Theoretical Dynamic Model of the Piezoelectric Actuator with Multi-DOF Loading
Structure in Process of Microforging
Peng Hu1, Tegoeh Tjahjowidodo2, Sylvie Castagne3, Muhammad Taureza4
School of Mechanical and Aerospace Engineering, Nanyang Technological University, Singapore , 639798
1
Email: [email protected]
2
Email: [email protected]
3
Email: [email protected]
4
Email: [email protected]
Abstract
A piezoelectric actuator is applied in a microforging system which was designed specially by maximizing its structural
rigidity to study size effects in microforming. To generate accurate movement in micrometer/nanometer scale, a
dynamical model of the piezoelectric actuator is in need. A new approach which considers the effect of multi -degreeof-freedom (multi-DOF) loading structures on the dynamics of the actuators is proposed in this paper. Merging the
constitutive equations of the piezoelectric materials to the dynamics of single -DOF and two-DOF external structures,
theoretical models of the actuator are presented. By utilizing dynamic stiffness, the dynamical models are applicable
to the piezoelectric actuators with multi-DOF loading structures. A general dynamic model of the piezoelectric
actuator with multi-DOF loading structure will predict the behavior of the actuator. The linear model of the
piezoelectric actuator with multi-DOF external structure is the main contribution of this paper.
Keywords: piezoelectric actuator; dynamic model; multi-DOF loading structure; process of microforging
1. Introduction
High accuracy of micro-/nano-meter scale
manufacturing is required in the miniaturization of
various applications such as high precision optical
systems, micrometer/nanometer valves, high accuracy
positioning etc. A microforging system was designed,
fabricated and assembled to generate high load output
and high dimensional accuracy and this setup is the
optimal one which can achieve the best performance of
the actuator [1]. In this design, a piezoelectric actuator
is employed to provide accurate movement and large
force to actuate the system.
While having many advantages including large
force and accurate movement, piezoelectric actuators
suffer deleterious hysteresis nonlinearity which brings
inaccuracy and instability [2] if not properly controlled.
Models such as Bouc-Wen model [3], Preisach model
[4] and Prandtl-Ishlinskii model [4] are proposed to
characterize hysteresis nonlinearity in the piezoelectric
actuators. However, these models characterize smart
actuators without considering dynamic interaction
between the actuators and external structures although
loading effects will affect the dynamics of the
piezoelectric actuators.
In many literature of ultrasonic assisted
microforming [5], smart actuators were applied to
generate precise movement to microform. However,
the nonlinearity, especially hysteresis had not been
deeply studied and discussed. The effect of interaction
between the actuator and the external structure has
attracted many researchers to study and characterize
the dynamic performance and the interaction of the
whole systems consisting of the piezoelectric actuators
and loading structures. In [6], the electro-mechanical
transfer function models presented incorporate the
contact forces and predict the wheel system
performance. These derived formulations in this paper
are based on the general concept of the constitutive
equations governing piezoelectric materials which
permit the introduction of kinetic energy, electrical
energy, and geometric constraints relating to the
deformation variables. In the previous mentioned works
[6, 7], the loading structures are a drive rod with spring
loaded rollers and wheel system. However, in many
other applications, the loading structures are different
and various. In [8], the loading structure is treated as a
single-DOF mass-damper-spring system and the
piezoelectric actuator is linearized and analyzed based
on the constitutive equations along the longitudinal axis
of the piezoelectric material.
In this paper, dynamics of two-DOF and multi-DOF
loading systems will be combined with constitutive
equations to derive a linear dynamic model of the
piezoelectric actuator. Then, a general conclusion on
the dynamic model of the actuator with multi-DOF
loading structure is drawn. With the d ynamic model of
1
the piezoelectric actuator with multi-DOF structure
which characterizes the linear dynamics of the
piezoelectric actuator working in low frequency range,
control strategies of the actuator can be proposed to
improve the movement accuracy and further improve
the accuracy of microforging. These are the main
contributions of this paper and this setup.
2.
3.
Theoretical modeling
The loading structure is treated as either a singleDOF or a two-DOF mass-damper-spring system as
illustrated in Figs. 2 and 3. All parameters of the massdamper-spring systems can be evaluated from hammer
test or finite element analysis , and hence all
parameters of the loading structure are assumed to be
known. Therefore, the actuator with external structure
can be simplified in Fig. 2 where the loading structure
is characterized by m e, ce and k e.
Setup of the microforging system
A microforging system whose theoretical design
gives expected vertical deflection of maximum 7.6µm
under the piezoelectric actuator loading of 5kN was
designed, fabricated and assembled in-house [1]. And
the optimal deformation of the structure will be smallest
considering the influence of the actuator. This system
focuses on maximizing the structural rigidity and is
used to study size effects in microforging when
actuated by a piezoelectric actuator. In this study, our
interest is the dynamic model of the piezoelectric
actuator in the microforging system.
Fig. 2 Schematic of the interaction between an
induced-strain actuator and a single-DOF external
structure under dynamic conditions [9]
The full assembly drawing of the microforging
system and the photo of the assembled microforging
machine are presented in Fig. 1 (a) and (b). This
actuator is a stack actuator which comprises many
stack layers and can generate very large forces. The
schematic of the stack piezoelectric actuator and the
photo of this actuator are presented in Fig. 1 (c) and (d).
Definitions of all parameters in Fig.2 are listed in
Table1 as follows. Parameters which are defined and
used in the following equations are listed in Table 2.
Table 1 Definitions of parameters in Fig.2
Parameter
V(t)
F(t)
u(t)
me
ce
ke
Piezoelectric actuator, P025.200
Definition
Excitation voltage
Output force
Output displacement*
Static mass of external structure
Static damping of external structure
Static stiffness of external structure
Table 2 Parameters and their definitions
(a)
Parameter
S
T
E
D
s33E
d 33
ε
ρ
x
A
u(t)
(b)
*
(c)
(d)
Fig. 1 Setup of the microforging system [1] (a) Full
assembly drawing; (b) Picture of the fully assembled
microforging system ; (c) Schematic of the stack
piezoelectric actuator with pre-stress; (d) Picture of the
piezoelectric actuator.
Definition
Axial strain
Axial stress
Electric field
Electric displacement
Compliance
Piezoelectric coefficient
Dielectric permittivity
Density of piezoelectric material
Position of stack layer
Area of cross-section
*
Accumulated deformation
Note that u(t) is the total displacement of the loading
structure as well as the accumulated deformation of the
piezoelectric material.
2
Recall the 1-D constitutive equations:
Substituting the above two equations
constitutive relations gives in (4):
S  s33E T  d33 E
d 2u
d 2u
dE
E

s

 d33
33
2
2
dx
dt
dx
(1)
D  d33T   E
Considering Newton’s second law of motion
dT
dx
A  A
d 2u
dt
(4)
dE
0
dx
and strain-displacement compatibility
S
the
In uniform electric field,
(2)
2
into
(5)
then, a wave equation is obtained from (4)
du
dx
d 2u
d 2u
E

s

33
dx 2
dt 2
(3)
(6)
and its general solution is (7)
u( x, t )  [C1 sin( x)  C2 cos( x)]e jt  uˆ( x)e jt
With boundary conditions at x=0 and x=l,
the solution to the wave equation is
u (t , 0)  0
uˆ ( x) 
(8)
T (l ) A  ke ( )uˆ (l )
Thus, from the 1-dimensional constitutive
equations, Newton’s second law of motion and strain
displacement compatibility, the dynamics of the
u ( x,  , t )  uˆ ( x)e jt
where
Ke ( ) is
(7)
ˆ
d33 El
sin( x) (9)
 l cos( l )  r ( )sin( l )
piezoelectric actuator is in summarized by equation
(10).


ˆ

 jt
d33 E

sin(

x
)
 e
E
  cos( l )  s33 K sin( l )

e

A

(10)
the equivalent dynamic stiffness [10] of the loading system and is defined as the inverse of the
receptance as listed in Table 3.
Table 3 Various types of frequency response functions
according to response parameters [10]
Frequency response function
Response r
Standard
Inverse
Displacement X
Receptance
Dynamic stiffness
Velocity V
Mobility
Mechanical impedance
Acceleration A
Inertance
Apparent mass
Fig. 3 Schematic of the interaction between an
induced-strain actuator and a single-DOF elastic
structure under dynamic conditions
3
For the single-DOF systems, the dynamic stiffness is
Ke ()  [ke   2 me ]  jce  [1  ( / 0 )2  j 2 ( / 0 )]ke
(11)
For the two-DOF structures, the dynamic stiffness is
( j ) 4 
K e ( ) 
c1m1  c1m2  c2 m1
cc k m k m k m
c k c k
kk
( j )3  1 2 1 1 1 2 2 1 ( j )2  1 2 2 1 ( j )  1 2
m1m2
m1m2
m1m2
m1m2
m2
c c
k k
( j ) 2  1 2 ( j )  1 2
m1m2
m1m2
m1m2
(12)
Hence, when the loading structures are of
different DOF, dynamical models of the piezoelectric
actuators with various loading structures are available.
Derivation of the dynamic models of the piezoelectric
actuator with single-DOF and two-DOF e xternal
structures is presented in this part. Equation (10)
considers the loading effect of the e xternal structure
on the dynamics of actuators by merging dynamics of
the loading structures into the constitutive equations
of the piezoelectric material. Due to the linear property
of constitutive equations, the derived dynamical
models are also linear. Thus, the dynamic models are
applicable only in low frequency region where
hysteresis nonlinearity is negligible.
4.
References
Conclusions and discussion
This dynamic model of equation (10) is also
effective for a piezoelectric actuator with a multi-DOF
loading structure by introducing equivalent dynamic
stiffness [10], as it shows in Section 4. Due to the
equivalent d ynamic stiffness which is defined as the
inverse of receptance [10] of the loading structure, the
model can be applied to piezoelectric actuators with
multi-DOF loading structures. Thus, a dynamic model
of the piezoelectric actuator with kinds of external
structures is generalized, as equation (10) indicates.
With this linear dynamic model, a linear control
strategy based on this model, e.g. PID and
feedforward control, will be proposed and to be tested
on the microforging system to improve accuracy of
microforging. Besides, another modeling approach
which characterizes the piezoelectric actuator taking
account of hysteresis is simulating the dynamics of the
the actuator with multi-DOF structure by finite element
method.
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