The dynamics of fluid in the elements of a pipe robot is investigated in this paper. Plane vibrations of ideal compressible fluid are analyzed by taking account the constraint of non-rotation by means of the penalty method. Reduced order numerical integration of the penalty term is performed. Graphical representation of rotations at the points of reduced integration is proposed for validation of the calculated eigenmodes. Fluid flow through different cross sections of nano-holes in metal sheets separating different compartments of a pipe robot is important in the design of a pipe robot. Calculation of the mass flow rate is necessary when performing the comparison of different cross-sections. A two-dimensional model performing the discretization of the cross-section is used for the solution of this problem. The obtained results are used in the process of design of a pipe robo

A. Bubulis

A. Matuliauskas

B. Spruogis

K. Ragulskis

L. Ragulskis

V. Mištinas

English

Ragulskis, L.

Ragulskis, K.

Bubulis, A.

Spruogis, B.

Mištinas, V.

Matuliauskas, A.

Journal of Mechatronics and Artificial Intelligence in Engineering

Investigation of dynamics of fluid in the elements of a pipe robot

Journal of Mechanical Engineering, Automation and Control Systems

  VIBROMECHANIKA.  JOURNAL OF  VIBROENGINEERING.  JUNE   2008,   VOLUME  10,  ISSUE  2,  ISSN 1392-8716  209361. Investigation of dynamics of fluid in the elements of a pipe robot    L. Ragulskis1, K. Ragulskis2, A. Bubulis3, B. Spruogis4, V. Mištinas5, A. Matuliauskas6 1Vytautas Magnus University,    Vileikos 8-702, LT-44404 Kaunas, Lithuania,  2, 3 Kaunas University of  Technology,        Studentų 50-222, LT-51368 Kaunas, Lithuania       Phone: +37069822456;  Fax: +37037330446 4, 5, 6 Vilnius Gediminas Technical University,              Plytines 27, LT-10223 Vilnius, Lithuania E-mail:  1 l.ragulskis@if.vdu.lt   2 kazimieras3@hotmail.com,  3 algimantas.bubulis@ktu.lt                   4  bs@vgtu.lt,   5 vygantas@ti.vgtu.lt,   6 arvydas@ti.vgtu.lt  (Received 19 May 2008; accepted 13 June 2008)      Abstract. The dynamics of fluid in the elements of a pipe robot is investigated in this paper. Plane vibrations f ideal compressible fluid are analyzed by taking account the constraint of non-rotation by means of the penalty method. Reduced order numerical integration of the penalty term is performed. Graphical representation of rotations at the points of reduced integration is proposed for validation of the calculated eigenmodes. Fluid flow through different cross sections of nano-holes in metal sheets separating different compartmen s of a pipe robot is important in the design of a pipe robot. Calculation of the mass flow rate is necessary when performing the comparison of different cross-sections. A two-dimensio al model performing the discretization of the cross-section is used for the solution of this problem. The obtained results are used in the process of design of a pipe robot.  Keywords: pipe robot, ideal compressible fluid, eigenmode, penalty method, reduced integration, finite elements, fluid                      flow, cross section, mass flow rate.  Introduction  In this paper the dynamics of fluid in the elements of a pipe robot is investigated. The plane vibrations of ideal compressible fluid are analyzed by using the displacement formulation [1, 2] and taking into account the constraint of non-rotati n by means of the penalty method [3, 4]. Reduced order numerical integration of the penalty term is performed [3, 4]. Graphical representation of rotations at the points of reduced order numerical integration is proposed for validation of the calculated eigenmodes. Fluid flow through different cross sections of nano-holes in metal sheets separating different compartments of a pipe robot is important in the design of a pipe robot. Calculation of the mass flow rate is necessary when performing the comparison of different cross-sections. Related problems of analysis of fluid flow are presented in [5, 6, 7]. A two-dimensional model performing the discretization of the cross-section is used for the solution of this problem [5]. The pipe robots of various types and their dynamical motions as well as the motions of their elem nts are investigated and described in [8] and other papers. This paper presents the continuation of investigations pre ented in the previous papers. The obtained results are used in the process of design of the elements of a pipe robot. Finite element model of the element of a pipe robot  The nodal variables for the analyzed two dimensional problem of vibrations of the fluid are the displacements u and v in the directions x and y of the orthogonal Cartesian system of coordinates. It is assumed that the angular frequency of excitation coincides with the eigenfrequency of the appropriate eigenmode and thus the eigenproblem is analyzed. For the calculation of the eigenmodes the expressions for the mass and stiffness matrices are necessary.  The mass matrix of the fluid is:  [ ] [ ] [ ] ,∫= dxdyNNMT ρ     (1)  where ρ is the density of the fluid in the status of equilibrium, [N] is the matrix of the shape functions determined by the following relationship:  361. INVESTIGATION OF DYNAMICS OF FLUID IN THE ELEMENTS OF A PIPE ROBOT.  L. RAGULSKIS1, K. RAGULSKIS2, A. BUBULIS3, B. SPRUOGIS4, V. MIŠTINAS5, A. MATULIAUSKAS6    VIBROMECHANIKA.  JOURNAL OF  VIBROENGINEERING.  JUNE   2008,   VOLUME  10,  ISSUE  2,  ISSN 1392-8716  210 [ ]{ },δNvu=                    (2)  where {δ} is the vector of nodal displacements. Explicitly:  [ ] ,00002121=LLNNNNN    (3)  where Ni are the shape functions of the analyzed finite element. The stiffness matrix of the fluid is:  [ ][ ] [ ] [ ] [ ]∫ ∫+==,~~2 dxdyBBdxdyBcBKTT λρ (4)  where c is the speed of sound in the fluid, λ is the penalty parameter for the constraint of non-rotation. The matrix [B] relating the volumetric strain with the displacements is defined from:  [ ]{ }.δByvxu=∂∂+∂∂     (5)  Explicitly:  [ ] .2211 ∂∂∂∂∂∂∂∂= LyNxNyNxNB  (6)  The matrix [B~] is used to characterize the rotation and is defined from:  [ ]{ }.~ δBxvyu=∂∂−∂∂     (7)  Thus:  [ ].~2211∂∂−∂∂∂∂−∂∂==LxNyNxNyNB(8)  The calculation of the second integral of the stiffness matrix is performed using numerical integration rule of reduced order. The rotations for the eigenmode are calculated at the points of reduced integration order of the second integral of the stiffness matrix:  [ ]{ }.~21δω B−=      (9)  Results of analysis of vibrations of the fluid in the element of a pipe robot   The problem in the rectangular domain is analyzed. The length of the analyzed structure is equal to twice of its dimension in the direction of the y axis. The displacements normal to the boundaries are assumed equalto zero.  As it is assumed that the angular frequency of excitation coincides with the eigenfrequency of the appropriate eigenmode and the excitation is not orthogonal to the eigenmode, the excitation is not specified explicitly and vibrations according to the eigenmode are analyzed. The rotations for the eigenmode are calculated at the points of reduced integration order and represent d in a circle around this point as a black angle from the positive direction of the x axis. This angle is equal to the rotation multiplied by a large constant. This constant is chosen for each eigenmode in order to achieve visible representatio  of the results.  The first eigenmode is presented in Fig. 1, the second eigenmode in Fig. 2, … , the tenth eigenmode in Fig. 10. The second and the third eigenmodes are multiple and also the eighth and the ninth eigenmodes are multiple.    Fig. 1. The first eigenmode    Fig. 2. The second eigenmode    Fig. 3. The third eigenmode  361. INVESTIGATION OF DYNAMICS OF FLUID IN THE ELEMENTS OF A PIPE ROBOT.  L. RAGULSKIS1, K. RAGULSKIS2, A. BUBULIS3, B. SPRUOGIS4, V. MIŠTINAS5, A. MATULIAUSKAS6    VIBROMECHANIKA.  JOURNAL OF  VIBROENGINEERING.  JUNE   2008,   VOLUME  10,  ISSUE  2,  ISSN 1392-8716  211 Fig. 4. The fourth eigenmode   Fig. 5. The fifth eigenmode   Fig. 6. The sixth eigenmode   Fig. 7. The seventh eigenmode   Fig. 8. The eighth eigenmode   Fig. 9. The ninth eigenmode  Figure 10. The tenth eigenmode  This graphical representation indicates the quality of satisfaction of the constraint of non-rotation for the eigenmode over the whole structure and thus serves for the validation of the results of calculations. It should be noted that  for this problem in several eigenmodes there is no rotational motion because of the displacements parallel to the axes of coordinates.  Analysis of mass flow rate between different elements of a pipe robot  Cartesian coordinate system is used with the z axis parallel to the axis of the tube. The cross-section of the tube does not vary with the z coordinate. The velocity of fluid flow in the direction of the z axis is the function of the coordinates of the cross-section: ( )yxww ,= . The components of velocity in the plane of the cross-section are equal to zero. The stresses are: ,pzyx −=== σσσ  ,0=xyσ  ,ywyz ∂∂= µσ  ,xwzx ∂∂= µσ  where p is the pressure and µ is the viscosity of the fluid. The equilibrium equation in the direction of the z axis has the form:  ,0=+∂∂−∂∂∂∂+∂∂∂∂gzpywyxwxρµµ            (10)  where ρ is the density of the fluid, g is the acceleration of gravity, zp∂∂ is the gradient of pressure in the direction of the z axis and it is assumed constant.  The boundary condition at the wall has the form:  ,wnwαµ =∂∂−                    (11)  where n is the outward normal to the boundary of the cross section of the flow, α is the coefficient of slippage between the fluid and the surface of the tube.  The stiffness matrix has the form:  [ ] [ ] [ ] [ ] [ ]∫ ∫+= ,dsMMdxdyBBKTT αµ            (12)  361. INVESTIGATION OF DYNAMICS OF FLUID IN THE ELEMENTS OF A PIPE ROBOT.  L. RAGULSKIS1, K. RAGULSKIS2, A. BUBULIS3, B. SPRUOGIS4, V. MIŠTINAS5, A. MATULIAUSKAS6    VIBROMECHANIKA.  JOURNAL OF  VIBROENGINEERING.  JUNE   2008,   VOLUME  10,  ISSUE  2,  ISSN 1392-8716  212 where s is the boundary line of the cross-section of the flow, [B] is the matrix of the derivatives of the shape functions (the first row with respect to x and the second row with respect to y), [M] is the row vector of the shape functions on the boundary of the cross-section.  The loading vector has the form:  { } [ ]∫ ∂∂−= ,dxdyzpgNF T ρ   (13)  where [N] is the row vector of the shape functions in the cross-section of the flow.  The mass flow rate is found:  ( )∫= ,, dxdyyxwQ ρ    (14)  where ( )yxw ,  are calculated from the obtained solution of the system of linear algebraic equations by using the shape functions of the appropriate finite elements.  Cross-section is assumed to be a circle and one fourth of it is analyzed. The characteristics of the fluid:   ,4004. cPmmsg==µ     ,04.2smmg=α    ,001.3mmg=ρ         .122smmggzp=+∂∂− ρ    Radius of the cross section is 10 mm. The obtained mass flow rate .sg 254.864=Q   Conclusions   The dynamics of a fluid in the element of a pipe robot is investigated. Vibrations of ideal compressible fluid are analyzed by taking into account the constraint of nn-rotation by the penalty method using reduced order numerical integration of the penalty term. Graphical representation of rotations at the points of reduced order numerical integration is proposed for validation of the calculated eigenmodes. This graphical representation shows the quality of satisfaction of the constraint of non-rotation over the whole structure. The mass flow rate of the fluid through various cross-sections is determined. This is important in thedesign of separating elements between the different compartments of a pipe robot. The obtained results are used in the process of design of the elements of a pipe robot.   References  [1] Iershov N. F., Shahvierdi G. G. The Method of Finite Elements in the Problems of Hydrodynamics and Hydroelasticity. Leningrad: Sudostroienie, 1984. [2] Eibl J., Stempniewski L. Dynamic analysis of liquid filled tanks including plasticity and fluid interaction - earthquake effects. Shell and Spatial Structures: Computational Aspects, Proceedings of the International Symposium. July 1986. Leuven, Belgium, Berlin: Springer-Verlag, 1987, p. 261-269. [3] Reddy J. N. Applied Functional Analysis and Variational Methods in Engineering. New York: McGraw-Hill, 1986. [4] Bathe K. J. Finite Element Procedures in Engineering Analysis. New Jersey: Prentice-Hall, 1982.  [5] Rao S. S. The Finite Element Method in Engineering. Oxford: Pergamon Press, 1982. [6] Connor J. J., Brebbia C. A. Finite Element Techniques for Fluid Flow. Leningrad: Sudostroienie, 1979. [7] Zienkiewicz O. C. The Finite Element Method in Engineering Science. Moscow: Mir, 1975. [8] Matuliauskas A., Mištinas V., Spruogis B., Ragulskis K. Pipe crawling robots with vibratory drives. Journal of Vibroengineering 4(5). Vilnius, 2000. P. 9-14.  

Journal of Vibroengineering

  VIBROMECHANIKA.  JOURNAL OF  VIBROENGINEERING.  JUNE   2008,   VOLUME  10,  ISSUE  2,  ISSN 1392-8716 
 
209
361. Investigation of dynamics of fluid in the elements of a pipe robot 
 
 
 
L. Ragulskis1, K. Ragulskis2, A. Bubulis3, B. Spruogis4, V. Mištinas5, A. Matuliauskas6 
1Vytautas Magnus University,  
  Vileikos 8-702, LT-44404 Kaunas, Lithuania,  
2, 3
 Kaunas University of  Technology,  
      Studentų 50-222, LT-51368 Kaunas, Lithuania 
      Phone: +37069822456;  Fax: +37037330446 
4, 5, 6 Vilnius Gediminas Technical University,  
 
          
 Plytines 27, LT-10223 Vilnius, Lithuania 
E-mail: 1 l.ragulskis@if.vdu.lt   2 kazimieras3@hotmail.com,  3 algimantas.bubulis@ktu.lt 
                  4  bs@vgtu.lt,   5 vygantas@ti.vgtu.lt,   6 arvydas@ti.vgtu.lt 
 
(Received 19 May 2008; accepted 13 June 2008) 
 
 
 
 
 
Abstract. The dynamics of fluid in the elements of a pipe robot is investigated in this paper. Plane vibrations of ideal 
compressible fluid are analyzed by taking account the constraint of non-rotation by means of the penalty method. Reduced 
order numerical integration of the penalty term is performed. Graphical representation of rotations at the points of reduced 
integration is proposed for validation of the calculated eigenmodes. 
Fluid flow through different cross sections of nano-holes in metal sheets separating different compartments of a 
pipe robot is important in the design of a pipe robot. Calculation of the mass flow rate is necessary when performing the 
comparison of different cross-sections. A two-dimensional model performing the discretization of the cross-section is used 
for the solution of this problem. 
The obtained results are used in the process of design of a pipe robot. 
 
Keywords: pipe robot, ideal compressible fluid, eigenmode, penalty method, reduced integration, finite elements, fluid  
                    flow, cross section, mass flow rate. 
 
Introduction 
 
In this paper the dynamics of fluid in the elements 
of a pipe robot is investigated. 
The plane vibrations of ideal compressible fluid 
are analyzed by using the displacement formulation [1, 2] 
and taking into account the constraint of non-rotation by 
means of the penalty method [3, 4]. Reduced order 
numerical integration of the penalty term is performed [3, 
4]. 
Graphical representation of rotations at the points 
of reduced order numerical integration is proposed for 
validation of the calculated eigenmodes. 
Fluid flow through different cross sections of 
nano-holes in metal sheets separating different 
compartments of a pipe robot is important in the design of 
a pipe robot. Calculation of the mass flow rate is necessary 
when performing the comparison of different cross-
sections. Related problems of analysis of fluid flow are 
presented in [5, 6, 7]. A two-dimensional model 
performing the discretization of the cross-section is used 
for the solution of this problem [5]. 
The pipe robots of various types and their 
dynamical motions as well as the motions of their elements 
are investigated and described in [8] and other papers. This 
paper presents the continuation of investigations presented 
in the previous papers. 
The obtained results are used in the process of 
design of the elements of a pipe robot. 
 
Finite element model of the element of a pipe robot 
 
The nodal variables for the analyzed two 
dimensional problem of vibrations of the fluid are the 
displacements u and v in the directions x and y of the 
orthogonal Cartesian system of coordinates. 
It is assumed that the angular frequency of 
excitation coincides with the eigenfrequency of the 
appropriate eigenmode and thus the eigenproblem is 
analyzed. For the calculation of the eigenmodes the 
expressions for the mass and stiffness matrices are 
necessary.  
The mass matrix of the fluid is: 
 
[ ] [ ] [ ] ,∫= dxdyNNM
T ρ     (1) 
 
where ρ is the density of the fluid in the status of 
equilibrium, [N] is the matrix of the shape functions 
determined by the following relationship: 
 361. INVESTIGATION OF DYNAMICS OF FLUID IN THE ELEMENTS OF A PIPE ROBOT.  L. RAGULSKIS1, K. RAGULSKIS2, A. BUBULIS3, B. SPRUOGIS4, V. MIŠTINAS5, A. MATULIAUSKAS6 
 
 
 VIBROMECHANIKA.  JOURNAL OF  VIBROENGINEERING.  JUNE   2008,   VOLUME  10,  ISSUE  2,  ISSN 1392-8716 
 
210 
[ ]{ },δN
v
u
=






                    (2) 
 
where {δ} is the vector of nodal displacements. Explicitly: 
 
[ ] ,
00
00
21
21






=
L
L
NN
NN
N    (3) 
 
where Ni are the shape functions of the analyzed finite 
element. 
The stiffness matrix of the fluid is: 
 
[ ]
[ ] [ ] [ ] [ ]∫ ∫+=
=
,
~~2 dxdyBBdxdyBcB
K
TT λρ
 (4) 
 
where c is the speed of sound in the fluid, λ is the penalty 
parameter for the constraint of non-rotation. The matrix [B] 
relating the volumetric strain with the displacements is 
defined from: 
 
[ ]{ }.δB
y
v
x
u
=
∂
∂
+
∂
∂
     (5) 
 
Explicitly: 
 
[ ] .2211 





∂
∂
∂
∂
∂
∂
∂
∂
= L
y
N
x
N
y
N
x
N
B  (6) 
 
The matrix [ B~ ] is used to characterize the rotation and is 
defined from: 
 
[ ]{ }.~ δB
x
v
y
u
=
∂
∂
−
∂
∂
     (7) 
 
Thus: 
 
[ ]
.
~
2211






∂
∂
−
∂
∂
∂
∂
−
∂
∂
=
=
L
x
N
y
N
x
N
y
N
B
(8) 
 
The calculation of the second integral of the stiffness 
matrix is performed using numerical integration rule of 
reduced order. 
The rotations for the eigenmode are calculated at 
the points of reduced integration order of the second 
integral of the stiffness matrix: 
 
[ ]{ }.~
2
1
δω B−=      (9) 
 
Results of analysis of vibrations of the fluid in the 
element of a pipe robot 
 
 The problem in the rectangular domain is 
analyzed. The length of the analyzed structure is equal to 
twice of its dimension in the direction of the y axis. The 
displacements normal to the boundaries are assumed equal 
to zero.  
As it is assumed that the angular frequency of 
excitation coincides with the eigenfrequency of the 
appropriate eigenmode and the excitation is not orthogonal 
to the eigenmode, the excitation is not specified explicitly 
and vibrations according to the eigenmode are analyzed. 
The rotations for the eigenmode are calculated at 
the points of reduced integration order and represented in a 
circle around this point as a black angle from the positive 
direction of the x axis. This angle is equal to the rotation 
multiplied by a large constant. This constant is chosen for 
each eigenmode in order to achieve visible representation 
of the results. 
 The first eigenmode is presented in Fig. 1, the 
second eigenmode in Fig. 2, … , the tenth eigenmode in 
Fig. 10. The second and the third eigenmodes are multiple 
and also the eighth and the ninth eigenmodes are multiple. 
 
 
 
Fig. 1. The first eigenmode 
 
 
 
Fig. 2. The second eigenmode 
 
 
 
Fig. 3. The third eigenmode 
 361. INVESTIGATION OF DYNAMICS OF FLUID IN THE ELEMENTS OF A PIPE ROBOT.  L. RAGULSKIS1, K. RAGULSKIS2, A. BUBULIS3, B. SPRUOGIS4, V. MIŠTINAS5, A. MATULIAUSKAS6 
 
 
 VIBROMECHANIKA.  JOURNAL OF  VIBROENGINEERING.  JUNE   2008,   VOLUME  10,  ISSUE  2,  ISSN 1392-8716 
 
211
 
Fig. 4. The fourth eigenmode 
 
 
Fig. 5. The fifth eigenmode 
 
 
Fig. 6. The sixth eigenmode 
 
 
Fig. 7. The seventh eigenmode 
 
 
Fig. 8. The eighth eigenmode 
 
 
Fig. 9. The ninth eigenmode 
 
Figure 10. The tenth eigenmode 
 
This graphical representation indicates the quality 
of satisfaction of the constraint of non-rotation for the 
eigenmode over the whole structure and thus serves for the 
validation of the results of calculations. It should be noted 
that  for this problem in several eigenmodes there is no 
rotational motion because of the displacements parallel to 
the axes of coordinates. 
 
Analysis of mass flow rate between different elements 
of a pipe robot 
 
Cartesian coordinate system is used with the z 
axis parallel to the axis of the tube. The cross-section of 
the tube does not vary with the z coordinate. The velocity 
of fluid flow in the direction of the z axis is the function of 
the coordinates of the cross-section: ( )yxww ,= . The 
components of velocity in the plane of the cross-section are 
equal to zero. 
The stresses are: ,pzyx −=== σσσ  
,0=xyσ  ,y
w
yz ∂
∂
= µσ  ,
x
w
zx ∂
∂
= µσ  where p is the 
pressure and µ is the viscosity of the fluid. 
The equilibrium equation in the direction of the z 
axis has the form: 
 
,0=+
∂
∂
−





∂
∂
∂
∂
+





∂
∂
∂
∂ g
z
p
y
w
yx
w
x
ρµµ            (10) 
 
where ρ is the density of the fluid, g is the acceleration of 
gravity, 
z
p
∂
∂
 is the gradient of pressure in the direction of 
the z axis and it is assumed constant. 
 The boundary condition at the wall has the form: 
 
,w
n
w
αµ =
∂
∂
−                    (11) 
 
where n is the outward normal to the boundary of the cross 
section of the flow, α is the coefficient of slippage between 
the fluid and the surface of the tube. 
 The stiffness matrix has the form: 
 
[ ] [ ] [ ] [ ] [ ]∫ ∫+= ,dsMMdxdyBBK
TT αµ            (12) 
 361. INVESTIGATION OF DYNAMICS OF FLUID IN THE ELEMENTS OF A PIPE ROBOT.  L. RAGULSKIS1, K. RAGULSKIS2, A. BUBULIS3, B. SPRUOGIS4, V. MIŠTINAS5, A. MATULIAUSKAS6 
 
 
 VIBROMECHANIKA.  JOURNAL OF  VIBROENGINEERING.  JUNE   2008,   VOLUME  10,  ISSUE  2,  ISSN 1392-8716 
 
212 
where s is the boundary line of the cross-section of the 
flow, [B] is the matrix of the derivatives of the shape 
functions (the first row with respect to x and the second 
row with respect to y), [M] is the row vector of the shape 
functions on the boundary of the cross-section. 
 The loading vector has the form: 
 
{ } [ ]∫ 




∂
∂
−= ,dxdy
z
pgNF T ρ   (13) 
 
where [N] is the row vector of the shape functions in the 
cross-section of the flow. 
 The mass flow rate is found: 
 
( )∫= ,, dxdyyxwQ ρ    (14) 
 
where ( )yxw ,  are calculated from the obtained solution 
of the system of linear algebraic equations by using the 
shape functions of the appropriate finite elements. 
 Cross-section is assumed to be a circle and one 
fourth of it is analyzed. The characteristics of the fluid: 
 
 ,4004. cP
mms
g
==µ     ,04. 2 smm
g
=α   
 
,001. 3
mm
g
=ρ         .1 22 smm
gg
z
p
=+
∂
∂
− ρ  
 
 Radius of the cross section is 10 mm. The obtained mass 
flow rate .
s
g
 254.86
4
=
Q
 
 
 
Conclusions 
 
 
The dynamics of a fluid in the element of a pipe 
robot is investigated. 
Vibrations of ideal compressible fluid are 
analyzed by taking into account the constraint of non-
rotation by the penalty method using reduced order 
numerical integration of the penalty term. 
Graphical representation of rotations at the points 
of reduced order numerical integration is proposed for 
validation of the calculated eigenmodes. This graphical 
representation shows the quality of satisfaction of the 
constraint of non-rotation over the whole structure. 
The mass flow rate of the fluid through various 
cross-sections is determined. This is important in the 
design of separating elements between the different 
compartments of a pipe robot. 
The obtained results are used in the process of 
design of the elements of a pipe robot. 
 
 
References 
 
 
[1] Iershov N. F., Shahvierdi G. G. The Method of Finite 
Elements in the Problems of Hydrodynamics and 
Hydroelasticity. Leningrad: Sudostroienie, 1984. 
[2] Eibl J., Stempniewski L. Dynamic analysis of liquid filled 
tanks including plasticity and fluid interaction - earthquake 
effects. Shell and Spatial Structures: Computational Aspects, 
Proceedings of the International Symposium. July 1986. 
Leuven, Belgium, Berlin: Springer-Verlag, 1987, p. 261-269. 
[3] Reddy J. N. Applied Functional Analysis and Variational 
Methods in Engineering. New York: McGraw-Hill, 1986. 
[4] Bathe K. J. Finite Element Procedures in Engineering 
Analysis. New Jersey: Prentice-Hall, 1982.  
[5] Rao S. S. The Finite Element Method in Engineering. 
Oxford: Pergamon Press, 1982. 
[6] Connor J. J., Brebbia C. A. Finite Element Techniques for 
Fluid Flow. Leningrad: Sudostroienie, 1979. 
[7] Zienkiewicz O. C. The Finite Element Method in 
Engineering Science. Moscow: Mir, 1975. 
[8] Matuliauskas A., Mištinas V., Spruogis B., Ragulskis K. 
Pipe crawling robots with vibratory drives. Journal of 
Vibroengineering 4(5). Vilnius, 2000. P. 9-14. 
 


JVE International

https://www.extrica.com/article/10192/pdf

Investigation of dynamics of fluid in the elements of a pipe robot

Abstract

Similar works

Full text

Available Versions

Journal of Mechatronics and Artificial Intelligence in Engineering

Journal of Mechanical Engineering, Automation and Control Systems

Journal of Vibroengineering

JVE International