The paper presents the results of numerical modelling and experiments of piezoelectric bimorph-type bending actuator for laser beam shutting system. Theoretical calculations are realized by using finite element method. The purpose of these calculations is to optimize geometric parameters of a piezoelectric bimorph, ensuring maximum resonant frequency of the first bending form and stability of oscillation amplitude. Experimental performance of piezoelectric actuators is compared with results of finite element simulations

G. Kulvietis

I. Tumasonienė

R. Bansevičius

S. Navickaitė

V. Jūrėnas

English

Journal of Vibroengineering

100 ? VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. MARCH 2013. VOLUME 15, ISSUE 1. ISSN 1392-8716  
925. Design and dynamic modeling of piezoelectric laser 
beam shutters 
R. Bansevičius, S. Navickaitė, V. Jūrėnas, G. Kulvietis, I. Tumasonienė 
925. DESIGN AND DYNAMIC MODELING OF PIEZOELECTRIC LASER BEAM SHUTTERS.  
R. BANSEVIČIUS, S. NAVICKAITĖ, V. JŪRĖNAS, G. KULVIETIS, I. TUMASONIENĖ 
R. Bansevičius1, S. Navickaitė2, V. Jūrėnas3, G. Kulvietis4, I. Tumasonienė5 
1, 2, 3Kaunas University of Technology, Donelaičio 73, LT-44029 Kaunas, Lithuania 
4, 5Vilnius Gediminas Technical University, Saulėtekio al. 11, LT-10223 Vilnius, Lithuania  
E-mail: 1ramutis.bansevicius@ktu.lt, 2sigita.navickaite@ktu.lt, 3vytautas.jurenas@ktu.lt, 
4Genadijus_Kulvietis@gama.vtu.lt, 5inga.tumasoniene@vgtu.lt 
(Received 17 May 2012; accepted 28 February 2013) 
Abstract. The paper presents the results of numerical modelling and experiments of 
piezoelectric bimorph-type bending actuator for laser beam shutting system. Theoretical 
calculations are realized by using finite element method. The purpose of these calculations is to 
optimize geometric parameters of a piezoelectric bimorph, ensuring maximum resonant 
frequency of the first bending form and stability of oscillation amplitude. Experimental 
performance of piezoelectric actuators is compared with results of finite element simulations. 
Keywords: laser beam shutter, piezoelectric bimorph, amplitude-frequency response. 
1. Introduction 
In this article we investigate a piezo-driven mechanical laser beam shutter, which can be 
used for modulation or the switching of the light intensity in various laser applications. Presently 
piezoelectric actuators are widely used in various mechatronic devices and systems, including 
high accuracy positioning systems, optical and laser scanning/deflecting devices, piezoelectric 
robots. Particular interest in using piezoelectric actuators was determined by their superior 
characteristics – high-energy efficiency and short response time. Piezoelectric actuators now 
belong to the most effective driving force or motion generation mechanisms; they operate both 
in quasi-static and resonance mode and in order to achieve the desired modal shape, it is 
necessary to optimize geometric parameters and obtain specific location of excitation zones. 
This fact is especially important for the ultrasonic transducers with several degrees of 
freedom [1-3]. 
The performance of piezoelectric devices strongly depends on the characteristics of the 
piezoelectric transducers, which are the main components of the piezomechanical system. The 
synthesis of specific oscillation areas of the piezoelectric actuator can be obtained by optimizing 
the geometrical parameters, the poling vector and the topology of excitation zones of the 
piezoelectric transducer. The piezoelectric effect plays an important role in the dynamical 
behaviour of these actuators [4]. So it is very important to know what modal shape will be 
excited when applying piezoelectric actuators for high resolution motion control [6]. 
Such piezoelectric bimorphs have very wide area of applications, such as sensor systems, 
printing technologies, scanners, ultrasonic atomizers, ultrasonic transducers, positioning 
transducers, viscosity measurement systems, flow measurement systems, energy harvesting, 
laser beam controlling systems, pneumatic valves for industrial and automotive applications, 
valves in medical applications, insulin pumps, braille keys for the blind, textile machinery, 
optical switches, oil exploration, machine and equipment monitoring, automotive engines, 
feedback sensors, high temperature accelerometers, rate & gyroscope sensors, intrusion alarms 
and many other devices. So development and investigations of such type piezoelectric actuators 
are very important. 
2. The beam laser shutter design 
The important problem in designing laser devices is controlling the intensity of laser beam 
power up to zero, effected by various laser beam shutters for safety and beam control 
925. DESIGN AND DYNAMIC MODELING OF PIEZOELECTRIC LASER BEAM SHUTTERS.  
R. BANSEVIČIUS, S. NAVICKAITĖ, V. JŪRĖNAS, G. KULVIETIS, I. TUMASONIENĖ 
 ? VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. MARCH 2013. VOLUME 15, ISSUE 1. ISSN 1392-8716 101 
applications. Here are considered the specifics of bimorph type piezoelectric transducer, 
generating bending motion of the double layer piezoelectric element. Investigated piezoelectric 
laser beam shutter consists of the bimorph type piezoelectric actuator 1, special element 2 for 
shutting the laser beam and holder 3 of the piezoelectric actuator (see Fig. 1 a). 
The model of a laser shutter is composed of two glued piezoelectric plates of rectangular 
shape, made of Pz26 piezoelectric material [5, 6]; the polarization vector is directed along the 
thickness of the plate. Seeking to cover laser beam, special element 2 is fixed at the tip of 
piezoelectric actuator 1. 
 
Fig. 1. The scheme of the laser shutter: ? and ? – width of the fixed and free ends of the plate respectively; 
? – height; ?? and ?? – lengths of the piezoceramic plate 1 and mass 2 respectively; 3 – holder 
The piezoelectric elements are electrically connected in series as shown in Fig. 2, in which 
two piezoelectric elements with opposite poling directions are directly bonded. When an electric 
field is applied across the thickness in such a way that one piezoelectric element contracts while 
the other expands, the free end of the bimorph  produces bending deformations of amplitude A. 
Bending deformations appears in actuator after bimorph is acted by voltage. Such electrodes 
placement is used for exciting the first bending mode in the piezoelectric actuator (Fig. 2). 
  
Fig. 2. Bending deformation of cantilever-mounted piezoelectric bimorph in series connection:  
? – poling vector; ? – amplitude of the bending deformation 
Geometric parameters of a piezoelectric bimorph have to be chosen in a way, ensuring the 
maximum of eigenfrequency of the first flexional mode and the stability of the oscillation 
amplitude ?. The geometrical parameters of piezoelectric actuators are shown in Table 1. 
Table 1. Geometric parameters of a bimorph  
Piezoelectric material PZ26 
Width ratio ? ? ? ?? ? ???? ?? ? ???? 
Length ??, m 0,050 0,050 0,050 
Length ??, m 0,005 0,005 0,005 
Width ?, m 0,01 0,01 0,01 
Width ?, m 0,01 0,005 0,0033 
Height ?, m 0,002 0,002 0,002 
925. DESIGN AND DYNAMIC MODELING OF PIEZOELECTRIC LASER BEAM SHUTTERS.  
R. BANSEVIČIUS, S. NAVICKAITĖ, V. JŪRĖNAS, G. KULVIETIS, I. TUMASONIENĖ 
102 ? VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. MARCH 2013. VOLUME 15, ISSUE 1. ISSN 1392-8716  
3. Modeling of bimorph 
A harmonic analysis of the piezoelectric actuator was made by using finite element method 
and also with experiments. Finite element method was used to perform numerical modelling of 
the actuator. It was used to carry out modal frequency and harmonic response analysis and to 
calculate trajectories of the free tip movements of bimorph-type cantilever. Driving force of the 
piezoelectric actuator is obtained from piezoceramic plate. Finite element discretization of this 
element usually consists of a few layers of finite elements. Therefore nodes coupled with 
electrode layers have known potential values in advance and nodal potential of the remaining 
elements are calculated during the analysis. Dynamic equation of piezoelectric actuator is 
derived from the principle of minimum potential energy by means of variational functionals and 
in this case can be expressed as follows [7]: 
?
??????? ? ? ??????? ? ? ??????? ? ????????? ? ???????? ? ????
????????? ? ?????????? ? ?????????? ? ?????
????????? ? ?????????? ? ?????????? ? ????
? (1) 
where ???, ???, ???, ???, ??? are matrices of mass, stiffness, electro elasticity, capacity, damping 
respectively; ???, {??, ??? are vectors of nodes displacements, potentials, structural mechanical 
forces and charge; ????, ???? are accordingly vectors of nodal potentials known in advance and 
calculated during numerical simulation. 
Natural frequencies and modal shapes of the actuator are derived from the modal solution of 
the piezoelectric system [7, 8].  
???????? ? ???????? ? ????? (2) 
where ????  is modified stiffness matrix and it depends on nodal potential values of the 
piezoelectric elements. 
Harmonic response analysis of piezoelectric actuator is carried out applying sinusoidal 
varying voltage on electrodes of the piezoelectric elements. Structural mechanical loads are not 
used in our case, so ??? ? ???. Equivalent mechanical forces are obtained, because of inverse 
piezoefect and can be calculated as follows [8]: 
??? ? ??????????? (3) 
here ???? ? ??????????? , where ???  is vector of voltage amplitudes, applied on the nodes 
coupled with electrodes. 
Results of structural displacements of the piezoelectric actuator obtained from harmonic 
response analysis are used for determining the trajectory of tip movement. 
For numerical analysis of piezoelectric actuators is used ANSYS software. The scheme for 
determining rational geometric parameters is shown in Fig. 3. 
Geometric parameters of the piezoelectric actuator must be chosen in such way, that the 
resonant frequency should get the values as high as it possible, ensuring short response time. 
The resulting design would satisfy system‘s technical characteristics and be rational from a 
technological standpoint. 
During analysis the dimensions of the free tip of piezoelectric bimorph-type cantilever have 
been changed (Table 1). In these bimorph models, it is imposed that one end of the cantilever is 
clamped in relation to mechanical boundary conditions. Hence, fixed constraint condition was 
applied for the vertical faces of the piezoelectric material layers. The other faces were 
unconstrained and free to move allowing the bending of the bimorphs under the application of 
voltage/force and due to the weight of the beam itself. The electroding of the top and bottom 
925. DESIGN AND DYNAMIC MODELING OF PIEZOELECTRIC LASER BEAM SHUTTERS.  
R. BANSEVIČIUS, S. NAVICKAITĖ, V. JŪRĖNAS, G. KULVIETIS, I. TUMASONIENĖ 
 ? VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. MARCH 2013. VOLUME 15, ISSUE 1. ISSN 1392-8716 103 
layers is required for the application of the voltage. However, the effects of the electrodes were 
not considered in the geometry because their mechanical behaviour can be neglected due to their 
thickness. The voltage of magnitude 19 volts is applied along the ?  direction. For each 
considered case the harmonic analysis was performed (Fig. 5).  
I
Integrating
Input
The begining of optimization
loop
Changing parameters
Eigen values
Calculation criteria of
optimization
End of optimization loop
Output
? ?? ? ? ?? ? ? ?? ? ? ?? ? ? ?? ?
? ? ? ? ? ?? ? ? ?QST
tRTKCM
T
k
??
????
??
???? ???
 
a) 
 
b) 
 
c) 
 
Fig. 3. The scheme for determining rational 
geometric parameters 
Fig. 4. FEM models of piezoelectric actuators with ratio of 
parameters a and b: a) a = b, b) a = b / 2, c) a = b / 3 
The largest bending displacements of the tip of piezoelectric actuator are in z direction. 
Calculated amplitude-frequency responses of piezoelectric actuators with different geometric 
parameters can be seen in Fig. 5. Theoretical results indicate, that the geometrical parameters of 
piezoelectric actuator have big influence for dynamic characteristics of it.  
 
Fig. 5. Amplitude-frequency responses of piezoelectric actuators with different geometric parameters 
(numerical simulation) 
With bigger ratio of parameters ? and ?, the resonant amplitude gets smaller values, but 
resonant frequency increases: ?? ? ? ? ? ?  68 μm, ? ?  320 Hz; ??? ?  2 – ? ?  55 μm, 
? ? 400 Hz; ??? ? 3 – ? ? 50 μm, ? ? 440 Hz. The deformations of piezoelectric bimorphs 
are in Fig. 6. 
925. DESIGN AND DYNAMIC MODELING OF PIEZOELECTRIC LASER BEAM SHUTTERS.  
R. BANSEVIČIUS, S. NAVICKAITĖ, V. JŪRĖNAS, G. KULVIETIS, I. TUMASONIENĖ 
104 ? VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. MARCH 2013. VOLUME 15, ISSUE 1. ISSN 1392-8716  
a)  b)  
c)  
Fig. 6. Deformations of piezoelectric actuators, when ratio of parameters ? and ?:  
a) ? ? ?, b) ? ? ???, c) ? ? ??? 
4. Experimental results  
The experimental setup is shown in Fig. 7 a). It consists of: 1 – power amplifier EPA-104; 
2 – signal generator Agilent 33220A; 3 – analog digital converter (ADC) “PicoScope-3424”;  
4 – laser displacement sensor LK-G82; 5 – laser sensor controller LK-G3001PV; 6 – Polytec 
OFV-5000 vibrometer controller; 7 – Polytec OFV-512 fiber interferometer; 8 – Polytec 
OFV-130-3 micro-spot sensor head; 9 – Polytec computer; 10 – computer; 11 – piezoelectric 
bimorph actuator. The piezoelectric actuators used in experiment are shown in Fig. 7 b). The 
dimensions of piezoelectric bimorphs are the same, as it is shown in Table 1. 
a)  b)  
Fig. 7. Experimental setup a) and piezoelectric bimorph actuators used in experiments b) 
As a result of experiments we have amplitude-frequency responses that are shown in Fig. 9. 
Experimental results confirm the results of numerical modelling of piezoelectric bimorph 
actuator. With bigger ratio of parameters a and b, the resonant amplitude gets smaller values, but 
resonant frequency increases: ??? ?  1 – ? ?  67 μm, ? ?  308 Hz, ??? ?  2 – ? ?  44 μm, 
? ? 384 Hz, ??? ? 3 – ? ? 38 μm, ? ? 419 Hz. 
Experimental results have confirmed, that the resonant frequency of the actuator depends on 
geometrical parameters of piezoelectric bimorph. The bending vibration resonant frequency of 
the cantilever increases approximately 1,3 times due to the decrease of the free end width of the 
cantilever by 3 times: that is from 308 Hz to 419 Hz. 
The shutter presented in this article can also be used as a laser beam intensity modulator, 
optimized for the resonant mode of operation. Large amplitude sinusoidal modulation at the 
resonant frequency of the cantilever bimorph is generated, but additional air-cooling of the 
device must be provided, due to heating of the piezoelectric actuator. 
925. DESIGN AND DYNAMIC MODELING OF PIEZOELECTRIC LASER BEAM SHUTTERS.  
R. BANSEVIČIUS, S. NAVICKAITĖ, V. JŪRĖNAS, G. KULVIETIS, I. TUMASONIENĖ 
 ? VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. MARCH 2013. VOLUME 15, ISSUE 1. ISSN 1392-8716 105 
 
Fig. 9. Amplitude-frequency responses of piezoelectric actuators with different ratio of parameters a and b 
(experimental results) 
5. Conclusions 
According to numerical calculations and experimental results, the geometrical form of 
piezoelectric bending actuator, has big influence to its dynamic characteristics. With bigger ratio 
of ???, the resonant amplitude gets smaller values, but resonant frequency of the cantilever 
increases approximately 1,3 times due to the decrease of the free end width of the cantilever by 
3 times: that is from 308 Hz to 419 Hz. Such optimization of geometrical parameters for 
piezoelectric bending actuators can be also used for piezoelectric laser beam intensity 
modulators. 
Acknowledgements 
This work has been supported by Research Council of Lithuania, Project MIP-075/2012 and 
Project Go-Smart, VP1-3.1-ŠMM-08-K-01-015. 
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Design and dynamic modeling of piezoelectric laser beam shutters

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