The mathematical model describing the interaction of a vibrating system and a system performing translational motion is proposed. Such motions take place in vibromotors. The main characteristics of the dynamic motion are defined. The model is investigated numerically and a number of graphical relationships are obtained. They are used in the process of design of systems with vibromotors like a walking inpipe robot

A. Matuliauskas

B. Spruogis

K. Ragulskis

L. Patašiene

M. Mištinas

English

Journal of Vibroengineering

 © VIBROMECHANIKA.  JOURNAL OF  VIBROENGINEERING.  2007 OCTOBER/DECEMBER,   VOLUME  9,  ISSUE  4,  ISSN 1392-8716 
 
77 
325. Interaction of Vibrating and Translational Motions 
 
 
 
A. Matuliauskas1, M. Mištinas1, L. Patašienė2, K. Ragulskis2, B. Spruogis1 
1Vilnius Gediminas Technical University, Faculty of Transport Engineering, Lithuania 
E-mail: bs@tti.vtu.lt; 
2Kaunas Technological University, Department of Mathematical Research in Systems, 
Studentų 50-222, Kaunas, Lithuania, E-mail: minvydas.ragulskis@ktu.lt 
 
(Received 30  September 2007; accepted 14 December 2007) 
 
 
 
 
 
 
 
 
 
 
 
 
Abstract. The mathematical model describing the interaction of a vibrating system and a system performing translational 
motion is proposed. Such motions take place in vibromotors. The main characteristics of the dynamic motion are defined. 
The model is investigated numerically and a number of graphical relationships are obtained. They are used in the process 
of design of systems with vibromotors like a walking inpipe robots. 
Keywords: vibromotor, dynamical system, characteristics of motion, numerical integration,  robot. 
 
 
 
Introduction 
 
The mathematical model describing the 
interaction of a vibrating system and a system performing 
translational motion is proposed in the form of the matrix 
differential equation describing the motion of a nonlinear 
vibrating system with two degrees of freedom. Such 
motions take place in vibromotors [1, 2]. 
The main characteristics of the dynamic motion 
are defined. The model is investigated by using numerical 
methods [3]. The time histories of motion and the 
graphical relationships representing the characteristics of 
motion are obtained.  
The obtained graphical relationships are used in 
the process of design of systems with vibromotors. 
 
Model of the system 
 
The dynamics of interaction of a vibrating system 
and a system performing translational motion is described 
by the matrix differential equation of the second order: 
 
[ ] [ ] [ ] { },
2
1
2
1
2
2
2
2
1
2
F
u
u
K
dt
du
dt
du
C
dt
ud
dt
ud
M =
⎭⎬
⎫
⎩⎨
⎧+
⎪⎭
⎪⎬
⎫
⎪⎩
⎪⎨
⎧
+
⎪⎪⎭
⎪⎪⎬
⎫
⎪⎪⎩
⎪⎪⎨
⎧
   (1) 
where u1 is the displacement of the first mass and u2 is the 
displacement of the second mass, t is the time variable, the 
mass matrix has the form: 
 
[ ] ,
0
01 ⎥⎦
⎤⎢⎣
⎡= µM                    (2) 
 
where µ is the mass of the second element of the analyzed 
system, the damping matrix has the form: 
 
[ ] ,
0
0
2
1 ⎥⎦
⎤⎢⎣
⎡=
h
h
C                    (3) 
 
where h1 is the coefficient of viscous friction of the first 
element of the analyzed system and h2 is the coefficient of 
viscous friction of the second element of the analyzed 
system, the stiffness matrix has the form: 
 
[ ] ,
00
021 ⎥⎦
⎤⎢⎣
⎡= nK                            (4) 
 
where n1 is the stiffness parameter of the first element of 
the analyzed system, the loading vector has the form: 
 325. INTERACTION OF VIBRATING AND TRANSLATIONAL MOTIONS. A. MATULIAUSKAS1, M. MIŠTINAS1, L. PATAŠIENĖ2, K. RAGULSKIS2, B. SPRUOGIS1 
 
 
© VIBROMECHANIKA.  JOURNAL OF  VIBROENGINEERING.  2007 OCTOBER/DECEMBER,   VOLUME  9,  ISSUE  4,  ISSN 1392-8716 
 
78 
{ } ,sin
2 ⎭⎬
⎫
⎩⎨
⎧= f
ta
F
ω
                   (5) 
 
where a is the amplitude of vibration excitation, ω is the 
frequency of vibration excitation and f2 is the constant 
force. 
 When: 
 
,
21
dt
du
dt
du >        (6) 
 
the following matrix is added to the damping matrix: 
 
[ ] ,ˆ ⎥⎦⎤⎢⎣⎡− −= λλ λλC                     (7) 
 
where λ is the large coefficient of viscous friction. Thus the 
constraint of motion of the two masses together is taken 
into account approximately by the penalty method. The 
numerical integration of the equations is performed by the 
method of constant average acceleration of Newmark. The 
acceptable large value of λ depends on the time step of 
numerical integration. 
 
Characteristics of motion of the system  
 
 The most important characteristics of  dynamical 
motion of the analyzed system are: 
1) the average velocity of the second mass: 
 
∫=
T
dt
dt
du
Tdt
du
,
1 22
     (8) 
 
where the overbar denotes time averaging, the integral is 
over the period of steady state motion and the period is 
expressed as: 
 
;
2
ω
π=T       (9) 
 
2) the useful work: 
 
∫=
T
n
dt
dt
dufA ;22                   (10) 
 
3) the work of the driving forces: 
 
∫=
T
v
tdt
dt
du
aA ;sin1 ω    (11) 
 
4) the coefficient of useful operation: 
 
.
v
n
A
A=η      (12) 
Results of analysis 
 
 The following parameters of the analyzed system 
are assumed: 
 
,1.01 =h ,1.02 =h ,2=µ ,11 =n ,5.0=a ,001.2 =f .100=λ
   (13) 
 
 The initial conditions are taken as: 
 
( ) ,001 =u ( ) ,002 =u ( ) ,05.001 =dt
du ( ) .15.002 =
dt
du
       (14) 
 
When ω=0.1 the time history of motion of the 
first element of the system is presented in Fig. 1a and of 
the second element in Fig. 1b. The corresponding 
velocities are presented in Fig. 1c and Fig. 1d. The quantity 
dt
du
dt
du 21 −  is presented in Fig. 1e. 
 
 
a) 
 
b) 
 
c) 
 325. INTERACTION OF VIBRATING AND TRANSLATIONAL MOTIONS. A. MATULIAUSKAS1, M. MIŠTINAS1, L. PATAŠIENĖ2, K. RAGULSKIS2, B. SPRUOGIS1 
 
 
© VIBROMECHANIKA.  JOURNAL OF  VIBROENGINEERING.  2007 OCTOBER/DECEMBER,   VOLUME  9,  ISSUE  4,  ISSN 1392-8716 
 
79 
 
d) 
 
 
 
e) 
 
Fig. 1. When ω=0.1: a) u1, b) u2, c) du1/dt, d) du2/dt,  
  e) du1/dt-du2/dt 
 
 
When ω=3 the time history of motion of the first 
element of the system is presented in Fig. 2a and of the 
second element in Fig. 2b. The corresponding velocities 
are presented in Fig. 2c and Fig. 2d. The quantity 
dt
du
dt
du 21 −  is presented in Fig. 2e. 
 
 
 
a) 
 
b) 
 
c) 
 
d) 
 
e) 
Fig. 2. When ω=3: a) u1, b) u2, c) du1/dt, d) du2/dt, 
           e) du1/dt-du2/dt 
 
 325. INTERACTION OF VIBRATING AND TRANSLATIONAL MOTIONS. A. MATULIAUSKAS1, M. MIŠTINAS1, L. PATAŠIENĖ2, K. RAGULSKIS2, B. SPRUOGIS1 
 
 
© VIBROMECHANIKA.  JOURNAL OF  VIBROENGINEERING.  2007 OCTOBER/DECEMBER,   VOLUME  9,  ISSUE  4,  ISSN 1392-8716 
 
80 
Further the main characteristics of motion as 
functions of the frequency of excitation are presented. The 
average velocity of the second mass is presented in Fig. 3. 
The useful work is presented in Fig. 4. The work of the 
driving forces is presented in Fig. 5. The coefficient of 
useful operation is presented in Fig. 6. 
 
 
 
Fig. 3. Averaged du2/dt as a function of ω 
 
 
 
 
Fig. 4. An as a function of ω 
 
 
 
 
Fig. 5. Av as a function of ω 
 
 
Fig. 6. η as a function of ω 
 
 
 
Construction Working Principle and characteristics of 
motion of the Robot vibromotor drive 
  
 
A drive of the wall press walking inpipe robot 
with pneumatic vibromotor drive [4,5] is illustrated in Fig. 
7 and Fig. 8. Robots drive consists of two locking blocks 1, 
which can turn by hinges 2 small angles one in point of 
other and interconnected, and propelled by two-way 
working pneumatic cylinder 3. 
 
 
 
 
 
Fig. 7. The scheme of the walking inpipe robot with vibromotors: 
1 − compressor, 2 − robots drive, 3 − operating block, 4 − pipes 
 
 
 
As on moving pneumatic cylinder as on locking 
blocks pneumatics cylinders are fixed pneumatic-magnetic 
operating sensors 4. They sending pneumatic signals to 
operating block and processed in operating block signals 
feed to cylinders of pneumatic drive. 
 
 325. INTERACTION OF VIBRATING AND TRANSLATIONAL MOTIONS. A. MATULIAUSKAS1, M. MIŠTINAS1, L. PATAŠIENĖ2, K. RAGULSKIS2, B. SPRUOGIS1 
 
 
© VIBROMECHANIKA.  JOURNAL OF  VIBROENGINEERING.  2007 OCTOBER/DECEMBER,   VOLUME  9,  ISSUE  4,  ISSN 1392-8716 
 
81 
 
 
Fig. 8. A drive scheme of the wall press walking inpipe robot 
with vibromotor: 1 − locking block, 2 − hinge, 3 − pneumatic 
cylinder, 4 − operating sensors 
 
Principle of robot working based on walking 
movement, periodically pressing the wall of pipeline with 
supporting elements of locking block. Pressured air gets 
into two-way working cylinder of locking block and its 
drawing stock in contact between supporting elements and 
pipeline wall fix a locking block. Than one of two locking 
blocks is fixed, pneumatic cylinder of other locking block 
working in reverse way. Supporting elements of second 
locking block is releasing by going out stock of cylinder. 
After stock of shift pneumatic cylinder going out/or in 
depend on which locking block contact with pipeline 
inside wall. In that way the wall press walking inpipe robot 
make a move forward. 
Having a physical experiment of this inpipe robot 
vibromotor drive pneumatic cylinder 1 presented in Fig. 9  
was obtained the dynamical parameters. Average velocity 
and acceleration is presented in Fig. 10 and Fig. 11 .  
 
 
Fig. 9. A scheme of the pneumatic vibromotor drive: 
1, 2 − pneumatic cylinder 
 
Fig. 10. The velocity of the pneumatic vibromotor drive over the 
time 
 
 
Fig. 11. The acceleration of the pneumatic vibromotor drive over 
the time 
 
Conclusions 
 
The mathematical model describing the interaction of a 
vibrating system and a system performing translational 
motion is proposed in the form of the matrix differential 
equation describing the motion of a nonlinear vibrating 
system with two degrees of freedom. This model is 
convenient for performing the numerical integration of the 
equations of motion. 
The main characteristics of the dynamic motion are 
defined. The time histories of motion and the graphical 
relationships representing the characteristics of motion are 
obtained. 
The presented results are used in the process of design 
of systems with vibromotors. 
The scheme of the construction of the walking inpipe 
robot with pneumatic vibromotor drive is given. The 
presented results of this drive physical experiment of 
dynamical parameters. 
  
 
References 
 
1. Ragulskis K., Bansevicius R., Barauskas R., Kulvietis G. 
Vibromotors for Precision Microrobots. New York: 
Hemisphere, 1987. 
2. Barauskas R., Kulvietis G., Ragulskis K. Calculation and 
Design of Vibromotors. Leningrad: Mashinostroienie, 1984. 
3. Bathe K. J. Finite Element Procedures in Engineering 
Analysis. New Jersey: Prentice-Hall, 1982.  
4. Matuliauskas, A.; Mištinas, M.; Spruogis. B.; Ragulskis, K. 
Pipe crawling robots with vibratory drives. Journal of 
Vibroengineering, No 4 (5), Vilnius, 2000, p. 9−14. 
5. Matuliauskas, A., Spruogis. B., Pikūnas A. Wall press 
walking in-pipe robot. Solid State Phenomena Vol. 113 (2006) 
pp. 296-300.  
 
 
 
 
 
 
 


Interaction of vibrating and translational motions

Matuliauskas, A.

Mištinas, M.

Patašiene, L.

Ragulskis, K.

Spruogis, B.

Journal of Mechatronics and Artificial Intelligence in Engineering

 © VIBROMECHANIKA.  JOURNAL OF  VIBROENGINEERING.  2007 OCTOBER/DECEMBER,   VOLUME  9,  ISSUE  4,  ISSN 1392-8716  77 325. Interaction of Vibrating and Translational Motions    A. Matuliauskas1, M. Mištinas1, L. Patašienė2, K. Ragulskis2, B. Spruogis1 1Vilnius Gediminas Technical University, Faculty of Transport Engineering, Lithuania E-mail: bs@tti.vtu.lt; 2Kaunas Technological University, Department of Mathematical Research in Systems, Studentų 50-222, Kaunas, Lithuania, E-mail: minvydas.ragulskis@ktu.lt  (Received 30  September 2007; accepted 14 December 2007)             Abstract. The mathematical model describing the interaction of a vibrating system and a system performing translational motion is proposed. Such motions take place in vibromotors. The main characteristics of the dynamic motion are defined. The model is investigated numerically and a number of graphical relationships are obtained. They are used in the process of design of systems with vibromotors like a walking inpipe robots. Keywords: vibromotor, dynamical system, characteristics of motion, numerical integration,  robot.    Introduction  The mathematical model describing the interaction of a vibrating system and a system performing translational motion is proposed in the form of the matrix differential equation describing the motion of a nonlinear vibrating system with two degrees of freedom. Such motions take place in vibromotors [1, 2]. The main characteristics of the dynamic motion are defined. The model is investigated by using numerical methods [3]. The time histories of motion and the graphical relationships representing the characteristics of motion are obtained.  The obtained graphical relationships are used in the process of design of systems with vibromotors.  Model of the system  The dynamics of interaction of a vibrating system and a system performing translational motion is described by the matrix differential equation of the second order:  [ ] [ ] [ ] { },2121222212FuuKdtdudtduCdtuddtudM =⎭⎬⎫⎩⎨⎧+⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧+⎪⎪⎭⎪⎪⎬⎫⎪⎪⎩⎪⎪⎨⎧   (1) where u1 is the displacement of the first mass and u2 is the displacement of the second mass, t is the time variable, the mass matrix has the form:  [ ] ,001⎥⎦⎤⎢⎣⎡=µM                    (2)  where µ is the mass of the second element of the analyzed system, the damping matrix has the form:  [ ] ,0021⎥⎦⎤⎢⎣⎡=hhC                    (3)  where h1 is the coefficient of viscous friction of the first element of the analyzed system and h2 is the coefficient of viscous friction of the second element of the analyzed system, the stiffness matrix has the form:  [ ] ,00021⎥⎦⎤⎢⎣⎡=nK                            (4)  where n1 is the stiffness parameter of the first element of the analyzed system, the loading vector has the form:  325. INTERACTION OF VIBRATING AND TRANSLATIONAL MOTIONS. A. MATULIAUSKAS1, M. MIŠTINAS1, L. PATAŠIENĖ2, K. RAGULSKIS2, B. SPRUOGIS1   © VIBROMECHANIKA.  JOURNAL OF  VIBROENGINEERING.  2007 OCTOBER/DECEMBER,   VOLUME  9,  ISSUE  4,  ISSN 1392-8716  78 { } ,sin2 ⎭⎬⎫⎩⎨⎧=ftaFω                   (5)  where a is the amplitude of vibration excitation, ω is the frequency of vibration excitation and f2 is the constant force.  When:  ,21dtdudtdu>        (6)  the following matrix is added to the damping matrix:  [ ] ,ˆ ⎥⎦⎤⎢⎣⎡−−=λλλλC                     (7)  where λ is the large coefficient of viscous friction. Thus the constraint of motion of the two masses together is taken into account approximately by the penalty method. The numerical integration of the equations is performed by the method of constant average acceleration of Newmark. The acceptable large value of λ depends on the time step of numerical integration.  Characteristics of motion of the system    The most important characteristics of  dynamical motion of the analyzed system are: 1) the average velocity of the second mass:  ∫=TdtdtduTdtdu,1 22      (8)  where the overbar denotes time averaging, the integral is over the period of steady state motion and the period is expressed as:  ;2ωπ=T       (9)  2) the useful work:  ∫=Tn dtdtdufA ;22                   (10)  3) the work of the driving forces:  ∫=Tv tdtdtduaA ;sin1 ω    (11)  4) the coefficient of useful operation:  .vnAA=η      (12) Results of analysis   The following parameters of the analyzed system are assumed:  ,1.01 =h ,1.02 =h ,2=µ ,11 =n ,5.0=a ,001.2 =f .100=λ   (13)   The initial conditions are taken as:  ( ) ,001 =u ( ) ,002 =u ( ) ,05.001 =dtdu ( ) .15.002 =dtdu       (14)  When ω=0.1 the time history of motion of the first element of the system is presented in Fig. 1a and of the second element in Fig. 1b. The corresponding velocities are presented in Fig. 1c and Fig. 1d. The quantity dtdudtdu 21 −  is presented in Fig. 1e.   a)  b)  c)  325. INTERACTION OF VIBRATING AND TRANSLATIONAL MOTIONS. A. MATULIAUSKAS1, M. MIŠTINAS1, L. PATAŠIENĖ2, K. RAGULSKIS2, B. SPRUOGIS1   © VIBROMECHANIKA.  JOURNAL OF  VIBROENGINEERING.  2007 OCTOBER/DECEMBER,   VOLUME  9,  ISSUE  4,  ISSN 1392-8716  79  d)    e)  Fig. 1. When ω=0.1: a) u1, b) u2, c) du1/dt, d) du2/dt,    e) du1/dt-du2/dt   When ω=3 the time history of motion of the first element of the system is presented in Fig. 2a and of the second element in Fig. 2b. The corresponding velocities are presented in Fig. 2c and Fig. 2d. The quantity dtdudtdu 21 −  is presented in Fig. 2e.    a)  b)  c)  d)  e) Fig. 2. When ω=3: a) u1, b) u2, c) du1/dt, d) du2/dt,            e) du1/dt-du2/dt   325. INTERACTION OF VIBRATING AND TRANSLATIONAL MOTIONS. A. MATULIAUSKAS1, M. MIŠTINAS1, L. PATAŠIENĖ2, K. RAGULSKIS2, B. SPRUOGIS1   © VIBROMECHANIKA.  JOURNAL OF  VIBROENGINEERING.  2007 OCTOBER/DECEMBER,   VOLUME  9,  ISSUE  4,  ISSN 1392-8716  80 Further the main characteristics of motion as functions of the frequency of excitation are presented. The average velocity of the second mass is presented in Fig. 3. The useful work is presented in Fig. 4. The work of the driving forces is presented in Fig. 5. The coefficient of useful operation is presented in Fig. 6.    Fig. 3. Averaged du2/dt as a function of ω     Fig. 4. An as a function of ω     Fig. 5. Av as a function of ω   Fig. 6. η as a function of ω    Construction Working Principle and characteristics of motion of the Robot vibromotor drive    A drive of the wall press walking inpipe robot with pneumatic vibromotor drive [4,5] is illustrated in Fig. 7 and Fig. 8. Robots drive consists of two locking blocks 1, which can turn by hinges 2 small angles one in point of other and interconnected, and propelled by two-way working pneumatic cylinder 3.      Fig. 7. The scheme of the walking inpipe robot with vibromotors: 1 − compressor, 2 − robots drive, 3 − operating block, 4 − pipes    As on moving pneumatic cylinder as on locking blocks pneumatics cylinders are fixed pneumatic-magnetic operating sensors 4. They sending pneumatic signals to operating block and processed in operating block signals feed to cylinders of pneumatic drive.   325. INTERACTION OF VIBRATING AND TRANSLATIONAL MOTIONS. A. MATULIAUSKAS1, M. MIŠTINAS1, L. PATAŠIENĖ2, K. RAGULSKIS2, B. SPRUOGIS1   © VIBROMECHANIKA.  JOURNAL OF  VIBROENGINEERING.  2007 OCTOBER/DECEMBER,   VOLUME  9,  ISSUE  4,  ISSN 1392-8716  81   Fig. 8. A drive scheme of the wall press walking inpipe robot with vibromotor: 1 − locking block, 2 − hinge, 3 − pneumatic cylinder, 4 − operating sensors  Principle of robot working based on walking movement, periodically pressing the wall of pipeline with supporting elements of locking block. Pressured air gets into two-way working cylinder of locking block and its drawing stock in contact between supporting elements and pipeline wall fix a locking block. Than one of two locking blocks is fixed, pneumatic cylinder of other locking block working in reverse way. Supporting elements of second locking block is releasing by going out stock of cylinder. After stock of shift pneumatic cylinder going out/or in depend on which locking block contact with pipeline inside wall. In that way the wall press walking inpipe robot make a move forward. Having a physical experiment of this inpipe robot vibromotor drive pneumatic cylinder 1 presented in Fig. 9  was obtained the dynamical parameters. Average velocity and acceleration is presented in Fig. 10 and Fig. 11 .    Fig. 9. A scheme of the pneumatic vibromotor drive: 1, 2 − pneumatic cylinder  Fig. 10. The velocity of the pneumatic vibromotor drive over the time   Fig. 11. The acceleration of the pneumatic vibromotor drive over the time  Conclusions  The mathematical model describing the interaction of a vibrating system and a system performing translational motion is proposed in the form of the matrix differential equation describing the motion of a nonlinear vibrating system with two degrees of freedom. This model is convenient for performing the numerical integration of the equations of motion. The main characteristics of the dynamic motion are defined. The time histories of motion and the graphical relationships representing the characteristics of motion are obtained. The presented results are used in the process of design of systems with vibromotors. The scheme of the construction of the walking inpipe robot with pneumatic vibromotor drive is given. The presented results of this drive physical experiment of dynamical parameters.    References  1. Ragulskis K., Bansevicius R., Barauskas R., Kulvietis G. Vibromotors for Precision Microrobots. New York: Hemisphere, 1987. 2. Barauskas R., Kulvietis G., Ragulskis K. Calculation and Design of Vibromotors. Leningrad: Mashinostroienie, 1984. 3. Bathe K. J. Finite Element Procedures in Engineering Analysis. New Jersey: Prentice-Hall, 1982.  4. Matuliauskas, A.; Mištinas, M.; Spruogis. B.; Ragulskis, K. Pipe crawling robots with vibratory drives. Journal of Vibroengineering, No 4 (5), Vilnius, 2000, p. 9−14. 5. Matuliauskas, A., Spruogis. B., Pikūnas A. Wall press walking in-pipe robot. Solid State Phenomena Vol. 113 (2006) pp. 296-300.         

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Interaction of vibrating and translational motions

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