Numerical model for the investigation of dynamics of a thermal vibro-transporter is proposed. The model is based on the one dimensional linear bar finite element and the stopping mechanism is being taken into account by the penalty method. Results of performed calculations of vibro-transportation are presented. From the presented results the effect of thermal vibro-transportation is clearly seen

B. Spruogis

K. Ragulskis

L. Patašiene

P. Paškevicius

English

Journal of Vibroengineering

 636. DYNAMICS OF A THERMAL VIBRO-TRANSPORTER.  
P. PAŠKEVIČIUS, L. PATAŠIENö, K. RAGULSKIS, B. SPRUOGIS 
 
 VIBROENGINEERING. JOURNAL OF VIBROENGINEERING.   JUNE 2011. VOLUME 13, ISSUE 2. ISSN 1392-8716     308 
636. Dynamics of a thermal vibro-transporter 
 
P. Paškevičius1, L. Patašien÷2, K. Ragulskis3, B. Spruogis4 
1
 Kaunas University of Technology, Kaunas, Lithuania 
2
 Kaunas University of Technology, Kaunas, Lithuania 
E-mail: laima.patasiene@ktu.lt 
3
 Kaunas University of Technology, Kaunas, Lithuania 
E-mail: kazimieras3@hotmail.com, kazimieras3@yahoo.com 
4 Vilnius Gediminas Technical University, Plytin÷s 27, LT-10223, Vilnius, Lithuania 
E-mail: bs@ti.vgtu.lt 
 
(Received March 4 2011; accepted May 15 2011) 
 
Abstract. Numerical model for the investigation of dynamics of a thermal vibro-transporter is 
proposed. The model is based on the one dimensional linear bar finite element and the stopping 
mechanism is being taken into account by the penalty method. Results of performed 
calculations of vibro-transportation are presented. From the presented results the effect of 
thermal vibro-transportation is clearly seen. 
 
Keywords: vibromotor, thermal motor, vibro-transporter, finite elements, thermal load, 
numerical integration, stopping mechanism, penalty method. 
 
 
Introduction 
 
Numerical model for the investigation of dynamics of a thermal vibro-transporter is 
proposed on the basis of the material presented in [1, 2, 3]. The model is based on the one 
dimensional linear bar finite element. Excitation is taken into account assuming the temperature 
distribution in the bar to be given and using the thermal load vector. Numerical integration of 
the equations is performed. The stopping mechanism is being taken into account by the penalty 
method. Results of performed calculations are presented. 
There are a number of vibromotors [4 - 11] which find wide practical applications. It is 
expected that the proposed thermal motor will find useful application in engineering. 
 
Numerical model of the thermal vibro-transporter 
 
 The analyzed bar is assumed to be in the direction of the x axis of coordinates from –Tx/2 to 
Tx/2, here Tx is the length of a periodic part of the structure. The only variable is the longitudinal 
displacement of the bar u. The displacement of the first node is assumed to be equal to the 
displacement of the last node. 
 The mass matrix has the form: 
 
[ ] [ ] [ ] ,TM N h N dxρ= ∫              (1) 
where ρ is the density of the material of the bar, h is the thickness of the bar and: 
[ ] [ ]1 2 ,N N N=              (2) 
where Ni are the linear shape functions of the one dimensional finite element. 
The stiffness matrix has the form: 
[ ] [ ] [ ]2 ,1
T EK B h B dx
ν
=
−∫             (3) 
 
 636. DYNAMICS OF A THERMAL VIBRO-TRANSPORTER.  
P. PAŠKEVIČIUS, L. PATAŠIENö, K. RAGULSKIS, B. SPRUOGIS 
 
 VIBROENGINEERING. JOURNAL OF VIBROENGINEERING.   JUNE 2011. VOLUME 13, ISSUE 2. ISSN 1392-8716     309 
where E is the modulus of elasticity of the bar, ν is the Poisson’s ratio of the bar and: 
 
[ ] 1 2 .dN dNB
dx dx
 
=   
             (4) 
 
The loading vector has the form: 
 
{ } [ ] 2 ,1
T EF B h Tdxα
ν
=
−∫             (5) 
 
where α  is the coefficient of thermal expansion and T is the temperature. The variation of 
temperature is assumed to be given as: 
 
sin sin 2 ,
x
x uT T t
T
ω π
+
=              (6) 
 
where T  is the amplitude of temperature variation, ω is the frequency of temperature variation 
and t is the time variable. 
The damping matrix has the form: 
[ ] [ ] [ ],C M Kα β= +              (7) 
where α is the coefficient of external damping and β is the coefficient of internal damping, 
except for the case described further the damping matrix takes into account the stopping 
mechanism and is more complicated. If the following condition is satisfied: 
1 1 2 20 ,x u x u+ ≤ < +              (8) 
where xi are the nodal coordinates of the finite element and ui are the nodal displacements of the 
finite element, then the following shape functions are calculated: 
 
2 2
1
2 2 1 1
,
x u
N
x u x u
+
=
+ − −
             (9) 
1 1
2
2 2 1 1
.
x u
N
x u x u
− −
=
+ − −
           (10) 
 
They are represented as a row vector of shape functions: 
 
1 2 .N N N   =               (11) 
 
The displacement is calculated as: 
1
2
.
u
u N
u
 
 =   
 
            (12) 
If the following condition is satisfied: 
,pu u<              (13) 
where pu  denotes the value of u  for the previous moment of time (from the previous 
integration step), then the damping matrix has the form: 
 
[ ] [ ] [ ] ,TC M K N Nα β λ   = + +               (14) 
 
where λ is the penalty parameter of the stopping mechanism. 
 636. DYNAMICS OF A THERMAL VIBRO-TRANSPORTER.  
P. PAŠKEVIČIUS, L. PATAŠIENö, K. RAGULSKIS, B. SPRUOGIS 
 
 VIBROENGINEERING. JOURNAL OF VIBROENGINEERING.   JUNE 2011. VOLUME 13, ISSUE 2. ISSN 1392-8716     310 
Results of analysis of the thermal vibro-transporter 
 
 The displacements during the first period of excitation are presented in Fig. 1. The 
displacements during the first two periods of excitation are presented in Fig. 2. The 
displacements during the first four periods of excitation are presented in Fig. 3. 
 
 
 
 
Fig. 1. Displacements during the first period of excitation 
 
 
 
 
Fig. 2. Displacements during the first two periods of excitation 
 636. DYNAMICS OF A THERMAL VIBRO-TRANSPORTER.  
P. PAŠKEVIČIUS, L. PATAŠIENö, K. RAGULSKIS, B. SPRUOGIS 
 
 VIBROENGINEERING. JOURNAL OF VIBROENGINEERING.   JUNE 2011. VOLUME 13, ISSUE 2. ISSN 1392-8716     311 
 
Fig. 3. Displacements during the first four periods of excitation 
  
 
From the presented results the effect of thermal vibro-transportation is clearly seen. 
 
Conclusions 
 
Investigation of dynamics of a thermal vibro-transporter is performed. Excitation is taken 
into account by using the thermal load vector. The operation of the stopping mechanism is 
analyzed. Results of performed calculations graphically showing the displacements of the 
structure during the several first periods of excitation are presented. From the obtained results 
the effect of thermal vibro-transportation is clearly seen. 
 
References 
 
[1] Segerlind L. J. Applied Finite Element Analysis. Moscow: Mir, 1979. 
[2] Zienkiewicz O. C. The Finite Element Method in Engineering Science. Moscow: Mir, 1975.  
[3] Bathe K. J. Finite Element Procedures in Engineering Analysis. New Jersey: Prentice-Hall, 1982. 
[4] Lučinskis R., Mažeika D., Hemsel T., Bansevičius R. The experimental research of piezoelectric 
actuator with two vectors of polarization direction. –Mechanika. –Kaunas: Technologija, 2010, No2 
(82), p. 50-58. 
[5] Bakšys B., Baskutien÷ J., Kilikevičius S., Chadarovičius A. Experimental analysis of the 
robotized assembly applying vibrations. Journal of Vibroengineering, Vol. 12, Issue 4, p. 572 – 581. 
[6] Juška V., Svilainis L., Dumbrava V. Analysis of piezomotor driver for laser beam deflection. 
Journal of Vibroengineering, Vol. 11, Issue 1, p. 17 – 26. 
[7] Mažeika D., Vasiljev P., Kulvietis G., Vaičiulien÷ S. New linear piezoelectric actuator based on 
traveling wave. Journal of Vibroengineering, Vol. 11, Issue 1, p. 68 – 77. 
[8] Ragulskis K., Bansevičius R., Barauskas R., Kulvietis G. Vibromotors for precision microrobots 
– N. Y., Hemisphere, 1987. – 326 р. 
[9] Ragulskis K., Spruogis B., Ragulskis M. Transformation of rotational motion by inertia couplings 
– Vilnius: Technika, 1999. – 236 p. 
[10] Ragulskis K. (Editor) Wave Mechanical Systems. Proceedings of International Seminar – Kaunas: 
Kaunas University of Technology, 1994 – 148 p., 1996 – 144 p. 
[11] Ragulskis K. and a number of other authors. Vibroengineering – Science and Arts of Lithuania. 
Book 19. Vilnius: Lithuanian Academy of Sciences, 1998. – p. 572. 


Dynamics of a thermal vibro-transporter

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