The analysed system has three degrees of freedom in a plane: translation motion in the direction of wave propagation, oscillatory motion in the direction orthogonal to the direction of wave propagation, and the rotation motion. The dynamical parameters of the system are approximated for steady state motions. It is shown that the particle can be translated by the velocity much lower that the velocity of wave propagation. At the same time it can experience high angular frequency rotation motion. The condition of existence of such type of regime of motion is determined. Such modes of motion can exist in gaseous of liquid environment whenever an unbalanced particle is subjected to propagating longitudinal wave

K. Ragulskis

M. Ragulskis

English

Journal of Vibroengineering

 © VIBROMECHANIKA.  JOURNAL OF  VIBROENGINEERING.  2007 APRIL/JUNE,   VOLUME 9,  ISSUE 2,  ISSN 1392-8716 
 
72 
276. Slow Translation and Fast Rotation Motions of an Unbalanced  
        Particle Subjected to Propagating Wave 
 
 
K. Ragulskis, M. Ragulskis 
Kaunas University of  Technology,  
Studentu st. 50, LT - 51368 Kaunas, Lithuania 
E-mail:  minvydas.ragulskis@ktu.lt,     
 
(Received  21  May  2007;  accepted 15 June  2007) 
 
 
 
 
Abstract. The analysed system has three degrees of freedom in a plane: translation motion in the direction of wave 
propagation, oscillatory motion in the direction orthogonal to the direction of wave propagation, and the rotation motion. 
The dynamical parameters of the system are approximated for steady state motions. It is shown that the particle can be 
translated by the velocity much lower that the velocity of wave propagation. At the same time it can experience high 
angular frequency rotation motion. The condition of existence of such type of regime of motion is determined. Such modes 
of motion can exist in gaseous of liquid environment whenever an unbalanced particle is subjected to propagating 
longitudinal waves.  
 
Keywords: unbalanced particle, propagating wave, translation and rotation motion 
 
1. Introduction 
 
Rotation motion occurs in different unbalanced systems 
subjected to waves and vibrations [1 – 4]. Transportation 
and manipulation of objects by propagating waves is 
analysed in a number of studies [5, 6]. Conveyance of 
particles and bodies by propagating waves is an important 
scientific and engineering problem with numerous 
applications. Particle segregation in suspensions subject to 
ac electric fields [7], transport of sand particles and oil 
spills in coastal waters [8], powder transport by 
piezoelectrically excited ultrasonic waves [9], 
transportation of thin films in biomedical applications [10] 
are just few examples of problems involving interaction 
between propagating waves and the transported objects. 
Conveyance of unbalanced rotors by propagating waves is 
an important problem. The object of this paper is to 
analyse steady state motions of such systems when its 
translation velocity in the direction of wave propagation is 
slow and rotation velocity is fast.  
 
2. The model of the system 
 
The schematic diagram of the system is presented in 
Fig. 1. Coordinates of points A and B are: 
 ( )
( )ϕϕη
η
cos,sin
,
ryruB
yuA
−++
+
    (1) 
 
where ABr = ; A is the axis of rotation of the rotor; B is 
the location of the concentrated unbalanced mass m; ϕ is 
the angle of deflection of the unbalanced mass from y-axis; ( )tu,ηη =  describes the longitudinal propagating wave.  
 
The kinetic energy and dissipative system’s function 
are: 
 
( ) ( ) ( )( )
( )( )
( )
22 2 2
0 0
2 2 2
2
2 cos sin
2 1
x y
T I mr m m u y
mr u y
D H u H y Hϕ
= + ϕ + + +η + +
+ +η ϕ+ ϕ
′= +η + + ϕ
? ?? ?
? ??
? ? ?
  (2) 
 
where 0m  is the mass of the particle without the 
unbalance; xH , yH  and ϕH  are viscous damping 
coefficients according to respective coordinates; top dot 
denotes full derivative by time; 
 
u∂∂=′ ηη ; tt ∂∂=′ ηη .       (3) 
 
The governing differential equations of motion take the 
following form: 
 ( ) ( )( )
( )( ) ( ) ( )
( ) ( ) 0cossin 1sincos
sincos
2
0
2
0
2
0
=++++
=′++−+++
=+++++
yHmrymm
FuHmrumm
MHyumrmrI
y
xx



ϕϕϕϕ
ηϕϕϕϕη
ϕϕϕηϕ ϕϕ
  (4) 
 
where ϕM  is external force moment according to variable 
ϕ; xF  is external force according to coordinate x;  
 
ttut
t
uuu
u
ηηηηη
ηηη
′′+′′+′′+′=
′+′=


22
      (5) 
 
The following notations  
 276. SLOW TRANSLATION AND FAST ROTATION MOTIONS OF AN UNBALANCED PARTICLE SUBJECTED TO PROPAGATING WAVE.  K RAGULSKIS, M. RAGULSKIS 
 
 
© VIBROMECHANIKA.  JOURNAL OF  VIBROENGINEERING.  2007 APRIL/JUNE,   VOLUME 9,  ISSUE 2,  ISSN 1392-8716 
 
73 
 
r
u=ψ ; 
r
ηα = ; 
r
y=β ; 
mm
Hh xx += 0
; 
 
 
mm
H
h yy += 0
; 2
0 mrI
H
h +=
ϕ
ϕ ;  
 
2
0 mrI
M
m +=
ϕ
ϕ ; ( )rmm
Ff xx += 0
;  
 
mm
m
+= 0
µ ; 2
0
2
mrI
mr
+=ν    (6) 
 
reduce eq. (4) to: 
 
Bh
h
h
y
x
=+
Ψ=+
Φ=+
ββ
ψψ
ϕϕ ϕ



     (7) 
 
where  
 
( )( )
( )( )ϕϕϕϕµ ψαϕϕϕϕµα
ϕβϕαψν ϕ
cossin
sincos
sincos
2
2



+−=
+−−−−=Ψ
+++−=Φ
B
fh
m
xx
  (8) 
 
In case of harmonic propagating wave: 
 ( )
( )
r
A
a
krta
kutA
=
−=
−=
ψωα
ωη
cos
cos
    (9) 
 
Further we will analyse steady-state motion with slow 
average velocity: 
 
kr
v
ωψ <<= .     (10) 
 
Equations (7 – 9) are transformed to the following form: 
 
( )
ψεωψω
εψα
ε
krtkrt
vhfhvhf xxxxx
−=−
−≈′−−
Φ=Φ
    (11) 
 
where ε is a small parameter. The change of variables: 
 
ψψ ~+= vt      (12) 
 
where ψ~  is motion which average in time equals to zero 
helps to construct the solution in the form of power series: 
 
…
…
…
++=
++=
+++=
10
10
1
~~
εβββ
εψψψ
εϕϕδϕ t
    (13) 
 
where  
 
krv−= ωδ ;     (14) 
 
ϕ  is constant quantity in time; iϕ  (i=1,2,…); jψ~ , jβ  are 
functions which time averages are equal to zero.  
 
The first approximations of differential equations from 
(7, 8, 10-13) take the following form: 
 
( )( )
( )
( )ϕδµδββ
ψδ
αψψψ
ϕδµδαψψ
δϕβϕαψνϕϕ
ψ
ϕϕϕ
+−=+
+−+⎟⎠
⎞⎜⎝
⎛
∂
∂=+
++−=+
−+++−=+
=
th
vhf
t
krh
th
hmh
y
xxx
x
sin
~~
~~
sin~~
sincos
2
00
1
0
011
2
000
0000011
0




 (15) 
 
where ta δδα cos20 −= ; and zero order approximations 
are: 
 
( ) ( ) ( )( )( )
( ) ( )( )ϕδδϕδµδ
λβ
ϕδδϕδµδδδδ
λψ
+++=
+++−+−=
tth
tththta
y
xx
x
sincos
sincossincos~
0
0
                   (16) 
 
top line denotes time average; time average of function 1~ψ  
is equal to zero; 
 
22
2
x
x h+= δ
δλ ; 22
2
y
y h+= δ
δλ .               (17) 
 
The two unknowns δ and ϕ  are determined exploiting the 
condition of periodicity of 1ϕ  and 1~ϕ : 
 
0
0
0 =−Ψ
=−Φ
vh
h
x
δϕ
     (18) 
 
or: 
 
( )( ) ( )
( ) δλϕϕδµλ
δλλνµδδϕδλϕλδν ϕϕ
xxxxxx
yxxxxx
hkrafvhshkra
hmhha
25.0cossin5.0
5.0cos1sin5.0
−−=−
−+−=−+− (19) 
 
Eq. (19) produces the condition of existence of the 
analysed regime of motion: 
 ( ) ( ) ( )( )
( ) 1215.0
5.0
2
222
<−
−−+−+−
δλµν
δλµλµννδ ϕϕ
x
yxxxx
kra
hakrsfvhmhkr
 
 
                                                                                       (20) 
where v is determined from Eq. (14).  
Only one unknown variable δ is present in inequality 
(20) and the regions of the existence of the analysed 
regimes of motion can be easily identified from this 
inequality.  
 
 276. SLOW TRANSLATION AND FAST ROTATION MOTIONS OF AN UNBALANCED PARTICLE SUBJECTED TO PROPAGATING WAVE.  K RAGULSKIS, M. RAGULSKIS 
 
 
© VIBROMECHANIKA.  JOURNAL OF  VIBROENGINEERING.  2007 APRIL/JUNE,   VOLUME 9,  ISSUE 2,  ISSN 1392-8716 
 
74 
 
3. Conclusions 
 
It is shown that unbalanced particle can be slowly 
conveyed in the direction of wave propagation in the field 
of propagating waves and at the same time it can have a 
large spin. The condition of existence of such regime of 
motion is determined exploiting approximate analytical 
techniques.  
 
References 
 
[1] Ragulskis K. Mechanisms on Vibrating Foundation: 
Problems of Dynamics and Stability. IEE Lithuanian 
Academy of Sciences, Kaunas, 1963. 
[2] Ragulskis K. Dynamics of angular motion of 
mechanism at vibrating tool. Theory of Machines and 
Mechanisms, USSR Academy of Sciences, Moscow, 1963, 
p. 96-97. 
[3] Kumpikas A., Ragulskis K. Drakbewegungen eines 
mechanizmus auf schwingungen Fundament. 
Wessunchaftliche Zutschrift: Technische Hochschule Karl-
Marx-Stadt, 1972, 4.2. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
[4] Ragulskis K. Schwingungforchung in Politechnischen 
Institut Kaunas. Feingerwetetechnik. Berlin, 1980, No. 7.  
[5] Ragulskis K., Bansevicius R., Barauskas R., 
Kulvietis G. Vibromotors for Precision Microrobots. 
Hemisphere, New York, 1987. 
[6] Vibroengineering. Vilnius: Lithuanian Academic 
Scentiarum, Vilnius, 1998. 
[7] Dusiaud A. D., Khusid B., Acrivos B. A. Particle 
segregation in suspensions subject to high-gradient ac 
electric fields, Journal of Applied Physics, 88, 2000, 
p.5463-5473. 
[8] Wael N. H., Ribberink J. S. Transport processes of 
uniform and mixed sands in oscillatory sheet flow,” 
Coastal Engineering 52( 9), 2005, p.745-770. 
[9] Mracek M., Wallaschek J. A system for powder 
transport based on piezoelectrically excited ultrasonic 
progressive waves, Materials Chemistry and Physics, 
90(2-3), 2005, p.378-380. 
[10] Ragulskis M., Koizumi K. Applicability of attractor 
control techniques for a particle conveyed by a propagating 
wave, Journal of Vibration and Control, 10, 2004, p.1057-
1070. 


Slow translation and fast rotation motions of an unbalanced particle subjected to propagating wave

Ragulskis, K.

Ragulskis, M.

Journal of Mechatronics and Artificial Intelligence in Engineering

 © VIBROMECHANIKA.  JOURNAL OF  VIBROENGINEERING.  2007 APRIL/JUNE,   VOLUME 9,  ISSUE 2,  ISSN 1392-8716  72 276. Slow Translation and Fast Rotation Motions of an Unbalanced          Particle Subjected to Propagating Wave   K. Ragulskis, M. Ragulskis Kaunas University of  Technology,  Studentu st. 50, LT - 51368 Kaunas, Lithuania E-mail:  minvydas.ragulskis@ktu.lt,      (Received  21  May  2007;  accepted 15 June  2007)     Abstract. The analysed system has three degrees of freedom in a plane: translation motion in the direction of wave propagation, oscillatory motion in the direction orthogonal to the direction of wave propagation, and the rotation motion. The dynamical parameters of the system are approximated for steady state motions. It is shown that the particle can be translated by the velocity much lower that the velocity of wave propagation. At the same time it can experience high angular frequency rotation motion. The condition of existence of such type of regime of motion is determined. Such modes of motion can exist in gaseous of liquid environment whenever an unbalanced particle is subjected to propagating longitudinal waves.   Keywords: unbalanced particle, propagating wave, translation and rotation motion  1. Introduction  Rotation motion occurs in different unbalanced systems subjected to waves and vibrations [1 – 4]. Transportation and manipulation of objects by propagating waves is analysed in a number of studies [5, 6]. Conveyance of particles and bodies by propagating waves is an important scientific and engineering problem with numerous applications. Particle segregation in suspensions subject to ac electric fields [7], transport of sand particles and oil spills in coastal waters [8], powder transport by piezoelectrically excited ultrasonic waves [9], transportation of thin films in biomedical applications [10] are just few examples of problems involving interaction between propagating waves and the transported objects. Conveyance of unbalanced rotors by propagating waves is an important problem. The object of this paper is to analyse steady state motions of such systems when its translation velocity in the direction of wave propagation is slow and rotation velocity is fast.   2. The model of the system  The schematic diagram of the system is presented in Fig. 1. Coordinates of points A and B are:  ( )( )ϕϕηηcos,sin,ryruByuA−+++     (1)  where ABr = ; A is the axis of rotation of the rotor; B is the location of the concentrated unbalanced mass m; ϕ is the angle of deflection of the unbalanced mass from y-axis; ( )tu,ηη =  describes the longitudinal propagating wave.   The kinetic energy and dissipative system’s function are:  ( ) ( ) ( )( )( )( )( )22 2 20 02 2 222 cos sin2 1x yT I mr m m u ymr u yD H u H y Hϕ= + ϕ + + +η + ++ +η ϕ+ ϕ′= +η + + ϕ  (2)  where 0m  is the mass of the particle without the unbalance; xH , yH  and ϕH  are viscous damping coefficients according to respective coordinates; top dot denotes full derivative by time;  u∂∂=′ ηη ; tt ∂∂=′ ηη .       (3)  The governing differential equations of motion take the following form:  ( ) ( )( )( )( ) ( ) ( )( ) ( ) 0cossin1sincossincos202020=++++=′++−+++=+++++yHmrymmFuHmrummMHyumrmrIyxxϕϕϕϕηϕϕϕϕηϕϕϕηϕ ϕϕ  (4)  where ϕM  is external force moment according to variable ϕ; xF  is external force according to coordinate x;   ttuttuuuuηηηηηηηη′′+′′+′′+′=′+′=22      (5)  The following notations   276. SLOW TRANSLATION AND FAST ROTATION MOTIONS OF AN UNBALANCED PARTICLE SUBJECTED TO PROPAGATING WAVE.  K RAGULSKIS, M. RAGULSKIS   © VIBROMECHANIKA.  JOURNAL OF  VIBROENGINEERING.  2007 APRIL/JUNE,   VOLUME 9,  ISSUE 2,  ISSN 1392-8716  73  ru=ψ ; rηα = ; ry=β ; mmHh xx +=0;   mmHhyy +=0; 20 mrIHh+= ϕϕ ;   20 mrIMm+= ϕϕ ; ( )rmmFf xx +=0;   mmm+=0µ ; 202mrImr+=ν    (6)  reduce eq. (4) to:  Bhhhyx=+Ψ=+Φ=+ββψψϕϕ ϕ     (7)  where   ( )( )( )( )ϕϕϕϕµψαϕϕϕϕµαϕβϕαψν ϕcossinsincossincos22+−=+−−−−=Ψ+++−=ΦBfhmxx  (8)  In case of harmonic propagating wave:  ( )( )rAakrtakutA=−=−=ψωαωηcoscos    (9)  Further we will analyse steady-state motion with slow average velocity:  krvωψ <<= .     (10)  Equations (7 – 9) are transformed to the following form:  ( )ψεωψωεψαεkrtkrtvhfhvhf xxxxx−=−−≈′−−Φ=Φ   (11)  where ε is a small parameter. The change of variables:  ψψ ~+= vt      (12)  where ψ~  is motion which average in time equals to zero helps to construct the solution in the form of power series:  ………++=++=+++=10101~~εβββεψψψεϕϕδϕ t    (13)  where   krv−= ωδ ;     (14)  ϕ  is constant quantity in time; iϕ  (i=1,2,…); jψ~ , jβ  are functions which time averages are equal to zero.   The first approximations of differential equations from (7, 8, 10-13) take the following form:  ( )( )( )( )ϕδµδββψδαψψψϕδµδαψψδϕβϕαψνϕϕψϕϕϕ+−=++−+⎟⎠⎞⎜⎝⎛∂∂=+++−=+−+++−=+=thvhftkrhthhmhyxxxxsin~~~~sin~~sincos20010011200000000110 (15)  where ta δδα cos20 −= ; and zero order approximations are:  ( ) ( ) ( )( )( )( ) ( )( )ϕδδϕδµδλβϕδδϕδµδδδδλψ+++=+++−+−=tthtththtayxxxsincossincossincos~00                   (16)  top line denotes time average; time average of function 1~ψ  is equal to zero;  222xxh+=δδλ ; 222yyh+=δδλ .               (17)  The two unknowns δ and ϕ  are determined exploiting the condition of periodicity of 1ϕ  and 1~ϕ :  000 =−Ψ=−Φvhhxδϕ      (18)  or:  ( )( ) ( )( ) δλϕϕδµλδλλνµδδϕδλϕλδν ϕϕxxxxxxyxxxxxhkrafvhshkrahmhha25.0cossin5.05.0cos1sin5.0−−=−−+−=−+− (19)  Eq. (19) produces the condition of existence of the analysed regime of motion:  ( ) ( ) ( )( )( )1215.05.02222<−−−+−+−δλµνδλµλµννδ ϕϕxyxxxxkrahakrsfvhmhkr                                                                                          (20) where v is determined from Eq. (14).  Only one unknown variable δ is present in inequality (20) and the regions of the existence of the analysed regimes of motion can be easily identified from this inequality.    276. SLOW TRANSLATION AND FAST ROTATION MOTIONS OF AN UNBALANCED PARTICLE SUBJECTED TO PROPAGATING WAVE.  K RAGULSKIS, M. RAGULSKIS   © VIBROMECHANIKA.  JOURNAL OF  VIBROENGINEERING.  2007 APRIL/JUNE,   VOLUME 9,  ISSUE 2,  ISSN 1392-8716  74  3. Conclusions  It is shown that unbalanced particle can be slowly conveyed in the direction of wave propagation in the field of propagating waves and at the same time it can have a large spin. The condition of existence of such regime of motion is determined exploiting approximate analytical techniques.   References  [1] Ragulskis K. Mechanisms on Vibrating Foundation: Problems of Dynamics and Stability. IEE Lithuanian Academy of Sciences, Kaunas, 1963. [2] Ragulskis K. Dynamics of angular motion of mechanism at vibrating tool. Theory of Machines and Mechanisms, USSR Academy of Sciences, Moscow, 1963, p. 96-97. [3] Kumpikas A., Ragulskis K. Drakbewegungen eines mechanizmus auf schwingungen Fundament. Wessunchaftliche Zutschrift: Technische Hochschule Karl-Marx-Stadt, 1972, 4.2.                                      [4] Ragulskis K. Schwingungforchung in Politechnischen Institut Kaunas. Feingerwetetechnik. Berlin, 1980, No. 7.  [5] Ragulskis K., Bansevicius R., Barauskas R., Kulvietis G. Vibromotors for Precision Microrobots. Hemisphere, New York, 1987. [6] Vibroengineering. Vilnius: Lithuanian Academic Scentiarum, Vilnius, 1998. [7] Dusiaud A. D., Khusid B., Acrivos B. A. Particle segregation in suspensions subject to high-gradient ac electric fields, Journal of Applied Physics, 88, 2000, p.5463-5473. [8] Wael N. H., Ribberink J. S. Transport processes of uniform and mixed sands in oscillatory sheet flow,” Coastal Engineering 52( 9), 2005, p.745-770. [9] Mracek M., Wallaschek J. A system for powder transport based on piezoelectrically excited ultrasonic progressive waves, Materials Chemistry and Physics, 90(2-3), 2005, p.378-380. [10] Ragulskis M., Koizumi K. Applicability of attractor control techniques for a particle conveyed by a propagating wave, Journal of Vibration and Control, 10, 2004, p.1057-1070. 

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Slow translation and fast rotation motions of an unbalanced particle subjected to propagating wave

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