispersing, the control equation of the active
vibration control analysis of the piezoelectric intelligent beam is established by the spline finite-point method. Two
typical examples are studied and the results are given.
Key words: intelligent constitutive relation; instantaneous variational principle; spline finite-point method; active
vibration control
INTRODUCTION
Intelligent structures are showing wide application prospects in aerospace, civil aerospace, automobile, shipbuilding,
civil engineering and hydraulic engineering et al., the study of intelligent structures is the leading edge in disaster
reduction and disaster prevention. Piezoelectric material, with its unique piezoelectric effect and converse
piezoelectric effect [1], is widely applied in intelligent structures. It can be used as sensors, actuators and
sensors/actuators in intelligent structures. In recent years, the electro-mechanical analysis of piezoelectric intelligent
structures is researched by many scholars at home and abroad. Spline finite point method, the new method for active
vibration control analysis of piezoelectric intelligent beam, is presented in the paper.
GENERAL CONCEPT
feedback
skeleton
structure
muscle
actuator
nerve
sensor
brain
actuator
structure
sensor
controller
controller
Figure 1 Bionics modal of intelligent structures
Fig. 1 gives the bionics modal of intelligent structures [9]. Intelligent structure, which integrates sensors, actuators and
controllers together, is a bionics structural system rooted in bionics. Comparing with biont, structure is skeleton,
actuator is muscle, sensor is nerve and controller is brain in intelligent structure. Sensing the deformation of the
intelligent structure, the determined potential accrued on the surface of the sensors. The potential could be transformed
to voltage and fed back to the actuators via the action of the proper controller. The actuators deform, by the action of
the voltage above, to reduce the deformation of the intelligent structures. Sensors, actuators and controller are made of
intelligent materials in intelligent structures.
⎯ 1194 ⎯
GENERAL THEORY
1. Displacement modal For the flexible piezoelectric laminated beam with small deformation, the displacement
component format of any point in the beam is given as below:
⎧
4 z 3 ∂w
4z3
=
−
+
−
u
(
x
,
z
,
t
)
u
(
z
)θ x
⎪ 1
3h 2 ∂x
3h 2
⎨
⎪u ( x, z , t ) = w
⎩ 3
(1)
where u1 is the displacement component of any point along coordinate axis x, u3 the displacement component along
z, u the displacement of any point in the neutral axis along x, w the displacement along z, θ x the rotation in the x-z
plane.
2. Geometric equations The strain component of any point is presented as:
ε11 =
∂u1
∂x
ε 33 =
∂u3
∂z
ε 31 =
∂u3 ∂u1
+
∂x
∂z
(2)
Because the inter-laminar shear stiffness of the laminated beam is relatively much smaller, the influence of transverse
shear deformation is great and should not be omitted.
3. Intelligent linear elasticity piezoelectric constitutive relation
⎧σ = Dε − eT E
⎨
⎩ B = eε + ζ E
(3)
where B, σ , ε , E is the electric displacement vector, the stress vector, the strain vector and the electric field
strength vector respectively; and D, e, ζ the elasticity matrix, the piezoelectric coefficient matrix and the
dielectric constant matrix.
4. Relations between electric field strength and potential
⎧ ∂ξ ⎫
⎪⎪ ∂x ⎪⎪
⎧E ⎫
E = ⎨ x ⎬ = − ⎨ ⎬ = − H {ξ }
⎩ Ez ⎭
⎪ ∂ξ ⎪
⎪⎩ ∂z ⎪⎭
(4)
where E is the electric field strength and ξ the potential. For E = 0 in the basic beam, we obtain:
σ = Dε
(5)
5. Intelligent variational principle
l
l
1
⎡⎣ε T Dε − ε T eT E − E T eε − E T ζ E − 2U T ( F − cU& − ρU&& ) ⎤⎦dV − ∫ U 0T qbdx + ∫ ξ Qbdx
∫
V
0
0
2
excluding interior charge, where
Π=
(6)
⎧⎪U = [u1 u3 ]T U 0 = [u w θ x ]T q = [ qx qz mx ]T
⎨
T
T
T
⎪⎩ X = F − CU& − ρU&& F = [ Fx Fz ] σ = [σ x τ zx ] ε = [ε11 ε 31 ]
(7)
MODELING METHOD
The active vibration control model of piezoelectric intelligent beam can be established by spline finite point method
and spline subdomains method. The active vibration control arithmetic of piezoelectric laminated beam (Figure 2) is
established by spline finite point method in this paper [2-4].
1. Spline dispersing The beam is dispersed by spline basic function along coordinate axis x:
0 = x0 < x1 < x2 < L < xN = l xi = x0 + ihx hx = xi +1 − xi = l / N
The displacement function is given below:
u = [φ ]{u} w = [φ ]{w} θ x = [φ ]{θ x } ξ = [φ ]{ξ }
(8)
⎯ 1195 ⎯
U 0 = [ N 0 ]{V } U = [ N ]{V } ε = [ A]{V } E = − [ H ]{ξ }
(9)
The concrete format of [φ ] , [ϕ3i ] , [Q ] , U 0 , [ N 0 ] , {V } , U , [ N ] , {V } and
[ A]
are in reference [2].
t
0
1
2
N
h
x
t
l
b
z
Figure 2: Schematic of a piezoelectric laminated beam
2. Establishing intelligent spline dispersing functional The intelligent beam is composed of three layers. The upper
layer and the lower layer are made of piezoelectric material, acting as actuator and sensor respectively. The middle
layer is the structural layer, made of elastic material. The spline dispersing functional of intelligent beam can be
derived from equation (8) and equation (9), excluding physical force:
Π=
.
..
1
T
T ⎛
⎞
{V } ⎡⎣ K ∗ ⎤⎦ {V } + {V } ⎜ [C ]{V }+ [ M ]{V } ⎟
2
⎝
⎠
T
T
T
T
1
T
+ {V } ⎡⎣ K p ⎤⎦ {ξ p } + {ξ p } ⎡⎣ K p ⎤⎦ {V } − {ξ p } ⎡⎣ K% p ⎤⎦ {ξ p }
2
T
1
T
T
+ {V } ⎡⎣ K S ⎤⎦ {ξ S } + {ξ S } ⎡⎣ K ⎤⎦ {V } − {ξ S } ⎡⎣ K% S ⎤⎦ {ξ S }
2
(
(
)
− {V } { P} + {ξ p }
T
T
)
(10)
{ f } + {ξ } { f }
T
p
S
S
where
⎧[ M ] = [ M a ] + ⎡ M p ⎤ + [ M S ]
⎣
⎦
⎪
⎪
⎨[C ] = [Ca ] + ⎡⎣C p ⎤⎦ + [CS ]
⎪ ∗
*
*
*
⎪⎩ ⎡⎣ K ⎤⎦ = ⎡⎣ K a ⎤⎦ + ⎡⎣ K p ⎤⎦ + ⎡⎣ K s ⎤⎦
⎧[ M ] = [ N ]T ρ [ N ] dV
a
⎪ a ∫Va
⎪
T
⎪ ⎡⎣C p ⎤⎦ = ∫ [ N ] c p [ N ] dV
Vp
⎪
T
⎪ *
⎪⎪ ⎡⎣ K a ⎤⎦ = ∫Va [ A] Da [ A] dV
⎨
l
T
⎪{ P} = ∫ [ N ] qbdx
0
⎪
T
⎪ %
⎪ ⎡⎣ K p ⎤⎦ = ∫V p [ H ] ζ [ H ] dV
⎪
⎪ f = − l φ T Q bdx
∫0 [ ] p
⎩⎪{ p }
(11)
[Ca ] = ∫V [ N ] ca [ N ] dV
⎡⎣ M p ⎤⎦ = ∫ [ N ] ρ p [ N ] dV
Vp
[ M s ] = ∫V [ N ]
[Cs ] = ∫V [ N ] cs [ N ] dV
T
a
ρ s [ N ] dV
T
s
T
T
s
⎡⎣ K *p ⎤⎦ = ∫ [ A] D p [ A] dV
Vp
T
⎡ K p ⎤ = ∫ [ A] eT [ H ] dV
⎣ ⎦ Vp
T
⎡⎣ K s* ⎤⎦ = ∫ [ A] Ds [ A] dV
Vs
T
⎡ K s ⎤ = ∫ [ A] eT [ H ] dV
⎣ ⎦ Vs
T
(12)
⎡⎣ K% s ⎤⎦ = ∫ [ H ] ζ [ H ] dV
Vs
T
T
{ f s } = − ∫0 [φ ] Qsbdx
l
where suffixes a, p, s indicates the basic structure, the actuator and the sensor respectively; Va , V p , Vs the area
of the basic structure, the actuator and the sensor respectively; q, F , Q the surface force vector, the physical force
.
..
vector and the charge density; c, ρ the damping coefficient and the mass density of material; U , U , U the
displacement vector, the velocity vector and the acceleration vector.
3. Establishing spline dispersing dynamic model of intelligent beam By instantaneous variational principle, we
obtain:
[ M ]{V }+ [C ]{V }+ ⎡⎣ K * ⎤⎦ {V } + ⎡⎣ K p ⎤⎦ {ξ p } + ⎡⎣ K s ⎤⎦ {ξ s } = {P}
..
.
⎯ 1196 ⎯
(13)
{ξ } = ⎣⎡ K%
p
−1
p
⎤⎦ ⎣⎡ K p ⎦⎤ {V } + ⎡⎣ K% p ⎤⎦
T
−1
{ f } {ξ } = ⎡⎣ K% ⎤⎦
p
s
s
−1
⎡⎣ K s ⎤⎦ {V } + ⎡⎣ K% s ⎤⎦
T
−1
{ fs}
(14)
By substituting equation (14) into equation (13), we obtain:
..
.
[ M ]{V }+ [C ]{V } + [ K ]{V } = { f }
(15)
Eq. (15) is the intelligent spline dispersing dynamic equation, where:
T
T
−1
−1
∗
⎧
⎡ ⎤ %
⎡ ⎤ ⎡ ⎤ %
⎡ ⎤
⎪[ K ] = ⎡⎣ K ⎤⎦ + ⎣ K p ⎦ ⎡⎣ K p ⎤⎦ ⎣ K p ⎦ + ⎣ K s ⎦ ⎡⎣ K s ⎤⎦ ⎣ K s ⎦
⎨
−1
−1
⎪{ f } = { P} − ⎡ K p ⎤ ⎡⎣ K% p ⎤⎦ { f p } − ⎡ K s ⎤ ⎡⎣ K% s ⎤⎦ { f s }
⎣
⎦
⎣
⎦
⎩
(16)
4. Active vibration control of intelligent beam Because the applied voltage value of the piezoelectric sensor is
commonly 0 volt and expressed as
Qs = Qs = 0 .
{ f } = {0}
s
{ f2 }
is derided from equation (12). When accepting closed-loop feedback control rule, the charge load vector
is the feedback control force, is given by:
{ f 2 } = ⎡⎣ K p ⎤⎦ ⎡⎣ K% p ⎤⎦ {ξ p }
−1
(17)
where {ξ p } is the output voltage function of the piezoelectric sensor and is determined by the feedback control rule:
{ξ } = −G
p
d
−1
−1
⎡⎣ K s ⎤⎦ ⎡⎣ K% s ⎤⎦ {V } − Gv ⎡⎣ K s ⎤⎦ ⎡⎣ K% s ⎤⎦ {V& }
(18)
where Gd and Gv is the displacement feedback gain coefficient and the speed feedback gain coefficient respectively.
The vibration control equation of intelligent structures is derived by substituting equation (18) into equation (17):
[ M ]{V }+ ([C ] + [Cb ]){V }+ ([ K ] + [ Kb ]){V } = { f }
..
.
(19)
where
T
⎧
% −1
% −1
⎪[ K b ] = ⎣⎡ K p ⎦⎤ ⎣⎡ K p ⎦⎤ Gd ⎣⎡ K s ⎦⎤ ⎡⎣ K s ⎤⎦
⎨
−1
−1
T
⎪[Cb ] = ⎡⎣ K p ⎤⎦ ⎡⎣ K% p ⎤⎦ Gv ⎡⎣ K% s ⎤⎦ ⎡⎣ K s ⎤⎦
⎩
(20)
Equation (19) indicates that, when the geometric parameters of intelligent structure are unchanged, changing the
feedback gain coefficient can change the total damp and total rigidity of intelligent structure, change the dynamic
characteristics and transient response of intelligent structure, and to achieve the active control purposes.
EXAMPLES
1. Bimorph cantilever beam Figure 3 shows the bimorph cantilever beam made up of two PVDF films with opposite
polarization direction. Thus the bending deformation accrued in the beam because the applied electric field induces the
simple bending moment. The material parameters of the bimorph beam are: E = 2.0 GPa , e31 = e32 = 0.046 C / m 2 ,
e33 = 0.072 C / m 2 , g 11 = g 22 = g 33 = 10.62 nF / m . The geometrical parameters of the beam are: l = 0.1 m, b = 5 mm, the
thickness of each film h is 0.5 mm. The analytic solution is given in conference [5]. The author compiled a C language
program according to the modeling method above [6] and the results are given in Table 1 and Table 2.
0 1 2
h
N
x
l
h
b
z
Figure 3: Bimorph cantilever beam
⎯ 1197 ⎯
2. Active vibration control of intelligent beam Figure 4 shows the intelligent cantilever beam: made up of two layers,
the upper is PVDF piezoelectric film and the lower steel. Basic structure parameters: a = 0.146 m , b = 1.27 cm ,
h1 = 0.381m m , elastic modulus E = 2.10 G Pa , the mass fasten on beam end m = 0.673 kg . PVDF parameters:
a = 0.146 m , b = 1.27 cm , h2 = 2.8 × 10 − 3 m m , elastic modulus E% = 2.0 GPa , density ρ = 1800 kg / m 3 , piezoelectric
constant d 31 = 22 × 10 − 12 m / V . The initialization deflection of beam end is 2cm and the beam becomes to vibrate.
What we do is to determine the beam end deflection and the feedback voltage value.
PVDF
h1
h2
steel beam
Figure 4: Schematic of an intelligent beam
The author of reference [8] did the active vibration control experiment of intelligent beam. The active vibration control
was researched by FEM method in reference [9]: the beam is divided into 10 generalized conforming plate elements
and the Bailey damping coefficient α = 0, β = 0.865 × 10 − 5 . The authors in the paper do the active vibration control
by spline finite point method. All the results are listed in Table 3 and Table 4.
Table 1 The deflection along the coordinate axis x subjected to the specific voltage (unit: 1×10−7m)
x(mm)
Analysis solution [5] Reference [7] Spline finite point method
0.00
0.000
0.000
0.000
0.02
0.138
0.137
0. 137
0.04
0.552
0.548
0.549
0.06
1.242
1.233
1.234
0.08
2.208
2.192
2.196
0.10
3.450
3.424
3.432
Table 2 The deflection of the beam end subjected to variable voltages (units: 1×10−5 m)
Voltage (V) Analysis solution [5]
0
0.000000
−4
Reference [7]
Spline finite point method
0.000000
0.000000
0.171221×10
−4
0.171585×10−4
50
0.172500×10
100
0.345000×10−4
0.342441×10−4
0.343169×10−4
150
0.517500×10−4
0.513662×10−4
0.514754×10−4
200
0.690000×10−4
0.684883×10−4
0.686339×10−4
Table 3 The deflection of the end point of the intelligent cantilever beam (cm)
Time/s
0
5
10
15
20
25
30
35
40
45
50
Finite point method 2.000 1.471 1.092 0.801 0.606 0.435 0.330 0.244 0.183 0.153 0.116
FEM
2.000 1.450 1.071 0.800 0.590 0.433 0.321 0.242 0.177 0.140 0.099
Experimental value 2.000 1.484 1.102 0.803 0.612 0.438 0.332 0.245 0.186 0.154 0.120
Table 4 The feedback voltage of the end point the intelligent cantilever beam (cm)
Time/s
Finite point method
FEM
Experimental value
0
100
100
100
5
74.92
72.61
75.74
10
54.96
54.35
55.02
15
39.61
39.21
40.15
20
31.61
31.86
31.22
⎯ 1198 ⎯
25
21.96
21.45
22.87
30
16.20
16.11
16.27
35
11.57
11.57
12.21
40
9.38
8.77
9.65
45
7.36
6.92
7.67
50
6.12
5.63
6.20
CONCLUSIONS
The spline finite point method, applied to analysis the piezoelectric active vibration control in the paper, have some
merits: concise and clear physical concept, logical clarity, convenient handling for stiffness matrix, load matrix
including the electro-mechanical matrix, simple calculation format and precise calculations. In a word, this paper
represents a good way to analysis the intelligent structures.
Acknowledgements
The support of own work by National Natural Science Fund of China (Project 19872020) and National Natural Science
Fund of Guangxi (Project 0339013) is greatly acknowledged.
REFERENCES
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⎯ 1199 ⎯