This paper describes a micro-robot with the vibrational piezoelectric actuator that was developed and tested using finite element method. Dependency of the robot motion speed on vibration frequency and coefficient of friction of the actuating elements and bearing surface has been investigated. The principle for motion control of the robot has been described and potential domains for its application have been discusse

A. Bubulis

Ck. Zimmerman

D. A. Stepanenko

I. A. Gishkeliuk

P. Kundu

R. Bansevicius

V. T. Minchenia

English

Journal of Vibroengineering

 559. SIMULATION OF VIBRATIONAL PIEZOELECTRIC ACTUATORS OF THE MICRO-ROBOT USING THE FINITE ELEMENT METHOD. 
V. T. MINCHENIA, D. A. STEPANENKO, CK. ZIMMERMAN, R. BANSEVIČIUS, A. BUBULIS, P. KUNDU, I. A. GISHKELIUK 
 
 
 
 VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. SEPTEMBER 2010. VOLUME 12, ISSUE 3. ISSN 1392-8716 
314 
559. Simulation of vibrational piezoelectric 
actuators of the micro-robot using the finite 
element method 
 
 
V. T. Minchenia1, D. A. Stepanenko1, CK. Zimmerman3, 
R. Bansevičius4, A. Bubulis4, P. Kundu2, I. A. Gishkeliuk2 
1 Belarusian National Technical University (BNTU), 
  220027, Minsk, Nezovisimosti avenue 65,  
E-mail: vlad_minch@mail.ru 
2 The International Institute on Radio Ecology of A. D. Saharova,  
  220027, Minsk, Dolgabrodskaja street 23 
3 The Ilmenau University of Technology, PF 100565, D-98684, Ilmenau 
4 Kaunas University of Technology, Lithuania 
 
(Received 07 July 2010;  accepted 10 September 2010) 
 
 
Abstract. This paper describes a micro-robot with the vibrational piezoelectric actuator that 
was developed and tested using finite element method. Dependency of the robot motion speed 
on vibration frequency and coefficient of friction of the actuating elements and bearing surface 
has been investigated. The principle for motion control of the robot has been described and 
potential domains for its application have been discussed. 
Keywords: vibration, piezoelectric actuators, micro-robot, finite element method. 
Introduction.  
Recently, vibrational piezoelectric actuators-based (VPA-based) moving systems have 
become highly prevalent in fields of engineering and medicine as such actuators feature an 
extremely high coefficient of efficiency, low energy consumption, and can be easily 
miniaturized [9]. The following represent the main domains for application of VPAs: 
development of precision systems for micro- and nano-positioning of different objects, 
development of mobile micro-robots and micro-automated devices intended for different 
technological task-solving. The object of current research paper – a single VPA element-based 
micro-robot. Fig. 1 illustrates conceptual diagram of the robot structure as well as images of the 
actual prototype. 
The robot under consideration consists of bimorph piezoelectric component 1, that 
generates bending vibrations, and spatially curved cane segments 2 (actuating elements or 
“legs”), attached to the lateral surface of the piezoelectric element that transform its vibrations 
depending on amplitude and direction. In result of transformation of vibrations depending on 
direction, ends of “legs” that interact with the bearing surface 3, shall represent a complex 
trajectory, i.e., elliptical, while ensuring movement and relocation of the robot in relation to the 
bearing surface. Increase in vibration amplitude of the actuating elements allows for increasing 
speed of such movement. In order to add more steadiness to its operation, it is additionally 
provided with the bearing element 4 (“tail”). Control of robot movement is accomplished 
through asymmetry in its “legs” operation, i.e., through assigning different geometric 
parameters. Due to asymmetry in its operation, “legs” of the robot will have different resonant 
frequencies, and will generate near-resonant vibrations with different amplitudes. Direction of 
the robot movement will change depending on the ratio of these amplitudes.  
 
 559. SIMULATION OF VIBRATIONAL PIEZOELECTRIC ACTUATORS OF THE MICRO-ROBOT USING THE FINITE ELEMENT METHOD. 
V. T. MINCHENIA, D. A. STEPANENKO, CK. ZIMMERMAN, R. BANSEVIČIUS, A. BUBULIS, P. KUNDU, I. A. GISHKELIUK 
 
 
 
 VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. SEPTEMBER 2010. VOLUME 12, ISSUE 3. ISSN 1392-8716 
315 
 
                                                            a) 
 
                                                             b) 
Fig. 1.  VPA-based micro-robot: a) diagram of structure; b) photo image of the actual device: 1 – 
bimorph piezoelectric component, 2 – curved cane segments, 3 – bearing surface, 4 – bearing 
element 
 
Similar principle of mobile robot control is described in paper [2]. In order to ensure high 
speed of the robot relocation in relation to the bearing surface and enable the robot to move on 
rough bearing surface, it uses the actuating elements that are realized in the form of lever-type 
four-link chains with the flexible hinges and ensure increase in vibrations amplitude. Actuating 
elements are made with different specific vibration frequencies, which in turn allows for 
controlling direction of the robot relocation with respect to the bearing surface by changing 
vibration frequencies. VPA operation involves two unimorphic piezoelectric elements. A phase-
shift appears between vibrations of piezoelectric elements which allows generation of elliptical 
trajectory of the movement of actuating elements ends that are interacting with the bearing 
surface.         
The main disadvantage of the robot is a complex structure of its actuating elements that in 
turn limits its miniaturization chances. Another widely known structure of the micro-robot 
involves robot control by means of external vibration field [3]. The robot is present on the 
vibrating surface and has bearing elements with different resonant frequencies resulting in 
vibration of bearing elements at different amplitudes, the ratio of which defines direction of the 
robot movement. 
In the context of selecting the optimal parameters for the structure of the robot under 
consideration, and development of algorithms for its movement control, its mathematical and 
computerized simulation is of high interest. Given the fact that simulation of robot operation 
represents an extremely complex task involving recording of phenomena such as piezoelectric 
effect, contact interaction of actuating elements with bearing surface, as well as transformation 
and amplification of vibrations in a body by actuating elements, the finite element method was 
 559. SIMULATION OF VIBRATIONAL PIEZOELECTRIC ACTUATORS OF THE MICRO-ROBOT USING THE FINITE ELEMENT METHOD. 
V. T. MINCHENIA, D. A. STEPANENKO, CK. ZIMMERMAN, R. BANSEVIČIUS, A. BUBULIS, P. KUNDU, I. A. GISHKELIUK 
 
 
 
 VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. SEPTEMBER 2010. VOLUME 12, ISSUE 3. ISSN 1392-8716 
316 
selected for the purpose of simulation as it represents a flexible tool applicable for the research 
of complex  technical systems, and is successfully implemented in several computer 
programmes available on the market, for example ANSYS Multiphysics and COMSOL 
Multiphysics [8,10].  
 
 
a) 
 
 
b) 
Fig. 2.  Effects of vibration frequency and coefficient of friction on micro-robot movement speed were 
examined 
 
Simulation methodology 
 
Since research of 3D finite element model of the robot under consideration requires 
significant computational time, a 2D model was analyzed as the first approximation (Fig. 2). 
Simulation was performed by using module of Structural Mechanics in COMSOL 
Multiphysics. Body and actuating elements of the robot under consideration were introduced as 
area A1, and bearing surface – as area A2. The lower border-line B1 of the area A2 was fixed at 
all the possible levels of its looseness in order to avoid potential movement of it as a solid body. 
Lateral border-lines B2 and B3 of the area A1 were subjected to periodically varying forces 
F1(t) and F2(t) that cause deformation in the robot body while simulating a piezoelectric effect. 
Two different types of a model were analyzed. In the model of type 1 (Fig. 2a), the area A1 
featured symmetric geometry, whereas force action was asymmetric in nature, and conformed 
with the following laws of force variation in time: 
 
F1(t) = F0sin(2πft)φ(sin((2πft)),  (1) 
 559. SIMULATION OF VIBRATIONAL PIEZOELECTRIC ACTUATORS OF THE MICRO-ROBOT USING THE FINITE ELEMENT METHOD. 
V. T. MINCHENIA, D. A. STEPANENKO, CK. ZIMMERMAN, R. BANSEVIČIUS, A. BUBULIS, P. KUNDU, I. A. GISHKELIUK 
 
 
 
 VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. SEPTEMBER 2010. VOLUME 12, ISSUE 3. ISSN 1392-8716 
317 
F2(t) = F0sin(2πft)(1–φ(sin((2πft))),  (2) 
 
where F0 – amplitude of the force action, f – frequency of the force action, t – temporary 
variable, φ(·) – Heaviside unit step function. 
In the model of type 2 (Fig. 2b), the area A1 featured asymmetric geometry, and force 
action was symmetric in nature, i.e.: 
 
F1(t) = – F2(t) = F0sin(2πft).  (3)      
    
Solution of the dynamic task was reduced to the solution of sequence of static problems, 
for the purpose of which a temporal discretization was performed, and at every level of 
discretization, consistent with the step of loading, a static load was applied to the model, 
consistent with the value of initial dynamic load at the beginning of the interval. Since 
magnitude of load at every step of loading was dependent on the parameter representing a 
temporal variable, a parametric segregated solver was used for the solution. Obtained results of 
task solution for every step of loading were used as initial conditions for obtaining solution in 
the next step. Mathematically, reduction of the dynamic task to the sequence of static tasks is 
adequate to elimination of inert members from (in) equations of equilibrium. 
 
Results and discussion of findings 
Based on results of computation, graphs were produced representing a temporal 
dependency of the actuating elements of the robot in a direction of axis Ox that was parallel to 
the border-line B1 of the area A2. Fig. 3 offers an example of such graphical representation for 
the coefficient of friction µ=0.2 and vibration frequency f = 104/2πHz. Computation was made 
for the model of type 2 (Fig. 2b). 
Upper curve represents displacement of the contact surface of the right (short) leg of the 
robot, whereas lower curve – displacement of the contact surface of the left (long) leg of the 
robot. As it is obvious from graphical schemes, displacement of actuating elements occurs in 
anti-phase. Analogous results were observed for the model of type 1 (Fig. 2a), however in the 
case of models with symmetric geometry and symmetric force action, displacement was not 
observed. All of this leads to conclusion that in order to make the robot move, it is necessary to 
create asymmetry of any kind, for example asymmetry in force or geometry. In order to 
determine speed of the robot movement, results of computation were subjected to numerical 
differentiation using MathCad that resulted in periodical function of temporal variable with the 
nonzero average value. This average value was considered to represent speed of robot 
movement. Dependencies of robot movement speed on the coefficient of friction µ (Fig.4) and 
vibration frequency f have also been examined.  
Graph presented in Fig. 4 was obtained for the vibration frequency of f = 104/2πHz. Its 
analysis revealed that dependence of robot motion speed on the coefficient of friction possesses 
an extreme character achieving maximum at µ ≈ 0.2. According to the results of simulation, 
dependence of movement speed on the vibration frequency possesses a linear character. In real 
physical experiments, a resonant character of dependence of robot movement speed on the 
vibration frequency is commonly observed [1]. The disagreement between findings of our 
simulation and real experiment might be explained by the fact that inert members have been 
eliminated from equations of equilibrium of the system, whereas resonant frequencies of the 
system are dependent upon its inertial features (density of the material). Consequently, 
reduction of the dynamic task to the sequence of static tasks resulted in loss of information 
regarding resonant features of the system and associated phenomena. Inability of the model to 
take into account resonant phenomena might also explain rather low robot movement speeds, 
obtained in a result of computation. In the future, a transient analysis might be used in order to 
 559. SIMULATION OF VIBRATIONAL PIEZOELECTRIC ACTUATORS OF THE MICRO-ROBOT USING THE FINITE ELEMENT METHOD. 
V. T. MINCHENIA, D. A. STEPANENKO, CK. ZIMMERMAN, R. BANSEVIČIUS, A. BUBULIS, P. KUNDU, I. A. GISHKELIUK 
 
 
 
 VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. SEPTEMBER 2010. VOLUME 12, ISSUE 3. ISSN 1392-8716 
318 
make model take into consideration resonant phenomena, including inertial effects. Simulation 
using different software (for example, ANSYS Multiphysics that proved to be sufficiently 
successful in simulation of VPA) is also possible. 
 
 
Fig. 3. Temporal variation of displacement of robot actuating elements 
 
 
Fig. 4. Dependence of robot movement speed on the coefficient of friction 
 
Curves represented in Fig. 3 are in agreement with the data obtained through experimental 
research of linear VPA [5]. They obviously demonstrate robot relocation to have a step-wise 
character. Even in the case when speed of movement is low (from tenths to tens of nm/s, in the 
model under consideration) VPA represent a high practical interest, for example in the field of 
precision positioning of objects in micro- and nano-technologies. In particular, a concept exists 
for developing a nano-plant on the basis of many interacting nano-robots with VPA, fitted with 
operating and measuring tools, for example a probe for scanning tunneling microscopy [6]. 
 
 
 
 
D
isp
la
ce
m
en
t, 
m
 
M
o
v
em
en
t s
pe
ed
, 
m
/s 
 
Coefficient of friction 
 
Time, s 
 
 559. SIMULATION OF VIBRATIONAL PIEZOELECTRIC ACTUATORS OF THE MICRO-ROBOT USING THE FINITE ELEMENT METHOD. 
V. T. MINCHENIA, D. A. STEPANENKO, CK. ZIMMERMAN, R. BANSEVIČIUS, A. BUBULIS, P. KUNDU, I. A. GISHKELIUK 
 
 
 
 VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. SEPTEMBER 2010. VOLUME 12, ISSUE 3. ISSN 1392-8716 
319 
Actuating elements of the robot might be directly used as sensor elements as their resonant 
frequencies are dependent upon physico-mechanical features of the bearing surface that serves 
to perform function of acoustic load. Dependence of the resonant characteristics of vibrational 
framed structures on features of acoustic load is well-known and widely used for ultrasound 
hardometers and resonant tactile sensors for examination of resilience qualities of biological 
tissues [7]. 
Consequently, the developed model might serve as a useful tool for the research of effect of 
different parameters on the efficiency of VPA operation. It would be expedient to undertake 
further development and elaboration of this model, as well as its additional verification by 
introducing results obtained through experimental studies, and results of simulations using 
different methods and software. 
Conclusions 
1.  A finite-element model of VPA-based micro-robot was developed by using software 
COMSOL Multiphysics. 
2. Effects of vibration frequency and coefficient of friction on micro-robot movement speed 
were examined. 
3. Based on findings and results of simulation, some disadvantages of model under 
consideration were revealed, namely its inability to take into consideration resonant 
phenomena that contributed to identification of potential course for further research and 
elaboration of the developed model. 
Acknowledgement 
 
This investigation was realized under Project AUT 09/2010 funded by Research Council of 
Lithuania. 
 
References 
[1] Abaza K. Ein Beitrag zur Anwendung der Theorie undulatischer locomotion auf  mobile Roboter. 
Evaluierung theoretischer Ergebnisse an prototypen; dissertation zur Erlangung des akademischen 
Grades Doktor-Ingenieur. Technische Universitat Ilmenau, 2007. S.126. 
[2] Goldfarb M., Gogola M., Fischer G., Garcia E. Development of a piezoelectrically – actuated 
mesoscale robot quadruped // Journal of micromechatronics. Vol. 1. 2001. P.205-219. 
[3] Yasuda T., Shimoyama I., Miura H. Microrobot actuated by a vibration energy field// Sensor and 
Actuators. Vol. 43. 1994. P.366-370. 
[4] Sharp S. L. Desing of a linear ultrasonic piezoelectric motor; M.Sc. Thesis. Brigham Yong 
University, 2006. 162 p. 
[5] Son J., Kartik V., Wickert J. A., Sitti M. An ultrasonic standing-wave-actuated nanopositioning 
walking robot // Journal of Vibration and Control. Vol.12. 2006. P.1293-1309. 
[6] Martel S., Hunter I. Nanofactories based on a fleet of scientific instrument configured as miniature 
autonomous robots // Journal of Micromechatronics. Vol. 2. 2004. P.201 – 214. 
[7] Murayama Y., Omata S. Fabrication of micro tactile sensor for the measurement of micro-scale 
local elasticity // Sensor and Actuators. Vol. 109. 2004. P.202-207. 
[8] Ragulskis M., Aleksa A., Ragulskis M., Bubulis A. Calculation of vibrations of a  single degree of 
freedom system // Journal of Vibroengineering / Vibromechanika. ISSN 1392-8716. 2007, Vol. 9, 
No. 4. p. 73-76. 
[9] Ragulskis K., Bansevicius R., Barauskas R., Kulvietis G. Vibromotors for precision microrobots. 
Hemisphere publishing Corp., 1988, USA,  ISBN 0-89116054905. 310p. 
[10] Bansevicius R., Drukteiniene A., Kulvietis G. Movement trajectory planning algorithm of 
rotating mobile piezorobot. Solid State Phenomena. Vol.164 (2010), P.371-376.  


Simulation of vibrational piezoelectric actuators of the micro-robot using the finite element method

Minchenia, V. T.

Stepanenko, D. A.

Zimmerman, Ck.

Bansevicius, R.

Bubulis, A.

Kundu, P.

Gishkeliuk, I. A.

JVE International

 559. SIMULATION OF VIBRATIONAL PIEZOELECTRIC ACTUATORS OF THE MICRO-ROBOT USING THE FINITE ELEMENT METHOD. V. T. MINCHENIA, D. A. STEPANENKO, CK. ZIMMERMAN , R. BANSEVIČIUS, A. BUBULIS, P. KUNDU, I. A. GISHKELIUK     VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. SEPTEMBER 2010. VOLUME 12, ISSUE 3. ISSN 1392-8716 314 559. Simulation of vibrational piezoelectric actuators of the micro-robot using the finite element method   V. T. Minchenia1, D. A. Stepanenko1, CK. Zimmerman3, R. Bansevičius4, A. Bubulis4, P. Kundu2, I. A. Gishkeliuk2 1 Belarusian National Technical University (BNTU),   220027, Minsk, Nezovisimosti avenue 65,  E-mail: vlad_minch@mail.ru 2 The International Institute on Radio Ecology of A. D. Saharova,    220027, Minsk, Dolgabrodskaja street 23 3 The Ilmenau University of Technology, PF 100565, D-98684, Ilmenau 4 Kaunas University of Technology, Lithuania  (Received 07 July 2010;  accepted 10 September 2010)   Abstract. This paper describes a micro-robot with the vibrat onal piezoelectric actuator that was developed and tested using finite element method. Dependency of the robot motion speed on vibration frequency and coefficient of friction f the actuating elements and bearing surface has been investigated. The principle for motion control of the robot has been described and potential domains for its application have been discus ed. Keywords: vibration, piezoelectric actuators, micro-robot, finite element method. Introduction.  Recently, vibrational piezoelectric actuators-based (VPA-based) moving systems have become highly prevalent in fields of engineering and medicine as such actuators feature an extremely high coefficient of efficiency, low energy consumption, and can be easily miniaturized [9]. The following represent the main domains for application of VPAs: development of precision systems for micro- and nano-positioning of different objects, development of mobile micro-robots and micro-automated devices intended for different technological task-solving. The object of current research paper – a single VPA element-based micro-robot. Fig. 1 illustrates conceptual diagram of the robot structure as well as images of the actual prototype. The robot under consideration consists of bimorph piezoelectric component 1, that generates bending vibrations, and spatially curved cane segments 2 (actuating elements or “legs”), attached to the lateral surface of the piezoelectric element that transform its vibrations depending on amplitude and direction. In result of ransformation of vibrations depending on direction, ends of “legs” that interact with the bearing surface 3, shall represent a complex trajectory, i.e., elliptical, while ensuring movement and relocation of the robot in relation to the bearing surface. Increase in vibration amplitude of the actuating elements allows for increasing speed of such movement. In order to add more steadin ss to its operation, it is additionally provided with the bearing element 4 (“tail”). Control of robot movement is accomplished through asymmetry in its “legs” operation, i.e., through assigning different geometric parameters. Due to asymmetry in its operation, “legs” of the robot will have different resonant frequencies, and will generate near-resonant vibrations with different amplitudes. Direction of the robot movement will change depending on the ratio of these amplitudes.    559. SIMULATION OF VIBRATIONAL PIEZOELECTRIC ACTUATORS OF THE MICRO-ROBOT USING THE FINITE ELEMENT METHOD. V. T. MINCHENIA, D. A. STEPANENKO, CK. ZIMMERMAN , R. BANSEVIČIUS, A. BUBULIS, P. KUNDU, I. A. GISHKELIUK     VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. SEPTEMBER 2010. VOLUME 12, ISSUE 3. ISSN 1392-8716 315                                                         a)                                                          b) Fig. 1.  VPA-based micro-robot: a) diagram of structure; b) photo image of the actual device: 1 – bimorph piezoelectric component, 2 – curved cane segments, 3 – bearing surface, 4 – bearing element  Similar principle of mobile robot control is described in paper [2]. In order to ensure high speed of the robot relocation in relation to the bearing surface and enable the robot to move on rough bearing surface, it uses the actuating elements that are realized in the form of lever-type four-link chains with the flexible hinges and ensure increase in vibrations amplitude. Actuating elements are made with different specific vibration frequencies, which in turn allows for controlling direction of the robot relocation with respect to the bearing surface by changing vibration frequencies. VPA operation involves two unimorphic piezoelectric elements. A phase-shift appears between vibrations of piezoelectric elem nts which allows generation of elliptical trajectory of the movement of actuating elements ends that are interacting with the bearing surface.         The main disadvantage of the robot is a complex structu e of its actuating elements that in turn limits its miniaturization chances. Another widely known structure of the micro-robot involves robot control by means of external vibration field [3]. The robot is present on the vibrating surface and has bearing elements with different resonant frequencies resulting in vibration of bearing elements at different amplitudes, the ratio of which defines direction of the robot movement. In the context of selecting the optimal parameters fo  the structure of the robot under consideration, and development of algorithms for its movement control, its mathematical and computerized simulation is of high interest. Given the fact that simulation of robot operation represents an extremely complex task involving recording of phenomena such as piezoelectric effect, contact interaction of actuating elements with bearing surface, as well as transformation and amplification of vibrations in a body by actuating elements, the finite element method was  559. SIMULATION OF VIBRATIONAL PIEZOELECTRIC ACTUATORS OF THE MICRO-ROBOT USING THE FINITE ELEMENT METHOD. V. T. MINCHENIA, D. A. STEPANENKO, CK. ZIMMERMAN , R. BANSEVIČIUS, A. BUBULIS, P. KUNDU, I. A. GISHKELIUK     VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. SEPTEMBER 2010. VOLUME 12, ISSUE 3. ISSN 1392-8716 316 selected for the purpose of simulation as it represents a flexible tool applicable for the research of complex  technical systems, and is successfully implemented in several computer programmes available on the market, for example ANSYS Multiphysics and COMSOL Multiphysics [8,10].    a)   b) Fig. 2.  Effects of vibration frequency and coefficient of friction on micro-robot movement speed were examined  Simulation methodology  Since research of 3D finite element model of the robot under consideration requires significant computational time, a 2D model was analyzed as the first approximation (Fig. 2). Simulation was performed by using module of Structural Mechanics in COMSOL Multiphysics. Body and actuating elements of the robot under consideration were introduced as area A1, and bearing surface – as area A2. The lower border-line B1 of the area A2 was fixed at all the possible levels of its looseness in order to avoid potential movement of it as a solid body. Lateral border-lines B2 and B3 of the area A1 were subjected to periodically varying forces F1(t) and F2(t) that cause deformation in the robot body while s mulating a piezoelectric effect. Two different types of a model were analyzed. In the model of type 1 (Fig. 2a), the area A1 featured symmetric geometry, whereas force action was asymmetric in nature, and conformed with the following laws of force variation in time:  F1(t) = F0sin(2πft)φ(sin((2πft)),  (1)  559. SIMULATION OF VIBRATIONAL PIEZOELECTRIC ACTUATORS OF THE MICRO-ROBOT USING THE FINITE ELEMENT METHOD. V. T. MINCHENIA, D. A. STEPANENKO, CK. ZIMMERMAN , R. BANSEVIČIUS, A. BUBULIS, P. KUNDU, I. A. GISHKELIUK     VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. SEPTEMBER 2010. VOLUME 12, ISSUE 3. ISSN 1392-8716 317 F2(t) = F0sin(2πft)(1–φ(sin((2πft))),  (2)  where F0 – amplitude of the force action, f – frequency of the force action, t – temporary variable, φ(·) – Heaviside unit step function. In the model of type 2 (Fig. 2b), the area A1 featured asymmetric geometry, and force action was symmetric in nature, i.e.:  F1(t) = – F2(t) = F0sin(2πft).  (3)          Solution of the dynamic task was reduced to the solution of sequence of static problems, for the purpose of which a temporal discretization was performed, and at every level of discretization, consistent with the step of loading, a static load was applied to the model, consistent with the value of initial dynamic load at the beginning of the interval. Since magnitude of load at every step of loading was dependent on the parameter representing a temporal variable, a parametric segregated solver was used for the solution. Obtained results of task solution for every step of loading were used as initial conditions for obtaining solution in the next step. Mathematically, reduction of the dynamic task to the sequence of static tasks is adequate to elimination of inert members from (in) equations of equilibrium.  Results and discussion of findings Based on results of computation, graphs were produced representing a temporal dependency of the actuating elements of the robot in a direction of axis Ox that was parallel to the border-line B1 of the area A2. Fig. 3 offers an example of such graphical representation for the coefficient of friction µ=0.2 and vibration frequency f = 104/2πHz. Computation was made for the model of type 2 (Fig. 2b). Upper curve represents displacement of the contact surface of the right (short) leg of the robot, whereas lower curve – displacement of the contact surface of the left (long) leg of the robot. As it is obvious from graphical schemes, displacement of actuating elements occurs in anti-phase. Analogous results were observed for the model of type 1 (Fig. 2a), however in the case of models with symmetric geometry and symmetric force action, displacement was not observed. All of this leads to conclusion that in order to make the robot move, it is necessary to create asymmetry of any kind, for example asymmetry in force or geometry. In order to determine speed of the robot movement, results of computation were subjected to numerical differentiation using MathCad that resulted in periodical function of temporal vriable with the nonzero average value. This average value was considered to represent speed of robot movement. Dependencies of robot movement speed on the coefficient of friction µ (Fig.4) and vibration frequency f have also been examined.  Graph presented in Fig. 4 was obtained for the vibration frequency of f = 104/2πHz. Its analysis revealed that dependence of robot motion speed on the coefficient of friction possesses an extreme character achieving maximum at µ ≈ 0.2. According to the results of simulation, dependence of movement speed on the vibration frequency possesses a linear character. In real physical experiments, a resonant character of dependence of robot movement speed on the vibration frequency is commonly observed [1]. The disagreement between findings of our simulation and real experiment might be explained by the fact that inert members have been eliminated from equations of equilibrium of the system, whereas resonant frequencies of the system are dependent upon its inertial features (density of the material). Consequently, reduction of the dynamic task to the sequence of static asks resulted in loss of information regarding resonant features of the system and associ ted phenomena. Inability of the model to take into account resonant phenomena might also explain rather low robot movement speeds, obtained in a result of computation. In the future, a transient analysis might be used in order to  559. SIMULATION OF VIBRATIONAL PIEZOELECTRIC ACTUATORS OF THE MICRO-ROBOT USING THE FINITE ELEMENT METHOD. V. T. MINCHENIA, D. A. STEPANENKO, CK. ZIMMERMAN , R. BANSEVIČIUS, A. BUBULIS, P. KUNDU, I. A. GISHKELIUK     VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. SEPTEMBER 2010. VOLUME 12, ISSUE 3. ISSN 1392-8716 318 make model take into consideration resonant phenomea, including inertial effects. Simulation using different software (for example, ANSYS Multiphysics that proved to be sufficiently successful in simulation of VPA) is also possible.   Fig. 3. Temporal variation of displacement of robot actuating elements   Fig. 4. Dependence of robot movement speed on the coefficient of friction  Curves represented in Fig. 3 are in agreement with the data obtained through experimental research of linear VPA [5]. They obviously demonstrate robot relocation to have a step-wise character. Even in the case when speed of movement is low (from tenths to tens of nm/s, in the model under consideration) VPA represent a high practic l interest, for example in the field of precision positioning of objects in micro- and nano-technologies. In particular, a concept exists for developing a nano-plant on the basis of many interacting nano-robots with VPA, fitted with operating and measuring tools, for example a probe f r scanning tunneling microscopy [6].     Displacement, m Movement speed, m/s  Coefficient of friction  Time, s   559. SIMULATION OF VIBRATIONAL PIEZOELECTRIC ACTUATORS OF THE MICRO-ROBOT USING THE FINITE ELEMENT METHOD. V. T. MINCHENIA, D. A. STEPANENKO, CK. ZIMMERMAN , R. BANSEVIČIUS, A. BUBULIS, P. KUNDU, I. A. GISHKELIUK     VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. SEPTEMBER 2010. VOLUME 12, ISSUE 3. ISSN 1392-8716 319 Actuating elements of the robot might be directly used as sensor elements as their resonant frequencies are dependent upon physico-mechanical features of the bearing surface that serves to perform function of acoustic load. Dependence of the resonant characteristics of vibrational framed structures on features of acoustic load is well-known and widely used for ultrasound hardometers and resonant tactile sensors for examination of resilience qualities of biological tissues [7]. Consequently, the developed model might serve as a useful tool for the research of effect of different parameters on the efficiency of VPA operation. It would be expedient to undertake further development and elaboration of this model, as well as its additional verification by introducing results obtained through experimental sudies, and results of simulations using different methods and software. Conclusions 1.  A finite-element model of VPA-based micro-robot was developed by using software COMSOL Multiphysics. 2. Effects of vibration frequency and coefficient of friction on micro-robot movement speed were examined. 3. Based on findings and results of simulation, some disadvantages of model under consideration were revealed, namely its inability to take into consideration resonant phenomena that contributed to identification of potential course for further research and elaboration of the developed model. Acknowledgement  This investigation was realized under Project AUT 09/2 10 funded by Research Council of Lithuania.  References [1] Abaza K. Ein Beitrag zur Anwendung der Theorie undulatischer locomotion auf  mobile Roboter. Evaluierung theoretischer Ergebnisse an prototypen; dissertation zur Erlangung des akademischen Grades Doktor-Ingenieur. Technische Universitat Ilmenau, 2007. S.126. [2] Goldfarb M., Gogola M., Fischer G., Garcia E. Development of a piezoelectrically – actuated mesoscale robot quadruped // Journal of micromechatronics. Vol. 1. 2001. P.205-219. [3] Yasuda T., Shimoyama I., Miura H. Microrobot actuated by a vibration energy field// Sensor and Actuators. Vol. 43. 1994. P.366-370. [4] Sharp S. L. Desing of a linear ultrasonic piezoelectric motor; M.Sc. Thesis. Brigham Yong University, 2006. 162 p. [5] Son J., Kartik V., Wickert J. A., Sitti M. An ultrasonic standing-wave-actuated nanopositioning walking robot // Journal of Vibration and Control. Vol.12. 2006. P.1293-1309. [6] Martel S., Hunter I. Nanofactories based on a fleet of scientific instrument configured as miniature autonomous robots // Journal of Micromechatronics. Vol 2. 2004. P.201 – 214. [7] Murayama Y., Omata S. Fabrication of micro tactile sensor for the measurement of micro-scale local elasticity // Sensor and Actuators. Vol. 109. 2004. P.202-207. [8] Ragulskis M., Aleksa A., Ragulskis M., Bubulis A. Calculation of vibrations of a  single degree of freedom system // Journal of Vibroengineering / Vibromechanika. ISSN 1392-8716. 2007, Vol. 9, No. 4. p. 73-76. [9] Ragulskis K., Bansevicius R., Barauskas R., Kulvietis G. Vibromotors for precision microrobots. Hemisphere publishing Corp., 1988, USA,  ISBN 0-89116054905. 310p. [10] Bansevicius R., Drukteiniene A., Kulvietis G. Movement trajectory planning algorithm of rotating mobile piezorobot. Solid State Phenomena. Vol.164 (2010), P.371-376.  

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Simulation of vibrational piezoelectric actuators of the micro-robot using the finite element method

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