universitatea "constantin brâncuşi" din târgu-jiu exista/4.pdf · 2014-06-11 ·...

Fiabilitate si Durabilitate - Fiability & Durability No 1/ 2014 Editura “Academica Brâncuşi” , Târgu Jiu, ISSN 1844 – 640X

1

CONTENTS

Pag.

1. Iulian POPESCU, Ludmila SASS - MODELLING THE MOVEMENT OF MECHANISMS WITH

THREE DYADS OF RRT TYPE. GENERAL CASE

3

2. Ludmila SASS, Iulian POPESCU - MODELLING THE MOVEMENT OF MECHANISMS

WITH THREE DYADS OF RRT TYPE. PARTICULAR CASES

10

3. Liliana LUCA, Iulian POPESCU - KINEMATICS OF A SCISSORS MECHANISM 18

4. Ion TĂTARU, Cosmin-Mihai MIRIŢOIU, Dan ILINCIOIU - THE VALIDATION OF SOME

EXPERIMENTAL RESULTS USING A NUMERICAL METHOD WITH 3D MESHING ELEMENTS

27

5. Mariana PǍTRAŞCU, Doina TǍRǍBUŢǍ, Simona IONESCU, Constantin D. STǍNESCU -

RESEARCH ON EXTRACTION PIPES OF DEWAXING PROBES

33

6. Mariana PǍTRAŞCU, Doina TǍRǍBUŢǍ, Simona IONESCU, Constantin D. STǍNESCU -

RESEARCH ON EQUIPMENT FOR MINING EQUIPMENT, DEWAXING PROBES

43

7. Cristian PIRGHIE, Ana-Camelia PIRGHIE - CHARACTERIZING THE BEHAVIOR OF THE

LUBRICANT FILMS USING MOLECULAR DYNAMICS SIMULATIONS

49

8. Sebastian Marian ZAHARIA, Cristin Olimpiu MORARIU - STATISTICAL PROCESSING OF

CENSORED DATA UNDER ACCELERATED RELIABILITY TESTING FOR RADIAL BALL

BEARING

57

9. Marin NEACSA, George ADÎR, Victor ADÎR, Ancuta ADÎR - DYNAMIC STUDY OF THE R-RTT

MECHANISM ASSISTED BY AUTODESK INVENTOR

64

10. Stan Marius - THE FAILURE MODES AND THEIR REMEDIATION PROGRESSIVE CAVITY

PUMPS USED IN OIL PRODUCTION

71

11. Gheorghe MARC, Maria Loredana BOCA - USING LOGIC PROGRAMMING FOR IMPROVE

AND INCREASE THE RELIABILITY OF TOOLS AND EMBEDDED MACHINE TO AVOID SOME

“MISSION CRITICAL „ IN FLEXIBLE MANUFACTURING LINES

78

12. Anastase PRUIU, Traian FLOREA, Daniel MĂRĂȘESCU, Adriana SPORIȘ -

CONSIDERATIONS IN DETERMINING ANALYTIC GRAPHICS FUNCTIONAL PARAMETERS OF

MARINE PROPULSION ENGINES

84

13. Anastase PRUIU, Traian FLOREA, Daniel MĂRĂȘESCU,Adriana SPORIȘ - ABOUT ENGINE

ROOM VENTILATION ON MERCHANT VESSELS

91

14. Ion BULAC , MATHEMATICAL MODEL FOR DETERMINING KINEMATIC PARAMETERS OF

THE CARDAN JOINT MECHANISM WITH TECHNICAL (GEOMETRICAL) DEVIATIONS

97

15. Ion BULAC, THE NUMERICAL STUDY OF THE INFLUENCE OF TECHNICAL

(GEOMETRICAL) DEVIATIONS OVER THE KINEMATIC PARAMETERS OF THE CARDAN JOINT

MECHANISM

103

16. Răzvan Bogdan ITU, Iosif DUMITRESCU, Vilhelm ITU - CORRELATING 2K-52MU CUTTING

AND LOADING MACHINE WITH TR-5 SCRAPER CONVEYER

110

17. Răzvan Bogdan ITU, Iosif DUMITRESCU, Vilhelm ITU - STUDY OF STABILITY OF 2K-52MU

CUTTING-LOADING MACHINE ON TR-5 CONVEYER IN FACES WITH INDIVIDUAL SUPPORT

120

18. Mădălina DUMITRIU - INFLUENCE OF THE SUSPENSION PARAMETERS UPON THE

HUNTING MOVEMENT STABILITY OF THE RAILWAY VEHICLES

129

19. Mădălina DUMITRIU - THE DYNAMIC BEHAVIOUR OF THE RAILWAY VEHICLES IN

CROSSING AN ISOLATED NIVELMENT DEFECT

137

20. GEORGE Novac - CRANK WEB DEFLECTIONS OF MARINE DIESEL ENGINES 145

21. GEORGE Novac - EXHAUST VALVE WEARS OF MARINE DIESEL ENGINES 151

22. Traian FLOREA, Ligia-Adriana SPORIȘ, Corneliu MOROIANU, Traian Vasile FLOREA,

Anastase PRUIU - GRAPHO-ANALYTICAL METHOD FOR CALCULLATING IRREVERSIBILITY

PROCESSES WITH FINITE SPEED

157


2

23. Traian FLOREA, Corneliu MOROIANU, Traian Vasile FLOREA , Ligia Adriana SPORIȘ,

Anastase PRUIU - THE COEFFICIENT OF REGENERATIVE LOSSES IN STIRLING MACHINES

165

24. Adina TĂTAR - GAUSSIAN MODEL 174

25. Corneliu MOROIANU, Ligia Adriana SPORIȘ, Traian FLOREA - THE DETERMINATION OF

THEORETICAL COMBUSTION TEMPERATURE OF HEAVY FUELS CONSIDERING THE

DISSOCIATION OF WATER VAPOURS FROM THE BURNING GASES

178

26. Corneliu MOROIANU, Traian FLOREA, Ligia Adriana SPORIȘ - MATHEMATICAL MODEL

FOR BURNING THE MARINE DIESEL FUEL DROP IN A HOT OXIDIZING ENVIRONMENT

183

27. Ligia-Adriana SPORIȘ, Traian FLOREA, Corneliu MOROIANU - ASUPRA UNUI SISTEM

KOROVKIN ÎNTR-UN CON DE FUNCŢII PONDERATE

190

28. Ligia-Adriana SPORIȘ, Corneliu MOROIANU, Traian FLOREA - ASUPRA UNOR ASPECTE

CALITATIVE ALE CONVERGENŢEI ÎN SPAŢII LINIARE ORDONATE TOPOLOGICE

193

29. Monica BÂLDEA, Mihaela ISTRATE - PROGRAM FOR THE CALCULATION OF GEOMETRIC

OPTIMIZATION OF PRIMARY SEALS

195

30. Iuliana Carmen BĂRBĂCIORU - A NOTE ON (α,β)-CUT IN INTUITIONISTIC FUZZY SETS

THEORY

200

31. Iuliana Carmen BĂRBĂCIORU , Viorica Mariela UNGUREANU - LYAPUNOV TYPE

OPERATORS ON ORDERED BANACH SPACES

207

32. Mădălina Roxana BUNECI - RANDOMLY GENERATED SUBGROUPOIDS OF X×Z×X 213

33. Mădălina Roxana BUNECI - USING MAPLE FOR VISUALIZATION OF TOPOLOGICAL

SUBGROUPOIDS OF X×Z×X

220

34. Elisabeta Mihaela CIORTEA, Mihaela ALDEA - ASPECTS OF A LINEAR PROGRAMMING

MODEL DEDICATED TO THE TRANSPORT SYSTEM 227

35. Miodrag IOVANOV - AN EXTREMAL PROBLEM FOR UNIVALENT FUNCTIONS 234

36. Constantin P. BOGDAN, Olimpia PECINGINA - BOOLEAN NORMED ALGEBRAS 240

37. Constantin P. BOGDAN, Olimpia PECINGINA - EXTENSION OF AN ADDITIVE FUNCTIONS

NUMARABILE

247

38. Olimpia PECINGINA Constantin P. BOGDAN, - ABN AND METRIC STRUCTURES SPACES OF

MEASURES

254

39. Cătălina IANĂŞI - INCREASING RESISTANCE OF STRUCTURAL ELEMENTS WITH CFRP

REINFORCEMENTS

261


3

MODELLING THE MOVEMENT OF MECHANISMS

WITH THREE DYADS OF RRT TYPE. GENERAL CASE

Professor Iulian POPESCU,

University of Craiova, member of Romanian Academy of Technical Sciences,

[email protected]

Associate Professor PhD Ludmila SASS,

University of Craiova, Faculty of Mechanics, [email protected]

Abstract. The paper deals with the modeling of mechanisms with three dyads of type RRT and

rotating leading element. Generated trajectories are provided, along with the sliders’ laws of

motions. The mechanism’s operational angular range is limited to the subinterval 00…180

0

because the Grashof conditions are disobeyed. Diagrams depicting the variation of coordinates

corresponding to the points presenting practical interest are presented.

Keywords: mechanism with three RRT dyads, trajectories, movement laws

1. INTRODUCTION

The movement of one of the 1,234,620 possible mechanisms with three dyads dyads [2]

is modeled, in order to reveal its kinematics. It relies on three RRT dyads. As far as we know,

no specialty studies on this mechanism were issued. Instead, mechanisms with 5 and 7 bar

presented interest for scientists. For example [3] presents the structured synthesis of the afford

mentioned mechanism, based on orthogonal trajectories. Distortions and couplers‘ mobility

are considered. [1] includes studies on many mechanisms with rotation couplers and glides,

used for research dedicated spatial vehicles. Further on we will study a mechanism based on

three dyads of RRT type.

2. THE STUDIED MECHANISM

The movement of the mechanism depicted by Fig. 1 was modeled.

Fig. 1. The studied mechanism

mailto:[email protected]


4

It consists of a leading element with a rotation movement AB, the first dyad BCD,

linked to the elements 1 and 0 (base), the second dyad EFG of type RRT connected to the 2-

nd and 3-rd element and the 3-rd RRT dyad HKL, connected to the 4-th and 5-th elements.

The mechanism‘s structural formula [2] is consequently: R-RRT-1+0- RRT-2+3-RRT-4+5.

The mobility degree is: M=3n – 2C5-C4=3.7-2.10=1.

Using the contours‘ method, the following equations can be written:

sin.

cos.

AByy

ABxx

AB

AB (1)

sin.sin.

cos.cos. 3

DCyBCyy

DCSBCxx

DBC

BC (2)

sin.

cos.

BEyy

BExx

BE

BE (3)

sin.sin.sin.

cos.cos.cos.

2

23

GFSyEFyy

GFSSEFxx

DEF

EF (4)

cos.cos.

sin.sin.

3 GFSEFx

GFyEFytg

E

DE

(5)

1 (6)

sin.

cos.

EHyy

EHxx

EH

EH (7)

2 (8)

sin.

cos.

2

23

Syy

SSx

DG

G (9)

sin.sin.sin.

cos.cos.cos.

5

5

LKSyHKyy

LKSxHKxx

GHK

GHK (10)

cos.cos.

sin.sin.

LKxHKx

LKyHKytg

GH

GH

(11)

3. RESULTS

The dimensions considered by our study are: Ax =18; Ay =22; AB=64; BC=81; DC=70;

EF=58; GF=90; HK=47; LK=38; yD=15; EH=26; BE=40; HM=24; 1 =75; 2 =115; =98;

1 .

Fig. 2 depicts the mechanism‘s position for =70o. Fig. 3 depicts the mechanism‘s

subsequent positions, considering that the element AB does not perform full rotations owing

to the adopted sizes which do not meet the Grashof conditions.


5

Fig. 2. The mechanism’s position for =70

o

Fig. 3. The mechanism’s subsequent positions

Fig. 4, presenting the trajectories of points B and C, reveal that B describes only a part

from the circle and C‘s race is small.

Fig. 4. The trajectoris of points B and C

The trajectories of points E and D are given in Fig. 4. D moves along a line whilst the

trajectory followed by E is an open rod-type curve.


6

Fig. 5. The trajectories of points E and D

The trajectories described by the points G (in the right side) and F (in the left side) are

depicted by Fig. 6. Both fall in the category of rod-type curves with loops.

Fig. 6. The trajectories described by the points G and F

Fig. 7 depicts the trajectories of points L, H and K. They are similar, but shifted.

Fig. 7. The trajectories of points L, H and K

Fig. 8 is used to reveal a comparison between the trajectories of the points K and M

from the element EF. They are similar, the one corresponding to M being left-shifted to that

corresponding to K.

Fig. 8. A comparison between the trajectories of the points K and M


7

The variations of traces S3 and S2 are provided by Fig. 9. One can see that the

mechanism can operate only for angles within the range 0o…180

o. For angles >180

o

segments appear in the diagram. The program used for simulation joins the ranges‘ limits by

means of lines.

Fig. 9. The variations of traces S3 and S2

Fig. 10 present the variation of the trance S5 with respect to . Also is revealed that the

mechanism cannot operated for o180

Fig. 10. The variation of the trance S5 with respect to

The variations for the coordinates of points E and G, at the input of the 2nd

dyad, given

by Fig. 11, represent a new proof for the mechanism‘s blocking when >180o. The curves

are normal for the rest of the values.


8

Fig. 11. The variations for the coordinates of points E and G, at the input of the 2nd dyad

Similarly in Fig. 12 one presents the variations corresponding to the coordinates of the

points H and L from the input of the 3-rd dyad. Conclusions identical to those from above can

be drawn.

Fig. 12. The variations corresponding to the coordinates of the points H and L

from the input of the 3-rd dyad

The diagrams for the coordinates of the points F, K and M (Fig. 13) reveal the already

mentioned blocking for >180o. An interesting aspect is related to the similarity of the

curves xi and yi respectively.

Fig. 13. The diagrams for the coordinates of the points F, K and M

0.0 100. 200. 300. 400.

Fi [ grd]

-50.

0.0

50.

100.

150.

200.

X EY EX GY G

0.0 100. 200. 300. 400.

Fi [ grd]

-50.

0.0

50.

100.

150.

X HY HX LY L

0.0 100. 200. 300. 400.

Fi [ grd]

-100.

-50.

0.0

50.

100.

150.

X FY FX KY KX MY M


9

4. CONCLUSIONS

- Studies on the mechanism R-RRT-1+0- RRT-2+3-RRT-4+5 were performed.

- Owing to the disobeying of Grashof conditions, the mechanism can operate only for

=0o…180

o.

- The curves generated by the points of interest are open rod curves. Although being

characterized by a high degree, they are not spectacular.

- The variations of the glides‘ traces variations were also represented. Normal diagrams were

obtained.

REFERENCES

1. Brink Jeffrey S. – Reverse kinematic analysis and uncertainty analysis of the space suittle

aft propulsion system (APS) pod lifting fixtura. A thesis, University of Florida, 2005.

2. Popescu Iulian. – Mecanisme. Noi algoritmi şi programe, Reprografia Universităţii din

Craiova, 1997.

3. Shih -Hsi Tong - Design of High-Stiffness Five-Bar and Seven-Bar Linkage Structures by

Using the Concept of Orthogonal Paths. J. Mech. Des. 128(2), pp. 430-435, Jun. 23, 2005.

http://mechanicaldesign.asmedigitalcollection.asme.org/searchresults.aspx?q=Shih%20-Hsi%20Tong&p=1&s=19&c=0&t=


10

MODELLING THE MOVEMENT OF MECHANISMS

WITH THREE DYADS OF RRT TYPE. PARTICULAR CASES

Associate Professor PhD Ludmila SASS,

University of Craiova, Faculty of Mechanics, [email protected]

Professor Iulian POPESCU,

University of Craiova, member of Romanian Academy of Technical Sciences,

[email protected]

Abstract. The paper deals with the modeling of mechanisms with three dyads of type RRT and

rotating leading element. Generated trajectories are provided, along with the sliders’ laws of

motions. Diagrams depicting the variation of coordinates corresponding to the points presenting

practical interest are presented. Two particular cases related to certain sizes of some elements are

studied.

Keywords: mechanism with three RRT dyads, particular cases, trajectories, movement laws

1. INTRODUCTION

This paper deals with the general case of the mechanism with three RRT dyads. [2]

provides a study of a mechanism with 6 bars of a press with certain elements having

adjustable lengths. The kinematics of the mechanism obtained through re-sizing are studied.

[3] presents the structural synthesis of the mechanisms with 5 and 7 bars, based on orthogonal

trajectories, when distortions and couplers‘ mobility are considered. In order to know the

kinematics possibilities of certain variants of the mechanism with three RRT dyads (namely

when some elements are zero sized), the movements of certain particular cases are modeled.

2. STUDIED MECHANISM

The starting point consist in the general case of the studied R-RRT-1+0- RRT-2+3-RRT-

4+5 mechanism [1], depicted by Fig. 1.

Fig. 1. Studied mechanism




11

Particular cases were obtained varying the sizes of certain elements.

3. RESULTS

The initial values for mechanism‘s sizes are: xA=18; yA=22; AB=64; BC=81; DC=70;

EF=58; GF=90; HK=47; LK=38, yD=15; EH=26; BE=40; HM=24; 1=75; 2=115; =98;

= + 1.

The first particular case corresponds to the case when AB is a rod. Repeated tests

yielded the following modifications: AB=30, DC=35. Fig. 2 depicts a mechanism obtained for

=700. One can see that the point G goes downward the rod from D.

Fig. 2. A mechanism obtained for =70

o

Fig. 3 depicts successive positions of the mechanism, revealing that the mechanism is

operational for the entire cycle.

Fig. 3. Successive positions of the mechanism

Fig. 4 reveals that B moves around a full circle and the trace of C is short.


12

Fig. 4. B moves around a full circle and the trace of C is short

The curves from Fig. 5 present the variations of the coordinates of B specific to a circle.

xC has a nonlinear variation whilst yC is constant, its translation being imposed by the glide

from D. The trajectories of E and G, at the input of the 2- nd dyad, represent rod-type curves

and are given by Fig. 6.

Fig. 5. The variations of the coordinates of B specific to a circle

Fig. 6. The trajectories of E and G

The variations for the coordinates of these points are given by Fig. 7. The corresponding

curves are continuous, with symmetric features.

0.0 100. 200. 300. 400.

Fi [ grd]

-50.

0.0

50.

100.

150.

X BY BX CY C


13

Fig. 7. The variations for the coordinates of the points E and G

The trajectories depicted by Fig. 8 correspond to the points H and L, from the input of

the 3-rd dyad, whilst Fig. 9 presents the variations corresponding to their coordinates.

Fig. 8. The trajectories described by the points H and L

The trajectory of L is a rod-type curve, egg-shaped, whilst the trajectory of H reveals

non-symmetric features and some disturbances in the up-right corner. The curves from Fig. 9

present symmetric features.

Fig. 9. The variations for the coordinates of the point L

0.0 100. 200. 300. 400.

Fi [ grd]

0.0

25.

50.

75.

100.

125.

X EY EX GY G

0.0 100. 200. 300. 400.

Fi [ grd]

-20.

0.0

20.

40.

60.

80.

X HY HX LY L


14

The trajectories described by the points F, K and M are depicted by Fig. 10. The up

curve exhibits the same abnormal behavior as the above discussed case for the trajectory of K.

The curves have a high degree and their shape make them fall into the same category with

similar rod-type curves corresponding to other mechanisms.

Fig. 10. The trajectories described by the points F, K and M

Fig. 11 presents the variations for the coordinates of the points F, K and M. One can see

the partial overlapping of the curves xK and xM and a quasi-parallelism for the curves yK and

yM. It means that the points from the element HK have similar trajectories.

Fig. 11. The variations for the coordinates of the points F, K and M

The traces S3, S2 and S5 present the variations from Fig. 12. One can notice the trace S5

with some irregularities for =0o…170

0. Traces with similar shapes are sometimes

associated to simpler mechanisms.

0.0 100. 200. 300. 400.

Fi [ grd]

-50.

0.0

50.

100.

150.

X FY LX KY KX MY M


15

Fig. 12. The traces S3, S2 and S5

Another particular case can be obtained when considering zero lengths for the elements

3, 5 and 7. Actually the sizing DC=0 cannot be accomplished because the sliding route of G

should vanish and a condition GF=0 should involve no guidance for L. The only possible

alternative is therefore KL=0. The rest of sizes are preserved.

Fig. 13 depicts the mechanism obtained for =700. Fig. 14 presents the subsequent

positions of this mechanism.

Fig. 13. The mechanism obtained for =70

o

Fig. 14. The subsequent positions of the mechanism

The trajectories for the points B, C, D, E, G, H and L will be identical to those presented

above, the only different trajectories corresponding to the points K and M. They are depicted

by Fig. 15 and are rod-type curves similar to other familiar ones.

0.0 100. 200. 300. 400.

Fi [ grd]

0.0

25.

50.

75.

100.

125.

150.

S3S2S5


16

Fig. 15. The trajectories described by the points K and M

The variations of the coordinates of K and M are depicted by Fig. 16. Symmetries and

similarities are detected for the curves corresponding to xK and xM, whilst some non-

uniformities are included by the yK and yM curves.

Fig. 16. The variations of the coordinates of K and M

The shape for the trace S5 is depicted by Fig. 17. It has symmetries and a jump for

>1800. The movement low is interesting and very few mechanisms describe it.

Fig. 17. The shape for the trace S5

4. CONCLUSIONS

- Studies were made concerning the modeling of the movement corresponding to the R-RRT-

1+0- RRT-2+3-RRT-4+5 mechanism for particular cases;

- When searching for particular cases, two of the elements linked to glides cannot be reduced

to zero length, because other glides should remain without guides;

- The first particular case assumed the determination of mechanism‘s sizes such as to make

the leading element to describe full rotations;

- The second particular case involved a null length for an element;

0.0 100. 200. 300. 400.

Fi [ grd]

-25.

0.0

25.

50.

75.

100.

125.

X KY KX MY M

0.0 100. 200. 300. 400.

Fi [ grd]

20.

22.5

25.

27.5

30.

32.5

S5


17

- Trajectories were build and variations for the couplings of interest were provided;

- The resulted trajectories are rod-type curves with high degree whose shape are common,

excepting two cases where some abnormal behaviors were detected;

REFERENCES

1. Popescu Iulian – Mecanisme. Noi algoritmi şi programe, Reprografia Universităţii din

Craiova, 1997.

2. Ren-Chung Soong – An adjustable six-barmechanism with variable input speed for

mechanical forming presses. Transactions of the CSMEIde fa SCGM Vol. 32, No. 3-4,2008.

3. Shih -Hsi Tong - Design of High-Stiffness Five-Bar and Seven-Bar Linkage Structures by

Using the Concept of Orthogonal Paths. J. Mech. Des. 128(2), pp. 430-435, Jun. 23, 2005.

http://mechanicaldesign.asmedigitalcollection.asme.org/searchresults.aspx?q=Shih%20-Hsi%20Tong&p=1&s=19&c=0&t=


18

KINEMATICS OF A SCISSORS MECHANISM

Prof. PhD. Liliana LUCA, Constantin Brancusi University of Targu-Jiu,

[email protected]

Prof. PhD. Iulian POPESCU, University of Craiova,

[email protected]

Abstrac:. We study the kinematics of a scissors mechanism composed of two conductive elements with related

movements and a RTR type dyad. They are written the relations based on contours method and they are given

the results in tables and diagrams.

Keywords: mechanisms for scissors, kinematic analysis, two conductive elements.

1 . INTRODUCTION

The mechanisms from the scissors of debitting metals have been studied over time by

various methods. Many of them were built empirically, on summary calculations. Computers

and new analytical analysis methods allow more detailed studies, which led to improving the

performance of these mechanisms. In the literature, studies continue to show this theme .

Thus, in [1] it is studied the kinematics of a scissors mechanism with a triad, which is

intended for cutting of steel products. They are given the analytical relations based on

contours method and numerous resulted diagrams. In a doctoral thesis [2] they are studied in

detail the mechanisms that ensure shear cutting branches of trees in order to clean them. They

are studied different variants of mechanisms, by modeling them . A detailed dynamic study on

a shear mechanism is given in [3]. The mechanism consists of two dyads .They are calculated

the positions, velocities , accelerations and reactions of couplings .

2. INITIAL DATA

We left from the kinematic scheme of a mechanism given in [4] and shown in Fig. 1.

Items 1 and 4 are both leading, with movements linked by a gear, cog belts, chain

Galle chain or other system. In E and F points are the tips of two knives that run the shear,

point F being on the element 2 which is having a flat movement and point E belongs to

element 3, which also has a flat movement. The symmetry properties of the mechanism allow

for a position of the mechanism the knives to cut the blank (sheet) 5.

3. THE MECHANISM STRUCTURE

The structural diagram of the mechanism is given in Fig. 2. The mobility degree is:

M=3n-2C5-C4=3.4-2.5=2, the mechanism having two conductive elements and a BCC dyad of

RTR type.


19

Fig. 1 Fig. 2

4. THE MECHANISM KINEMATICS

The correlation between angles and is obtained (Fig. 3), through the relations:

Fig. 3


20

θ - α= 90 (1)

Ψ+ α= 270 (2)

Ψ+ θ - 90=270 (3)

Ψ=360- θ (4)

For the kinematic analyze, they are written the relations:

xB =xA+ABcosθ (5)

yB =yA+Absinθ (6)

xC =xD+CDcosψ (7)

YC =YD+CDsinψ (8)

S=yC-yB (9)

S2=yB + BF (10)

S3=yC-CE (11)

xF=xB-a (12)

yF=yB+BF (13)

xE=xC-a (14)

yE=yC-CE (15)

We have adopted the following initial values:

XA = 400: XD = 400: YD = 700: BF = 50: CE = BF: AB = 300 CD = AB: A = BF / 2.

5.THE OBTAINED RESULTS

In the FIG. 4 it is shown the mechanism for = 120 degrees. The image is similar to

that of FIG. 1, so the program is done correctly. The two points of the figure are the E and F

points, that is the tips of the knives for this position.

The successive positions of the mechanism are shown in Fig. 5 for .0...120


21

Fig. 4 Fig. 5

It is noted that while element 1 rotates clockwise, item 4 rotates counterclockwise.

The figure also shows the trajectories of E and F points, thus the trajectories of knives peaks.

At a full rotation they result the successive positions of Fig. 6.

The trajectories of tops knives are circles and race S, meaning the distance between C

and B, is variable (fig. 7 for .0...120 ).

Fig. 6 Fig. 7


22

The complete trajectories of E and F points are shown in Fig. 8, and they are circles

with centers that are different of A and D (Fig. 9), as E and F are located close to the C and B

but offset to their left. In the figure also appear and C and B circles.

Tangent circles of FIG. 9 are described by E and F.

Fig. 8 Fig. 9

The variations of S, S2 and S3 races to the position of the mechanism are shown in

Fig. 11. It is observed that the minimum S, S2 and S3 races are equal to 90 degrees.

0.0 100. 200. 300. 400.

Fi [ grd]

-500.0

0.0

500.

1000.

1500.

SS2S3

Fig. 10


23

Referring to FIG. 11, it is shown that diagrams for coordinates variation of the F and B

points are very close, the shifts being determined by the fact that E and F are also staggered

relative to the BC element.

0.0 100. 200. 300. 400.

Fi [ grd]

-400.0

-200.0

0.0

200.

400.

600.

800.

X BY BX FY F

Fig. 11

The same observation applies to the E and C coordinates in Fig. 12.

0.0 100. 200. 300. 400.

Fi [ grd]

0.0

200.

400.

600.

800.

1000.

X CY CX EY E

Fig. 12


24

For the drawer, it is interesting the contact area between the blade tips. To this, they

were plotted in FIG. 13 the coordinates of E and F points, observing that for of about 90

degrees, YE and YF curves are tangent, meaning the E and F points coincide, this is the

shearing time.

0.0 100. 200. 300. 400.

Fi [ grd]

-250.0

0.0

250.

500.

750.

1000.

X EY EX FY F

Fig. 13

In Table 1 they are also given the numerical results for this area of the operating cycle of the

mechanism.

Table 1

Fi XE YE XF YF

80 427.0933 354.5575 427.0948 345.4423

81 421.9293 353.6933 421.9307 346.3065

82 416.7509 352.9194 416.7523 347.0804

83 411.5598 352.236 411.5611 347.7638

84 406.3575 351.6433 406.3589 348.3565

85 401.1457 351.1415 401.1471 348.8584

86 395.926 350.7307 395.9273 349.2692

87 390.6999 350.4111 390.7012 349.5889

88 385.4688 350.1827 385.4702 349.8172

89 380.2347 350.0457 380.2361 349.9543

90 374.999 350 375.0003 350

91 369.7633 350.0457 369.7647 349.9544

92 364.5291 350.1828 364.5305 349.8173

93 359.2982 350.4112 359.2996 349.5889

94 354.072 350.7308 354.0734 349.2692

95 348.8523 351.1417 348.8537 348.8584


25

96 343.6405 351.6436 343.6419 348.3566

97 338.4383 352.2363 338.4396 347.7639

98 333.2472 352.9197 333.2485 347.0805

99 328.0687 353.6937 328.0701 346.3066

100 322.9046 354.5578 322.9059 345.4424

It appears that indeed, at = 90, YE = YF.

It has been enlarged the diagram of FIG. 13 in the area of interest, finding fig. 14 and

15, where it is clear the tangency of the two circles and the equality of the two ordinates.

80. 85. 90. 95. 100.

Fi [ grd]

300.

325.

350.

375.

400.

425.

450.

X EY EX FY F

Fig. 14

80. 85. 90. 95. 100.

Fi [ grd]

340.

345.

350.

355.

X EY EX FY F

Fig. 15


26

6. CONCLUSIONS

- The studied mechanism satisfies the condition of blank shear.

- Although blades are having a flat movement, they have trajectories that become tangent

when shearing.

- From constructive point of view, these knives can be even on BC element, without the

offset.

- The mechanism is cleverly designed.

REFERENCES

[1]. Berghian, A. B. , Vasiu, Th. , Kinetics study on laboratory model of the mechanisms of

parallel gang sheoars’ type assigned for cutting metallurgical products. Journal of

Engineering annals of Faculty of Engineering Hunedoara, tome V, 2007, fasc. 3.

[2]. Maglioni, C. ,Analysis of reciprocating single blade cutter bars. Tezi di Dottorato.

Universita di Bologna, 2009.

3]. Tyagi, R. K., Verma, M., Borah, S. , Dynamic analysis of a shaper machine cutting tool

and crank pin. Journal of Enviromental Science, Computer Science and Engineering &

Technology, sept.- nov. 2012, vol. 1 no.3, pp. 372-380.

[4]. Kojevnikov, S. N., Esipenko, Ia. I., Raskin, Ia. M. , Mehanizmî. Sparvocinâe posobie. Izd.

Maşinostroenie, Moskva, 1976.

[5] Popescu, I., Luca, L., Cherciu, M., Structura şi cinematica mecanismelor. Aplicaţii. Editura

Sitech, Craiova, 2013.


27

THE VALIDATION OF SOME EXPERIMENTAL RESULTS USING A

NUMERICAL METHOD WITH 3D MESHING ELEMENTS

1Eng. Ion TĂTARU, University of Craiova, Faculty of Mechanics, Department of Applied

Mechanics and Civil Constructions, Calea Bucuresti Street, no. 107, Craiova,Code 200512,

Romania, [email protected] 2Assistant Phd. Eng. Cosmin-Mihai MIRIŢOIU, University of Craiova, Faculty of

Mechanics, Department of Vehicles, Transports and Industrial Engineering, Calea Bucuresti

Street, no. 107, Craiova,Code 200512, Romania, [email protected] 3Prof. phd. eng. Dan ILINCIOIU, University of Craiova, Faculty of Mechanics, Department

of Applied Mechanics and Civil Constructions, Calea Bucuresti Street, no. 107, Craiova,Code

200512, Romania, [email protected]

Abstract. In this paper we present the validation of some experimental results obtained in [1], where there was

presented a device for bars and plates bending which works with strain gauges attached, by using a numerical

method – the finite element analysis. There will be used the same loading variants as in [1]. The structure

analysis was made in Ansys with two types of meshing techniques: map mesh with Brick 8 Node 45 finite element

and auto mesh with Tet 10 Node 187. In the end, we will make comparisons between the used methods and

extract the errors that appear.

Keywords: metallic structure, finite element analysis, mesh, brick elements, tetrahedral elements

Contents:

1. Introduction

2. The previously studied problem

3. Finite element analysis. Meshing and loading cases

4. Conclusions

5. Acknowledgement

1. INTRODUCTION In this paper, starting from the experimental results determined in Miriţoiu (2012)[1],

we will present a finite element analysis validation method with three dimensional elements used for meshing: map mesh with Brick 8 Node 45 finite element and auto mesh with Tet 10 Node 187. In the end we will make comparisons between the results and determine the errors.

According to Călbureanu (2011)[2], the finite element method has appeared and rapidly developed because of the necessity to have a powerful, quick and simple method to solve the complex stress and displacement problems from various engineering areas, like: mechanics, aeronautics, civil engineering, nuclear engineering and marine engineering. This method can be successfully applied also for solving of some problems like: heat transfer, dynamic analysis, fluid mechanics, and so on.

In Quin (2010)[3] the finite element method was used for a complete modeling and calculation for steady state viscoelastic stress analysis. It was made an algorithm formulation of one-dimensional case and extended to a generalized one. The numerical examples were given for two cases: one-dimensional and two-dimensional. There are studied the effects of


28

the load speed, material properties and pressure distributions at the contact surface. In Karunakaran (2011)[4] is presented a finite element procedure for thermal analysis in pulsed current gas tungsten arc welding (abbreviated PCGTAW) of Az 31B magnesium alloy sheets. The studied material can be used in aircraft, automobile and high-speed train components. The software Ansys was used for finite element analysis and the results obtained were compared with experimental ones. The conclusion of the study was that the finite element analysis using Ansys can be effectively used to model PCGTAW process for finding temperature distribution.

In El-Asfoury (2009)[5], the finite element analysis was used for a static and dynamic study of pelvic bone. The bone was subjected to quasi-static and dynamic loading conditions simulating the effect of both weight gain and impact.

The mechanism of damping in welded structures was studied in Singh (2010)[6]. The

study emphasized the theoretical investigation of slip damping in layered and jointed welded

cantilever structures using finite element approach. The developed finite element model

shows that the damping capacity of such structures is influenced by a number of vital

parameters, such as: pressure distribution, kinematical coefficient of friction and micro-slip at

the interfaces, amplitude, vibration frequency, specimen length and thickness.

2. THE PREVIOUSLY STUDIED PROBLEM In [1], a device for metallic structures (bars, plates) bending and stress measurement

was studied. The studied metallic structure is presented in fig. 1 and the used device is presented in fig. 2. According to [1], the device works with strain gauges (shown in fig. 3 and 4).

Fig. 2. The studied device [1]

Fig. 1. The studied metallic structure [1]

Fig. 3. The first half bridge [1]

Fig. 4. The second half bridge [1]


29

Two loading variants were considered: variant 1- P= 701,55 daN, variant 2- F= 507,56 daN. The stresses obtained, for each loading case, are shown in fig. 5 and 6.

3. FINITE ELEMENT ANALYSIS. MESHING AND LOADING CASES For the first loading case, there was used a mapped mesh, with Brick 8 Node 45 finite

elements (fig. 7). The stress distribution for the whole structure is presented in fig. 7 and in fig. 8 the stress distribution in the area of the first half bridge is presented. From fig. 8 it can be seen that the stress distribution from the second half bridge area is almost 0. The stress distribution from the first half bridge is presented in fig. 9 for variant 1. The mesh type in variant 2 (with Tet 10 Node 187 meshing elements) is presented in fig. 10. In fig. 11 we have presented the stress distribution for the whole structure and in fig. 12 the stress distribution in the first half bridge area.

Fig. 5. The stress values in the first Fig. 6. The stress values in the second

loading case [1] loading case [1]

Fig. 7. Mesh type (variant 1) Fig. 8. Stress distribution (variant 1)


30

Important remark: in the area of the second half bridge the stress values are very

small (according to fig. 8. and 11) and will be approximated being zero in the following parts of the paper (the real values are: for variant 1- 0,12∙10

-4 and for variant 2- 0,006736).

Fig. 9. Stress distribution (first half bridge) Fig. 10. Mesh type (variant 2)

Fig. 11. Stress distribution (variant 2) Fig. 12. Stress distribution (first half bridge)


31

4. CONCLUSIONS In the table 1 we have listed the results obtained with the three-dimensional meshing

elements for the considered cases.

Table 1. The results obtained with the finite element analysis

Stress type Value [MPa]

Method Finite element analysis

Loading variants First loading variant Second loading variant

First half bridge 32,8125 69,125

Second half bridge 0,12∙10-4

0,006736

The results obtained with the experimental method from [1] are listed in table 2.

Table 2. Stress results from [1]

Experimental method

Stress type First half bridge Second half bridge First half bridge Second half bridge

Loading variant 1 2 1 2

Stress Value

[MPa] 34,269 0,0042094 72,871 0,0079752

We have determined the errors between the experimental and numerical method with

relation (1).

In (1) we have marked with: ε1 – the stress obtained with the experimental method, ε2

– the stress obtained with the numerical method and with εmax the maximul stress. The errors obtained are listed in table 3.

Table 3. Errors obtained between the experimental and numerical methods

Numerical method/ Experimental method Stress type First half bridge Second half

bridge First half bridge Second half

bridge Loading variant 1 1 2 2

Error [%] 4,25 0 5,141 0

From the table 3 we can extract the next conclusions: - the experimental method from [1] gives similar results like the numerical method, so

the strain gauges were glued well on the structure; - the errors are very small (under 6%); - the stress values obtained from the second half bridge are 0 (because the bars are not

loaded); - the stress values obtained from the first half bridge are different from zero, because

the area is loaded by the forces considered from the two loading variants;

100%max

21

(1)


32

- both methods can be successfully used for metallic structures stress calculus; - the errors are higher when the tetrahedral elements are used, because this type of

elements insert higher errors than the hexahedral ones (used in the first loading variant), fact that was expected before using this type of elements.

4. ACKNOWLEDGEMENT

This work was supported by the strategic grant POSDRU/159/1.S/S/133255, Project

ID 133255 (2014), co-financed by the European Social Fund within the Sectorial Operational

Program Human Resources Development 2007-2013.

REFERENCES Miriţoiu, C., M., (2012) A Simple but Accurate Device and Method Used for Bending and Stress Measurement of Metallic Structures, IOSR Journal of Engineering (IOSRJEN), 2(6), 1334-1339 Călbureanu, M., (2011) Introduction to finite element analysis, Universitaria Publishing House Quin, F., Yu, Y., Rudolphi, T., (2010) Finite Element Modeling of Viscoelastic Stress Analysis under Moving Loads, International Journal of Mechanical and Materials Engineering, 4(1), 226-233 Karunakaran, N., Balasubramanian, V., (2011) Multipurpose Three Dimensional Finite Element Procedure for Thermal Analysis in Pulsed Current Gas Tungsten Arc Welding of AZ 31B Magnesium Alloy Sheets, International Journal of Aerospace and Mechanical Engineering, 5(4), 267- 274 El-Asfoury, El-Hadek, M., A., (2009) Static and Dynamic Three-Dimensional Finite Element Analysis of Pelvic Bone, International Journal of Engineering and Applied Sciences, 5 (5), 315-321 Singh, B., Nanda, B., K., (2010) Mechanism of Damping in Welded Structures using Finite Element Approach, International Journal of Information and Mathematical Sciences, 6(2), 138-142 Ajovalaist, A., Zucarello, B., (2005) Local Reinforcement Effect of a Strain Gauge Installation on Low Modulus Materials, The Journal of Strain Analysis for Engineering Design, 40 (7), 643-653 Atanackovic, T., (2000) Theory of Elasticity for Scientists and Engineers, Published by Birkhauser Boston Avalle, M., Goglio, L., (1997) Static lateral compression of aluminium tubes: Strain gauge measurements and discussion of theoretical models, The Journal of Strain Analysis for Engineering Design, 32 (5), 335-343 Huttelmaier, H., P., Glockner, P., G., (1985) Stresses and displacements due to underground mining using a finite element procedure, Geotechnical and Geological Engineering, 3(1), 49-63 Korayem, M., H., Heidari, A., Nikoobin, A., (2008) Maximum allowable dynamic load of flexible mobile manipulators using finite element approach, The International Journal of Advanced Manufacturing Technology, 36 (5,6), 606-617 Kulkarni, S., D., Kapuria, S., (2007) A new discrete Kirchhoff quadrilateral element based on the third-order theory for composite plates, Computational Mechanics, 39 (3), 237-246 Mao, S., Shi, Z., (2009) High accuracy analysis of two nonconforming plate elements, Numerische Mathematik, 111 (3), 407-443

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33

RESEARCH ON EXTRACTION PIPES OF DEWAXING PROBES

Drd. Eng Mariana PǍTRAŞCU (ANTONESCU); e-mail: [email protected]

Drd.Eng Doina TǍRǍBUŢǍ (ENE);e-mail: [email protected];

Matemat.drd.Simona IONESCU: e-mail: [email protected]

Prof.univ.Emerit. Dr.eng. Constantin D. STǍNESCU e-mail:[email protected]

Polytechnic University of Bucharest;

Abstract: In this paper I present dewaxing methods of extraction wells pipes. Dewaxing tubing of the

probes is done by mechanical, thermal, chemical Research undertaken on cleaning oil pipelines and networks as

for transport crude oil and petroleum products shows that within them is deposited drilling mud and paraffin

and cerezima which reduces pipe diameter and fluid flow velocity .In this context it is necessary, periods of

cleaning these pipes with solutions and special devices.

Keywords : dewaxing , pipeline, rehabilitation

1. INTRODUCTION

During operation of a hydrocarbon reservoir, and, in the probe and surface facilities is

submitted, a large amount of particles as a solid.

Paraffin wax or oil is the formula CnH2n +2 solid phase respectively C16H34 to

C64H130., Respectively, a mixture of liquid components, solid products (paraffin,

microcrystalline wax) as fine crystals, which add substance asphalt, resins sand, shale, clay.

After weight content of paraffin oils from Romania are divided into three categories:

- Waxy crude oils containing less than 2% paraffin;

- Semiparafinoase crude oils with a content of 1-2% paraffin;

- The wax crude oils containing less than 1% paraffin.

Separation of oil and paraffin deposition is greatly influenced by temperature and

pressure.

By lowering the temperature to reach a crystallization onset temperature of the wax,

and the lower part of pressure oil out of the solution, so that the dissolution capacity of the

solids falls.

Beginning of crystallization temperature is between 35 € - 38 ¤ C, as corresponds to

the paraffin deposition depths between 600 ... 1000 m as geothermal gradient and oil quality.

Paraffin oil is separated from the small crystals, which, due to movement of the fluid in

contact with each other, aglomerandu being around a nucleus, which may be a foreign body

such as sand, shale or fine metal particles resulted from the These clusters of corrosion

phenomena paraffin crystals are deposited on the walls of tubing, a phenomenon accentuated

tubing roughness.

Paraffin deposition is accentuated intermittently producing wells due to repeated oil

leaks on the interior walls of tubing.



34

Areas where deposition occurs parafinii as conditions are:

- Layer pores - in the area immediately surrounding the hole made by the probe;

- The exit of the column operating layer of shallow wells

- Inside the column tubing the wells sucker for great depths;

- Inside the plant surface and the mixing pipe.

Paraffin deposits produce reducing production capacity wells, reducing fluid flow through

section tubing.

Methods to reduce and control paraffin deposition are:

- Prevention methods which prevent or delay the precipitation and deposition of paraffin;

- Methods for cleaning and removal of paraffin deposited in waxy crude equipment by

circulating

2. DEWAXING MECHANICAL TUBING WELLS ERUPTING

Dewaxing operation is performed using mobile winch and consists of the following

phases:

• Install the hard pole connector (which can attach a jar) and paraffin cleaner type A or

type B;

• insert the pipe assembly dewaxing;

• pressure mounts bronze rings and rubber seals;

• Install the downspout dewaxing glands, which is mounted above the last valve head

rash;

• Link to drain the casing;

Fig 1. Hydro hook

Fig 2. banana Guy

brake cleaner

Fig 3. brake cleaner

with variable

diameter and blade

furniture

Fig 4. brake cleaner with

furniture blade with straight

pins and brake cleaner with

spiral blades


35

After the installation has been installed and controlled, open upper valve head slowly

eruption, noting if the hose is not leaking. Cleaner in the tubing down to the depth of paraffin

deposition.

After the cleaning tubing, pull the pipe cleaner, close the upper valve head eruption,

reduce pressure inside the casing, remove the stuffing and extracted cleaner.

To clean wax from the tubing walls or columns operation, use the lower scale and

other non-standard cleaners such as column hook dewaxing operation (Fig. 1) banana cleaner

(Fig. 2 ) with variable diameter and blade cleaner furniture (fig. 3), furniture cleaner blade and

guide pins, washers spiral blades (fig. 4)

3. DEWAXING MECHANICAL TUBING WELLS IN ARTIFICIAL

ERUPTION

Dewaxing column tubing wells operating in artificial eruption can use the same probes

used dewaxing natural eruption, but it can also use a special cleaner compressed acted

requiring installation of a special device at the head of eruption . Cleanser paraffin acted

compressed (Fig. 5) is made of two metal plates welded to the cross, which is welded a series

of fins arranged inclined. On top cleaner has a reducer fitted externally with radial notches to

be trapped and removed.

Equipment needed for the use of cleaner requires compressed, schematic in Fig. 5

consists of a launching tube made from a 3 1/2 in the upper part provided with a bumper

cover, the side with the connecting pipes 2, and a socket stop flap at the bottom and another A

lockable side of half the drain pressur.

Fig. 5. Diagram of dewaxing acted with cleaner compressed gas

1 separator, 2 spring detent 3-cap Silenced

In column tubing under parafining depth, a reduction seat mounts, fitted with silencer

to stop cleaner requires.

Dewaxing operation with this cleaner as follows:


36

Release device is mounted on the upper valve head eruption. Remove its cap and

insert cleaner who sits on the flip stop.

It will stop the injection of gas through the annular space of the well by closing the

valve I, II then closes and valve through which fluid from the probe mixture. III then opens

valve at the top of the head of the eruption and acts on the throttle stops, allowing cleaner

requires the probe to descend through tubing under its own weight.

After the release pipe cleaner requires the injection of compressed gas begins by

opening the valve IV stick to push cleaner requires to seat reducer. In this time of scraped

cleaner wax deposited on the tubing wall.

It will stop the injection of gas through the tubing when the cleaner came in

termination point by closing the valve IV and will switch to normal injection gas through the

annular space by opening ventiluluui I.

- Evacuation probe mixture oil and gas and directing it to the separator by launching

device, opening the valve V.

Raising cleaner requires the device launch probe fluid. Cleanser flap switch lid damper stop

kicking, but can not return to the probe because the flap returns immediately closed position

under the action of a spring.

Open valve II to return oil to the normal route through arm rash head and close valves

III and V. To remove the head cleaners of dewaxing release gas will leak through the opening

below the 1/2 inch and then remove the lid of the launch.

4. DEWAXING MECHANICAL TUBING AND SUCKER RODS IN PUMPING

WELLS WITH PIPES

Dewaxing tubing in pumping wells with pipes is accomplished by means of coil

cleaners called scrapers (fig. 6).

A scraper has a cylindrical body of steel with a diameter of less than 4-5 mm than the

inside diameter of the tubing is inserted. On the body are processed three cutter wrapped by

the left propeller. Ends is properly threaded plugs that are inserted pole size.

Fig. 6. Cleaner helical

Cleaners are inserted between two poles socket instead of pumping the entire length of

the paraffin deposition zone.


37

Cleaning paraffin column tubing, in this case, is made continuously by moving up and

down lining sucker for a length equal to the pumping stroke length during normal operation,

or periodically throughout the the paraffin deposition by executing a maneuver to seal pipes

of pumping over a distance greater than the distance between two cleaners.

Wells equipped with pumps that are included with sucker rods, not bobbing pump

mounting location during these maneuvers using a coupling device - called bayonet coupling

Dontov (Fig. 7.a), interlaced in the rod string at a depth greater than that which is deposited

paraffin.

The device consists of two parts:

- Barrel or bayonet scabbard fitted inside two parallel channels of special shape (Fig. 7

b);

- Hanging rod entering the bayonet scabbard and fix this with two wings, which engages

in the channels formed in the bayonet (Figure 7.c)

Fig 7 Bayonet Dontov

a-bayonet coupled device; b- bayonet scabbard; c-hanger rod with wings.

To leave a bayonet displacement of the weight of the ram pipes on the device rotates

gasket and then pull right up. To leave bayonet coupling down and automatically fins hanging

rod sliding on sloping channels and stop the clogged portion thereof.

In pumping wells equipped with pumps that are included with tubing not need this

device because of the pump piston can be lifted by sucker rods without leaking pipes.

A better cleaning tubing paraffin, along with cleaning rod pumping paraffin seal is

achieved by extracting sucker for changing pump P (R) or change piston pumps type T.

Dewaxing sucker is by direct scraping and extracting them from the probe.


38

For scraping usually use a wire that wraps around the pipe and you stretched with two

rings (handles) of a probe while the probe is extracted from the pumping rod string. This

process is disadvantageous, presenting fire and some of the wax flows into the probe.

Instead use wire clippers dewaxing, which have two blades each provided with a

semicircular notch. When the blades are tight form an opening equal to the pipe section.

Were constructed and used sporadically site some special cleaners to clean wax from the

sucker rods.

Fig.8 Cleaner for pipes of pumping rubber wipers Fig.9 Tables of sucker rods with metal knife.

Figure 8 shows schematically a device for cleaning the wiper rubber mounts instead of

the polished rod stuffing box. This device has two semi-circular or helical rubber tiles that

sucker rubs during extraction of the probe. Grated paraffin flow through two side arms. Fins

pressed by a spring prevents penetration of wax scraped tubing. In Figure 9 is outlined wax

cleansing device on sucker rods with a metal knife.

5. THERMAL METHODS DEWAXING EXTRACTION PIPES FROM WELLS

This method is achieved by raising the temperature in the deposition of wax, that it

dissolve and be entrained in the upward fluid probe.

Heat to melt the wax is obtained by:

- Circulation of heat in the probe;

- Using an electric heating tubing.

Dewaxing tubing by circulating a heating using steam as heat, which is introduced into the

annular space and out through tubing with heated crude oil.


39

Wells in artificial eruption, steam is introduced with the probe gas is injected in

ascension.

Pumping wells in leak pressure column and then steam generator is connected to the column.

Dewaxing pipes these probes are conducted in four phases:

1) steam is injected into the column, keeping the pump in operation for about 15

minutes;

2) is continuous steam injection, the probe stops for 15 minutes to heat the oil and

melt the wax resting on the walls of the pipes;

3) Replace the probe in use for 15 minutes, along with steam injection;

4) The new probe stops about. 15 minutes for heating oil at rest. Steam injection stops

and the probe is passed in continuous operation.

Probe can be inserted into another fluid heated by direct circulation or reverse.

Using the hot fluid has the advantage that it does not change the volume by giving

stored heat, condensation from steam, its volume shrinks more traffic. The heat used is crude

semiparafinos glazed or heated to 60-800 C. Water, although it has better thermal capacity is

not recommended as it can have a harmful influence on productivity layer exploited.

Dewaxing tubing with electrical heating is electrical energy conversion in heat energy.

Heaters are two types:

- Electric resistance heater;

- Electric heater induction.

Electric resistance heater (Fig. 10) consists of two conductors connected in series, with

different resistant. These conductors are column tubing equipped with plugs isolated column

operation.

Tubing heat from the electrical current will be higher than in column operation, due to

the difference of section (A Pipe > A column → then Rt> Rc) and the law Joule - Lenz result

Qt> Qc or:

where Q is the amount of heat that is released (in pipes Qt, Qc column);

I - the electric current;

R - the electrical resistance of the column tubing;

R - the electrical resistance of the column operation;

t - the amount of electrical current passing through the two conductors.

When using such an electric heater for dewaxing must ensure the necessary elements

insulation tubing from column exploitation insulating sleeves and a contact device at the

lower end of the pipe to close the electrical circuit between the pipes and columns.

Operated wells in pumping circuit consists of pipes and tubing, the focal point being the

piston pump shallow wells or springs contact device mounted on poles in depth parafining.

(1)


40

Fig.10 Scheme dewaxing tubing with electric heater Fig.11 Scheme dewaxing tubing with induction heater

Induction heater (fig. 11) consists of a cylindrical body of metal (tubing), wearing

insulating material over which a coil is wound with copper wire, covered in turn with

insulating material and the entire assembly is locked in a shell.

The lower end of the coil are welded to the metal body, and the upper contact device

connects the column operation. The heater is mounted in the column pipe at a certain depth,

the power will be done through the column tubing and operating connected to a power source.

Warming is caused by induced AC.

6. CHEMICAL METHODS OF DEWAXING EXTRACTION PIPES

These methods consist in placing the tubing of a solvent, either pure or dissolved in a

liquid.

Type of solvent required for each probe, the amount required for treatment, the

proportion from transport agent, duration and frequency of treatment is determined

experimentally by taking samples of raw wax clean the tubing walls and reviewing the

corresponding solubility of different solvents in the same conditions.

To dissolve the wax can be used: carbon disulfide, carbon tetrachloride, methylene

chloride, chloroform, butane, either as a single component or as a mixture of several solvents.

For the transport of solvent into the wellbore may be used: gasoline, kerosene, diesel oil

glazed.

Dewaxing pipes eruptive wells following sequence:

- Insert the solvent extraction pipes, where allowed time of 3 - 4 o'clock maintaining probe

closed to dissolve the paraffin.

- Opens probe for a short time for cleaning.

- Close the probe and insert again solvent.

The operation is repeated several times in succession mentioned, then restore normal

function probe.


41

Introducing the solvent with a higher density than the fluid in the probe is made by

The lubrication.

Pumping wells in traffic enforcement is indicated solvent.

It increases the flow probe increasing pumping elements so reduce submergenta.

Solvent is introduced into the annular space operating column - tubing is then sucked by the

pump with fluid discharged in the probe and tubing.

When the solvent reaches the pump head are directed into the annular space and

movement is mix oil - cleaning solvent to obtain paraffin deposition in production tubing.

Before reinserting the circuit oil mixture - solvent was removed from the probe will be a

separation of paraffin involved in the probe.

Dewaxing pipe mixture can be made about: mechanical, thermal, chemical.

Mechanical dewaxing paraffin pipeline is made using special cleaners called godevile.

Pig (Fig. 12) is a device composed of a central rod with one or more hinges, which is fixed to

a group of scraper wings, levers, having rollers at both ends needle for guide and some

supports for the fitting some of seals.

The joints allow Pig to pass easily through pipe bends. Seals made of leather or

synthetic rubber are designed to set in motion Pig in the fluid and push the wax scraped from

the pipe.

Gone are some blades scraping steel wax cleansing role. Gears, made of steel guides

cleaner and prevents its rotation. Launching and receiving godevilelor be done some special

connection pipes mixture, called the bypass pipe, fitted with valves to direct fluid required a

short insertion and removal of these cleaners.

Wheel steering:

Fig. 12. Godevil

Dewaxing about termicaa pipes is done by injecting superheated steam or hot oil.

Dewaxing chimica pipes are made with solvents that establish quantitative and

qualitative experimentally. Prevention of paraffin deposit pipe is cleaned by introducing

periodical liquid stream of pure solvent plugs.


42

7. CONCLUSIONS

Research undertaken on cleaning oil pipelines and networks as for transport crude oil

and petroleum products shows that within them is deposited drilling mud and paraffin and

cerezima which reduces pipe diameter and fluid flow velocity. In this context it is necessary,

periods of cleaning these pipes with solutions and special devices.

REFERENCES

[1.] Branzan Ovidiu,Studiul prelungirii durabilitatii conductelor de transport produse

petroliere ,2007

[2.] Luerenich Sren-Untersuchun gen zur Bildung Korrosiver Belage inolge feuerten

Crasturbinen,Hachen 2009

[3.] Schutze,Michael-Conrosion and enviconmental degradation NewYork 2000


43

RESEARCH ON EQUIPMENT FOR MINING EQUIPMENT,

DEWAXING PROBES

Drd. Eng Mariana PǍTRAŞCU (ANTONESCU); e-mail: [email protected]

Drd.Eng Doina TǍRǍBUŢǍ (ENE);e-mail: [email protected];

Matemat.drd.Simona IONESCU: e-mail: [email protected]

Prof.univ.Emerit. Dr.eng. Constantin D. STǍNESCU

e-mail:[email protected]

Polytechnic University of Bucharest;

Abstract :In thisprezentswouldrather on general aspects of the art equipment for dewaxingequipment

extraction wells Research undertaken on cleaning oil pipelines and networks as for transport crude oil and

petroleum products shows that within them is deposited drilling mud and paraffin and cerezima which reduces

pipe diameter and fluid flow velocity

In this context it is necessary, periods of cleaning these pipes with solutions and special devices

Keyworbs : dewaxing probe, pipe

1 INTRODUCTION

During operation of a hydrocarbon reservoir in the probe and the surface facilities is

submitted, a large amount of solid particles appropriate comb.

Paraffin wax oil or solid phase represents the formulaCnH2n+2respectivelyC16H34 up

toC64H130.,respectively, a mixture of liquid components, solid products (paraffin,

microcrystalline wax) as fine crystals, which add substance asphalt, resins, sand, shale, clay

After the weight content of paraffin naphthas in Romania is divided into three

categories:

- Paraffin oils containing less than 2% paraffin;

- Semiparafinoase oils with a content of 1-2% paraffin;

- Glazed oils containing less than 1% paraffin.

Separation and paraffin deposition in oil is much influenced by temperature and pressure.

The depression of the temperature reaching a crystallization onset temperature of the

wax, and the depression of a part of the pressure oil out of the solution, so that the dissolution

capacity of the solids falls.

Beginning of crystallization temperature is between 35 ¤ - 38

¤ C, corresponds to the

paraffin deposition depths between 600 ... 1000 m as geothermal gradient and quality of crude

oil.

Paraffin oil is separated from the small crystals, which, due to movement of the fluid

in contact with each other, making the being around a nucleus, which may be a foreign body

such as sand, shale or fine metal particles resulted from the corrosion phenomena paraffin

these clusters of crystals are deposited on the walls of tubing, a phenomenon accentuated

tubing roughness.



44

Filing paraffin is sharp wells producing intermittent, due to repeated oil spills on the

inner walls of the tubing.

Areas where deposition occurs parafinii under the conditions specified are:

- The pores layer - the immediate area of hole made by Probe;

- The exit of the column operating layer of shallow wells

- Inside the column tubing wells sucker for great depths;

- The surface and inside of the mixing pipe.

Produce paraffin deposits decreasing production capacity wells, reducing fluid flow

through section tubing.

Methods for reducing and combating parafinǎ deposits are:

- Methods of prevention which prohibits evitǎ or paraffin precipitation and deposition;

- Clean and methods of paraffin removal of the equipment submitted that go paraffin oil.

2. PREVENT SETTLING PARAFFIN.

Keeping her training paraffin matter and appropriate comb crystals or agglomerates of

crystals on the surface and solubility in paraffin oil flow regime depends on the mixture and

thermodynamic regime of the probe .

For these reasons the following means to prevent shows of cellulose:

a) Maintenance of gas in solution by choosing a suitable operating rate higher

pressure and saturation pressure by providing the lowest possible pressure drop in the

tubing.

b) Avoid sudden pressure change by FOLLOWING measure by:

- Avoid using bottom nozzles;

- Avoid using columns telescopic tubing;

- Avoid possible to start using valves in the pipes parafining extraction;

- Carefully controlling the tightness of plugs and body tubing.

c) Influence of temperature conditions by:

- Loss prevention caldurǎ the path of the oil;

- Heating the oil before it reaches the foot of the probe with a paraffin deposit

temperaturǎ favorable views.

d) Avoiding flicker and pulsations in the operation of the probe.

e) The use of surface active agents such as paraffin retardants, working in meaning to

preventing accumulation of paraffin crystals by maintaining the suspension of

large amounts of fine crystals.

f) Ensure smooth movement by covering the inside of the tubing with special paints

or plastics to prevent adherence of paraffin crystals. In these lakes or plastic

coatings, even if not achieved total prevention of paraffin deposition, the

deposition of wax inside tubing is long overdue and it is relatively easy cleaning

due to poor adhesion of the wax film plastic.

g) The use of a pulsed ultrasound generator, the effect of the accumulation of gas

bubbles on the interior walls of the pipe, which change the structure of the

molecule will influence the shrinkage of the wax crystal formation temperature.


45

Methods for removal of wax deposited on the inner walls of the column Probe tubing,

sucker rods and on the internal walls of the mixing pipes, consisting of:

cleaning about Mechanics;

Thermal cleaning;

chemical cleaning.

Each of these methods they differ depending on the operating system of the probes and

the nature of their equipment.

3. INSTALLATION DEWAXING MECHANICALLY.

Mechanical methods consist of scraping paraffin dewaxing to be submitted during the

operation of paraffinic hydrocarbons on metal surfaces through which they travel, using

special devices called knives.

Mechanical cleaning wax knives are inserted tubing periodically erupting wells

operated natural and artificial hives that are envisaged with special dewaxing plant.

Dewaxing assembly consists of the following elements:

dewaxing knife;

a heavy stick (or rod) mounted above the blade to ensure its descent producing

wells with high flow or bursts;

a special connection cable or wire;

cable or wire cutter launch and maneuver;

adewaxing pipe (head or head pistonaredewaxing);

a guide pulley cable or wire;

winch.

After the dimensions and construction components are three main types of dewaxing:

dewaxing heavy type;

dewaxing medium type;

dewaxing of light type;

Dewaxing of heavy type (Fig. 1) consists of: purifying tubular type A or type B

lamellar hard, heavy pole, fixed connection cable diameter 12-16 mm piston head and a winch

column intervention. Guidance cable is over crownblock tower production.

Dewaxing medium type (Fig. 2) consists of: Paraffin Cleaner easily laminated type C

or type D curǎţitor knives, heavy stick, removable connector, cable diameter 7 ... 8 mm, type

A head start for dewaxingdewaxing and mobile hoist.

Dewaxing easily type consists of: Cleaner easily laminated type C, heavy stick,

removable pipe, wire diameter 1.9 ... 2.2 mm, head wire B launch, over whose roll is guiding

wire and a hand winch type Yakovlev and Halliburton, which typically are used at different

probe measurements. Dewaxing mechanical eruption tubing wells is carried out using

standard paraffin cleaners which are schematically in Figure 3. and 4.

These knives have the upper thread with a pin which is screwed to a heavy pole.


46

Heavy pole is a steel cylindrical Scheduled to a head pin and the other to jack the

thread sucker. Its role is to ensure the weight of the descending knife NECESSARY,

especially in wells with high pressure natural erupting respectively wells producing large

volumes or in bursts.

Fig. 1.Plant type dewaxing heavy

1 - Clean tubular or tubular type B, 2-pole heavy 3-

permanent connection, 4-wire 12 ... 16mm diameter., 5-

head piston column, 6 - winch intervention

Fig. 2 medium type dewaxing plant

1 Paraffin Cleaner easily laminated type C or type

D washers knives, 2 - heavy pole, 3-removable

connector, 4-wire with diameter 7 ... 8 mm, 5-head

type dewaxing launch, 6 - winch dewaxing phone.

a b c d

Fig. 3. Paraffin cleaners for wells in eruption

a) cleaners tubular type –A; b) cleaners hard lamellar type- B ; c.)cleaners easily slide type- C;

d.) cleaners with knives type- D.

-cleaner tube (sheath) - A (fig.3.a.) consists of a tubular body with a longitudinal cut,

with larger diameter at the bottom (a few mm less than the inside diameter of the pipe decǎt

which introduces ). The terminal body 450 is cut to provide the advancement easier, but also a

better scraping paraffin on the walls of the tubing.


47

At the top is welded reducer for heavy pole now.

-cleaner hard lamellar type B (Fig. 3.b.) consists of a steel blade of 15 mm and length

of 1200 mm wind the strand in a sense the opposite upper and lower side, avoiding the

possibility rotation of the pipe cleaner.

-cleaners easily laminated type C (Figure 3.c) consists of a steel blade with a thickness

of 7 mm and a length of 500 mm. Body surface cleaners, twisted up and down in ways

contrary, are envisaged some windows that can easily switch from oil extraction pipe.

-D Cleanser knives (butterfly) (Figure 3.d) consists of a steel rod welded on three

floors of four knives with a special form. From one floor to another settlement knife is offset

by an angle of 300, so the knives to ensure the entire circumference, which provide a

complete cleaning wall tubing

Cleaner requires guidance is provided by four curved blades welded to the bottom of

the plunger body.

a ) b) c)

Fig. 4. Connectors, cables and wires

a)-fixed connection cable; b)- removable for cable connection; c)- removable connector for wire.

In Figure 4 presents the cable connections. From the point of view of the attachment of

the cable or wire is distinguished these types of connections:

a) fixed connection cable diameter 12 ... 18 mm (fig. 4 .. a). It is constructed from a piece

of steel pipe which is threaded for screwing the bottom heavy stick, and the upper and

lower diameter is provided on the outside with circular grooves to be caught in case

there Corunca probe . Inside the connector has a cylindrical hole that is half the length,

and the other half is conical. For this type of connection cable can not be removed

except by cutting;


48

b) Connection removable cable diameter 7 ... 8 mm (fig. 4 .. b) does not require cutting

cable disassembly. Inside the steel body is fixed to a steel composed of two pieces and

a galvanized metal sheath, which assembles over the cable node.

c) Removable connection wires 1.7 ... 2.2 mm is shown in Figure 4.c. In this connection,

the head bends the wire around a pin having a diameter of 8 mm and a length of 20

mm. The free end of the wire head rotates several times around the wire. This knot

with a pin is retained until entering the coupling body.

Pistonare downspouts and dewaxing (Figure 5) is mounted at the head of the probe rash

introduction cleaners of paraffin. This casing has a diameter of 31/2 - 4 m and the length of

the downpipe 12, the upper part has a socket in which is screwed a particular sealing cap, and

the lower part has a thread in which is screwed a flange connecting the upper valve head

eruption. The body casing at the bottom of a pipe welding 23/8 which is mounted in the drain

pipe in the basement that serves the oil escaping from the well tubing by ferries, in the top of

the device casing.

Special head seal from the top of the casing is called oil removal. It consists of a lower

body, an intermediate body and a cap inside the bodies are aflǎ rubber gaskets for sealing

cable and pressing rings (two pieces) made of bronze.

Tighten the cable seals by rotating the cap with levers that is Scheduled.

Fig.5 Pipe dewaxing

1-cup, 2-lower body, 3 rubber seals, 4-socket 23/8 ", 5-ring compression,

6-body casing, 7-spacer, 8-body intermediate

4.CONCLUSIONS

Research undertaken on cleaning oil pipelines and networks as for transport crude oil

and petroleum products shows that within them is deposited drilling mud and paraffin and

cerezima which reduces pipe diameter and fluid flow velocity In this context it is necessary,

periods of cleaning these pipes with solutions and special devices

REFERENCES

[1.] BranzanOvidiu,Studiulprelungiriidurabilitatiiconductelor de transport

produsepetroliere ,2007

[2.] LuerenichSren-

UntersuchungenzurBildungKorrosiverBelageinolgefeuertenCrasturbinen,Hachen 2009

[3.] Schutze,Michael-Conrosion and enviconmental degradation NewYork 2000


49

CHARACTERIZING THE BEHAVIOR OF THE LUBRICANT

FILMS USING MOLECULAR DYNAMICS SIMULATIONS

Lecturer Ph.D. Cristian PIRGHIE, Stefan cel Mare University of Suceava,

Department of Mechanics and Technology, [email protected]

Assistant lecturer Ph.D. Ana-Camelia PIRGHIE, Stefan cel Mare University of

Suceava, Department of Mechanics and Technology, [email protected]

Abstract. When large industries develop products, the size and complexity of it impose

major challenges. This is the case for nano-devices, which are widely present in engineering

applications. As we know, in nanotechnology the matter must be manipulated at this miniscule

scale, the nanotechnology playing by different rules. The laws we know for large systems do not

necessarily apply at the nanoscale. Doubtless, the nanotechnology developments are connected to

a correct understanding of micro- and nanotribological processes. The components used in micro-

and nanostructures are light and operate under loads, and generally lubricated with molecularly

thin films. The tribology field is evident interdisciplinary, involving scientists from many different

disciplines, including physicists, chemists, engineers, and biologists. Development of the

micro/nanotribology field has contributed to the fundamental understanding of friction and wear

processes, these being dependent on the surface interactions. The experiment results for lubricant

nanofilms have highlighted interesting properties, their comprehensive analysis being possible by

computer simulations, in the last decade an exponential increase in computing power simulation

techniques taking place. Giving the operating conditions lubricants are subjected to in practical

applications, in this paper the thin-film lubrication at sliding surfaces is considered. In this

respect, we are investigated confined nanofilms, between two walls, thin film lubrication being

simulated using non-equilibrium molecular dynamics. To impose shear on the fluid, the upper wall

is moved at different constant sliding velocities, in the same time supporting different constant

loads. Therefore, we provide a clearer understanding of the influence of molecular architecture on

lubricant viscous behavior.

Keywords: nanotribology, molecular dynamics, computer simulation, boundary lubrication

INTRODUCTION

In our days the requirements in engineering are continuously increasing, since the nano-

devices are present everywhere. We know that in magnetic storage systems,

microelectromechanical / nanoelectromechanical systems and other industrial applications,

the nanotribology is present. The magnetic storage industry continuously develops, since in

1956 the areal density was 2000 bit/in2, reaching in 2005, 1 Tbit/in

2. The present disk drives

use a lubricant film on magnetic hard disks, ensuring a longer durability, given that a relative

motion between magnetic medium and a read-write magnetic head is present.

In this situation, the tribology was replaced by micro and nanotribology, a research field

that enjoys the attention of increasingly more scientists around the world. As a consequence

of this, the traditional analysis methods are no longer valid, new and challenging one being

present. The results of experiments have revealed intriguing properties of the lubricant

nanofilms, but their analysis is not complete.


50

The film lubricant properties are not the same at different scales. Viscosity is one of the

significant of these differences. Therefore, to gain a better fundamental understanding of the

involved phenomena, the molecular dynamic simulations are widely used to characterize the

lubricant film properties [1-3]. Thin film lubricants being subject to large shear rates, the

effect of shear thinning is significant. The shear rate is varied by maintaining constant film

thickness and changing the wall speed. But important is the fact that viscosity at a given shear

rate (for a film thickness and wall speed) may not be the same to the same shear rate, but for

other film thickness and wall speed. In this paper the viscosity, or better say, shear viscosity,

is calculated for seven wall speed and six pressures.

SIMULATION METHOD

In these days, searching into the molecular phenomena of confined thin lubricant films,

the molecular dynamics simulations has been proved to be an effective tool. Given the many

applications of the perfluoropolyether lubricants, our attention are focused on PFPE-Z, film

confined between two parallel solid gold walls, with an fcc lattice [4]. The system

temperature (300 K), the total number of particles and the total kinetic energy are

conservative.

In this paper, the molecular structure was obtained using the HyperChem software (Fig.

1) [5]. The analysis must be very carefully, because there are three types of atoms, carbon,

fluorine and oxygen. Consequently, we need to consider the bond bending component into

COC, CCO, OCF, FCF, and CCF terms, the bond stretching contribution into CC, CO, and

CF terms, and the bond torsional component into FCCO, OCCO, COCF, COCC, and FCCF

terms.

Fig. 1. The molecular model of PFPE-Z (C8F18O4).

Carbon atoms are gray, fluorine is white, and oxygen atoms are black.

The present simulations of thin film lubrication were run using non-equilibrium

molecular dynamics (NEMD). Molecular dynamics tracks the motion of atoms and molecules

as a function of time. For this motion we are using a set of coupled differential equation, this

being

2

2( ), 1, ,i i i tot i

dm r E r i n

dt (1)


51

where im is mass, ir

is position of atom i, Etot is potential energy and n is the number of atoms

in system [6]. The potential energy gives the information about intermolecular and

intramolecular interactions. The interactions between the atoms of molecules are described by

the AMBER (Assisted Model Building and Energy Refinement) force field, namely [4]

2 2

0 0 0 12 6( ) ( ) [1 cos( )] .

2

ij ij i jntot r

bond angle dihedrals i j ij ij ij

A B q qVE K r r K n

R R R

(2)

The wall-molecule potential is given by Lorentz-Berthelot mixing rule [7]. The

molecular simulations were run with the Large Atomic/Molecular Massively Parallel

Simulator (LAMMPS) [8]. The simulation model of the thin lubricant film confined between

two solid walls is viewed in the Fig. 2.

Fig. 2. Snapshot of the simulation cell

The ―structure‖ of the walls is stratified. One wall contains three ―layers‖, the upper,

the middle and the upper layer. We apply the load on the upper layer, the lowest layer being

in contact with the thin lubricant film.

The layer between them (the middle one) is keep at constant temperature (Langevin

thermostat). Both walls, the upper and the lower one, exhibit the same structure. In order to

impose shear on the fluid, only the upper wall is moved at different constant velocity [9].

Seven different wall speeds were evaluated: 3, 4, 5, 6, 7, 8 and 9 m/s. So, the sliding

simulations are performed by keeping the lower wall fixed.

In order to investigate the both effects, of load and velocity respectively, on frictional

behavior of the thin lubricant film, the upper wall is supporting different constant loads (0.1,

0.5, 1.0, 1.5, 2.0, 2.5 GPa) for each mentioned velocity [10]. This constant pressure is applied

to the rigid part of the top wall in the z-direction (Fig. 2). The thicknesses of the investigated

films were 2.1 nm (50 PFPE-Z molecules) and 1.5 nm (25 PFPE-Z molecules).


52

RESULTS AND DISCUSSION

Focusing on the velocity profiles of lubricant films (Fig. 3), we are able to say that

during shearing, the lubricant has different behaviors at the walls. First of all we encounter the

stick and slip phenomenon, which is determined by load and sliding velocity.

Fig. 3 Velocity profile: (a) 8 m/s sliding velocity, 1.5 nm lubricant film thickness;

(b) 4 m/s sliding velocity, 2.1 nm lubricant film thickness

For low load (L=0.1 GPa), the velocity profile (Fig. 3) show that the thin lubricant film



For low load (L=0.1 GPa), the velocity profile (Fig. 3) show that the thin lubricant film

present two layers adsorbed on the walls. More precisely, the lubricant near the walls (3-4 Å)

became like a solid (frozen state), while the lubricant between these layers is fluid, with

Newtonian behavior.

This behavior is confirmed by the density profile (Fig. 4) where we can observe a

stratification of the film lubricant, confined between the walls. Increasing the load applied on

thin lubricant film, being now 1.0 GPa, we notice the same behavior for the lubricant film, in

addition occurring a strong slip near the walls (Fig. 3a). The velocity discontinuity between

the adjacent lubricant molecules and wall indicates that slip appear at this interface.

(a) (b)


53



By examining the radial distribution function (RDF), we obtain valuable information

about the state of the confined thin lubricant film whose thickness is 2.1 nm (Fig. 5). We

represent RDF of the lubricant at the beginning of the experiment and during the experiment.

We notice that in RDF pattern during the experiment appears a second peak, meaning that in

lubricant appear an ordinate structure, produced by the strong interaction between the

lubricant atoms and wall atoms. We suppose that the ordered structure of lubricant which

appears during the experiment is responsible for the viscosity changes.

Fig. 5 Radial function distribution for 1.5 GPa load and 6 m/s velocity profile

Information stated above is reinforced by the x-position of the lubricant atoms (Fig. 6),

confirming the stick and slip phenomena at the both wall.

(a) (b)


54

Fig. 6. x-position of lubricant atoms as a function of time: (a) 2.1 nm, v=6 m/s, L=1.0 GPa; (b) 1.5 nm, v=6 m/s,

L=1.0 GPa

Going further, our attention is focused on shear viscosity [4], being evaluated as the

ratio of shear stress, xz , to effective shear rate, ,

xz .

(3)

In order to calculate the viscosity, the slope of linear part of velocity profile was used,

where we have a Newtonian behavior for the lubricant film.

Fig. 7 Shear viscosity as a function of pressure, 2.1 nm thickness

(a) (b)


55

Fig. 8 Shear viscosity as a function of shear rate (shear thinning), 2.1 nm thickness

As we can see, on low load, the Barus low is respected, p

0e , while at high

pressure does not work properly (Fig. 7). The dependence of shear viscosity on shear rate is

given by power law, , where , generally lower than one, is a constant depending on

system, conditions and type of lubricant molecules (Fig. 8).

CONCLUSION

In this paper we investigated confined PFPE-Z thin film using molecular dynamics

simulations. The attention was focused on two nanometric thicknesses for the lubricant film,

only the upper wall being moved at constant shear velocity, while the bottom wall was fixed.

In the same time, the upper wall was supporting constant load (pressure). The simulations

were done for different shear velocity of the upper wall and different loads, but during one

simulation this parameters were constant. Our investigation reveals a film lubricant

stratification and appearance of a second peak during the experiment, meaning the presence of

an ordinate structure. Further, we evaluated shear viscosity, first, as a function of pressure

and then as a function shear rate.

ACKNOWLEDGMENT The authors would like to thanks to PrivDoz. Dr. rer. nat. Dipl. -Phys. András Vernes,

DI Dr. Stefan Eder, and Univ.-Prof. Dipl.-Ing. Dr. Friedrich Franek from Austrian Center of

Competence for Tribology (AC2T research GmbH), for their support and assistance.

REFERENCES

[1] A. Jabbarzadeh, P. Harrowell, R.I. Tanner, Very Low Friction State of a Dodecane

Film Confined between Mica Surface, Physical Review Letters, 94, 2005, 126103

[2] A. Martini, Y. Liu, R.Q. Snurr, Q.J. Wang, Molecular dynamics characterization of

thin film viscosity for EHL simulation, Tribology Letters, 21 (3), 2006, 217-225

[3] N.V. Priezjev, Shear rate threshold for the bou ndary slip in dense polymer films,

Physical Rewiew E, 80(3), 2009, 031608


56

[4] C. Pirghie, A.C. Pirghie, S. Eder, F. Franek, PFPE-Z Lubricant Thin Films in

Molecular Dynamics Simulations – Shear thining and Friction law, Optoelectronics and

Advanced Materials – Rapid Communications, 7 (5-6), 2013, 434-438

[5] HyperChem is distributed by Hypercube, Inc.

[6] K. Tanaka, T. Kato, Y. Matsumoto, Molecular Dynamics Simulation of Vibrational

Friction Force Due to Molecular Deformation in Confined Lubricant Film, Journal of

Tribology, 125, 2003, 587-591

[7] A. R. Leach, Molecular Modeling Principles and Applications 2nd ed., Pearson

Education Limited, Harlow, 2001

[8] S. J. Plimpton, Journal of Computational Physics, 117, 1, 1995

[9] C. Pirghie, S. Eder, G. Vorlaufer, F. Franek, Viscosity oh Highly Confined PFPE-Z

Nanofilms with Molecular Dynamics, WTC 2013, 5th

World Tribology Congres, Torino,

Italy, 8-13 September 2013

[10] C. Pirghie, A.C. Pirghie, Investigations of Nanotribological Systems using

Molecular Dynamics Simulations, International Conference on EHD Lubrication and

Traction, 25-27 octombrie 2012, Proceedings of VAREHD, vol. 16, 2012, 53-60


57

STATISTICAL PROCESSING OF CENSORED DATA

UNDER ACCELERATED RELIABILITY TESTING FOR RADIAL

BALL BEARING

Lect. PhD.Eng. Sebastian Marian ZAHARIA, Department of Manufacturing Engineering,

Technological Engineering and Industrial Management Faculty, Transilvania University of

Brasov, Romania, e-mail: [email protected]

Assoc. PhD.eng. Cristin Olimpiu MORARIU, Department of Manufacturing Engineering,

Technological Engineering and Industrial Management Faculty, Transilvania University of

Brasov, Romania, e-mail: c.morariu@unitbv. ro

Abstract: The organizing and the execution out of the reliability testing of the products represent complex

activities from an organizational standpoint and are also big resources consumers. Reliability test

elements, conditions and other factors are determined based on customer needs for reliability, by clarifying

the environmental and time conditions under which devices will be used and the failure definitions The

paper is targeted at the study of the reliability parameters and the lifetime of radial ball bearings using

reliability testing. For the case study it used ALTA 7 software what provides a complete array of analysis

tools for data from reliability tests. The case study is represented by a set of data from accelerated tests

using censored data type.

Keywords: reliability, ball bearings, life, reliability testing, censored data.

1. INTRODUCTION

Reliability test is the general term for reliability determination tests and reliability

compliance tests. In other words, reliability characteristics values (failure rate, reliability,

mean life, MTTF, etc.), which are scales representing the time-dependent quality of products,

are estimated and verified statistically from the test data. These tests also play an important

role in improving reliability by analyzing failures which occur during tests and clarifying

these failure mechanisms. Reliability tests provide the greatest effects when statistics and

failure physics function reciprocally [1].

The reliability tests have a great importance, aiming either to determine, either to check

the reliability characteristic of a product, if this is established in a predictive way. The

reliability tests are extremely necessary and they have a decisive role in improving the

technical solutions and in increasing the performances. The essential problem of reliability

tests is the testing duration, which is generally comparable with the product‘s useful life time

[2]. A too long testing duration renders the reliability test useless. These reliability tests help

us with: the determination of the statistical pattern of the operating time without failures; the

estimation of the pattern‘s parameters; the realization of predictions on the reliability of the

products. The reliability tests precisely analyze the phenomenon of failure, quantified either in

operating time, either in number of cycles. By reliability test we understand an experiment

organized in order to determine the reliability parameters for a well-defined product. The

organization of reliability tests represents a problem that requires on the specialist‘s part a

good theoretical background and a vast practical experience.



58

In order to realize reliability tests we must take into account the following aspects:

a) A previously determined n number of products that will be subjected to testing

b) A testing plan that includes the following aspects: the selection of stress parameters,

which will determine the failure mechanisms specific to the product; the determination of

environmental conditions in which the experiment takes place, that has to take into account

the actual conditions in which the products will function in real operation

c) Instructions regarding the adequate type of test and the methods of calculation, in

order to estimate the reliability parameters

d) A test chart where the experimental data are recorded and the statistical calculations,

as well as the chronological recording of observations and interventions, are made

e) Testing stands, testing equipment, auxiliary materials and qualified staff in order to

realize the test.

The most used reliability tests are the following [3]:

Complete tests (type n out of n) - in these tests n products of the same kind, the

experiment being considered finished when all of the n products have failed. This

kind of tests can‘t be applied in all the situations, because in the case of expensive

products is uneconomical, and in the case of products that have a relatively long

life time by nature, the experiment will take too long. Nevertheless, this kind of

experiment has the advantage that the obtained information is complete, meaning

that at the end the failure times for all the products subjected to testing are

determined;

Censored tests (type r out of n) - are commonly used and they consist of subjection

to testing of n products of the same type, the experiment being considered finished

after the failure of r<n tested products; obviously, the r number is previously

determined, usually by technical, economical and statistical considerations. At this

type of experiments, the testing duration is random, because it‘s unknown when

the r-th product will fail; the information here is incomplete, because in the end we

will have only r, instead of n experimental data

Truncated tests (with a fixed testing time) - a n number of products are subjected

to testing, but the experiment doesn‘t stop according to the number of failed

elements, but according to a tr time, previously set, a period during which the

testing takes place. After this testing time (previously set) has passed, the

experiment is considered finished.

The testing methodology about to be used has a direct economical impact, because in

every test the following terms intervene: the cost of the tested product; the total cost of the

experiment; the time consumed for testing and for the statistical processing of the data

resulted from testing. Therefore, the selection of a specific type of test is a managerial

decision that has to be taken by a responsible authority. Also, in the follow-up of the results of

the statistical processing, we will propose certain corrective technical and economical actions,

aimed directly at the quality and reliability of the product in question [4].


59

2. CASE STUDY

Single Row Radial Ball Bearings - 6004 2RS is a high quality bearing used in axial fans,

motors, drive axles, clutch, idler wheels, HVAC, snow mobiles and many other industrial

applications. Bearings are made of chrome steel, pre-lubricated with grease and have rubber

seals on both sides to protect the bearings from dust or any other types of possible

contamination. In the table 1 is described the geometric and constructive aspects of the radial

ball bearings type 6004, used for the accelerated reliability testing. The radial ball bearings

type 6004 is described in figure 1 [5,6,7].

Tab. 1. The parameters of the ball bearings type 6004

Fig. 1. The radial ball bearings type 6307

2.1. Experimental data

Using the testing stand of the ball bearings, we realized accelerated reliability tests with

the purpose of reducing the testing duration and the material costs associated with these tests.

The results obtained at the accelerated reliability testing (table 2) of the ball bearings for the 3

accelerated stress regimes are: for the first accelerated stress regime 8000 N; for the medium

accelerated stress level 9000 N and for the maximum accelerated stress level 10000 N. Tab. 2. Number of cycles until failure – bearing load

No. State

F or S

Time to

F or S

Bearing Load

[N]

1. F 1234 8000

2. F 1423 8000

3. F 1567 8000

4. F 1790 8000

5. S 2000 8000

6. S 2000 8000

7. F 789 9000

8. F 897 9000

9. S 1000 9000

10. S 1000 9000

11. F 456 10000

12. F 490 10000

13. F 543 10000

14. F 589 10000


60

2.2. Statistical processing

Once you have selected an underlying life distribution (Weibull) and life-stress

relationship (inverse power law) model to fit your accelerated test data, the next step is to

select a method by which to perform parameter estimation. Simply put, parameter estimation

involves fitting a model to the data and solving for the parameters that describe that model. In

our case, the model is a combination of the life distribution and the life-stress relationship

(model). Available methods for estimating the parameters of a model include the graphical

method, the least squares method and the maximum likelihood estimation method. ReliaSoft ALTA 7 - www.ReliaSoft.com

Life vs Stress

Beta=6,861; K=3,603E-26; n=5,677

Bearing Load

Lif

e

7000 1400010000100

10000

1000

Life

Data 1Inverse Power LawWeibull7000F=10 | S=4

Eta L ine8000

Stress Level PointsEta PointImposed Pdf

9000Stress Level PointsEta PointImposed Pdf

10000Stress Level PointsEta PointImposed Pdf

sebastian zahariaUniversitatea Transilvania

Mean life of ball bearing

is 3846 hrs

ReliaSoft ALTA 7 - www.ReliaSoft.comFailure Rate vs Time

Beta=6,861; K=3,603E-26; n=5,677

Time

Fa

ilu

re

R

ate

0 200004000 8000 12000 160000

10

2

4

6

8

Failure Rate


Failure Rate L ine

sebastian zahariaUniversitatea Transilvania Brasov

a) Life vs. Stress plot b) Failure rate vs. Time plot

Fig. 2. Reliability parameters

Once you have calculated the parameters to fit a life distribution and a life-stress

relationship to a particular data set, you can obtain the same plots and calculated results that

are available from standard life data analysis such as:

Reliability Given Time - the probability that a unit will operate successfully at a

particular point in time under normal use conditions;

Probability of Failure Given Time – the probability that a unit will be failed at a

particular point in time under normal use conditions. Probability of failure is also

known as "unreliability" and it is the reciprocal of the reliability;

Mean Life - The average time that the unit in the population are expected to operate at

a given stress level before failure (figure 2.a). This metric is often referred to as "mean

time to failure" (MTTF) or "mean time before failure" (MTBF). A variety of life

characteristics, such as B(10) life or eta, can be displayed on the plot. This plot

demonstrates the effect of a particular stress on the life of the product;

Failure Rate: The number of failures per unit time that can be expected to occur for the

product at a given stress level (figure 2.b);


61

Warranty Time - the estimated time when the reliability will be equal to a specified

goal at a given stress level;

B(X) Life - the estimated time when the probability of failure will reach a specified

point (X%) at a given stress level (figure 3.a);

Acceleration Factor - a unit less number that relates a product's life at an accelerated

stress level to the life at the use stress level (figure 3.b);

Probability Plot - a plot of the probability of failure over time. This can display either

the probability at the use stress level or, for comparison purposes, the probability at

each test stress level;

ReliaSoft ALTA 7 - www.ReliaSoft.comAcceleration Factor vs Stress

Beta=6,861; K=3,603E-26; n=5,677

Bearing Load

Ac

ce

lera

tio

n F

ac

tor

7000 100007600 8200 8800 94000

10

2

4

6

8

Acceleration Factor


AF L ine


a) BX Life b) Acceleration Factor vs. Bearing Load


Reliability vs. Time Plot: A plot of the reliability over time at a given stress level. A

similar plot, unreliability vs. time, is also available (figure 4.a,b);

Probability density function plot - represents a plot of the probability density function

(pdf) at a given stress level. The pdf plot shows the probability density function of data

over time at the specified use stress level. This allows you to visualize the distribution

of the data set;

The Cox-Snell Residuals plot is similar to the standardized residuals plot, except the

line is plotted on an exponential probability plotting paper and is on the positive

domain (figure 5.a);

Residuals Plots - plots of the residual values that have been assigned, via regression

analysis, to each point in a data set. These plots provide a tool to assess the adequacy

of the model used to analyze the data set (figure 5.b).


62

ReliaSoft ALTA 7 - www.ReliaSoft.comReliability vs Time

Beta=6,861; K=3,603E-26; n=5,677

Time

Re

lia

bil

ity

0 60001200 2400 3600 48000

1

0,2

0,4

0,6

0,8

Reliabil ity


Data PointsReliabil ity L ine


ReliaSoft ALTA 7 - www.ReliaSoft.comUnreliability vs Time

Beta=6,861; K=3,603E-26; n=5,677

TimeU

nre

lia

bil

ity

0 60001200 2400 3600 48000

1

0,2

0,4

0,6

0,8

Unreliabil ity


Data PointsUnreliabil ity L ine


a) Reliability function b) Unreliability function


ReliaSoft ALTA 7 - www.ReliaSoft.com

Cox-Snell Residuals

Beta=6,861; K=3,603E-26; n=5,677

Residual

Pro

ba

bil

ity

0 102 4 6 81

99

50

90

95

Cox-Snell Residuals

Data 1Inverse Power LawWeibull

Residual L ine8000F=4 | S=2

Residuals9000F=2 | S=2


Residuals


ReliaSoft ALTA 7 - www.ReliaSoft.com

Standardized vs Fitted Value

Beta=6,861; K=3,603E-26; n=5,677

Eta

Re

sid

ua

l

100 100001000-10

10

-6

-2

2

6

0

Standard - Fitted




Residuals


a) Cox Snell Residuals b) Standardized vs. Fitted Value

Fig. 5. Reliability plots

3. CONCLUSION

In accelerated reliability testing, products are exposed to stress levels higher than

those at normal use in order to obtain information in a quickly time. In this paper we expand

the limits of performing the accelerated life tests on a product from the automotive field (ball

bearing). The implementation of accelerated reliability tests on the products from the

automotive field has produced a significant reduction of the testing time.


63

We can observe that, by using the accelerated life tests, the testing time has been

reduced by 8 times. Given the fierce competition existing on the automotive industry market

and considering the reduction in testing time and therefore the optimizing of the products‘ life

cycle, many companies will implement and develop various methods of obtaining data as fast

as possible regarding the reliability and quality of the products.

4. REFERENCES

[1] Zaharia S.M, Martinescu I., Reliability tests, Transilvania University Press, Brasov,

2012.

[2] Nelson W. Accelerated Testing: Statistical Models, Test Plans, and Data Analysis, Wiley,

New Jersey, 2004.

[3] Klyatis L.M., Accelerated Reliability and Durability Testing Technology, Wiley, New

Jersey, 2012.

[4] Collins J.A., Busby H.R., Staab G.H., Mechanical Design of Machine Elements and

Machines, John Wiley & Sons, Wiley, 2010.

[5] Guangbin Y. Life Cycle Reliability Engineering, Wiley, New Jersey, 2007.

[6] Harris T.A., Kotzalas M.N., Advanced Concepts of Bearing Technology, Rolling Bearing

Analysis, CRC Press, New York, 2006.

[7] Bertsche B., Reliability in Automotive and Mechanical Engineering, Springer, 2008.


64

DYNAMIC STUDY OF THE R-RTT MECHANISM

ASSISTED BY AUTODESK INVENTOR

Assoc. Prof. Dr. Marin NEACSA, University POLITEHNICA of Bucharest,

[email protected],

Assoc. Prof. Dr. George ADÎR, University POLITEHNICA of Bucharest,

[email protected]

Assoc. Prof. Dr. Victor ADÎR, University POLITEHNICA of Bucharest,

[email protected]

Eng. Ancuta ADÎR, Grigore Cerchez Technological College, Bucharest

[email protected]

Abstract: In this paper is introduced the way of using of the Dynamic Simulation Module from

Autodesk Inventor Professional, for the dynamic analysis of an R-RRT mechanism . As example, it

was chosen a mechanism used to operate refrigerators compressors.

Keywords: Dynamic study , mechanism, refrigerator compressor, Autodesk Inventor Professional.

STRUCTURE AND OPERATION OF THE MECHANISM STUDIED

In the manufacturing process of refrigerators there are used R-RTT mechanisms. The

piston has a translation motion inside the compressor‘s cylinder. The leading element (1) of

the mechanism, is the rotor of the electric motor that drives the compressor. This one is joined

to the engine‘s stator (this joint is marked by R, the first letter from the structural relationship

R – RTT). The rotor has (at one of the ends) a crank, as leading element, connected to dyad

RTT, made by 2 elements, namely the piston and a rock slide.

The rock slide is articulated on the crank an is gliding in a piston guide, positioned

perpendicular on its translation axle inside the compressor‘s cylinder. The flow of the cooler

agent in the compressor‘s cylinder is ordered by 2 valves, one for intake and the other for

discharge. The distance between the 2 joints of the crank is 0,0065 m. The translation axle of

the piston in cylinder is passing through the revolute axle of the crank, against the engine‘s

stator, against base and the chassis of the electric motor. In Fig. 1 is presented the most

important piece of the mechanism, namely the piston with the guiding system perpendicular

on the cylinder‘s axle.

The rotation speed of the driving element (motor‘s rotor) is considered as constant:

n1= 2690 rot/min.

The elements of the mechanism are made by steel and the masses and the mechanical

inertia moments of the elements are automatically established by AUTODESK INVENTOR.



65

Fig. 1 The piston and the guiding system

The phases of the theoretical diagram, Vp , of the compressor are [1]:

1-2 polytropic compression;

2-3 discharge;

3-4 recoil of the fluid in the dead space;

4-1 aspiration

For the analysis of the mechanism R-RTT of a compressor with freon, there are

considered the following data:

- aspiration pressure in installation: 182805afp [N/m2 ];

- medium pressure loss for aspiration: afaf pp )1.005.0( ;

- discharge pressure in installation: 850050refp [N/m2 ];

- medium pressure loss for discharge: refref pp )15.01.0( ;

- polytropic exponent at rebound: 08.1dn ;

- polytropic exponent at compression: 1.1cn ;

- the technical compressor with piston has a dead space 0V ; the relative dead space ,

is:

05.0/0 sVV (1)

where sV is the aspiration volume (space volume necessary to the piston‘s displacement):

ABd

Vs

4

22

(2)

d - piston‘s diameter

AB - crank‘s radius


66

0V , the volume of the dead space, is calculated with the relation:

ABd

VV S

4

22

0 (3)

- the discharge pressure in compressor is: refrefr ppp ; it is taken into

consideration refref pp 1.0 ;

- aspiration pressure in compressor: afafa ppp ; it is considered

afaf pp 05.0 ;

- Ecuation of the polytropic curve, at rebound:

dn

rx

AB

xpp

2

(4)

where x represents the piston‘s stroke from the upper dead point to the lower dead point;

- stroke x4, corresponding to the end of rebound, is established by the relation:

124 dn

a

r

p

pABx (5)

- Ecuation of the polytropic curve, at compression:

cn

ax

AB

xpp

2

1 (6)

- stroke x2, coresponding to the end of compression, is established by the relation:

cn

r

a

p

pABx )1(22 (7)

Displacements 321 , , xxx and 4x , necessary to establish the pressure variation diagram

against stroke, are:

ABx 21 (8)


67

cn

r

a

p

pABx )1(22 (9)

03 x (10)

124 dn

a

r

p

pABx (11)

The corresponding pressures to each operating field are:

- for ],[ 43 xxx ,

dn

rx

AB

xpp

2

(12)

- for ],[ 14 xxx , ax pp (13)

- for ],[ 21 xxx ,

cn

ax

AB

xpp

2

1 (14)

- for ],[ 32 xxx , rx pp (15)

THE DYNAMIC STUDY OF THE MECHANISM ASSISTED BY AUTODESK

INVENTOR PROFESSIONAL

To study the dynamics of the R-RTT analyzed mechanism, it is necessary to follow

the stages:

1) Modeling in Inventor of the mobile elements of mechanism, namely: rotor of the

electric motor (with the crank fixed on it), piston and the rock slide. For these elements, is

important to know the mass and the mechanical inertia moments, automatically established by

Inventor. It is not relevant the rigorously modeling of the base, of the fixed part (the engine‘s

chassis, compressor‘s cylinder etc).

2) Modeling of the assembly of the elements of mechanism

3) Dynamic simulation of the mechanism with Dynamic Simulation Module from

Autodesk Inventor Professional.

Further on, we shall refer to the last stage, namely to the dynamic simulation of

mechanism. After modeling the assembly of elements, it is selected from Environments Tab

the icon Dynamic Simulation.


68

So, it appears the window depicted in Fig. 2, in which it is selected the button marked

by 1, for editing the properties of couples. If the restrictions imposed when modeling the

assembly of the elements of mechanism were right, the mechanism‘s couples are

automatically and right modeled in Dynamic Simulation Module. To edit the properties of

the revolute couple between the leading element and the engine‘s stator, it is selected in

browser the respective couple and it is pushed the right button of the mouse, operation marked

by BdM further on [2].

Fig. 2 Window Simulation Player

It appears the window from Fig. 3. To specify the revolute speed, that in Inventor is

considered in deg / s, it is selected the button dof 1(R), marked by 1, then it is selected the

button 2 and then there are selected the buttons marked by 3 and 4 on the figure. After that, it

is possible to specify the revolute speed in the window marked by 5.

To edit the properties of the translation couple, between the piston and the

compressor‘s cylinder, it is selected in browser the respective couple and it is pushed the right

button of the mouse. It appears the window depicted in Fig. 4. To specify the necessary data

to calculate the forces in the translation couple between piston and cylinder, it is selected the

icon marked by 1, then is ticked the button marked by 2, it is specified the friction coefficient

in couple in the window marked by 3 ands then is pressed twice, succesively, the left button

of the mouse, with the prompter put on the window marked by 4.

As a result, it appears the window depicted in Fig. 5. It is selected the icon marked by

1, that allows to import the values of the pressure force against the piston, from a text file.


69

The values of the pressure force picked up are present in the window marked by 2 and

the variation diagram of the pressure force against the piston‘s position is presented in the

window marked by 3.

Fig. 3 Specifying of the revolute speed Fig. 4 Editing of data for calculation forces

in the couple piston – cylinder

Fig. 5 Import of the values of the pressure force against piston

To obtain the variation of the balancing moment in the crank‘s joint at the base,

corresponding to the imposed rotation speed, it is selected from the menu Output Grapher. It

appears the window presented in Fig. 6; it is selected the revolute couple between the crank

and base, indicated in Fig. 1 and then it is ticked the button marked by 2,corresponding to the

balancing moment. The diagram obtained can be exported in MS Excel.


70

Fig. 6 Selecting of the balancing moment

in the joint of the crank at the base

CONCLUSIONS

The dynamic modeling of mechanisms can be relatively easy studied by using the

Dynamic Simulation Module from Autodesk Inventor Professional. As a result, after this

modeling, it is posible to optimize the design of these mechanisms to increase their

competitiveness.

REFERENCES

1) Moise, V., ş.a., Analiza mecanismelor aplicate, Editura PRINTECH, 2007

2) Stăncescu, C., Modelare parametrică şi adaptivă cu Inventor, volumul 2, Editura FAST,

Bucureşti, 2010

3) http://knowledge.autodesk.com/support/inventor-products/learn-

explore/caas/CloudHelp/cloudhelp/ENU/123112/files/inventor-2014-tutorials-html.html


71

THE FAILURE MODES AND THEIR REMEDIATION PROGRESSIVE

CAVITY PUMPS USED IN OIL PRODUCTION

Lecturer PhD STAN Marius , Petroleum - Gas University of Ploiesti

[email protected]

Abstract: The primary objective for this project was to design and select a suitable PC pumping system to

meet the applications production and lift requirements and proposing solutions to improve the operation of

the entire extraction system. Information contained in the “PC Pump Design Form” was used to develop a

simulated model with a specialized. The main difficulties that characterize the work of the probe are:

abrasion, corrosion, paraffin deposition, gas, oil viscosity, scale deposition and wellbore trajectory.

Keywords: analysis, elastomer, sustainability, modeling, reliability

1. INTRODUCTION

Today, primary heavy oil and bitumen applications are almost exclusively produced

with PC pumping systems.

Recent estimates characterize nearly 50% of the world's liquid hydrocarbon as having

a gravity of less than 20° API1 These reserves are generally referred to as bitumen and heavy

oil, and large deposits are abundant in Canada, Russia, Venezuela and China,

The most distinguishing characteristic of these fluids is their high viscosity which

typically ranges between 500 and 15,000 cp for heavy oil and may approach 100,000 cp for

bitumen,

These viscous crudes are commonly located in shallow reservoirs (300 to 600 depths)

and individual wells usually produce these fluids at low to moderates rates (I to 70 m3/day).

In order to achieve economic production rates, most wells must be pumped at low

bottom hole pressures which magnify the adverse effects of any produced gas, In addition,

heavy oil and bitumen are typically withdrawn from poorly consolidated reservoirs which are

prone to sand production that can exceed 30% by volume. The close well spacing required for

viable field development often leads to the preferential use of directional and horizontal wells.

One of the most important mining industry is the exploitation of deposits of

hydrocarbon fluids through wells.

The use of these natural resources as a source of energy, fuel, petrochemical

feedstock‘s or in the pharmaceutical industry has made a fluid hydrocarbon extraction

industry of prime importance.

Labor productivity growth in the oil industry called centralized monitoring wells in the

pumping operation, computer-aided data transmission to remote processes automation and

control devices adaptation of stopping and starting of electric motors, test probes, detection

and diagnosis of faults. Also in this area is remarkable development of complex computer

programs for optimization of the choice of equipment and operation of facilities.



72

Oil and gas industry is characterized generally by conveying large amounts of fluids:

oil and gas accompanied by salt water and sometimes hard particles (sand) which are

extracted from the wells productive strata, then separated and stored installation surface. [12].

The material they are made of deep and surface equipment for extraction of oil wells are

subjected to wear caused by fluids inside and outside the vehicle. Often these processes are

produced with high intensity and leads to considerable difficulties such as:

• Premature wear and decommissioning of equipment before normal time;

• Sudden interruption of processes that lead to loss of production due to accidents or

technical, requiring large expenditures for remediation;

• Breakage of tubular material (tubing, oil pipelines and salt water) that generates high

production and investment losses supplementary. [8]

The main difficulties that characterize the work of the probe are: abrasion, corrosion,

paraffin deposition, gas, oil viscosity, scale deposition and the drill hole.

2. FORMULATIONS The chemical Eastover define the applications for which a PCP is suitable. Rubber.

The most cannon lassoers are rubber formulations. The term ―Rubber‖ is usually applied to

formulations based on the isoprene and/or butadiene polymer chains.

This discussion is confined to butadiene formulations Rubbers still offer the best

combinations of mechanical properties and resistance to chemical attack for most

applications.

The additives are mechanically mixed with a kneading action intothe plastic,

elastomeric butadiene compound. The mixture is then moulded and vulcanized Vulcanization

is the ―cedcing‖ process which speeds up the cross-linking of the additives with butadiene.

Usually there are a dozen or more additives to butadiene to make a rubber. Some are

iinstilcient research has been it and make it a science.

Acrylonitrile, oflen referred to as ‗nitrile‖, is also a carbon chain. It increases some of

the mechanical properdes of rubber while decreasing others. Within limits, the properties

which are decreased are still within the desirable range, which is why acrylonitrile is the most

used additive. In addition to its on mechanical properties, acrylo-nitrile greatly enhances the

resistance of rubber to aromatic solvents. It will be know that the acrylonitrile sdditive has a

triple bond. However, acrylottitrile rubbers will pump high water cut oil wells because

rubbers are oilwettable and attract a protective Iiirn of oil which prevents absorption of water

even in low oil cut production.

HNBR Hydrogenated acrylonitrile butadiene is another elastomer which is being

offkred by the suppliers of PCP ‗s.

To make HNBR, hydrogen is added to nitrile rubber to reduce or eliminate the

residual unsaturation. Research in the nuclear industry has shown that HNBR is better than

rubber at resisting H$. This elastomer has better resistance to temperature also. A problem

with HNBR is that there is not as large a body of knowledge of additives to make it as

resistant to the aromatic solvents as high acryIonitrile rubber. More research is required to

fmd additives which will provide the same aromatic solvent resistance as high nitrile rubber.


73

Fluorocarbon and Perfluorocarbon. There are elastomers in the fluorocarbon and

perfluorocarbon groups which are excellent at resisting chemical attack and high

temperatures. However, the practical problem with these elastomers is that they have never

been successfully moulded in large quantities.

The moulding difficulty with fluorocarbon can be overcome by making a copolymer

with butadiene. This wmpromises the chemical resistance of tluomcwbm but results in an

elastomer with higher chemical and temperature resistance than rubber.

Polyethylene chlorosulphide is an elastomer with excellent abrasion resistance and

other mechanical properties similar to rubber. However, it has no resim to light hydrocarbon

liquids, although it does have good resistance to CO. Us resistance to water is superior to

rubber.

2. PROBLEMS ENCOUNTERED IN PUMPING HELICAL

Three most common pc pump failures in wells, [13]:

Overheating

Cause – Gas entering the pump causes the Elastover to swell and overheat. Pumping

the well off, or running dry.

Result - Can result in hardening, then shearing of the elastomer. Gas may also cause

blistering of the stator or surface cracking of the rotor. Any of these issues may cause damage

to the seal between the rotor and stator, even before failure.

Solids

Cause - High amounts of solids going through the pump or solids settling back

through the discharge or tubing.

Result - A plugged pump resulting in broken rods or rotors.

Incorrect operation of the pump

Cause - Due to the fluid, gas, and solids, it is critical to operate the pump within the

recommended parameters.

Result - Can cause additional stress to the system.

The presence of free gas at pump suction directly affects its efficiency by reducing the

volume of fluid pumped through the pump. [9,10]

Spiral pump can pump the fluid in limited quantities of product gas is taken to the

surface, thus overcoming the friction pressure loss.

The lack of adequate amounts of fluid gives rise to conditions of "dry run", which

results in burning of the surface of the elastomer and the occurrence of cracks, cracks, etc..

In Figure 1 is shown that one can suffer wear elastomer under "dry run".


74

Fig.1 Wear elastomer.

Materials and combinations of materials given in Table II for . metal parts , such as

pump housing , rotor shafts , etc and Table I pt . stator are made of any metallic material by

conventional processes .

Table I Stator are made of any material and recomandations for use

Helical type pumps

BORNEMAN

Materials for helical screw pump

stators

(Facts)

Recommendations for use

Code Title Temp

max

adm. 2)

Possible to use the:3) Not be used to:

702 NR natural Rubber 70

Aqueous media, abrasive organic

acids, alcohols, cetane, aldehydes,

organic fertilizer, cement paste,

grape marc

Ozone concentrated acids, fats, oils,

hydrocarbons.

718/731

SBR/B

R

Buna (Buna S)

Styrene-

butadiene

80 Water, aqueous media,

water sanitation, liquid ammonia, li

quid manure, someinorganic acids.

Like natural rubber, in addition to

ketone and ester.

703/729

(NBR)

Perbunan N 1)

Rubber nitrile

90 Some hydrocarbons, hydraulic

fluids, weak acids and weak acid

solutions, gasoline, mineral, animal

and vegetable oils and fats.

Chlorinated aromatic hydrocarbons,

ester, aldehyde, ozone strong acids.

704 CR Neopren 1)

Rubber

cloroprenic

STAS 10297-

75

90 Ozone, grease, oils, paraffinic or

naphthenic, aliphatic

hydrocarbons, reducing agents,

aqueous salt solutions.

Strong oxidizing acids

732/705

11R/CJ-

11R

Rubber butilic 110 Animal and vegetable fats and oils,

ozone, strong oxidizing chemicals,

acids and acid solutions.

Oil

707

CSM

Hypalon 1)

Rubber sulfonil

polyethylene

100 Alcohols, salt solutions, ammonia,

hydrogen peroxide, oxidizing

chemicals, oils, water, pulp,

chromic acid.

Concentrated acids oxidants


75

716

EPDM

Rubber EPDM

Rubber

ethylene-

propylene

110 Ozone hot water, sulfuric acid,

animal and vegetable oils and fats.

As butyl rubber 11 R

724

ACM

Rubber

polyacrylic

STAS 10577-

76

120 Oils at high temperatures, oils

edible animals, latex, PVC pastes.

Water, aromatic hydrocarbons,

ketones, alkaline solutions, acids.

708

FPM

Vitan

Rubber

fluorocarbon

180 Aliphatic, aromatic or halogen

groups, oils, grease, acids.

Almost all liquids!

Ketones, chlorinated hydrocarbons,

acids, oxidizers.

712

HDPE

Polyethylene

High density

(HDPE)

90 Water, inorganic salt solutions,

acids and weak acid solutions,

aliphatic oils, fats

Petrol, chlorinated hydrocarbons,

acids, oxidizers

720

PA

Polyamide

(Nylon)

90 Bentin, petrol, some solvents,

hydrocarbons, water, alcohols,

ester, ketones, oils and fats,

minerals, anorganic salt solutions.

Acids

715

PUR

(AU)

Polyurethane

NPR

HNPR

70 Ozonated oils and fats; strongly

abrasive environments

Neoprene

719

PTFE

Teflon 1)

Polytetrafluoro

ethylene

250 Almost all liquids! Fluorine

1) The name or trade name registered.

2) The maximum temperature of the working fluid (standard values)

3) Only indicative information. For carry fluids seek further details.

Materials and combinations of materials given in Table I for . metal parts , such as

pump housing , rotor shafts , etc and Table II pt . stator are made of any metallic material by

conventional processes .

• Apart from standard materials such as gray cast iron or stainless steel resistant to

oxidation and acid ( major CrNi 18-8 or 18-10 ) special materials specific properties can be

used , eg stainless steel with molybdenum , titanium , niobium or copper nickel alloys with

high purity , such as brass , alpacana , titanium, bronze or light alloys , etc. .

• It is possible to use plastic or hard rubber in the construction of the pump body and

intake chambers . Actionarepot impeller or shaft as hard chrome or covered with various

materials in the connection .

It should be borne in mind that the Cr -Ni steels resistant to corrosion and acid can not

be hardened . The rotors of these materials require a hard layer of chromium which protects

them if you carry abrasive or corrosive environments .

As materials for stator and protection shirts cardan links offer data in Table II. Stator

can be used more plastics or elastomers , you can use any material with specific properties

depending on the particular case of operation. Usually black elastomer used in the food

industry use white elastomers .


76

Table II is basically a guide for the use of different materials . stator and offers only

informative data . A better choice can be made using tables of strength and experience. In

case of doubt, have sent a sample of our material testing laboratory .

Table II - Materials elicoidal eccentric rotor pumps ( data information )

1 )

standard

material

Pump body and

intake, exhaust

flange

Rotor Shafts Drive

housing

sub

assembly

Standard version 1) by

type

H L R S K

V

M V B V

B

1 IRON GGL 25 Steel

17% Cr

1.4122

Steel

13% Cr

1.4021

IRON

GGL 25

®

®

®

1A STEEL 1.0308 Otel

1.0308

®

®

®

4 IRON GGL 25 Steel Cr-Ni-Mo-Ti

1.4571

IRON

GGl 25

®

®

4A HDPE (polyethylene) SteelCr-Ni

1.4301

HDPE

(polyethyl

ene)

®

10 STEEL 1.0308 Steel

17% Cr

1.4122

Steel

Cr-Ni

1.4301

IRON

GGL 25

®

®

11 Cr-Ni 1.4301 Steel Cr-Ni-Mo-Ti

1.4571

®

®

®

®

12 Cr-Ni 1.4408

13 Cr-Ni 1.4308 Steel Cr-Ni

1.4301

®

®

®

15 Brass C 2.4602

18 STEEL Cr-Ni-Mo

1.4408

Brass 2.4602

19 G-Al si 7 Mg

3.2374.6

SteelCr-Ni 1.4301

®

1 ) standard material that is in every model sold . Using other materials than those tabulated

may be made to almost all models to the pump. Combinations of materials of different parts

are possible in most cases.

The presence of solid particles in the fluid pump products reduce the battery usage and

efficiency of the pump coil.

The rotation of the rotor inside the stator rough and smooth soft and elastic leads to a

very good tolerance to pump sand and abrasive particles. Solids are suspended in cavities and

forward through the pump once the fluid displaced. Any particle of sand trapped between the

rotor and stator is pressed within elastic elastomer without damaging the pump. Figure 3 are

the wear on careo data may suffer rotor abrasion conditions.


77

CONCLUSIONS

The study presented aims to warn of problems encountered in practice. PCP pumps

shows the same type of loss of durability regardless of the company that produces them. It is

significant correlation between the factors and factors of influence exploatatre, materials,

manufacturing technology, speed drive.

PCP pumps are extensively applied in operation. The choice of materials is very

important to take into account compatibility with the fluid in the tube. It is necessary to know

the physical and chemical properties of petroleum vehicle. You know the content of sand and

elastomer compatibility of gases or water .. u poatearde and stator is necessary to choose the

correct speed drive. Gases can diffuse into the elastomer due to aromatics. Pentre rotor speed

is chosen depending on the viscosity of the oil and sand content.

BIBLIOGRAPHY

1.*** Catalog Geremia.

2.*** Catalog Griffin-Legrand.

3.*** Catalog PCM Moineau Oilfield.

4.***Catalog Weatherford Eastern Europe SRL.

5.*** Composite Catalog of Oil Field Equipment and Services 1998-1999, World Oil a Gulf

Publishing Company Publication.

6.***Documentaţie Premium Artificial Lift Systems Ltd.

7.*** Documentaţie Weatherford Eastern Europe SRL.

8.Dumitru V., "Contribuţii la tehnologia extracţiei petrolului prin sonde în pompaj de

adâncime cu pompe cu construcţie specială".

9.Gabor Takacs., "Modern Sucker-Rod Pumping", Penn Well Publishing Company, Tusla,

1993.

10.Nelik, L., Brennan, J., Progressing cavity pumps, downhole pumps and mudmotors, Gulf

Publishing Company, Houston, 2005.

11. Stan Marius, Fiabilitatea sistemelor si aplicatii, Editura Universităţii Petrol – Gaze din

Ploieşti, ISBN 978-973-719-249-3, 2010

12. Stan Marius, SUSTAINABILITY ASPECTS OF PROGRESSIVE CAVITY PUMPS

USED IN OIL PRODUCTION, Fiabilitate si Durabilitate - Fiability & Durability Supplement

no 1/ 2013, Editura ―Academica Brâncuşi‖ , Târgu Jiu, ISSN 1844 – 640X

13. ***http://blog.kudupump.com, Three Most Common Pcp Failures In A Cbm Well

Posted By Kerri Clarke On Tue, May 06, 2014 @ 02:14 Pm

http://blog.kudupump.com/


78

USING LOGIC PROGRAMMING FOR IMPROVE AND INCREASE

THE RELIABILITY OF TOOLS AND EMBEDDED MACHINE TO

AVOID SOME “MISSION CRITICAL „ IN FLEXIBLE

MANUFACTURING LINES

Lecturer PhD Gheorghe MARC, ―1 Decembrie 1918‖ University of Alba Iulia,

[email protected],

Asist PhD Maria Loredana BOCA, ―1 Decembrie 1918‖ University of Alba Iulia,

[email protected]

Abstract: The management of automation systems, including control system for tools machines can be

done in three ways:

-logical wired

-logic programming

-and mixed, both of them, wired and logical programming

Nowadays, we still use in manufacturing lines logical wired, automation made by using relays and

contactors. In this way the reliability is very low and leads to a high rate of defects and therefore breaks

the functionality of the entire manufacturing process, sometimes with significant losses in productions. This

entire possible problem can lead to “mission critical” situations in the manufacturing lines.

In this paper we try to design some solutions to grow the reliability of flexible manufacturing lines by using

microcontroller, PLC - Programmable Logic Controller and other embedded tools.

Keywords: Embedded, automation, manufacturing lines, mission critical

INTODUCTION

Nowadays, in the manufacturing lines still exist many machines whose command is made

in wired logic, automation being done with relays and contactors. Their low reliability leads

to a high rate of defects and therefore disruptions in operations, sometimes with significant

losses in production [1].

Re-designing of the power switchboards, control and plants automation by using PLC‘s,

microcontrollers, replacing of limitation and sensors with automation tools, usually without

switching tools, leads to a significant increase of working reliability.

In practice the aim is to maximize the effectiveness of equipment and technological lines

by maintenance works, respectively by proper maintenance with repairs at specify time, and

always using spare parts and maintenance materials of a good quality [3],[4].

Depending on the number of operating hours, the planned repairs are classifeid in the

following categories:

technical review (RT)

Current repairs I (RC1)

Current repairs II (RC2)

Total repairs (RK).

The main objectives of maintenance operations are:

Avoiding accidents and accidental disruptions of the production flow




79

Reducing of production losses through accidental interruption of the technological

equipments

Avoiding wastage as result of disturbance and incorrect adjustments of

measurements

The possibility of estimations cost of repairs and maintenance[6]

As is already known, this type of preventive maintenance is done with certain costs,

which generally depend on the intervention tips, and the age of the machine.

DESCRIPTION OF TOOL MACHINE IN TECHNOLOGICAL FLOW

Aanalyzing of maintenance operations cost related to the repair cycle may cause a

decision to modernize equipments and technological lines, which lead to a more reliability of

the that equipments [7][8].

Driving classic electrical equipment of a tool is the most exposed to accidental falls that

can cause unexpected disruptions to the technological process. This can happened because of

a large number of switching elements, such as contactors, relays, pushbuttons, limit switches,

position sensors on the one hand, and on the other hand, operation conditions of the industrial

environment as dust, high humidity, etc [9],[10].

The aim of this paper is automation of a tool for protective foil of a pallet bagging line in

manufacturing of construction materials.

Figure 1 present a graphical panel of technological flow which shows the execution tool

where we make some interventions.

Fig. 1 - The graphical panel of the technological flow


80

The next figure (figure 2) shows one part of electrical panel scheme of the manufacturing

tool.

Fig. 2 - Electrical panel scheme of the manufacturing tool

To achieve power and automation scheme we establish the placing bags operation

flowchart of the machine shown in Figure 3, from which we can obtain the wired logic

program with PLC [1],[2].


81

Analyzing the complexity of operating scheme related to operation hours and the

matching of the tool in the current technological process, we chose to use a programmable

equipment from Moeller Easy PLC 700 series and replacing all position sensors and limit

switch with inductive and optical sensors.

Fig. 3 - The placing bags operation flowchart

Automation part is composing of sensors, PLC and control scheme achieved 719 AB

Easy. RC is a module of Easy700 series. Commander SK Inverter is a product of Emerson-

Control Techniques used to control AC motor. We used inverter to have effective control of

motor speed required in precise settlement of the film on the pallet, clamping system,

handling and fixing film.

This requires a high accuracy due to requirements imposed by the small size less than 1

mm foil and low stiffness.


82

In this system we used small pneumatic components, filters, dispensers having extended

facilities in technologies of precision mechanics and robotics.

Following the automation of equipment and replacement of power panel we present in

Figure 4 automation and control scheme made using programmable tool PLC Easy719.

Fig.4 - Automation and control scheme made using programmable tool PLC Easy719

CONCLUSIONS

Using wired logic by introducing PLC and variable speed drives, in rehabilitating and

upgrading of control machine and machine tool drives, leading on the one hand to increase the

reliability in operation, on the other hand the performance and increased efficiency.

Following the practical research and implementation of the solution presented, machine has

been running for over a year without any failures to register.

FUTURE WORK

Encouraging results give us the idea to expand our research to the design of modular

and flexible automation systems to obtain complex control and actuation systems with

minimal cost and in a short time.


83

These systems can be provided including self-diagnosis system for easy interventions

in case of failure, resulting in decreased time stationary and also with counting times of

running for a rigorous tracking of planning maintenance and repairs.

REFERENCES

[1] Allen-Bradlez , RON BLISS.Introducţion to IEC1131-3 Ladder Diagram

[2] Ardelean.I, Petrescu.L, Giuroiu.H, Circuite integrate CMOS, Editura Tehnica,

Bucuresti 1986,

[2] Balut Lucian , Circuite Electronice, Editura SIGMA TRADINGMETAFORA,

Constanta, 1999

[3] Balut Lucian, Componente si Dispozitive Electronic, EdituraLEDA, Constanta , 1997

[4] Festila L, Electronica digitala II , Circuite Logice Secventiale –LITO UTCN, 1994

[5] Haba C.G., Sisteme de comandă a maşinilor electrice, Ed.Gh.Asachi, Iaşi

[6] Hugh Jack. Automating Manufacturing Systems (with PLC,s), Ladder Logic,

Programing, PLC hardware, inputs and outputs, relays, logical sensors, OPTICAL

(Photoelectric) Sensors, pneumatics Components, 2005;

[7] Maghiar Teodor, Mircea Calugareanu, Constantin Stanescu, Karoly Bondor,

Electronica Industriala, Editura UNIVERSITATII DIN ORADEA, Oradea 2001

[8] Pasca S. , Tomescu N, Sztojanov I., Electronica Analogica si digitala (Vol I,II, III )

[9] ***, Comander SK- AC variable speed drive for 3 phase induction motors from 0,25 kw to

7,5 kw,0,33hp to 10 hp. Model sizes A,B,C, and D, 2008

[10] CONTROL TECHNIQUES WWW.controltechniques.com

http://www.controltechniques.com/


84

CONSIDERATIONS IN DETERMINING ANALYTIC GRAPHICS

FUNCTIONAL PARAMETERS OF MARINE PROPULSION ENGINES

Prof. univ. dr. ing. Anastase PRUIU, Academia Navală „Mircea cel Bătrân‖,

[email protected]

Prof. univ. dr. ing. Traian FLOREA, Academia Navală „Mircea cel Bătrân‖,

[email protected]

Instructor principal drd. ing. Daniel MĂRĂȘESCU, Academia Navală „Mircea cel Bătrân‖,

[email protected]

Lect. univ. dr. Adriana SPORIȘ, Academia Navală „Mircea cel Bătrân‖,

[email protected]

Abstract: In this paper the authors, present the main relationships between the functional parameters which

allow a correct interpretation of the functional parameters determined experimentally with their monitoring

programs.

Keywords: functional parameters, monitoring

1. RELATIONSHIP BETWEEN FUNCTIONAL AND GEOMETRICAL

PARAMETERS OF MARINE INTERNAL COMBUSTION ENGINES

Operation of machines and equipment in terms of naval safety is imposed by the

operating manual of the designer and manufacturer on and through compliance with the rules

of classification societies, these rules required by the IMO conventions and codes.

Mechanical engineers from the ship's complement, according to its competences, should

be on every day to determine the functional parameters of the propulsion engine to perform,

to make the necessary adjustments to reduce pollution of the marine environment and

efficient use of energy on the ship.

IMO certificates: Cargo Ship Safety Equipment Certificate; Certificates in compliance

with the International Safety Management Code for the Safe Operation of Ship and for

Pollution Prevention (ISM Code):

a) Document of Compliance (DOC)

b) Safety Management Certificate (SMC);

International Air Pollution Prevention Certificate;

International Energy Efficiency Certificate.

To maintain the validity of the statutory certificates issued by the Society, specified

surveys required conventions are to be conducted and endorsement by the society is to be

obtained.

The actual (effective) power of the engine:

kW60

12

4

2

e in

SD

pP mim

kW60

12

4

2

minen in

SD

pP nmn

rel.(1) Nominal

conditions






85

kW60

12

4

2

eop in

SD

pPop

miopmn

Operating

conditions

where:

Nominal conditions Operating conditions

The mechanical efficiency of the engine mn mop

The mean indicated pressure

2min

m

kNp

2m

kNmiopp

The speed of rotation nn [rev/min] nop [rev/min]

The number of cycles per second 60

12

nn

60

12

opn

The number of cycles per minute

nn2

opn2

The number of cycles per hour 602

nn 60

2

opn

D [m]- cylinder diameter (bore); S [m] - piston stroke; η- the number of strokes of the piston

or racing in the motor cycle is performed : η = 2 for engines 2 strokes; η = 4 for engine in 4

strokes;

The actual power of the engine can be determined with the relationship:

The effective power of the engine Nominal conditions Operating conditions

kW3600

eih

e

QCP

kW

3600en

ihnen

QCP

(2) kW

3600eop

ihop

eop

QCP

where:


The effective efficiency of the engine en eop

The fuel consumption per hour

h

kgfuelhnC

h

kgfuelhopC

kgfuel

kJiQ - The power lower calorific value (LCV);

The air consumption for

combustion per hour Nominal conditions Operating conditions

h

airkg

minaerchac mCC

h

airkg

minaerchnhnacn mCC (3.1

)

h

airkg

minaerchophopacop mCC


86

The air consumption per hour

for charger Nominal conditions Operating conditions

h

airkg

minaerchhach mCC

h

airkg

minaerchnhnachn mCC (3.2)

h

airkg

minaerchophopachop mCC

where:


The excess air coefficient for charger α ch α chn α chop

The excess air coefficient for combustion α c α cn α cop

α- The excess air coefficient; maermin

kgaer

kg cb

-the theoretical amount of air required to burn

one kilogram of fuel;

Results:

The fuel consumption per hour Nominal conditions Operating conditions

h

fuelkg

minaerc

ach

m

CC

h

fuelkg

minaercn

acnhn

m

CC

(4)

h

fuelkg

minaercop

acop

hopm

CC

The relationship (2) becomes:

The effective power of the engine

kW3600

min

ei

aerc

ace

Q

m

CP

kW3600

min

eni

aercn

acnen

Q

m

CP

(5) Nominal

conditions

kW3600

min

eopi

aercop

acop

eop

Q

m

CP

Operating

conditions

or:

The air consumption per hour for combustion

h

airkg3600min

ei

aerc

eacQ

mPC

h

airkg3600min

eni

aercn

enacnQ

mPC

(6) Nominal

conditions

h

airkg3600min

eopi

aercop

eopacopQ

mPC

Operating

conditions


S

aerc

iiVmi

m

Qp

min

2m

kN


87

Sn

aercn

iinVn

m

Qp

min

min

2m

kN

(7) Nominal

conditions

Sop

aercop

iiopVopmiop

m

Qp

min

2m

kN

Operating

conditions

where:

ηV - the coefficient of filling of the cylinder with air; ηi - the indicated efficiency; S

kgaer

m3

-

the air density for supercharger;

The compression of supercharging air is done after a trial polytrophic with exponent

polytrophic nc.

Either (po; To) the environmental parameters that determine air density.


3m

airkg

o

oo

TR

p

3m

airkg

n

nn

TR

p

(8)

3m

airkg

op

op

opTR

p

where:

2m

kNop - the ambient air pressure; R

kJ

kg K

- the constant air; To K - the ambient air

temperature.

Either (ps ; Ts) the air compressor parameters.

The air supercharger density: Nominal conditions Operating conditions

3m

airkg

s

ss

TR

p

3m

airkg

sn

snsn

TR

p (9)

3m

airkg

sop

sop

sopTR

p

Results:

p po on

s snc c v v (10) or p po

on

s

snc c

(11) and cn

o

sos

p

p1

(12)

Taking into account the relationship (12), relationship (7) becomes:


cn

o

so

aerc

iiVmi

p

p

m

Qp

1

min

2m

kN

cn

n

snn

aercn

iinVn

p

p

m

Qp

1

min

min

2m

kN

(13) Nominal

conditions


88

cn

op

sop

op

aercop

iiopVopmiop

p

p

m

Qp

1

min

2m

kN

Operating

conditions

By entering the relationship (13) in the relationship (1) is obtained:

The effective power

kW60

12

4

2

1

min

e in

SD

p

p

m

QP

cn

o

so

aerc

iiVm

kW60

12

4

2

1

min

en in

SD

p

p

m

QP n

n

n

snn

aercn

iinVnmn

c

(14) Nominal

conditions

kW60

12

4

2

1

min

eop in

SD

p

p

m

QP

opn

op

sop

op

aercop

iiopVopmop

c

Operating

conditions

2. SHIP PROPULSION POWERS [6]

For propulsion, the effective power of main engine is transmitted through an axial line

shaft and Stern propeller at hull of ship, at the movement speed Vs.

PeME –the effective power for main engine; PD –the power delivered to propeller; PT –

The thrust power; Ptow –the towing power;

PeME sh = PD; sh - the shaft efficiency

PD PR = PT; PR - the propeller and propulsive efficiency

PT H =Ptow;H - the hull efficiency

PeME HPRSH = Ptow

3. REFERENCE AMBIENT CONDITIONS

Functional parameters defined in certain circumstances prevail with the effective

nominal power at nominal speed nominal parameters are functional. It uses index \"n\".

REFERENCE CONDITIONS

1 2 3 4 5 6 7 8 9

Ambient air temperature (0C) 45 25 25 10 15.6 20 25 15 15

Ambient air pressure (bar) 1 1 1 1 1.01325 1.01325 1.01325 1.01325 1.01325

Sea water temperature (0C) 32 25 25 10 - - - - -

Relative humidity (%) 60 60 30 60 - - - 0 -

1 - IACS M28 (1978); Tropical conditions at sea level; 2 - ISO 3046-1-2002; 3 - ISO 3046-

1-2002(E); ISO 15550- 2002E; 4 - Winter ambient reference conditions; 5 - STP - Standard

Temperature and Pressure (USA); 6 - NTP- Normal Temperature and Pressure; 7 - SATP –

Standard Ambient Temperature and pressure; 8 - ISA- International standard Atmosphere; 9 -

ICAO-Standard Atmosphere International Civil Aviation Organization


89

For 1;2;3;4; on use the fuel with a power lower calorific value (LCV) equal to 42,70

kJ/kg. For specific fuel consumption values refer to brake power.

For propulsion engines, by contract, determine the functional parameters from the

manufacturer (manufacturer) or in the shipyard where the engine compartment mounting

follow the cars with the participation and under the supervision of designer companies for

classification of IACS (International Association of Classification Societies).

Ambient environmental conditions will be those of the inspections, and fuel-diesel fuel

supplied from analyses elementary and lower calorific corresponding rules for fabrication of

refineries.

Follows, according to the contract, the conditions for recalculation of the functional

parameters of reference.

The functional operating parameters are functional parameters other than nominal ones.

It uses the index: \"exp\" or \"op\".

Functional parameters defined in cylinder engine are called functional parameters

specify. (subscript\"i\"). Define the functional parameters of the coupling flanges of the motor

with an axial line are called actual functional parameters (index \"e\")

4. PERFORMANCE CHARACTERISTICS OF THE ENGINE

Performance characteristics of the MAN B&W L60MC-

C8engine. Performance curves without VIT [7]

Performance characteristics of the MAN B&W

L60MC-C8 engine. Performance curves with VIT [7]


90

5. CONCLUSIONS

1. The relationship (14) in order to determine the engine power allows to analyses the

influence of bands such as: quality of supercharger air route: The pipe of supercharger air

quality, quality of processes in engine cylinder, fuel quality, Elementary analysis of fuel, air-

fuel mixture, environmental parameters, turbocharger air pressure, compression ratio, load

indicator and use marine heavy fuel oil economy.

REFERENCES

[1] - Anastase PRUIU - Instalaţii energetice navale, Editura Muntenia, Constanţa, 2000

[2] - Alexandru DRAGALINA - Motoare cu ardere internă. Vol I şi II, Ed. Academia

Navală, 2003

[3] - Gh. DUMITRU – Motoare cu ardere interna, Universitatea din Galati

[4] - Nicolae BERECHET – Teza de doctorat, Academia Tehnică Militară 2007

[5] - Anastase PRUIU – MAI- Procese, caracteristici si supraalimentare, Universitatea

Maritima Constanta

[6] -Basic Principles of Ship Propulsion MAN diesel & Turbo

[7] -MAN B&W L60MC-C8 Project Guide

[8] - Marine Engine IMO Tier II Programme 2013


91

ABOUT ENGINE ROOM VENTILATION ON MERCHANT VESSELS

Prof. univ. dr. ing. Anastase PRUIU, Academia Navală „Mircea cel Bătrân‖,

[email protected]

Prof. univ. dr. ing. Traian FLOREA, Academia Navală „Mircea cel Bătrân‖,

[email protected]

Instructor principal drd. ing. Daniel MĂRĂȘESCU, Academia Navală „Mircea cel

Bătrân‖, [email protected]

Lect. univ. dr. Adriana SPORIȘ, Academia Navală „Mircea cel Bătrân‖,

[email protected]

Abstract: Engine room ventilation must ensure the quality and quantity of air for breathing, as well as the

amount of air necessary machines and thermal installations and for heat flow transmitted through the warm

surfaces of machines and naval installations. Through the assumptions and calculations, the authors allow those

interested in a quick and correct interpretation of the flow and temperature for supercharger of marine diesel

engines in all navigation conditions.

Keywords: engine room, ventilations

1. ATMOSPHERE OF EARTH

The earth's atmosphere has a mass of about 4,9 · 10e18 kg and the population

7 · 10e9 persons, so that each person has the a mass of air per person 7 · 10e8 kg/person.

Air composition Table 1

Nitrogen (N2) 78,0842% Helium (He) 0,000524%

Oxygen (O2) 20,9463% Methane (CH4) 0,0001745%

Argon (Ar) 0,93422% Krypton (Kr) 0,000114%

Carbon dioxide (CO2) 0,03811% Hydrogen (H) 0,000055%

Water vapor ≈ 1% Neon (Ne) 0,001818%

Air requirement and production CO2 Table 2

Consumer Power mc/day kg/day Kg CO2/day

MAN (0,1-0,5) kW/man 15 18 0,8

The population

of earth 2450000000 1.05E+11 1.26E+11 5600000000

Marine diesel

4T i=1-28V

(50-3000)

kW/cil 17 200 000 20 640 000 1 362 240






92

Marine diesel

2T i=1-14L

(200-7000)

kW/cil 20 000 000 24 000 000 1 584 000

Marine boilers

i=1-12

(200-50000)

kW/unit 24 000 000 28 800 000 5 184 000

Marine steam

turbines

i=1-4

(1000-70000)

kW/unit 28 500 000 34 200 000 6 155 000

Marine gas

turbines

i=1-6

(1000-80000)

kW/unit 195 000 000 234 000 000 10 530 000

Table 3

Ambient air

temperature

(0C)

Relative

humidity

(%)

Pressure of

scavenge air

(bar)

Temperat

ure after

the

scavenge

air (0C)

Water

content

(kgwater/

kg dry air)

Water

content

(kgwater/

kg dry

moist)

The flow of

water from the

humid air motor

consumption

Kg water/h

10 60% 3.5 16 0.009 0.00892 8920

25 40% 3.5 30 0.008 0.007937 7937

30 60% 3.5 40 0.015 0.014778 14778

35 60% 3.5 50 0.022 0.021526 21526

45 60% 3.5 59 0.035 0.033816 33816

Numerical application are made to environmental parameters in reference conditions,

engineers task is to make adjustments for ambient conditions in operations conditions.


93

The cooling occurs condensation of water vapor in the air, maximum water production is

reflected in the table above. It is necessary to periodically purging of air coolers (1-4 times on

watch = 4 hours), therefore is necessary automatic purging.

Health effects from lack of oxygen

Table 4

Health effects from lack oxygen

O2 level Effects

22% Oxygen enriched atmosphere

20.8% Normal level – Safe for Entry (±0.2%)

19.5% Oxygen deficient atmosphere

16% Impaired judgment and breathing

14% Rapid fatigue and faulty judgment

11% Difficult breathing and death in a few minutes

Low pressure fan p =60-100 mmcw; p =0.588- 0.98 kN/ m2

Table 5

Marine

Equipment Power Kg/day mc/day

Flow of fan

Ventilation

kg/day

Flow of fan

Ventilation

mc/day

Marine

diesel 4T

i=1-28V

(50-3000)

kW/cil 20640000 17200000

30960000

41280000

25 800 000

34 400 000

Marine

diesel 2T

i=1-14L

(200-7000)

kW/cil 24000000 20000000

36000000

48000000

30 000 000

40 000 000

Marine

boilers

i=1-12

(200-50000)

kW/unit 28800000 24000000

42000000

56000000

36 000 000

48 000 000

Marine

steam

turbines

i=1-4

(1000-70000)

kW/unit 34200000 28500000

51300000

64400000

42 750 000

57 000 000

Marine gas

turbines

i=1-6

(1000-80000)

kW/unit 234000000 195000000

351000000

468000000

292 500 000

390 000 000


94

Table 6

Marine

Equipment Power

Flow of

fan m3/s

Power el.motor

of fan kW

f =0.8; p =60

mmcw

p =0.588 kN/

m2

Power el.motor of

fan kW

f =0.8; p =100

mmcw

p =0.980 kN/ m2

Number

of fan

Marine diesel

4T i=1-28V

(50-3000)

kW/cil

298.6111

398.1481

216.4931

288.6574

365.7986

487.7315

6 – 8

Marine diesel

2T i=1-14L

(200-7000)

kW/cil

347.2222

462.963

251.7361

335.6481

425.3472

567.1296

6 – 8

Marine

boilers

i=1-12

(200-50000)

kW/unit

416.6667

555.5556

302.0833

402.7778

510.4167

680.5556

8 – 10

Marine

steam

turbines

i=1-4

(1000-70000)

kW/unit

494.7917

659.7222

358.724

478.2986

606.1198

808.1597

8 – 10

Marine gas

turbines

i=1-6

(1000-80000)

kW/unit

3385.417

4513.889

2454.427

3272.569

4147.135

5529.514

12 – 16

Where:

Power shaft of fan f

p

antF

FP

[kW]; (1)

Massic flow of fan

s

KgairCF ach)2..5.1(1 (2)

Flow of fan

s

mFF

air

3

1

(3)

Flow air of thermal machinery

s

Kgairm

CC airch

hf

ach min3600

(4)

The excess air coefficient for charger - αch; the theoretical amount of air required to burn one

kilogram of fuel -

f

airmkg

kg air

min; Efficiency of fan f = 0.7-0.85; Air density –

3m

kgair ; Fuel consumption of thermal machinery -

h

kgC

f

hf ; Pressure fan


95

p =0.588- 0.98 [kN/ m2];

Engine room temperature TER and difference ΔT

The engine room temperature TER and the engine room/ambient air temperature

difference ΔT are shown as functions of the ambient air temperature Tamb [1]

ISOSMCRscsc pp /. Bypass opens when scp exceeds offsetsomeISOSMCRscp /.

ISOscsc pp . Bypass opens when scp exceeds offsetsomeISOscp . [1]


96

Reference ambient conditions

Tabel 7

1 2 3 4

Ambient air temperature (0C) 45 25 25 10

Ambient air pressure (bar) 1 1 1 1

Sea water temperature (0C) 32 25 25 10

Relative humidity (%) 60 60 30 60

1 - IACS M28 (1978); Tropical conditions at sea level; 2 - ISO 3046-1-2002;

3 - ISO 3046-1-2002(E); ISO 15550- 2002E; 4 - Winter ambient reference conditions;

BIBLIOGRAPHY

[1] - Influence of Ambient Temperature Conditions on Main Engine Operation

of MAN B&W Two-stroke Engines

[2] - Caterpillar Marine Engines Application and Installation Guide

[3] - Engine Selection Guide Two-stroke MC/MC-C Engines

[4] - Marine and Machinery Systems and Equipment OCTOBER 2013

[5] - Wartsila Engines - Project Guide

[6] - INTERNATIONAL ASSOCIATION OF CLASSIFICATION SOCIETIES –

Requirements concerning MACHINERY INSTALLATIONS

[7] - MAN B&W L60MC-C8 Project Guide

[8] - Marine Engine IMO Tier II Programme 2013


97

MATHEMATICAL MODEL FOR DETERMINING KINEMATIC

PARAMETERS OF THE CARDAN JOINT MECHANISM WITH

TECHNICAL (GEOMETRICAL) DEVIATIONS

Ion BULAC

Doctor, University of Pitești, email: [email protected]

Abstract: The cardan joint mechanism is derived from the 4R spherical quqdrilateral mechanism

and a particular case of spatial quadrilateral mechanism RCCC where by C, R was noted the

cylindrical kinematic pair respectively the rotation kinematic pair. In this paper are deducted the

calculation relations , is established the mathematical model for determining the the kinematic

parameters of the cardan joint mechanism with technical (geometrical) deviations.

Keywords: cardan joint, kinematic pair, quadrilateral mechanism, cardan transmission .

1. INTRODUCTION

The technical (geometrical) deviations lead to the changine of kinematic parameters

of the cardan joint mechanism. This deviations is established in paper [3].

For determining the influence of both angular and axis deviations [1]., over the

kinematic parameters of the cardan joint mechanism, it is necessary to consider the cardan

joint, not as a spherical quadrilateral but as a particular case of RCCC mechanism. where by

C, R was noted the cylindrical kinematic pair respectively the rotation kinematic pair.

In this paper are deducted the calculation relations , is established the mathematical

model for determining the kinematic parameters of the cardan joint mechanism with technical

(geometrical) deviations.

2. THE POSITIONAL ANALYSIS OF THE RCCC MECHANISM

The RCCC mechanism (see Figure 1) is made of four elements noted with 1, 2, 3 and

4, the forth element (the base) being fixed and the elements being connected through the

kinematic pairs 321 ,, OOO , and 4O , the 1O being the rotation kinematic pair and 32 ,OO ,and

4O being the cylindrical kinematic pairs.

The axes of the kinematic pairs are noted with ,...2,1, izO ii and the following

perpendiculars are noted with 4,3,2,1,1 iOO ii ,point 5O being identical with point 1O .

One notates with i , i , 4,3,2,1i the length of the axes and the angle between them.

So it is chosen a local reference system iiii zyxO , 4,3,2,1i so that the axes ii xO to be

situated on the shared perpendicular of the axes ii zO , 11 ii zO . It is noted with is the

distances iiOO and with i the angle between the axes 11 ii xO , ii xO , 4,3,2,1i .



98

Fig. 1. RCCC Spatial Mechanism

In these conditions, the geometrical parameters 4,3,2,1,,, is iii being known, the

positional analysis for determining 432432 ,,,,, sss is based on the angle 1 .

From the equation of rotations closing, using the diagram ‖„ [7] and the order

3,4,1 and 2 is obtained the following equation:

0)()()( 13413413 CcBsA (1)

where :

2143114313

4114313

11313

)(

()(

)(

ccccscscC

cssccsB

sssA

( 2)

The trigonometrically functions cos, sin being noted with c, s. Through the

conventional derivate D of the relations (1), (2) having as basis the relations [7].

iiiiii cssDsscD )(;)( ( 3)

csDscD iiiii )(;)( ( 4)

is obtained the equation:

0433323131343 KHGFsFsD ( 5)

where :

The angle 4 is determined by solving the equation (1) and through the equation (5) is

known the parameter 4s .


99

With circular permutations the relations follows:

0)()()( 42342342 CcBsA ( 6)

0)()()( 31231231 CcBsA ( 7)

from where are determined, in order, the angles 3 and 2 and also the equations:

0322212424232 KHGFsEsD ( 8)

0211141313121 KHGFsEsD (9)

from which are determined the parameters 23, ss .

The expression of the coefficients iiiiiiiii KHGFEDCBA ,,,,,,,, , i=3,2,1 ,are given

in TABLE 1 from the paper [2].

In the initial position , 00 i the expressions are obtained

2413341333 )(),(,0 cccCssBA (10)

and it results that:

)(

)(

413

241304

ss

cccc (11)

3. THE TECHNICAL (GEOMETRICAL) DEVIATIONS OF CARDAN JOINT

The cardanic joint enables the transmission of the rotation movement from the shaft 3

through the cardanic cross 2.

The cardanic cross is tied to the brackets of the shafts 1 and 2 through the cinematic

rotation couples A , A and also B , B .

Fig. 2. Cardan joint mechanism

In the papers [1], [3] is established the technical (geometrical) deviations of cardan

joint mechanism.

A kinematic diagram that represents a mechanism with one cardan joint, with all

geometrical deviations possible [1], [3]., is presented in Figure. 3.


100

Fig. 3. Geometrical deviations.

These deviations are small and fulfill the condition:

iiii

ii

OOiOO

i

441

4

,2,1,

,3,2,1,2

(12)

The angularly deviation of the main shaft bracket is defined by the parameter 1 and

the smoothness deviations for the same bracket is given by the parameter 1 .

The angularly deviation of the cardanic cross 2 is given by the parameter 2 and also

the deviation from smoothness is given by the parameter 2 .

The angularly deviation of the driven shaft bracket 3 is given by the parameter 3

and the smoothness deviation is given by the parameter 3 .

The angularly deviation of the driven shaft 3 depending on the driving shaft 1 is given

by the parameter 4 .

As shown in the papers [1], [3] in default of shafts 1 and 3 are known the points

43322114 ,,,,,,, OOOOOOOO are overlaid with point O (see Figure. 3.) and the kinematic

cylindrical couples A , B and D (see Figure. 1.) become rotation kinematic couples (there are

no displacements 432 ,, sss along the axes 432 ,, OzOzOz .

The existence of technical deviations conducts to the displacements 4,3,2,1, isi and.

In order to determine these displacements it is first necessary to calculate the angularly

parameters 432 ,, variation depending on the angle 1 .

4. THE DETERMINING PARAMETERS 432 ,,

The angularly parameters 432 ,, is derermined from the system of equations

.3,2,1,0111 iCcBsA iiii (13)


101

To this purpose, one uses the Newton method [10], [11]. and with the notations

.,

4

3

2

4

3

2

(14)

.3,2,1,11 iCcBsA iiiiii (15)

.

3

2

1

(16)

.23211

33211

3311

ssscC

sscsB

cssA

(17)

.44322

44322

4422

ssscC

sscsB

cssA

(18)

4343

232323232

121212121

00

0

0

sBcA

CcBsAsBcA

CcBsAsBcA

J (19)

is obtained the matric equation

1

J (20)

from which results the variation { } for the known values of angles 4321 ,,, .

After the determining the angularly parameters 432 ,, variation depending on the angle 1

.is determined the parameters 432 ,, sss with the relations

)(1

43332313133

4 KHGFsED

s (21)

)(1

32221242422

3 KHGFsED

s (22)

)(1

21114131311

2 KHGFsED

s (23)

where the coefficients .3,2,1,,,,, iKHGFE iiiii are calculated from the TABLE 1. from

the paper [2].


102

5. CONCLUSIONS

The mathematical model presented in this paper, makes possible the study of technological

deviations influence over the kinematic parameters of the cardan joint mechanism.

Using this mathematical model, made possible the elaboration an calculation algorithm that

by numerical simulation can highlight the influence of different technical deviations that

appear in the mechanism elements.

REFERENCES

1. BULAC, I,. Contributions to the study of technical deviations over the dynamic response of

policardan transmissions, Doctoral Thesis, University of Piteşti, 2014.

2. BULAC, I,. Mathematical model for determining kinematic parameters of the spatial quadrilateral

mechanism RCCC (Submitted for publication), SISOM 2014, Bucharest, May 22-23, 2014.

3. BULAC, I,. The technical (geometrical) deviations of cardan joint mechanism (Submitted for

publication), Annals of the Oradea University, 2014.

4. DUDIŢĂ, FL., Cardan shafting, Technical Publishing House, Bucharest, 1966.

5. DUDIŢĂ, FL., FL., DIACONESCU, D., BOHN, CR., NEAGOE, M., SĂULESCU, R., Cardan

shafting, Transilvania Expres Publishing House, Brașov, 2003 .

6. DUMITRU, N., NANU, GH., VINTILĂ, DANIELA, Mechanisms and mechanic shafting, Didactic

and Pedagogical PublishingHouse, Bucharest, 2008.

7. PANDREA, N., Solid mechanics plucheriane coordinates, Romanian Academy Publishing House,

Bucharest, 2000.

8. PANDREA, N., POPA, D., Mechanism, Technical Publishing House, Bucharest, 2000.

9. RIPIANU, A., CRĂCIUN, I., Mechanism, Technical Publishing House, Bucharest, 2000.

10.STĂNESCU, D., PANDREA, N.,, Numerical methods, Didactic and Pedagocical Publishing

House, Bucharest, 2007.

11. TEODORESCU, P., STANESCU, D,. PANDREA, N., Numerical analysis with applications in

Mechanics and Engineering, Wiley, 2013.


103

THE NUMERICAL STUDY OF THE INFLUENCE OF TECHNICAL

(GEOMETRICAL) DEVIATIONS OVER THE KINEMATIC

PARAMETERS OF THE CARDAN JOINT MECHANISM

Ion BULAC

Doctor, University of Pitești, email: [email protected]

Abstract: The cardan joint mechanism is derived from the 4R spherical quqdrilateral mechanism

and a particular case of spatial quadrilateral mechanism RCCC where by C, R was noted the

cylindrical kinematic pair respectively the rotation kinematic pair. In the present paper are

established the connection relations between the technical (geometrical) deviationsl and of the

kinematic parameters of this mechanism .The results of the numerical solving of this problem will

be presented under the form of a diagram and will be commented.

Keywords: cardan joint, kinematic pair, quadrilateral mechanism, technical deviations .

1. INTRODUCTION

For determining the influence of both angular and axis deviations, over the kinematic

parameters of the cardan joint mechanism, it is necessary to consider the cardan joint, not as a

spherical quadrilateral but as a particular case of RCCC mechanism where by C, R was noted

the cylindrical kinematic pair respectively the rotation kinematic pair

In the previous papers [1], [2], [3]., there was elaborated a mathematical model and the

calculation algorithm for the kinematic parameters of the cardan joint mechanism with

technical (geomeyrical) deviations.

In the present paper are established the connection relations between the technical

(geometrical) deviationsl and of the kinematic parameters of this mechanism.

2. THEORETICAL ASPECTS

The RCCC mechanism (see Figure 1) is made of four elements noted with 1, 2, 3 and 4, the

forth element (the base) being fixed and the elements being connected through the kinematic

pairs 321 ,, OOO , and 4O , the 1O being the rotation kinematic pair and 32 ,OO ,and 4O being

the cylindrical kinematic pairs.

From the equation of rotations closing, using the diagram ‖„ [7] and the order

3,4,1 and 2 is obtained the following equation:

0)()()( 13413413 CcBsA (1)

and with circular permutations the relations follows:

0)()()( 42342342 CcBsA ( 2)

0)()()( 31231231 CcBsA ( 3)

the trigonometrically functions cos, sin being noted with c, s.



104

Fig. 1. RCCC Spatial Mechanism

Through the conventional derivate D of the relations (1),(2),(3) and of coefficients

3,2,1.,,, iCBA iii having as basis the relations [7].

iiiiii cssDsscD )(;)( ( 4)

csDscD iiiii )(;)( ( 5)

is obtained the equation:

0433323131343 KHGFsFsD ( 6)

0322212424232 KHGFsEsD ( 7)

0211141313121 KHGFsEsD (8)

The angle 4 is determined by solving the equation (1) and in order, the angles 3 and

2 by solving the equations (2),(3).

From ecuation (6) is determined the parameter 4s and also the equations (7), (8) is

determined the parameters 23, ss .

The expression of the coefficients iiiiiiiii KHGFEDCBA ,,,,,,,, , i=3,2,1 ,are given

in TABLE 1 from the paper [1].

In the paper [2], is established the technical (geometrical) deviations of cardan joint

mechanism.

A kinematic diagram that represents a mechanism with one cardan joint, with all

geometrical deviations possible [2]., is presented in Figure. 2


105

Fig. 2. Geometrical deviations.

These deviations are small and fulfill the condition:

iiii

ii

OOiOO

i

441

4

,2,1,

,3,2,1,2

(9)

The angularly deviation of the main shaft bracket is defined by the parameter 1 and

the smoothness deviations for the same bracket is given by the parameter 1 .

The angularly deviation of the cardanic cross 2 is given by the parameter 2 and also

the deviation from smoothness is given by the parameter 2 .

The angularly deviation of the driven shaft bracket 3 is given by the parameter 3

and the smoothness deviation is given by the parameter 3 .

The angularly deviation of the driven shaft 3 depending on the driving shaft 1 is given

by the parameter 4 .

3. NUMERICAL APLICATION

One considers a cardanic joint (see Figure 3) for which:

.4,3,2,1,0,0,0,001,0,0 1312 israd i (10)

Fig. 3. The cardan joint mechanism


106

On the basis of a calculation program realized in Excel with the mathematical model

presentea in the paper [4], and the algorithm presented in the paper [3], where obtained, in

the case where 0 , for the parameters 432 ,, the results from diagram from Figure. 4.,

Figure. 5., and Figure. 6.

Fig. 4.The variation of rotation angle θ3 Fig.5.The variation of rotation angle θ4

Fig. 6. The variation of rotation angle θ2

For a cardanic joint for which:

4,3,2,1,0,001,0,001,0,0 1 israd ii (11)

The variation graphs are presented in the Figures 7,8,9,10,11,12


107

Fig. 7.The variation of rotation angle θ4 Fig.8.The variation of rotation angle θ3

Fig. 9. The variation of rotation angle θ2 Fig. 10. The variation of displacement s4

Fig. 11. The variation of displacement s3 Fig. 12. The variation of displacement s2

For a cardanic joint for which:

4,3,2,1,001,0,001,0,001,0,0 1 israd ii (12)

The variation graphs are presented in the Figures 13,14,15,16,17,18


108

Fig. 13. The variation of rotation angle θ4 Fig. 14. The variation of rotation angle θ3

Fig. 15. The variation of rotation angle θ2 Fig. 16. The variation of displacement s4

Fig. 17. The variation of displacement s3 Fig. 18. The variation of displacement s2

4. CONCLUSIONS

For the normal cardan join with technical deviations with 0 and radi 001,0 ,

when 1 covers the interval 3600

the angle 4 varies between 45090 ;

the angle 3 varies between 27088,269 ;

the angle 2 varies between 06,9090 .

The influence of i and 1s deviations over the angles 234 ;; are insignificant in value.

The variation of angles i i=1,2,3 does no influence the displacements is , i=2,3,4.

The displacements is , i=2,3,4. are influenced only by the value of the i and 1s

parameters.


109

For 0 , the variation curves form of the kinematic parameters are alike.

REFERENCES 1. BULAC, I,. Mathematical model for determining kinematic parameters of the spatial quadrilateral

mechanism RCCC (Submitted for publication), SISOM 2014, Bucharest, May 22-23, 2014.

2. BULAC, I,. The technical (geometrical) deviations of cardan joint mechanism (Submitted for

publication), Annals of the Oradea University, 2014.

3. BULAC, I,. Calculation algorithm for determining kinematic parameters of the cardan joint

mechanism with technical (geometrical) deviations (Submitted for publication),

Annalsof the Oradea University, 2014.

4. BULAC, I,. Calculation algorithm for determining kinematic parameters of the cardan joint

mechanism with technical (geometrical) deviations (Submitted for publication),7tth

,Durability and Reliability of Mechanical Systems``,Târgu-Jiu, Romania, 23-24 May, 2014.

5. DUDIŢĂ, FL., Cardan shafting, Technical Publishing House, Bucharest, 1966.

6. DUDIŢĂ, FL., FL., DIACONESCU, D., BOHN, CR., NEAGOE, M., SĂULESCU, R., Cardan

shafting, Transilvania Expres Publishing House, Brașov, 2003 .

7. PANDREA, N., Solid mechanics plucheriane coordinates, Romanian Academy Publishing House,

Bucharest, 2000.

8. PANDREA, N., POPA, D., Mechanism, Technical Publishing House, Bucharest, 2000.

9. STĂNESCU, D., PANDREA, N.,, Numerical methods, Didactic and Pedagocical Publishing

House, Bucharest, 2007.


110

CORRELATING 2K-52MU CUTTING AND LOADING MACHINE

WITH TR-5 SCRAPER CONVEYER

PhD, Eng., Consulting Teaching Staff, Răzvan Bogdan ITU, Department of Industrial

Mechanical Engineering and Transport, University of Petrosani, [email protected]

PhD, Associate Prof. Eng., Iosif DUMITRESCU, Department of Industrial Mechanical

Engineering and Transport, University of Petroşani, [email protected]

PhD, Lecturer, Eng., Vilhelm ITU, Department of Industrial Mechanical Engineering and

Transport, University of Petroşani, [email protected]

Abstract: The necessity of mechanization of coal extraction in Jiu Valley mines, in the conditions

of the present economic crisis, involved the adaptation of the existing equipments to the new

conditions of reequipping of certain faces. Thus, within the program of reequipping of a longwall

in Lonea Mining Plant, the adaptation of 2K-52MU cutting and loading machine to TR-5 scraper

conveyer was required. The main problem of adaptation of the two equipments was in the design

and development of systems for affixing the ends of the feeding gear chain of the machine to the

driving and returning stations of the TR-5 scraper conveyer. In solving this problem, the focus was

on ensuring for all the elements of the attaching system of the ends of the machine chain to resist

to the machine’s maximum haulage force of 250 kN, and this strain not to be transmitted to the

metal structure of the stations and be taken over by the hydraulic prop of anchorage of the

conveyer station.

Keywords: correlating, cutting and loading machine, scraper conveyer

INTRODUCTION

To adapt the existing equipments in view of reequipping the faces, one should be deeply

familiar with the structure and technical characteristics of the equipments that are used and of

the way they are correlated with the mining method, the geological-mining conditions specific

to the layer exploited, and with observing the occupational health and safety norms.

By the implementation of a longwall reequipping program in Lonea Mining Plant in the

fourth trimester 2013, the classical method of extraction with a productivity of 6,8 ton/job was

abandoned in favor of longwall extraction method with individual support and cut with a

machine with an estimated productivity of 12 ton/job. A technical documentation was

required to be drawn up in this sense, regarding the adaptation of the affixing program for

the2K-52MU machine to TR-5 scraper conveyer stations. In order to see the correlation of the

equipments, an analysis of the framework extraction method was made for longwalls with

cutting and loading machines. Fig. 1 shows the working stages for a maximum 3,2 m high

longwall and 1,8 m high cut with the machine, as it follows: I – machine in the bay; II –

cutting the first strip; II – manual fixing of the place for mounting the articulated beam; IV –

machine advance in the bay and mounting of the fourth prop; V – cutting the second strip; VI

– manual cleaning up of the face and machine advance in the bay; VII – moving the back prop

under the first beam; VIII – ripping of the back prop and beam.


111

From the above, it results that an analysis of economic efficiency should be made,

regarding the correlation of the longwall height with the cutting height of the machine, with

the geological mining conditions and the occupational health and safety norms. The price of

the 2K-52MU machine, recovered from cassation should be considered, scrap iron – 2500

Euro, rehabilitated and put to operate with maximum 5000 Euro.

Fig. 1. Longwall extraction method with

individual support and machine cutting

The technical characteristics of the 2K-52MU machine were not correlated with those of

the TR-5 scraper conveyer, especially considering that the latter had not been designed for

faces, and they had been manufactured by different companies in different countries. The two

equipments are nevertheless compatible, at least from the following points of view:

- More than half of the theoretical cutting capacity of the machine, which is not achieved

in practice due to the correlation of the advance speed with the geological-mining conditions,

can be handled by the scraper conveyer;

- Size-wise, the machine can be mounted on the conveyer by adopting suitable

modifications, adapting TR-5 stations to chutes, TR-6 with the use of TR-7 laterals, and

modification of machine shoes without diminishing the characteristics of their resistance;

- The conveyer‘s robust design supports the approx. 12 …14 ton mass of the machine;

The design of the scraper conveyer allows coal to be loaded in good conditions by the

auger drums.


112

The main lack of correlation between the two equipments consists in the fact that the

actuating and return stations for the TR-5 conveyer are not foreseen with attaching

hangers/plates of the affixing device of the machine haulage chain end.

AFFIXING SYSTEM TO THE ACTUATING STATION

Fig. 2. Movement of the affixing system on

he actuating station [3]

Fig. 3. Design solution for the affixing system

of the chain end to the TR-5 actuating station[3]

Fig. 2 shows the design solution for the positioning of the Fig. 3. Design solution for the

affixing system of the chain end to the TR-5 actuating station[3] attachment system of the end of

the chain calibrated with 26x92 links of the 2K-52MU machine advance mechanism,

reference point 2, in the metal construction of the actuating station, reference point 1.

The design solution of the affixing system is shown in Fig. 3, which is made up of 1 –

hanger fixed on the station; 2 – lateral connecting plate; 3 – lateral blocking plate; 4 –

blocking bolt Φ50; 5 – blocking bolt Φ60; 6 – articulation bolt Φ60; 7 – device hanger; 8 –

haulage rod; 9 – wear ring; 10 – wear ring support; 11 – chain fixing strap; 12 – centering pin;

13 – chain calibrate with 26x92 links; 14 – M20x100 screw; 15 – M10 screw for greasing; 16

– distance ring; 17 – M42 low washer; 18 – M42 washer; 19 – M24x100 screw.

The affixing system for the chain end to the TR-5 actuating station shows the following

design improvements:

- the hanger fixed to the station, instead of the lifting hanger, leans on the framework of

the station by two distanced shoes and has its own support for the SVJ station anchoring

hydraulic prop, which improves stability of hanger and the way forces are transmitted to the

station;

- overlapping the prop supports avoids the error possibility in placing the anchoring prop

and allows interchangeability of the actuating station;


113

- parallel positioning of the lateral linking plate, reference point 2 (Fig. 3), and rigid

fixing on the hanger through the three bolts Φ50, reference point 4, and Φ60, reference point

5, made a movement outward the chain axis by 120 mm, which improved the movement of

the machine on the conveyer in the area of the actuation station;

- the distance between the wings of the device hanger, reference point 7, by inside

welding of the of two 20 mm thick attachments plates, which improved the strain on the

articulation bolt, reference point 6;

- axial rolling contact bearing with balls 51112 was replaced with a wear ring of cast

iron or bronze, reference point 9, its part being to take over the twist of the chain, and a screw

was provided, reference point 15, for greasing the contact area between the wear ring and the

ring support, reference point 10;

- transversal pins were provided, reference point 12, to center the two bridles in view of

improving the haulage force transmission between the haulage rod, reference point 8, and the

chain link, reference point 13;

- only 20 mm and 60 mm thick sheets were used, which were in stock in Lonea Mining

Plant;

Fig. 4. Calculation model for the connection system

at the actuation station[3]


114

Based on the design solution in Fig. 3, the calculation model was drawn up, with constructive

dimensions in view of dimensional verification of its elements given in Fig. 4, where: 1 –

partition of actuation station; 2 – hanger fixed to the station; 3 – device hanger; 4 –

articulation bolt Φ60x270; 5 – distance ring; 6 – wear ring; 8 – haulage rod Φ57x200; 9 –

bridle; 10 - M20x100 screw; 11 – centering pin Φ20x45; 12 - M24x100 screw; 13 – lateral

connecting plate; 14 – blocking bolt Φ60; 15 – blocking bolt Φ50.

Figure 5. Variation of safety coefficients

of connection system to the anchoring station[3]

Based on the calculation model in Fig. 4, a calculation breviary was drawn up in Math

CAD for the variation of the machine‘s haulage force Ftc between 160 and 250 kN, and the

value of safety coefficients are graphically shown in Fig. 5 for the following constructive

elements [1], [2], [3]:

- lateral connecting plate(Fig. 5a);

- hanger fixed to the station, 2, (Fig. 5b);

-hanger of the connecting device, 3,(Fig. 5c);

- articulation bolt Φ60x245, 4, (Fig. 5d);

- haulage rod Φ57x200, 8, (Fig. 5e);

- attachment bridle for the chain link, 9, (Fig. 5f).


115

Safety coefficients resulted by relating to mechanical characteristics of OL 37, 210

N/mm2 flow limit for sheets and improved OLC 45, 500 N/mm

2 flow limit for bolts. The

lowest values are with bending for bolts, reference point 4, Csib=1,28, and with shear of the

fixing bridle hanger of the chain, reference point 9, Csfub=1,29, these values can be amplified

1,7 times, if the relating is made at tear resistance.

Based on the documentation of execution the system of connection of the machine chain

to the actuating station was carried out in Lonea Mining Plant, Fig. 6.

Fig. 6. Chain connecting system to the actuating station carried out in Lonea Mine[3]

CONNECTING SYSTEM TO THE RETURN STATION

Fig. 7 shows the constructive positioning solution of the connecting system of the chain

end calibrated with 26x92 links of the advance mechanism of the 2K-52MY machine,

reference point 2, to the metal structure of the return station, reference point 1.

Fig. 8 shows the constructive solution of the connecting syste, made up of: 1 – hanger

fixed to the station; 2 – articulation bolt Φ60; 3 – chain connecting device; 4 - M30x100

screw; 5 –M30x200 screw.


116

Fig. 7. Connecting system positioning to

the return station[3]

Fig. 8. Design solution of the chain end

connecting system to the TR-5return station [3]

The chain end connecting system to the TR-5 return station shows the following

design improvements:

- the hanger fixed to the station, to the lifting bolt by a bolt Φ50 and four screws M30, has in

front a rigid threshold by two bolts Φ50, blocking the turning of the hanger against the station

partition;

- on the outside of the hanger the support for the station anchoring hydraulic prop SVJ

was placed, in a welded structure, solider than the one on the return station;

-by overlapping the prop supports, the error of wrong positioning of the anchoring prop

can be avoided;

- in case of changing the return station, part of the gussets for hardening the prop support

should be discharged with oxyacetylene flame, on the return station, on the hanger‘s

mounting part, and the two holes for screws M30x100 should be made in the lifting hanger.


117

Fig. 9. Calculation model for connecting system to return station[3]

Based on the design solution in Fig. 8, the calculation model with constructive

dimensions in view of dimensional verification of its elements, shown in Fig. 9, was drawn

up, where: 1 – station lifting hanger ; 2 – connecting system hanger; 3 – bolt Φ50x100; 4 -

bolt Φ50x120; 5 - screw M30x100; 6 – chain connecting device.

Based on the calculation model in Fig. 9, a calculation breviary in MathCAD was drawn up

for Ftc haulage force variation of the machine, between 160 and 250 kN, and the values of the

safety coefficients are graphically shown in Fig. 10 for the following design elements:

assembling by bolt Φ50x80, 4, and screws M30x100, 5, of the hanger on the station partition,

(Fig. 10a); station lifting hanger, 1, (fig. 10b). Since the values of sliding safety coefficient

Csai are low, an additional blocking of the hanger was carried out against the return station

lifting hanger by bolts Φ50x120, reference point 4, and a vertical rule.


118

Fig. 10. Safety coefficients variation of the connecting system to the return station[3]

Based on the execution documents, a connecting system of the machine chain to the

return station was carried out in Lonea Mine.

Fig. 11. Connecting system of the chain tothe return

station carried out in Lonea Mine[3]

Fig. 12. Mouting the machine

to the conveyer[3]


119

Fig. 11.To check the way of positioning and movement of the 2K-52MU machine on the

conveyer carried out of TR-5 stations, TR-5 chutes and lateral of TR-7A , the assembly in

Fig. 12 was carried out in Lonea Mine Shops, with new systems of connecting the chain ends

to the machines‘ advance mechanism.

CONCLUSIONS

Even if the cutting height of the machine is not correlated with the height of the face,

an improvement of work conditions results for a minimum reequipping of the face. By

adapting the 2K-52MU machine to a hybrid scraper conveyer, TR-5 stations, TR-6 chutes and

TR-7A laterals, experience was gained in the use of these equipments in the exploitation of

longwalls of individual support ant machine cut. In an endeavor to move the machine on the

conveyer, the connecting systems of the chain ends of the advance mechanism to the

conveyer stations were checked. The information obtained as a result of face reequipping in

Lonea Mine, with the advantages and disadvantages of the method application, will allow in

the future to achieve an optimization of the equipments correlation in a shortwall working.

REFERENCES

[1] Dalban, C., ş.a. – Construcţii metalice, Editura Didactică şi Pedagogică, Bucureşti, 1983.

[2] Găfiţeanu, M. şa - Organe de maşini, vol. I şi II, Editura Tehnică, Bucureşti, 1981 şi 1983.

[3] * * * - Documentaţie tehnică privind adaptarea sistemului de legare a combinei 2K-52

MY de staţiile transportorului tip TR-5 şi verificarea stabilităţii combinei pe transportor,

contract 713/ 26.08.2013, E.M. Lonea


120

STUDY OF STABILITY OF 2K-52MU CUTTING-LOADING MACHINE

ON TR-5 CONVEYER IN FACES WITH INDIVIDUAL SUPPORT

PhD, Eng., Consulting Teaching Staff, Răzvan Bogdan ITU, Department of Industrial

Mechanical Engineering and Transport, University of Petrosani, [email protected]

PhD, Associate Prof. Eng., Iosif DUMITRESCU, Department of Industrial Mechanical

Engineering and Transport, University of Petroşani, [email protected]

PhD, Lecturer, Eng., Vilhelm ITU, Department of Industrial Mechanical Engineering and

Transport, University of Petroşani, [email protected]

Abstract: Besides the problem of adapting 2K-52MU machine to la TR-5 scraper conveyer, the

study of stability of the machine on conveyer was required, in faces with individual supports. To

this end stability coefficients of the machine were determined in cutting and loading coal on the

conveyer, and even in blocking the drum at the face. The study was made for maximum 250 kN

traction force and variation of the inclination of the face in the range of 0 and 300 for three cases

of machine movement towards the face.

Keywords: correlating, cutting and loading machine, scraper conveyer

INTRODUCTION

To adapt the existing equipments in view of refurnishing the faces, we should be

familiar with the construction and characteristics of the equipments used and how to correlate

those with the coal extraction mining method, the geological-mining conditions specific to the

exploited layer and with respecting the occupational health and safety standards.

Fig. 1. Extraction method of the longwall with individual support and machine cut


121

By the implementation of a refurnishing program of a longwall in Lonea Mine during

the fourth trimester of the year 2013, the classical extraction method with a productivity of

6,8 ton/job, was replaced with longwall method with individual support and cut with the

machine with an estimated productivity of 12 tons/job.

CORRELATION OF THE EQUIPMENTS

To see how the equipments were correlated, an analysis of the framework method of

longwall method with cutting the coal with the machine was performed [5]. Fig. 1 shows the

work stages for a face of maximum 3,2m high and 1,8 high cut with the machine, thus: I –

machine in the butt pocket; II –cutting the first slice; III – manual execution of the seat to

mount the articulated beam; IV – advance of the machine in the butt pocket and mounting the

fourth prop; V – cutting the second slice. VI – manual clearing of the face and advance of the

machine in the pocket butt; VII – moving the back prop under the first beam; VIII – ripping

the prop and the back beam.

From the above, it results that an economic efficiency analysis should be made

regarding the correlation of the longwall height with the cutting height of the machine, with

the geological-mining conditions and the occupational safety standards. In this analysis, the

price of the machine should be taken into consideration, which was recovered from cassation,

scrap iron – Euro 2500, rehabilitated and put to use with maximum Euro 5000. The technical

characteristics of the 2K-52MU machine were not correlated with those of the TR-5 scraper

conveyer, especially considering that the latter had not been devised for longwalls, and they

were designed and manufactured by different companies in different countries. The two

equipments are however compatible, at least from the following points of view [1], [5]:

- more than half of the theoretical capacity of cutting of the machine, which is not

reached in practice due to the correlation of the heading speed with the geological-mining

conditions, can be supported by the scraper conveyer;

- dimensionally, the machine can be mounted on the conveyer by adopting suitable

modifications, adapting the TR-5 stations to TR-6 chutes by using TR-7A laterals and by

modifying the machine shoes, without diminishing their resistance characteristics;

- the conveyer‘s robust construction withstands the machine weight of approximately 12

…14 ton;

- the design of the scraper conveyer allows the coal loading by the auger drum in good

conditions;

The main non-correlation of the of the two equipments lies in the fact that the driving

and turning stations of the TR-5 scraper conveyers re not provided with fixing plates/hooks

for the linking device of the end of the traction chain end of the machine.

STUDY OF MACHINE STABILITY ON THE CONVEYE

Starting from the maximum transportation capacity of the TR-5 scraper conveyer of

4,1 ton/min, and knowing the width of the slice that has been cut of 0,63m and its thickness of

1,9 m, the maximum heading speed of 2,45 m/min was determined.


122

Knowing the design characteristics of the machine‘s auger drum (diameter, number of

cutting lines, number of bits per face and corner on a cutting line, the width of the cutting

edge of a CMR3 bit) and the 42,7 rpm rotation of the drum, the average depth of the cut was

determined, made by a 0,029 m face bit, and maximum 0,045 m [2], [4].

Fig. 2. Variation of the machine’s heading force and the motor power function

of the inclination angle of the longwall.

With the help of these data, the cutting forces on the auger drum of the machine were

calculated, of 18,9 kN for the drum cutting on the roof and in compact seam, and 10,78 kN for

the drum cutting the floor for 0,7 m high.

The variation of the machine‘s heading force, in kN, and the power required at the

electric driving motor, in kW, function of the inclination angle of the longwall, when it is

inclined towards the main gallery where the driving station ins found, is shown in Fig. 2,

where [5]: Facq – heading force in cutting towards the turning station, in the range of 84,1 -

161,6 kN; Facaq – advance force in cutting towards the driving station, in the range of 84,1 şi

-4,5 kN; Pmotq – necessary force at the electric driving motor in cutting towards the turning

station, in the range of 125,76 - 130,15 kW; Pmsaq – necessary power at the electric driving

motor in cutting towards the driving station, in the range of 125,76 - 120,75 kW.

Facq – heading force in cutting towards the turning station, in the range of 84,1 -161,6 kN;

Facaq – advance force in cutting towards the driving station, in the range of 84,1 şi -4,5 kN;

Pmotq – necessary force at the electric driving motor in cutting towards the turning station, in

the range of 125,76 - 130,15 kW; Pmsaq – necessary power at the electric driving motor in

cutting towards the driving station, in the range of 125,76 - 120,75 kW.

Knowing the technological forces acting on the machine, its design dimensions and how it is

positioned on the conveyer, shown in Fig. 3, the stability of the 2K-52MY machine was

verified, by developing a simplified calculation model for stability, shown in Fig. 4. In Fig. 1

it is noticed that in cutting towards the main gallery, towards the driving station, at angles

larger than 25°, the heading force is compensated by the component of the machine‘s weight,

which was 118,66 kN and a machine-conveyer friction coefficient of 0,25.


123

Fig. 3. Design dimensions of 2K-52MY machine and

its position of the conveyer.


124

Fig. 4. Simplified calculation model of 2K-52MY machine

stability on the conveyer

It results that for the machine‘s heading, 4 … 7% of the motor power is consumed,

and the greatest part, 90 … 95%, for cutting the coal. The motor power was estimated for a

spare coefficient of 1,25. In the case of the simplified calculation model of the 2K-52MY

machine stability on the conveyer, Fig. 4, points 1 and 2 represent the center of the support

skid of the machine on the longwall side lateral of the conveyer, and points 3 and 4 represent

the captive guiding of the machine on the back lateral of the conveyer [5]. The technological

forces on the auger drum of the machine are represented for two cases, noted with index 1

when it moves towards the driving station, and with index 2, when moving towards the

returning station, with:

Fts, Ftj – cutting force on the upper drum, at the roof, or down, on the floor;

Fas, Faj – force opposing to the machine movement up on the drum, at the roof, and

down on the floor;

Frs, Frj – machine‘s rejection force by the face on the upper drum, at the roof, or down on

the floor.

To determine the weight centre of the machine, 3D modeling of the machine‘s

component parts and their assembling was performed, shown in Fig. 5 [3].

Thus, for a 12095,82 kg weight of the machine, the coordinates of the weight centre

were determined, and the reference system is found in the assembling plan of the feeding

mechanism with the motor, in the horizontal central plane of the motor and on the outside of

the feeding mechanism on the longwall side, for three possible working cases, when the front

drum cuts up on the roof:

a) when the machine moves towards the driving station, the position shown;

b) when the machine moves towards the returning station;

c) when the machine moves with both drums down to clear the longwall.


125

Fig. 5. 3D model of 2K-52MY machine with positions

of the weight centre for three working cases


126

With the help of these data, the reactions in the leaning points of the machine on the

empty running conveyer were determined, thus in the trials performed in the Mechanical

Workshop in Lonea Mining Plant, the results were, per total for the three cases, high values

on the skids on the longwall side, 90,4 kN and 28,4 kN on the back guides, and the repartition

difference on the skids is 40,8 kN in point 1 and 49,6 kN in point 2, and 17,6 kN in point 3

and 10,6 kN in point 4, respectively. These differences between the reactions on the longwall

side and back are not more than 25%, as it is recommended in the literature of specialty. The

greater load on the skids on the face side, which was highlighted at the machine movement on

the conveyer at the functioning trials at the surface, are lower during the cutting of coal, due

to the technological forces acting on the machine. The maximum contact pressure between the

skid and the conveyer lateral is 3,76 N/mm2, very near the admissible pressure at the guides of

machine-tools, of 4 N/mm2.

Figure 6. Variation of stability coefficients

function of the longwall inclination angle, case 1

Figure 7. Variation module of stability coefficients

function of the longwall inclination angle, case 2

Variation module of stability coefficients function of the longwall inclination angle, case 2

four lines determined by the support points, whose variation depending on the angle of the

longwall inclination is shown in Fig. 6, where:

Cs14q – stability coefficient according to line 1-4, with values in the range of 2,95 and 2,48;



Cs34q – stability coefficient according to line 3-4, with values in the range 2,95 and 2,62;

Cs14bq – stability coefficient according to line 1-4 when the upper drum is blocked and the

traction force reaches maximum 250 kN, with values in the range of 1,53 and 1,51;

Cs23bq – stability coefficient according to line 2-3 when the lower drum is blocked and the

traction force reaches maximum 250 kN, with value in the range of 1,67 and 1,24.

Verification of the stability of 2K-52MY machine on the conveyer, during cutting

towards the return station, case 2, was made by calculation of the stability coefficients

according to the four lines determined by the support points, whose variation, depending on

the longwall inclination angle, is shown in Fig. 7, where:


127



Cs12q –stability coefficient according to line 1-2, with values in the range of 5,98 şi 5,5;

Cs34q – stability coefficient according to line 3-4, with values in the range of 2,95 şi 2,62;

Cs14bq – stability coefficient according to line 1-4, when the lower drum is blocked and the

traction force reaches maximum 250 kN, with values in the range of 1,24 and 1,22;

Cs23bq – stability coefficient according to line 2-3 when the upper drum is blocked and the

traction force reaches maximum 250 kN, with values in the range of 1,21 and 1,0.

Fig. 8. Variation of stability coefficients depending on the longwall inclination, case 3

Verification of the 2K-52MY machine stability on the conveyer during loading the coal

that has been cut towards the returning station, case 3, was made by the calculation of the

stability coefficients according to the four lines determined by the leaning points, whose

variation, depending on the longwall inclination angle is shown in Fig. 8, where:





Cs14bq – stability coefficient according to line 1-4 when the lower drum is blocked and the

traction force reaches maximum 250 kN, with values in the range of 1,25 and 1,2; Cs23bq –

stability coefficient according to line 2-3 when the lower drum is blocked and the traction

force reaches maximum 250 kN, with values in the range of 50 - 14,34.


128

CONCLUSIONS

The study of 2K-52MY machine stabilityon the conveyer showed that all the stability

coefficients have values higher than 1,2 as it is limited in the literature of specialty, even in

the case of blocking of a drum in the longwall, due to sterile intercalation and haulage of the

machine with maximum force of 250 kN, except for the movement during cutting towards the

return station, case 2, when in the event of the upper drum being blocked, the stability

coefficient according to line 2-3, has smaller values than 1,2 [5]. It is seen that in the two

cases of machine movement during cutting, case 1 and 2, when a drum is blocked due to

sterile intercalations, or due to other causes (wear or lack of bits for the cutting part) the

stability coefficients values reduce greatly, and great reactions result horizontally on the

machine‘s captive guides, of up to 100 kN, in case 1, much greater than the friction force

between the conveyer and the longwall floor, leading to tear of the conveyer and even to

breaking the articulations between the chutes and the back laterals, since the conveyer chutes

are not supported in the longwall as in the case of mechanized support. To avoid the situations

presented above, of conveyer tear horizontally, the following measures of exploitation for the

equipments are recommended:

- at the beginning of each shift or whenever required, the sterile intercalations should be

visually checked and localized, all along the longwall, and their derocking should be done

with the hammer; after cutting a slice with the machine, the condition of the bits is checked of

the cutting parts and the worn and broken ones are replaced and the missing ones are

completed;

- cutting a slice with the machine in the longwall is made by moving the machine

towards the turning station of the conveyer, case 2, when the reactions on the machine‘s

captive guides are not greater than 50 kN, this value being half of the one in case 1;

- reduction of the pressure value of the advance mechanism from 160 bar to 120 bar,

which reduces the maximum traction force from 250 kN to 187, kN, and the reactions

horizontally from 50 kN to 37,5 kN, by 25%, respectively.

REFERENCES

[1] Hirian, M., Kovacs, I., - Maşini miniere, Institutul de Mine Petroşani, Petroşani 1987;

[2] Iliaş, N., Zamfir, V., Kovacs, I., ş.a. - Maşini miniere. Exemple de calcul, Editura Tehnică,

Bucureşti, 1993.

[3] Muscă G. – Proiectarea asistată folosind Solid Edge, Editura Junimea, Iaşi, 2006.

[4] Posea, N., ş.a. – Rezistenţa materialelor, Probleme, Editura Ştiinţifică şi Enciclopedică,

Bucureşti, 1986.

[5] * * * - Documentaţie tehnică privind adaptarea sistemului de legare a combinei 2K-52

MY de staţiile transportorului tip TR-5 şi verificarea stabilităţii combinei pe transportor,

contract 713/ 26.08.2013, E.M. Lonea.


129

INFLUENCE OF THE SUSPENSION PARAMETERS UPON THE

HUNTING MOVEMENT STABILITY OF THE RAILWAY VEHICLES

Mădălina DUMITRIU

Department of Railway Vehicles, University Politehnica of Bucharest

[email protected]

Abstract: The hunting movement of the railway vehicles is essential for the safety, the comfort of

the passengers and the integrity of the transported goods. The dynamic performance of a railway

vehicle can be seriously damaged, due to the phenomenon of instability of the hunting motion that

occurs when a certain speed is exceeded – the critical one. An important role is played by the

suspension parameters in the extension of the stability interval during hunting motion at high

velocities. And this is what the paper draws attention to, based on the analysis of how the elasticity

and damping suspension characteristics influence the critical speed.

Keywords: railway vehicle, hunting, critical speed, suspension

1. INTRODUCTION

The hunting motion is specific for railway vehicles, as a result of the fact that the rolling

surfaces of the wheels have a reversed conicity, provided that they are rigidly fixed on the

axle. The hunting of the wheelsets is conveyed via the suspension elements to the entire

vehicle, thus generating vibrations that can affect the safety, comfort of the passengers or the

integrity of the transported goods [1, 2].

What is particular about the hunting movement is the dependence of its nature on the

velocity, whether it is stable or unstable. For relatively low velocities, the vehicle has a stable

hunting motion. Its movement occurs in the vicinity of the balance position and is maintained

by the geometrical irregularities of the track. Starting from a certain speed, called critical

speed, the hunting movement of the vehicle becomes unstable. The loss of stability is

manifest in the increase of the amplitude in the movements of wheelsets, which will reach to

hit the interior flanges of the rails. From a practical perspective, the phenomenon of the

instability in the hunting motion is essential, since it restricts the maximum velocity of the

vehicles.

The extension of the stability range of the hunting depends on a series of parameters of

the vehicle– the equivalent conicity of the wheel rolling surface, the wheelset mass and inertia

momentum, the bogie axle base, the bogie inertia momentum [3, 4, 5]. An important role is

also played by the vehicle suspension by its stiffness and damping characteristics [5, 6, 7].

This paper focuses on the influence of stiffness and damping on the transversal and

longitudinal direction of the suspension upon the stability of the hunting motion during the

vehicle running on the alignment.



130

To this purpose, the vehicle is modelled via a 21-degree-of-freedom system, which

includes all its basic components – the carbody, the suspended masses of the bogies and the

wheelsets, as well as the elastic and damping elements of the two levels of suspension.

Solving the issue of stability is based on the analysis of the system‘s matrix eigenvalues,

comprising the movement equations describing the free vibrations of the vehicle and resulting

into the establishment of the critical speed. The possibilities of an increase in the vehicle‘s

critical speed is pointed out at, via a better choice of the suspension parameters.

2. THE VEHICLE MECHANICAL MODEL AND MOVEMENT EQUATIONS

The mechanical model to study the horizontal-transversal vibrations of a 4-axle vehicle,

two levels of suspension, travelling at a constant speed V on a tangent track is shown in

figures 1 and 2. This contains 7 rigid bodies representing the carbody, the suspended masses

of the bogies and the 4 wheelsets, connected among them through Kelvin-Voigt systems, by

which the two levels of suspension are modelled.

The vehicle carbody is assimilated to a three-degree-of-freedom rigid, with the following

movements: lateral displacement yc, rolling c and yaw c. This is supported in four points by

the elements of the secondary suspension that have the transversal basis 2bc. It is about the

secondary suspension‘s springs of the vehicles, which can deform themselves in three

directions and whose rigidities are noted with kxc, kyc and kzc. Every bogie has a system with

an anti-rolling torsion bar, whose stiffness is kc. In the vertical direction, the secondary

suspension of a bogie has two dampers with the damping constant czc, and in the horizontal

direction, there is one damper whose damping constant is cyc. Similarly, there will be

considered the antiyawing dampers that are mounted on the lateral sides of the bogies, with

the damping constant cxc. The following carbody parameters are to be mentioned: mass mc

and the inertia moments around the longitudinal axis Jxc and the vertical axis Jzc, respectively.

The height of the carbody centre of mass compared to the secondary suspension plan is hc,

and the vehicle axlebase is 2ac.

The bogie carriage (the bogie suspended mass) is also considered a three-degree-of-

freedom rigid, namely lateral displacemeent ybi, rolling bi and yaw bi, with i = 1 or 2, as it

regards the front or the rear bogie. This carriage supports the elements of the secondary

suspension and is resting in four points on the wheelsets, via the elements of the primary

suspension. The main parameters of the bogie carriage are its mass mb, the inertia moment in

respect to the longitudinal axis Jxb, and the inertia moment in respect to the vertical axis Jzb.

The bogie mass center is located at the distance of hb1 from the primary suspension plan and

at hb2 from the secondary suspension plan. It should be mentioned that the bogie has the

axlebase of 2ab.


131

Fig. 1. The mechanical model of the vehicle – frontal view.

Fig. 2. The mechanical model of the vehicle – top view.


132

The elements of the primary suspension also operate after three directions and they have

the rigidities kxb, kyb, kzb and the damping constants cxb, cyb and czb. The Kelvin-Voigt systems

placed in the plan of the axles, at the distance hb1 from the mass center of the bogie suspended

side, model their elastic driving. The transversal basis of the primary suspension is 2bb.

In terms of the wheelsets, they are considered to be able to perform the following

independent movements: one movement of lateral translation, yoj,(j+1), and a rotation

movement – yaw, oj,(j+1), where j = 2i – 1, with i = 1 or 2, mentioning that the bogie i has the

axles j and j + 1. Similarly, the wheelset makes a rotation movement around its own axis, at

the angle speed oj,(j+1) = V/ro + oj,(j+1), where oj,(j+1) is the angle sliding velocity of the axle

compared to V/ro, and ro is the radius of the rolling circle when the axle takes the median

position on the track. The following parameters will be considered: mass mo and the inertia

moments Jyo and Jzo.

The movement equations of the vehicle are as below:

- the carbody movement equations (lateral displacement, rolling, yaw)

.0)()(22

)()()(2

21221

21221

bbbbbcccyc

bbbbbcccyccc

hyyhyk

hyyhycym (1)

0

)()()(22)(2)2(

)()()(2)(22

21221212

21221212

ccc

bbbbbccccycbbcczcc

bbbbbccccycbbcczccxc

ghm

hyyhyhkbkk

hyyhyhcbcJ

(2)

.0)()(22)](2[2

)()(2)](2[2

21221212

21221212

bbbbbcccycbbccxc

bbbbbcccycbbccxcczc

hyyaakbk

hyyaacbcJ (3)

- the bogie movement equations (lateral displacement, rolling, yaw), for i = 1, 2 and j = 2i – 1

0)()(22))1((2

)()(22))1((

)1(11

2

)1(11

12

joojbjbbjybcci

cccbibbiyc

joojbibbiybcci

cccbbbiycbib

yyhykahyhyk

yyhycahyhycym

(4)


133

02

4

)()(22))1((2

))(2(4)()(22

))1(()(2

1122

)1(111

22

22)1(11

122

2

bibbc

bzb

joojbibibbybcci

cccbibibbyc

cbiczccbibzbjoojbibibbyb

cci

cccbibibbyccbiczcbixb

mhm

hgbk

yyyhhkahyyhhk

bkkbcyyyhhc

ahyyhhcbcJ

(5)

.0)](2[2)](2[2

)(2)](2[2

)](2[2)(2

)1()1(2

2)1(

)1(22

joojbibbybjoojbibxb

cbicxcjoojbibbyb

joojbibxbcbicxcbizb

yyaakbk

bkyyaac

bcbcJ

(6)

- the axle movement equations (lateral displacement, yaw, rotation around its own axis), for

i = 1, 2 and j = 2i – 1

211)1(,

1)1(,)1(,

2

2

jjbibbibbijojyb

bibbibbijojybjojo

YYahyyk

ahyycym

(7)

).(

)(2)(2

2)1(,1)1(,

)1(,2

)1(,2

)1(,

jjjjoojo

yo

bijojbxbbijojbxbjojzo

XXer

VJ

bkbcJ

(8)

)( 2)1(,1)1(,)1(, jjjjojojyo XXrJ , (9)

where Xj,(j+1)1,2 represent the longitudinal forces, and Yj,(j+1)1,2 – the guiding forces acting upon

the axle j and (j+1) respectively, at the contact points of the wheels 1 or 2 with the rails.

Based on the analysis of the system matrix eigenvalues derived from the equations (1 -

9) turned into first-order differential equations, a quality-based evaluation can be done for the

stability in the vehicle movement, with the calculation of the critical speed during hunting as a

result.


134

3. NUMERICAL APPLICATION

This section deals with the influence of the elasticity and damping characteristics in the

longitudinal and transversal direction of the suspension upon the critical speed while hunting.

During the numerical simulations, the parameters have been appropriately adapted to a Y 32R

bogie-fitted passenger railway vehicle, with a maximum velocity of 200 km/h. For the

reference values of the vehicle, the critical speed of 261.56 km/h has been calculated.

Fig. 3. Influence of the secondary suspension upon the critical speed:

(a) influence of the elasticity characteristics; (b) influence of the damping characteristics.

(b)

Fig. 4. Influence of the primary suspension upon the critical speed:

(a) influence of the elasticity characteristics; (b) influence of the damping characteristics.

Figure 3 shows the critical speeds calculated for values of the secondary suspension

rigidities within the range of 104 N/m and 10

6 N/m and dampings between 10

3 Ns/m and 10

5

Ns/m.


135

In diagram (a), the high value of the critical speed is visible (374.89 km/h) for high

rigidities of the secondary suspension (kxc = kyc= 106 N/m). The increase of the critical speed

compared to the reference value is still possible should only one of the two suspension

parameters is modified. While the reference value (kyc= 1.7105 N/m) is maintained in the

transversal direction and the value of 106 N/m is adopted for kxc, the result is the critical speed

of 302.34 km/h. Upon considering the reference value kxc= 1.7105 N/m and increasing the

transversal stiffness of the suspension up to 106 N/m, the critical speed of 314.43 km/h is

obtained. The diagram (b) shows, on the one hand, the increase of the critical speed during the

raising of the secondary suspension damping in the transversal direction. Should the situation

corresponding to the damping reference value in the longitudinal direction cxc = 6.46104

is

looked at and the transversal damping is increased to 105 Ns/m, the critical speed of 319.22

km/h will be the result. On the other hand, the rise in the critical speed can be also obtained

by the increase in the secondary suspension damping in the longitudinal direction. Thus, if the

reference value of cyc = 1.52104 Ns/m is maintained and only the suspension longitudinal

damping is increased (cxc = 105 Ns/m), the critical speed can be raised up to 269 km/h. Finally,

should cxc = cyc = 105 Ns/m is adopted, the critical speed can be 320 km/h.

The diagram (a) in figure 4 shows the results concerning the critical speed for various

values of the primary suspension rigidities within the interval of (105 ... 10

8) N/m, and in

diagram (b) the critical speed is calculated for dampings that can have values from 103 Ns/m

to 105 Ns/m. The diagram (a) points out at a series of interesting aspects related to the way in

which the stiffness in the longitudinal and transversal directions of the primary suspension

affects the value of the critical speed. It is about the fact that an increase in the longitudinal

stiffness leads to a higher critical speed, but this only occurs for certain values of the stiffness

in the transversal direction. Should this range is exceeded, an opposite effect will result, i.e. a

decrease in the critical speed. Further, the diagram (b) of the figure 4 shows that a slightly less

increase of the critical speed is possible by lowering the damping in the primary suspension.

As an example, for cxb = cyb = 103 Ns/m, the critical speed rises from the reference value of

261 km/h to 269 km/h.

4. CONCLUSIONS The dynamic performance of the railway vehicles, defined in terms of safety, comfort of

the passenger and the integrity of the transported merchandise, can be seriously damaged, due

to the phenomenon of instability of the hunting motion that occurs at higher speed than the

critical one. The stability interval for the hunting can be extended at high velocities by

adopting certain parameters of the vehicle, among which the stiffness and the suspension

damping are essential.


136

This paper examines the possibilities to increase the critical speed by adopting the best

values of the vehicle suspension parameters. It has been proved that an increase of the

stiffness and damping in the longitudinal and transversal directions of the secondary

suspension, the critical speed of the vehicle can be higher. Likewise, it is demonstrated that

higher critical speeds can result from the correlation of the values of longitudinal and

transversal stiffness of the primary suspension. A less increase in the critical speed turns into

a lower damping in the primary suspension of the vehicle.

REFERENCES

[1] Sebeşan, I., Dinamica vehiculelor feroviare, Ed. MatrixRom, Bucureşti, 2011.

[2] Sebeşan, I., Mazilu, T., Vibraţiile vehiculelor feroviare¸ Ed. MatrixRom, Bucureşti,

2010.

[3] Polach, O., Influence of wheel/rail contact geometry on the behaviour of a railway vehicle

at stability limit, ENOC – 2005, Eindhoven, Netherlands, 7 – 12 aug. 2005, pp. 2203 –

2210.

[4] Dabin, C., Li, L., Jin, X., Xiao, X., Ding, J., Influence of vehicle parameters on critical

hunting speed based on Ruzicka model, Chinese Journal of Mechanical Engineering,

Jan., 2012, pp. 1-7.

[5] Serajian,R., Parameters’ changing influence with different lateral stiffnesses on

nonlinear analysis of hunting behavior of a bogie, Vibroengineering. Journal of

Measurements in Engineering, Vol. 1, Iss. 4, 2013, pp. 195-206.

[7] Yabuno, H., Okamoto T., Aoshima, N., Effect of lateral linear stiffness on nonlinear

characteristics of hunting motion of a railway wheelset, Meccanica, Vol. 37, 2002, pp.

555–568.

[8] Huang, C., Zeng, J., Liang, S., Carbody hunting investigation of a high speed passenger

car, Journal of Mechanical Science and Technology, Vol. 27, No. 8, 2013, pp. 2283-

2292.


137

THE DYNAMIC BEHAVIOUR OF THE RAILWAY VEHICLES IN

CROSSING AN ISOLATED NIVELMENT DEFECT

Mădălina DUMITRIU


[email protected]

Abstract: The vibrations in the vertical plan of the railway vehicles are mainly initiated by the

interaction with the track that can have nivelment defects due to its construction. The maximum

values of such defects correspond to the isolated defects that lead to a higher intensity of the

vibrations behaviour of the vehicle while crossing them and, hence, this situation needs to be

considered when evaluating the dynamic behaviour of the railway vehicle. The paper focuses on

the movements of the running gear and the maximum accelerations at the carbody level when

crossing an isolated nivelment defect, derived from numerical simulations.

Keywords: railway vehicle, dynamic behaviour, isolated defect, vertical vibrations

1. INTRODUCTION

The quality of the track can be affected by a series of deviations from the ideal

geometry, which come in the shape of certain defects of longitudinal or transversal nivelment,

alignment defects or gauge defects, plus the track torsion [1]. The quality levels of the track

are established in dependence on the extent of such defects [2, 3]. To evaluate the dynamic

behavior of the railway vehicles in terms of the safety, ride quality and the track fatigue

attention should be paid to the values of the standard deviations of the defects of longitudinal

nivelment and of the alignment of track, as well as the maximum values of the isolated

defects of this type [2].

The nivelment defects represent the main cause of the vertical vibrations in the vehicle

[4], for which the bouncing, pitch vibrations are essential for its dynamic behaviour, along

with the carbody bending vibrations [5 - 9]. During certain running conditions – the crossing

of an isolated defect in the nivelment at a high speed – the vibration behaviour of the vehicle

is much more amplified and the vertical accelerations at the carbody level can reach values to

damage the ride quality.

This paper examines the dynamic behavior in a railyway vehicle during a crossing at

speed of 200 km/h over an isolated defect of nivelment with the maximum amplitude

admitted for a track with a QN2 quality level. The movements of the running gear and the

accelerations at the carbody level in three reference points – at the centre and above the two

bogies – are being analyzed. In this context, the accelerations are shown to be lower at the

centre of the carbody and higher above the bogies, and maximum above the rear bogie.



138

The numerical simulations are based on a mechanical model of the vehicle, of a

discrete-continuous type, where the carbody is modelled via an Euler-Bernoulli beam and the

wheelsets and the suspended masses are assimilated to rigid bodies [8, 9]. This model allows

to take into consideration both the rigid vibration modes of the carbody and the bogies –

bounce and pitch, as well as the complex vibration modes of the carbody – the symmetrical

and antisymmetrical bending.

2. THE VEHICLE MECHANICAL MODEL AND MOVEMENT EQUATIONS

The mechanical model to study the dynamic behaviour of a two-level suspension railway

vehicle while crossing over an isolated defect of nivelment is shown in figure 1. It is about a

discrete-continuous model, where the vehicle carbody is modelled via an Euler-Bernoulli

beam of constant section and mass of uniform distribution, and the suspended masses of the

bogies and those four wheelsets are assimilated to rigid bodies. The two levels of suspension

of the vehicle are modelled by means of Kelvin-Voigt systems [8, 9].

Fig. 1. The mechanical model of the vehicle.

For the vehicle carbody, the first two natural modes of bending in the vertical plan

(symmetrical and antisymmetrical) are considered, as well as the rigid vibration modes,

namely the bounce zc and pitch c.


139

The following carbody parameters are mentioned: mc – the carbody mass; 2ac – the

carbody wheelbase; Jc – the inertia moment for the pitch movement; L – the carbody length;

EI – the bending module; - the coefficient of structural damping. The suspended masses of

the bogies can carry out bounce movements zbi, and pitch bi, with i = 1, 2. The mass of a

bogie is mb, its wheelbase is - 2ab, and the inertia moment for the pitch movement is Jb. The

characteristics of the secondary suspension elements pertinent to a bogie are represented by

the elastic and the damping constants 2kzc and 2czc; at the primary suspension level,

corresponding to a suspension, the elastic and the damping constants are noted with 2kzb and

2czb.

Should consider that the stiffness of the track is much higher than the stiffness of the

suspension levels, and the natural frequencies of the wheelsets are higher than the ones for the

vehicle, the hypothesis of the rigid track will be adopted.

As a result, the vertical movements of the wheelsets j (where j = 1...4) are equal with

the dimension of the isolated nivelment defect (xj) against them, and they can be calculated

from the below equations

0)( jj x , for xj < 0; )()( jjj xx , for xj ≥ 0, (1)

where xj represents the x-coordinate along the track and is given by the axle position in the

vehicle, as such

xx 1 ; baxx 2 ; caxx 23 ; cb aaxx 224 , where x = Vt. (2)

Upon applying the method of the modal analysis, the movement equations of the vehicle

can be written as follows:

- for the carbody bounce

; 0)()]()([22

)()]()([22

2122122

2122122

bbczc

bbczccc

zzlXlXTzk

zzlXlXTzczm (3)

- the carbody pitch

; 0)()]()([)]()([22

)()]()([)]()([22

211323312222

211323312222

bbccczc

bbccczccc

zzlXlXTlXlXTaak

zzlXlXTlXlXTaacJ (4)

- the carbody symmetrical bending

; 0)]()([)]()([)]()([2

)]()([)]()([)]()([2

2221212221

2222212

2221212221

2222212

222222

lXzlXzlXlXTlXlXzk

lXzlXzlXlXTlXlXzc

TkTcTm

bbczc

bbczc

mmm

(5)


140

- the carbody antisymmetrical bending

; 0)]()([)]()([)]()([2

)]()([)]()([)]()([2

2321312231

2331323

2321312231

2331323

333333

lXzlXzlXlXTlXlXak

lXzlXzlXlXTlXlXac

TkTcTm

bbcczc

bbcczc

mmm

(6)

- for the bounce of the bogies

; 0)()(2

)()(2

)2(2)2(2

32,1322,122,1

32,1322,122,1

4,23,12,14,23,12,12,1

TlXTlXazzk

TlXTlXazzc

zkzczm

cccbzc

cccbzc

bzbbzbbb

(7)

- for the pitch of the bogies

0)2(2)2(2 4,23,12,14,23,12,12,1 bbbzbbbbzbbb aakaacJ , (8)

where l1 and l2 are the positions of the leaning points of the carbody on the secondary

suspension, Tn is the time coordinate of the natural bending mode n (n = 2, 3), and

)cosh(coscoshcos

sinhsinsinhsin)( 2,12,12,12,12,1 ll

LL

LLlllX nn

nn

nnnnn

(9)

are the natural functions of the first two modes of the carbody vertical bending, where

EILmcnn /2 , n is the natural pulsation of the vibration mode n.

It is also mentioned that mm2,3, cm2,3 and km2,3 represent the masses, dampings and the

modal stiffnesses of the carbody symmetrical and antisymmetrical bending modes [8].

The solution of the movement equations in (1-8) is in the application of the Runge-Kutta

algorhythm.

3. NUMERICAL APPLICATION This section deals with the results of the numerical simulation of the dynamic behaviour

of the railway vehicle during crossing over an isolated nivelment defect of an amplitude o

and length whose analytical form is given by the equations below

)/(sin)( 2 xx o , for 0 ≤ x ≤

0)( x , for x < 0 or x > .


141

The maximum amplitude of the defect admitted by UIC leaflet 518 [2] for a track with

QN2 quality level and o = 9 mm will be considered and the defect length is established based

on the regulations concerning the track construction and maintenance, = 10.8 m [10]. The

parameters of the vehicle model are presented in table 1. Table 1

Carbody mass mc = 34320 kg

Bending module EI = 3.2109 Nm

2

Carbody length L = 26.4 m

Carbody wheelbase 2ac = 19 m

Carbody pitch inertia moment Jc = 7.6 kgm2

Carbody damping ratio m2,3 = 0.015

Vertical stiffness of the secondary suspension 4kzc = 2.4 MN/m

Vertical damping of the secondary suspension 4czc = 68.88 kNs/m

Bogie mass mb = 3200 kg

Bogie pitch inertia moment Jb = 2048 kgm2

Bogie wheelbase 2ab = 2.56 m

Vertical stiffness of the primary suspension 4kzb = 4.4 MN/m

Vertical damping of the primary suspension 4czb = 52.21 kNs/m

The figures 2 and 3 show the travelling of the running gear of the vehicle while crossing

the isolated nivelment defect at the speed of 200 km/h. There are also included the

movements of the wheelsets, of the bogie frames against the axles and of the centre of each

bogie. Generally speaking, the time history of the components in the two bogies are similar,

which proves that there is weak coupling among them. The rear bogie has a certain delay in

crossing the isolated defect, of 0.35 s, corresponding with the travelling of the distance

between the bogies at a given speed. Similarly, the second axle of each bogie crosses over the

track defect with a delay of circa 0.046 s behind the first axle. The moment the first axle

passes over the isolated defect, the bogie frame makes a combined movement between pitch

and bounce. The bounce comes from the movement of the bogie centre, which has a similar

path with the defect‘s. The pitch movement makes the point on the bogie frame above the rear

axle rise, while the point on the corresponding frame of the front axle come down.


142

Fig. 2. The movements of the axles and of the first bogie frame:

──, axle 1; ──, bogie frame above the axle 1; ──, axle 2;

──, bogie frame above axle 2; ──, bogie centre.

Fig. 3. The movements of the axles and of the second bogie frame:

──, axle 3; ──, bogie frame above the axle 3; ──, axle 4;

──, bogie frame above axle 4; ──, bogie centre.

Fig. 4. The carbody acceleration; : ──, at the centre; ──, over the front bogie;

──, over the rear bogie.


143

The presence of the two movements, pitch and bounce, are made also visible by the

paths of the three points on the bogie that are shifted. The bogie pitch movement is quickly

damped after crossing the defect, and so will tend the bounce to do. This aspect is clearly seen

in the fact that all three bogie points, at the centre and above the two wheelsets, practically

have the same movements.

Figure 4 shows the carbody acceleration at its centre and above the two bogies. The

excitation of the two movements, bounce and pitch, is visible here. The pitch movement can

be noticed in which the carbody is engaged when the front bogie touches the track irregularity

– the point above the rear bogie rises, while the point against the front bogie goes down. This

movement has a slow damping, which is obvious from the fact that the points above the

bogies vibrate in anti-phase, as it can be seen in the second part of the movement, after the

front bogie has also crossed over the defect area. On the other hand, the carbody bounce

damps more quickly, as noticed in the acceleration of the carbody centre.

Similarly, the influence of the carbody symmetrical bending can be identified in the

modulations imposed to the acceleration, more at the carbody centre and less above the

bogies. The maximum acceleration is recorded above the bogies, namely 0.66 m/s2 - above

the front bogie and 0.82 m/s2 – above the rear bogie.

4. CONCLUSIONS To have a numerical simulation of the dynamic behaviour of the four-axle railway

vehicle during crossing an isolated nivelment defect, this needs to be mechanically modelled

by a 8-degree-of-freedom system of a discrete-continuous nature. The model includes the

basic elements of the vehicle, such as the carbody – modelled by an Euler-Bernoulli beam, the

wheelsets and the suspended massed of the bogies – considered as rigid bodies, as well as the

elements of the suspension, modelled via Kelvin-Voigt systems.

The analysis of the movements in the vehicle running gear highlights the pitch and

bounce movements that are excited during the crossing over the isolated nivelment defect.

These movements are conveyed from the bogies farther to the vehicle carbody via the

elements of the secondary suspension, visible in the accelerations calculated at its level.

Likewise, the carbody symmetrical bending movement is identified, and it is stronger at its

centre and weaker above the two bogies.

The accelerations are also noticed to be lower at the carbody center and higher above the

bogies, where the maximum acceleration is recorded above the rear bogie.

REFERENCES [1] Mazilu, T., Dumitriu, M., Metode şi proceduri de cercetare experimentală şi încercare a

vehiculelor feroviare, Editura MatrixRom, Bucureşti, 2012.

[2] UIC Leaflet 518, Testing and approval of railway vehicles from the point of view of their dynamic

behaviour – Safety – Track fatigue – Running behaviour, 2009.

[3] EN 13848-1, Railway applications. Track geometry quality. Part 1. Characterization of track

geometry, 2006.


144

[4] Sebeşan, I., Mazilu, T., Vibraţiile vehiculelor feroviare¸ Editura MatrixRom, Bucureşti, 2010.

[5] Zhou, J., Goodall, R., Ren, L., Influences of car body vertical flexibility on ride quality of

passenger railway vehicles, Journal of Rail and Rapid Transit, Vol. 223, 2009, pp. 461-471.

[6] Cheli, F., Corradi, R., On rail vehicle vibrations induced by track unevenness: Analysis of the

excitation mechanism, Journal of Sound and Vibration, Vol. 330, 2011, pp. 3744–3765.

[7] Diana, G., Cheli, F., Collina, A., Corradi, R., Melzi, S., The development of a numerical model

for railway vehicles comfort assessment through comparison with experimental measurements,

Vehicle System Dynamics, Vol. 38, no. 3, 2002, pp. 165- 183.

[8] Dumitriu, M., Influence of the suspension damping on ride comfort of passenger railway vehicles,

UPB Scientific Bulletin, Series D: Mechanical Engineering, iss. 4, vol. 74, 2012, pp. 75-90.

[9] Dumitriu, M., Influence of damping suspension on the vibration eigenmodes of railway vehicles,

Mechanical Journal Fiability and Durability, Issue 1, 2013, pp. 109 – 115.

[10] Instrucţia de norme şi toleranţe pentru construcţia şi întreţinerea căii, nr. 314, MTTc, 1989.


145

CRANK WEB DEFLECTIONS OF MARINE DIESEL ENGINES

Eng. PhD. Candidate, NOVAC GEORGE

―Mircea cel Batran‖ Naval Academy, Constanta

[email protected]

Abstract: In order to establish the proper functioning of a marine diesel engine, respectively the

crankshaft bearing condition and uniformity of thermal processes taking place in each cylinder, a

study was made over the crankshaft of a MAN B&W 7K80 MC-C engine, over a period of about

four years by measuring the crank web deflections during overhauls or various repairing

operations on board. The study includes measurements made by the crew of the ship and related

graphics recording the crank web deflections evolution depending on the number of engine

operation hours.

Keywords: deflections, marine, crankshaft, engine, measurements.

1. INTRODUCTION

The entire chemical process that takes place inside the cylinder of the internal

combustion marine diesel engine is materialized by transforming into mechanical

work on crankshaft and further on the propulsin plant‘s propeller.

The translational motion of the piston is converted through the crank mechanism into

rotating motion of the crankshaft. The gas pressure forces produced in the combustion

chamber are thus discharged through intermediate components - piston, connecting

rod, cross head – in loads of journals and crankpin bearings.

Because of this, crankshaft deflections are measured periodically, ensuring that

journals and crankpin bearings are performing their lubrication role efficiently.

The crankshaft components are illustrated in Figure 2.3.

2. MEASUREMENTS

Overhaul and control operations of crankshaft deflection are performed by the

embarked crew on the merchant vessel during the performed international voyages.

Indications of making these deflection measurements are provided by engine

documentation manual [1], and shown in Figures 2.1, 2.2 [2].

The marine diesel engine included in this study is a MAN B&W 7K80 MC-C model,

and measurements are extended over a period of approximately four years.

Values to be taken into consideration for the final evaluation of the crankshaft

deflection are actually the ―V‖ (deflection of vertical misalignment) and ―H‖

(deflection of horizontal misalignment) values, where ―V‖ and ―H‖ are the differences

between the deflections measured in two of the five areas shown in Figure 2.2. More

specifically: V = T-B, and H=C-E [2].


146

Fig. 2.3. Crankshaft components:

1 – flange; 2,4 – screw; 3,11 – crankshaft; 5,6 – guard; 7 – nut; 8 – fitted bolt; 9 – split

pin; 10 – chain wheel; 12 – gasket; 13 – cover, 14 – screw; 15 – locking wire. [1]


147

Fig. 2.1

Fig. 2.2. Measurement indications

This study includes six measurements (M1, M2, M3, M4, M5, M6)

performed on the crankshaft, depending on the number of engine operating hours,

the crankcase temperature, date and place of the ship during measurements and

also ship‘s trim (Table 2.1) – [2].

Table 2.1. Measurements values [2]

Date: Crankcase

temp.:

Runing

hours: Place: Trim:

M1 30.05.2006 48⁰C 698 Road to Shanghai Fwd.-7,7m/Aft.-9.3m

M2 02.09.2008 40⁰C 16071 Yokohama (drifting) Fwd.9,0m/Aft.10.0m

M3 02.04.2009 39⁰C 20191 Hong-Kong

(anchored) Fwd.9,05m/Aft.9,5m

M4 23.02.2010 45⁰C 26250 Manzanillo Fwd.8,6m/Aft.10,55m

M5 25.11.2010 45⁰C 31255 Hong-Kong


M6 01.02.2010 45⁰C 31256 Hong-Kong


The following charts (Figures 2.4, 2.5) shows the evolution of the ―V‖ and

―H‖ values between 30.05.2006 and 01.12.2010.

At the top of each chart is schematically represented the crankshaft but only

for illustrative purposes in order to highlight the individual measurement value for

each of the seven crankshaft journals.


148

Fig. 2.4. Deflections evolution of vertical misalignment (“V”) [2]


149

Fig. 2.5. Deflections evolution of horizontal misalignment (“H”) [2]


150

3. CONCLUSIONS

The graphs above, correlated with other measures for each cylinder-piston

group in part, may reveal information relating to the proper functioning of the

internal combustion marine diesel engine.

Deflection measurements and checks have an important role in determining

the correct functioning of bearings that provide crankshaft lubrication, and thus

obtaining a maximum output by converting the fuel combustion energy into

mechanical work transmitted to the propulsion plant of the ship.

4. REFERENCES

[1]. MAN B&W Diesel 46-98 MC Edition 40F – Engine No. 16, 2006;

[2]. Ship‘s engine overhauls condition reports (on board official papers).


151

EXHAUST VALVE WEARS OF MARINE DIESEL ENGINES

Eng. PhD. Candidate, NOVAC GEORGE,

―Mircea cel Batran‖ Naval Academy, Constanta,

[email protected]

Abstract: Exhaust valve overhauls and wear registrations reveal information about the quality and efficiency of

thermal processes occurring in the combustion chamber,the fuel quality, the compression pressure inside the

cylinder or combustion gas leakages. The measurements included in this study are targeting the bottom piece,

the spindle and the housing of the exhaust valves, for a period of four years and aims the MAN B&W 7K80 MC-

C marine diesel engine. By correlating with other associated wears for each cylinder, these studies can help

improve engine thermal efficiency of processes, and therefore marine diesel engine reliability.

Keywords: exhaust, valve, marine, engine, wears.

1. INTRODUCTION

This paper studies the exhaust valves wears, of internal combustion marine diesel engine

SULZER 7K80MC-C model and the tabulated measurements extends over a period of about

three years (2007-2010).

The condition and wearing of exhaust valves, provide important information about the

quality and efficiency of the combustion process inside the combustion chamber.

A study over a long period would highlight any operational problems of each cylinder,

but also can generate a range of values within the dimensional and functional characteristics

of the valve will fit, during normal functioning wear depending on the operation hours of each

valve.

2. EXHAUST VALVE

The exhaust valve is presented in Figure 2.1.

Each cylinder is equipped with an exhaust valve, mounted in the center of the cylinder

head. The bottom of the casing is provided with tapered seat for the valve head made from a

heat-treated material.

The valve spindle bore is also equipped with a reinforced duct and the whole assembly is

water cooled. At the top of the spindle valve, are mounted two pistons: an air piston used to

close the exhaust valve and a hydraulic piston used to open the exhaust valve.

The exhaust valve is operated by a cam on the camshaft via a hydraulic transmission.


152

Figure 2.1. The exhaust valve:

1. lifting eye bolt; 2. screw; 3. lifting attachment; 4. gasket; 5. orifice; 6. disc; 7. oil cylinder; 8. stud; 9. nut; 10.

safety strap; 11. disc; 12. screw; 13. lock washer; 14. air cylinder; 15. valve housing; 16. cover; 17. gasket; 18.

surub; 19. valve spindle; 20. surub; 21. flange; 22. gasket; 23. stop screw; 24. O-ring; 25. valve seat; 26. O-

ring; 27. piston, complete; 28. piston; 29. sealing ring; 30. damper piston; 31. piston; 32. spring; 33. disc;

34. plug screw; 35. sealing ring; 36. sealing ring; 37. sealing ring; 38. spindle guide; 39. plug screw; 40.

gasket; 41. flange; 42. O-ring 43. screw; 44. cooling water connection; 45. gasket.


153

3. MEASUREMENTS

Measurements included in this study were taken by the embarked crew on the

commercial ship during overhauls or other operations involving disassembling the exhaust

valve.

Measurements are targeting the following dimensional characteristics of the valve

bottom piece:

- deposits quantity in chamber, extented on circumference (in mm);

- angular position of maximum deposit (in degrees – Figure 3.1);

- maximum deposit thickness in duc (in mm).

For the valve spindle the following dimensional characteristics will be inspected (Figure 3.3):

- number of dent marks larger than 7 mm in diameter;

- stem diameter above sealing area - d0 (in mm);

- minimun stem diameter - dmin (in mm);

- wear of stem sealing ring (%).

For the valve housing the following dimensional characteristics will be inspected (Figure 3.2):

- minimum spindle guide diameter;

- maximum spindle guide diameter;

Below are tabulated (Table 2.1) the measurements performed on six of the seven

engine‘s valves, in order of their functioning hours [2].

Indications concerning the measurement areas are provided by the marine diesel engine

documentation manual [1], and are shown in figures 3.1, 3.2, 3.3 [2].

Fig. 3.1. Angular position of maximum deposit (0⁰ =

port side) [2]

Fig. 3.2. Measuring of spindle guide diameter [2]


154

Table 2.1. Measurements results [2]

Valve no. 4 5 1 9 3 2

Valve hours: 10422 10620 17529 17529 18694 25547

Hours total: 29071 26252 17529 23036 29144 25547

Engine hours (total): 29249 26475 23359 24832 29144 29379

Valve dismount date:

09.0

8.2

010

11.0

3.2

010

09.0

8.2

008

09.1

2.2

007

01.0

8.2

010

21.0

8.2

010

Hours since last

overhaul: 18629 15632 16372 17569 18694 18563

Bottom piece

Amgular position of

max. deposit [0⁰]: 120 140 120 140 180 120

Spindle

No. of dent marks

larger than 7 mm: 3 3 5 4 6 5


155

Valve no. 4 5 1 9 3 2

Valve hours: 10422 10620 17529 17529 18694 25547

Hours total: 29071 26475 23359 24832 29144 29379

Engine hours (total): 29249 26475 23359 24832 29144 29379

Valve dismount date:

09.0

8.2

010

11.0

3.2

010

09.0

8.2

008

09.1

2.2

007

01.0

8.2

010

21.0

8.2

010

Spindle

Housing


156

Fig. 3.3. Stem diameter at sealing (dmin), and above sealing (d0) [1]

4. CONCLUSIONS

Traces of wear retrieved by these valve checks, reveals important information about the

quality and chemical content of the combustion fuel used, and other information related to the

combustion process, injection, combustion mixture formation or the gases exhaust efficiency.

So appropriate measures can be taken to minimize components wear of marine diesel

engine, to increase engine efficiency and reliability.

5. REFERENCES

[1]. MAN B&W Diesel 46-98 MC Edition 40F – Engine No. 16, 2006;

[2]. Ship‘s engine overhauls condition reports (on board official papers).


157

GRAPHO-ANALYTICAL METHOD FOR CALCULLATING

IRREVERSIBILITY PROCESSES WITH FINITE SPEED

Prof.Eng. Traian FLOREA, PhD1,

Lecturer.mat. Ligia-Adriana SPORIȘ, PhD1,

Assist.prof.Eng. Corneliu MOROIANU, PhD1,

Eng. Traian Vasile FLOREA Ph.D2,

Prof.Eng. Anastase PRUIU, PhD1

1 “Mircea cel Batran” Naval Academy of Constanta, ROMANIA

2 A.P.M. Agigea of Constanta, ROMANIA

Abstract: A technique for calculating the efficiency and power of Stirling machines is presented. This

technique is based on the First Law of Thermodynamics for processes with finite Speed. A new and novel pv/pX

diagram is presented that shows the effects of pressure losses due to friction, finite speed and throttling

processes in the regenerator of the Stirling engine. The method used for the analysis of this irreversible cycle

with finite speed involves the direct integration of equations based on the First Law for processes with finite

speed to directly obtain the cycle efficiency and power. This technique is termed the Direct Method.

The results predicted by this analysis were in good agreement with the actual engine performance data of twelve

different Stirling engines over a range of output from economy to maximum power. This provides a solid

verification that this analysis can accurately predict actual Stirling engine performance, particularly with

regard to efficiently and output power.

Keywords: power, finire speed, analysis

1. INTRODUCTION

A new technique for calculating the efficiency and power of actual operating Stirling

machines is presented. This technique is based on the First Law of Thermodynamics for

processes with finite speed [1 – 14] and a new method for determining the imperfect

regeneration coefficient [15] . The analytical results depend upon inclusion of two

calibration coefficients (y and z) based on experimental data to accurately predict

performance. The thermal efficiency is expressed as a product of the Carnot cycle

efficiency and second law efficiency [21, 22, 31, 32] as has been suggested by Bejan [23]:

iP,irrev,IIX,irrev,II

T,irrev,IICC

lnT/TP

P31

ln1

T/T1X1

T

T1

T

T1

LS,H1,

i

1

S,HL

1

S,H

L

S,H

Lirrev,IICCSE

(1)

with XirrevIISHL TT ,,,

, /1


158

3/11

The pressure losses and their effect on efficiency and power of the engine depend on the

piston speed and hence the speed of the engine.

The power output of the engine is:

SwmRTPower gHSESE 2/ln, (2)

The speed for maximum power may be determined since the power output is also a

function of the engine speed. Therefore the operating speed of a particular Stirling engine

can be selected for either maximum economy or for maximum power. Also, knowledge of

the nature of these losses can be effectively used in engine design.

A major loss in Stirling engines is caused by incomplete regeneration. This is expressed

by the coefficient of regenerative losses, X. An analysis for determining this loss has been

made by Petrescu et al. and Florea [15, 33]:

1

,21

,,ln/

/111

TcR

TTyXyX

v

SHL

XirrevII (3)

M

eMX

B

12

211 ,

M

eMX

B

12 (4-5)

RR

gvg

Cm

cmM

, ,

w

S

cm

hAMB

gvg

R ,

1 (6-7)

3/2576,0

576,0424,0

Pr1/4

11

/4395,0

R

mmPgLm

Ddb

TvTcwRTPh

(8)

where y is one of the adjusting coefficients of the method with the value of 0.72.

One objective of this paper is to make a more realistic analysis of the pressure losses

through use of a pv/pX diagram as will be described below and by Petrescu et al. [14].

Finally, the technique for calculating the efficiency and power of Stirling engines is

presented and the results predicted by this analysis are compared with performance data

taken on twelve actual Stirling engines over a range of operating conditions [24, 25, 26,

27, 28].

2. THE METHOD OF DETERMINING THE PERFORMANCE OF THE

STIRLING ENGINE

Computation of pressure losses, work losses, efficiency and power for the processes

shown on the new pv/pX diagrams [14, 15] are made using the first law of

thermodynamics for processes with finite speed [1-13]. The first law written to

specifically include these conditions is:


159

dVP

Pf

P2

Pb

c

aw1PQdU

i,m

f

i,m

throtti,m

(9)

The irreversible work is:

dVP

P

P2

Pb

c

aw1PW

i,m

f

i,m

throtti,mirrev

(10)

when applied to processes with finite speed, as shown on the pv/pX diagram.

Computing and summing all pressure losses of the Stirling engine cycle presented above,

the term PiirrevII ,, from eq. (1) becomes [14, 15, 33]:

ln

4

10w045,094,03N

w

w5

ln

ln1w

w

1,

4

52

L,S

,

L,SPi,irrev,II

(11)

The heat input during the expansion process is also irreversible due to finite speed. In

order to take account of this influence, an calibration coefficient z is introduced:

lnmRTzQ g,H34 (12)

Finally, the real power output of the engine, eq. (2) becomes:

S2/wlnzmRTPower g,HSEirrev,SE (13)

where the value of z was evaluated at 0,8 by comparison with available experimental data

for twelve Stirling engines [24-28].

3. DISCUSSIONS

The variation of the coefficient of regenerative losses, X, with the piston speed for several

values of the gas average pressure is shown in Fig. 2. It is an example of the results

obtained from the sensitivity study [15] for X. It shows an important increase of the

regenerative losses with the gas average pressure. Then, the results of computations based

on this analysis are compared to performance data taken from a number of operating

Stirling engines in Figs. 3-5 and Table I. The high degree of correlation between the

analytic and the operational data shown in these figures indicate that the analysis is

capable of accurately predicting Stirling engine performance under a wide range of

conditions.

This capability should be of considerable value in Stirling engine design and in predicting

the performance of a particular Stirling engine over a range of operating speed.


160

Table I. Comparison between analytical results

and actual engine performance data [24 - 29]

Fig.1. The pv/pX diagram for description of the

Stirling cycle processes

Sti

rlin

g

En

gin

e

Act

ual

Po

wer

[kW

]

Cal

cula

t

ed

Po

wer

[kW

]

Act

ual

Eff

icie

nc

y

Cal

cula

t

ed

Eff

icie

nc

y

NS-03M,

regime 1

(economy)

2.03 2.182 0.359 0.3392

NS-03M,

regime 1

(max.

power)

3.81 4.196 0.31 0.3297

NS-03T,

regime 1

(economy)

3.08 3.145 0.326 0.3189

NS-03T,

regime 1

(max.

power)

4.14 4.45 0.303 0.3096

NS-30A,

regime 1

(economy)

23.2 29.45 0.375 0.357

NS-30A,

regime 1

(max.

power)

30.4 33.82 0.33 0.3366

NS-30S,

regime 1

(economy)

30.9 33.78 0.372 0.366

NS-30S,

regime 1

(max.

power)

45.6 45.62 0.352 0.3526

STM4-120 25 26.36 0.4 0.4014

V-160 9 8.825 0.3 0.308

4-95 MKII 25 28.4 0.294 0.289

4 – 275 50 48.61 0.42 0.4119

GPU-3 3.96 4.16 0.127 0.1263

MP1002

CA

200

W

193.9

W 0.156 0.1536

Free Piston

Stirling

Engine

9 9.165 0.33 0.331

RE-1000 0.939 1.005 0.258 0.2285


161

4. CONCLUSION

The objective of this approach was to closely simulate the operation of actual

Stirling engines without losing insight to the mechanisms that generate the

irreversibilities. Pressure and work losses generated by finite speed of the actual processes

were computed as were the power and efficiency of engines. The first law of

thermodynamics for processes with finite speed was used to compute the power losses

generated by the pressure losses. The analysis presented was applied to specific operating

Stirling cycle engines and results were compared to the measured performance of the

engines. The strong correlation between the analytical results and actual engine

performance data indicates that the Direct Method of using the First Law for Finite Speed

is a valid method of analysis for irreversible cycles.

Fig. 2. Coefficient of regenerative losses Fig. 3. Comparison of the analysis results

versus the piston speed for several values with actual performance data

of the average pressure of the working gas for the NS-30S Stirling engine

(DC = 60 mm, b/d = 1.5, τ = 2, N = 700)


162

Fig. 4. Comparison of the analysis results Fig. 5. Comparison of the analysis

with actual performance data for the results with actual performance

4-275 Stirling engine data for the Free Piston Stirling engine

REFERENCES:

[1] Stoicescu, L., Petrescu, S., 1964a, The First Law of Thermodynamics for Processes

with Finite Speed, in Closed Systems, (German language), Bulletin of Polytechnic Institute

of Bucharest, Vol. XXVI, No. 5;

[2] Stoicescu, L., Petrescu, S., 1964b, Thermodynamic Processes Developing with

Constant Finite Speed, (German language), Bulletin of Polytechnic Institute of Bucharest,

Vol. XXVII, No. 6;

[3] Stoicescu, L., Petrescu, S., 1965a, Thermodynamic Processes Developing with

Variable Finite Speed, (German language), Bulletin of Polytechnic Institute of Bucharest,

Vol. XXVII, No. 1;

[4] Stoicescu, L., Petrescu, S., 1965b, Experimental Verification of the Processes with

Finite Speed, (German language), Bulletin of Polytechnic Institute of Bucharest, Vol.

XXVII, No. 2;

[5] Stoicescu, L., Petrescu, S., 1965c, Cycles with Finite Speed, (German language),

Bulletin of Polytechnic Institute of Bucharest, Vol. XXVII, No. 2;

[6] Petrescu, S., 1969a, Contribution to the Study of Interactions and Processes of non-

equilibrium in Thermal Machines, Ph.D. Thesis, Polytechnic Institute of Bucharest,

Romania;

[7] Petrescu, S., 1969b, The Determination of the Expression of the Work in a Process

with Finite Speed using the Phenomenological Thermodynamics of Reversible Processes,

Studii şi Cercetări de Energetică şi electrotehnică, Romanian Academy, Vol. 19, No. 2;


163

[8] Petrescu, S., 1971, Kinetically Considerations Regarding the Pressure on a Piston

Moving with Finite Speed, Studii şi Cercetări de Energetică şi Electrotehnică, Romanian

Academy, Vol. 1, No. 11;

[9] Petrescu, S., 1974, Experimental Study of the Gas-Piston Interaction with Finite

Speed in the Case of an Open System, Studii şi cercetări de Mecanică Aplicată, Romanian

Academy, Vol. 31, No. 5;

[10] Petrescu, S., 1991, Lectures on New Sources of Energy, Helsinki University of

Technology, Finland;

[11] Petrescu, S., Stanescu, G., Iordache, R., Dobrovicescu, A., 1992, The First Law of

Thermodynamics for Closed Systems, Considering the Irreversibility Generated by the

Friction Piston-Cylinder, the Throttling of the Working Medium and Finite Speed of the

Mechanical Interaction, Proc. of the Inter. Conf. on Efficiency, Costs, Optimization and

Simulation of Energy Systems, ECOS‘92, Zaragoza, Spain, edited by A. Valero and G.

Tsatsaronis, ASME, pp. 33-39;

[12] Petrescu, S., Harman, C., Florea, T., 1994, The Connection between the First and

Second Law of Thermodynamics for Processes with Finite Speed. A Direct Method for

Approaching and Optimization of Irreversible Processes, Journal of The Heat Transfer

Society of Japan, Vol. 33, No. 128;

[13] Petrescu, S., Zaiser, J., Valeria Petrescu, Florea, T., 1996, Lectures on Advanced

Energy Conversion, Bucknell University, Lewisburg, PA, USA;

[14] Petrescu, S., Harman, Florea, T., C., Costea, M., 2000a, Determination of the

Pressure Losses in a Stirling Cycle through Use of a PV/Px Diagram, paper accepted to

the Inter. Conf. on Efficiency, Costs, Optimization and Simulation of Energy Systems,

ECOS’2000, Entschede, Netherlands, July 5-7;

[15] Petrescu, S., Harman, C., Florea, T., Costea, M., 2000b, A Method for Calculating

the Coefficient for the Regenerative Losses in Stirling Machines, Proc. of 5th

European

Stirling Forum 2000, Ösnabruck, Germany, February 22-24;

[16] Petrescu, S., Stanescu, G., 1993, A Direct Method of the Study of Irreversible

Processes which are Developing with Finite Speed in Closed Systems, (Romanian

Language), Termotehnica, No. 1, Editura Tehnica, Bucharest;

[17] Petrescu, S., Stanescu, G., Florea, T., Costea, M., 1993a, The study of the

optimisation of the Carnot cycle which develops with finite speed, Proc. of the Inter. Conf.

on Energy Systems and Ecology, Cracow, Poland, edited by J. Szargut, Z. Kolenda, G.

Tsatsaronis andA. Ziebik, pp. 269-277;

[18] Petrescu, S., Stanescu, G., Costea, M., Florea, T., 1993b, A Direct Method for the

Optimization of Irreversible Cycles using a New Expression for the First Law of

Thermodynamics for Processes with Finite Speed, Proc. of the 1st Conference on Energy

ITEC‘93, Marrakesh, Morocco, pp. 650-653;

[19] Petrescu, S., Petrescu, V., Florea, T., Stanescu, G., Costea, M., 1993c, A

Comparison between Optimization of Thermal Machines and Fuel Cells based on New

Expression of the First Law of Thermodynamics for Processes with Finite Speed, Proc. of

the 1st Conference on Energy ITEC‘93, Marrakesh, Morocco, pp. 654-657;


164

[20] Petrescu, S., Harman, C., Bejan, A., Florea, T., 1994, The Carnot Cycle with

External and Internal Irreversibility, Proc. of FLOWERS’94 Symposium, Energy for The

21st Century: Conversion, Utilization and Enviromental Quality, Florence, Italy;

[21] Petrescu, S., Harman, C., 1996, Stirling Cycle Optimization Including the Effects of

Finite Speed Operation, Proc. of the Inter. Conf. on Efficiency, Costs, Optimization

Simulation and Enviromental Aspects of Energy Systems, ECOS‘96, Stockolm, Sweden,

edited by P. Alvfors, L. Edensten, G. Svedberg and J. Yan, pp. 167-173;

[22] Costea, M., Petrescu, S., Harman, C., Florea, T., 1998, The Effect of

Irreversibility’s on Solar Stirling Engine Cycle Performance, Proc. of the Inter. Conf. on

Efficiency, Costs, Optimization Simulation and Environmental Aspects of Energy Systems,

ECOS‘98, Nancy, France, edited by A. Bejan, M. Feidt, M.J. Moran and G. Tsatsaronis,

Nancy, France p.821-828;

[23] Bejan, A., 1988, Advanced Engineering Thermodynamics, Wiley, New York;

[24] Fujii, I., 1990, From Solar Energy to Mechanical Power, Harwood Academic

Publishers, New York;

[25] Allen, D.I., Tomazic, W.A., 1987, ―ot Piston Ring Tests, NASA TM-100256;

[26] Geng, S.M., 1987, Calibration and Comparison of the NASA Lewis Free-Piston

Stirling Engine Model Prediction with RE-1000 Test Data, NASA TM-89853;

[27] Stine, W.B., Diver, R.B., 1994, A Compendium of Solar Dish / Stirling Technology,

Sandia Laboratories Report;

[28] Farell, R.A. et al., 1988, Automotive Stirling Engine Development Program, NASA

CR-180839;

[29] Organ, J.A., 1992, Thermodynamics and Gas Dynamics of Stirling Cycle Machine,

Cambridge University Press, Cambridge;

[30] Walker, G., 1983, Cry coolers – Part 1: Fundamentals, Plenum Press, New York;

[31] Costea, M., Petrescu, S., Harman, C., 1999, The Effect of Irreversibility on Solar

Stirling Engines Cycle Performance, Energy Conversion & Management, Vol. 40, pp.

1723-1731;

[32] Costea, M., 1997, Improvement of heat exchangers performance in view of the

thermodynamic optimisation of Stirling machine; Unsteady-state heat transfer in porous

media, Ph.D. Thesis, P. U. Bucharest & U.H.P. Nancy 1;

[33] Florea, T., 1999, Grapho-Analytical Method for the Study of Irreversible Processes

in Stirling Engines, Ph.D. Thesis, Polytechnic University of Bucharest;


165

THE COEFFICIENT OF REGENERATIVE LOSSES

IN STIRLING MACHINES

Prof.Eng. Traian FLOREA, PhD1,

Assist.prof.Eng. Corneliu MOROIANU, PhD1,

Eng. Traian Vasile FLOREA Ph.D2,

Lecturer.mat. Ligia Adriana SPORIȘ, PhD1,

Prof.Eng. Anastase PRUIU, PhD1

1 “Mircea cel Batran” Naval Academy of Constanta, ROMANIA

2 A.P.M. Agigea of Constanta, ROMANIA

Abstract: The coefficient of regenerative losses, X, is the term that includes all of the losses due to heat transfer

in the regenerator. This parameter in turn depends on a large number of variables. Among these are piston

speed, cylinder dimensions, regenerator dimensions, materials internal to the regenerator, gas proprieties and

the range of operating conditions. These variables are employed in a new technique for calculating the

parameter X. The computed values of X were compared with estimated values of X based on experimental data

available in the literature. Agreement between these values was found to be excellent, indicating that the

technique for calculating X is accurate. This predictive capability should be a powerful tool in the design of

effective Stirling machines.

Keywords: irreversibility, losses, regeneration, Stirling, efficiency.

1. INTRODUCTION

This paper presents a new technique for calculating the efficiency and power of actual

operating Stirling machines. This technique is based on the First Law of Thermodynamics for

processes with finite speed (1-13) and is used in conjunction with a new and novel pv / pX

diagram (13,14) and a new method for determining the imperfect regeneration coefficient.

One of the objectives of this paper is to develop the method for determining the

imperfect regeneration coefficient X, and to use it for calculating the efficiency and the power

output of the Stirling engine.

Initially, the thermal efficiency is written as a function of three basis parameters.

,PiT,irrev,IICCIIirrevSESE (1)

where

S,H0CC T/T1 (2)

is the Efficiency of a Carnot cycle operating between the same temperature limits as the

Stirling engine.

The second law efficiency

S,HOT,irrev,II T/T1/1 (3)

takes into account the irreversibility due to the temperature difference between the heat source

and the gas in the engine. The second law efficiency

1

n

S,HO

IIirrevXl1-

T/T1X1

(4)


166

takes into account the loses in the regenerator due to incomplete regeneration through use of

the coefficient of losses, X.

The second law efficiency 1

n0S,H11,

ii

iIIirrevXlT/TVP

VP1P

(5)

with XirrevIISHTT ,,,0 /1, takes into account the irreversibility losses due to the

pressure drop caused by the finite piston speed. The power output of the engine is

S2/wlnmRTPower g,HSESE

(6)

where is the compression ratio, w is piston speed, S is the stroke of the piston, and is the

specific heat ratio.

A major loss in Stirling engines is caused by incomplete regeneration. An analysis for

determining this loss is the primary objective of this paper. A second objective is to make a

more realistic analysis of the pressure losses through use of a pv / pX diagram as will be

described below (for details, see 11). Finally, the power and efficiency, as determined by

this analysis which involves the computation of X, is compared with performance data taken

on twelve actual Stirling engines over a range of operating conditions 7-17.

2. DETERMINATION OF LOSSES, EFFICIENCY AND POWER OF THE STIRLING

ENGINE BASED ON AN INTUITIVE PV / PX DIAGRAM FOR DESCRIPTION OF

THE CYCLE PROCESSES

Computation of pressure losses, work losses, efficiency and power for the processes

shown on the new pv / pX diagrams 14, 15 are made using the first law of thermodynamics

for processes with finite speed 1-14. The first law written to specifically include these

conditions is:

dVPPfPPbcawPQdU imfimthrottim ,,, /2//1

(7)

The irreversible work then is:

dVPPP

Pb

c

awPW imf

im

throttimirrev

,

,, /

21 (8)

when applied to processes with finite speed.

The work expression for the finite speed isothermal irreversible compression process 12

(Fig.1) can be integrated using the Direct Method 5,6,10,12-17 to obtain:

2

1

dVi,cpr,mi,cpr,mf

i,cpr,m

thrott2

1

i,mirrev,12 PP/PP2

Pb

c

awdVPW

(9)


167

Fig. 1: The new pv/pX diagram of the ideal Stirling cycle

The work losses may be calculated for the compression process 12 by using eq. (9), 15:

12fthrott

i,cpr,mrev,12irrev,12losses,12 VVP2

PbP

c

awWWW

(10)

Computing and summing the losses due to finite speed of the pistons, throttling of the

gas through the regenerator 12,16, and mechanical friction 4,7 for the whole Stirling

engine cycle, and introducing them in eq. (5), it becomes 14:

(11)

The heat input during the expansion process is also irreversible due to finite speed. In order to

take account of this influence, an adjusting parameter z is introduced:

.lnmRTzQ g,H34 (12)

Finally, the real power output of the engine, eq.(6) becomes: .S2/wlnzmRTPower g,HSEirrev,SE (13)

ln/4

w045,094,03

w

wN5ln1

w

w1 ,

4

2

L,SL,SP,irrev,II i


168

Where the value of z was evaluated at 0,8 by comparison with available experimental

data for twelve Stirling engines 12-17.

3. A METHOD FOR CALCULATING THE COEFFICIENT OF REGENERATIVE

LOSSES, X, IN STIRLING ENGINE

The analysis resulted in differential equations that were then integrated. This integration

is based on either a lump analysis, which gives pessimistic results, X1, or on a linear

distribution of the temperature in the regenerator matrix and gas (see fig. 2), which gives

optimistic results, X2.

Fig. 2: Gas and matrix temperature distribution in the regenerator

The resulting expressions for are:

.

1;

12

2121

M

eMX

M

eMX

BB

(14)

where:

.1

;

,

,

w

S

cm

hAMB

cm

cmM

gvg

R

RR

gvg

(15)

3/2576,0R

576,0mmP

424,0g

L

m

PrD1d/b4

11

TvTcwRT

P4395,0

h

(16)


169

with mg is the mass of the passing through the regenerator, mR is the mass of the screens of the

regenerator, AR is the surface area of the wires in the regenerator, v is the viscosity of the

working gas, and h is the convective heat transfer coefficient in the regenerator (based on

correlation given in [17]).

The sensitivity of X1 and X2 to changes in operating variables such as the piston speed

was determined. The computed values of X1 and X2 were compared with values of X

determined from experimental data available in the literature [12-17]. The results based on the

theory were found to predict the values from experimental data by using the following

equation:

,1 21 XyyXX (17)

where y is an adjusting parameter with the value of 0,72.

The loss due to incomplete regeneration as determined through use of eq. (17) is the

final loss to be considered in the analysis. The second law efficiency due to irreversibility‘s

from incomplete regeneration is:

1021,, ln///128,072,01

TcRTTXX vHSXirrevII

(18)

In Fig. 3-5 the variation of the coefficient of regenerative losses with the piston speed is

represented for several values of the analysis parameters (d, S, porosity), and Fig. 6 illustrates

the convective heat transfer coefficient dependence upon the piston speed.

Fig. 3: Coefficient of regenerative losses Fig. 4: Coefficient of regenerative losses

versus the piston speed for several values of versus the piston speed for several value of the wire

diameter (DC = 60 mm, DR = 60 mm, the piston stoke (DC = 60 mm, DR = 50 mm,

Pm = 50 bar, S = 30 mm, N = 700, τ = 2) Pm = 50 bar, d = 0.05 mm, N = 700, τ = 2)


170

Fig. 5: Coefficient of regenerative losses Fig. 6: Convective heat transfer coefficient in

versus the piston speed for several values the regenerator versus the piston speed for

of matrix porosity (DC = 60 mm, DR = 50 mm, several values of the average pressure

Pm = 50 bar, S = 30 mm, d = 0.05 mm, of the working gas (DR=50,,,b/d =1.5, η = 2)

N = 700, η = 2)

4. COMPARISON OF ANALYTIC RESULTS WITH THE OPERATING

PERFORMANCE OF ACTUAL STIRLING ENGINES

The results of computations of efficiency and power output based on this analysis are

compared to performance data taken from twelve operating Stirling engines in Figs. 7-8 and

in Table I.

Fig. 7: Comparison of the analysis results

with actual performance data for the

STM4-120 Stirling engine [27]

Fig. 8: Comparison of the analysis results with actual

performance data for the V-160 Stirling engine [27]


171

Table 1: Comparison between the analytical results and actual engine performance data [24-29]

Stirling Engine Actual Power

[kW]

Calculated

Power [kW]

Actual

Efficiency

Calculated

Efficiency

NS-03M, regime 1

(economy) 2.03 2.182 0.359 0.3392

NS-03M, regime 1 (max.

power) 3.81 4.196 0.31 0.3297

NS-03T, regime 1

(economy) 3.08 3.145 0.326 0.3189

NS-03T, regime 1 (max.

power) 4.14 4.45 0.303 0.3096

NS-30A, regime 1

(economy) 23.2 29.45 0.375 0.357

NS-30A, regime 1 (max.

power) 30.4 33.82 0.33 0.3366

NS-30S, regime 1

(economy) 30.9 33.78 0.372 0.366

NS-30S, regime 1 (max.

power) 45.6 45.62 0.352 0.3526

STM4-120 25 26.36 0.4 0.4014

V-160 9 8.825 0.3 0.308

4-95 MKII 25 28.4 0.294 0.289

4 – 275 50 48.61 0.42 0.4119

GPU-3 3.96 4.16 0.127 0.1263

MP1002 CA 200W 193.9W 0.156 0.1536

Free Piston Stirling Engine 9 9.165 0.33 0.331

RE-1000 0.939 1.005 0.258 0.2285

This figures show that there is high degree of correlation between this analysis and the

operational data. This indicates that this analysis can be used to accurately calculate X and of

other losses. Therefore, this analysis can be used to accurately predicting Stirling engine

performance under a wide range of conditions. This capability should be of considerable

value in Stirling engine design and in the prediction the performance of a particular Stirling

engine over a range of operating speed.

The strong correlation between the analytical results and actual engine performance data

also indicates that the Direct Method of using the first law for processes with finite speed is a

valid method of analysis for irreversible cycles.


172

REFERENCES [1] Petrescu Stoian, Florea Traian, Harman Charles, Costea Monica: A Method for

Calculating the Coefficient for the Regenerative Losses in Stirling Machines, European

Stirling Forum 2000, Osnabrück, Germany, February 22-24, 2000, pag. 121-129;

[2] Petrescu Stoian, Florea Traian, Costea Monica, Harman Charles: Application of the

Direct method to Irreversible Stirling cycles with Finite speed, International Journal of

Energy Research, No.26, pag. 589-609,, May, 27, 2002, edited by John Wiley & Sons,

Ltd., USA;

[3] Florea Traian, Petrescu Stoian, Costea Monica, Feidt Michel: La methode directe

utilisée dans la thermodynamique a vitesse finie pour l’optimisation des machines

thermique, Energie, environnement, economie et thermodynamique, Université

―Politehnica‖ Bucarest, avril 2002, pag, 72-79, ISBN 973-8165-22-9;

[4] Petrescu Stoian, Florea Traian, Feidt Michel, Harman Charles, Costea Monica:

Optimization of the Irreversible Carnot Cycle Engine for Maximum Efficiency and

Maximum Power through Use of Finite Speed Thermodynamic Analysis, Int. ECOS‘2002

Conference, G. Tsatsaronis, M. Moran, F. Cziesla and T. Bruckner, eds., Berlin, Germany,

Vol. II pp.1361-1368;

[5] Petrescu Stoian, Florea Traian, Costea Monica, Petre Camelia, Feidt Michel: A

scheme of Computation, Analysis, Design and Optimization of Solar Stirling Engines,

ECOS-2003, Copenhagen,Denmark, June 30-July 2, 2003, Volume I, Editors: Niels

Houbac, Brian Elmegaard, Bjorn Qvale, Michael J. Moran, pag.1255- 1262,ISBN

9015763461;

[6] Florea Traian, Dragalina Alexandru, Costiniuc Corneliu, Florea Elisabeta, Florea

Traian Vasile: A Method for Calculating of the Coefficient for the Regenerative Losses in

Stirling Machines, COMEFIM‘8, The 8 th International Conference on Mechatronics and

Precision Engineering, Technical University of Cluj Napoca, June 8 th - 10 th , 2006,

pag.747-754, ISBN 1221-5872;

[7] Florea Traian, Dragalina Alexandru, Costiniuc Corneliu, Florea Elisabeta, Florea

Traian Vasile: A Method for Determinig the Performances of Stirling Machines Based on

the First Law for Processes with Finite Speed and using a pv/pX Diagram, COMEFIM‘8,

The 8 th International Conference on Mechatronics and Precision Engineering, Technical

University of Cluj Napoca, June 8 th - 10 th , 2006, pag.755-764, ISBN 1221-5872;

[8] Alexandru DRAGALINA, Traian FLOREA, Corneliu COSTINIUC, Constantin

DANCU: Control Optimisation and Load Prediction for Marine Diesel Engines Using a

Mean Value Simulation Model, Conferinta internationala NAV-MAR-EDU 2007,

Constanta, 15-17 noiembrie 2007, ISBN 978-973-8303-84-3;

[9] Florea Traian, Dragalina Alexandru, Florea Traian Vasile, Pruiu Anastase: The study

of the irreversibility of the operational process of the external combustion engines with

heat regenerators and increase of power and output for the internal combustion engines

using the experimental and graphoanalytical methods, Proceedings of the internationally

attended national conference on thermodynamics, Brasov, Romania, 21 – 22 May 2009,


173

Vol II, Series I, ISSN 2065-2119, Special Issue No.1 Vol.1 2009,pag. 347- 354, ISBN 978-

973-598-521-9;

[10] Florea Traian: Regenerarea cãldurii în masinile termice, Editura Leda&Muntenia,

Constanţa, 2001, 128 pag., ISBN 973-9286-96-8 şi ISBN 973-8082-51-X;

[11] Florea Traian: Regimurile optime de putere maximã şi grafice globale, sintetice pentru

optimizarea motoarelor cu ardere externã Stirling, Editura Academiei Navale, Constanţa,

2000, 176 pag., ISBN 973-99564-6-7;

[12] Florea Traian, Petrescu Stoian, Florea Elisabeta: Scheme de calcul pentru studiul

ireversibilitãţii proceselor funcţionale ale motoarelor cu ardere externã Stirling, Editura

Leda&Muntenia, Constanţa, 2000, 147 pag., ISBN 973-8082-07-2 şi ISBN 973-9286-55-0;

[13] Petrescu Stoian, Florea Traian, Harman Charles, Costea Monica: Advanced Energy

Conversion - volume I,(ed.rev.), Bucknell University, Lewisburg PA 17837, USA, January

2005, 469 pag., MECH 422/622;

[14] Petrescu Stoian, Florea Traian, Zaiser James, Harman Charles, Petrescu Valeria,

Costea Monica,Petre Camelia, Florea Traian Vasile: Advanced Energy Conversion -

volume II, (ed.rev.), Bucknell University, Lewisburg PA 17837, USA, February 2005, 497

pag., MECH 422/622;

[15] Petrescu Stoian, Florea Traian, Costea Monica,Florea Elisabeta, Florea Traian

Vasile: Thermodynamics and Heat Transfer, 769 pag., ENGR 204, Bucknell University,

Lewisburg PA 17837, USA, January 2006;

[16] Petrescu S., Harman C.: Stirling Cicle Optimization Including the Effects of Finite

Speed Operation, Proc. Of the Inter. Conf. On Efficiency, Costs, Optimization Simulation

and Environmental Aspects of Energy Systems, ECOS‘96, Stockholm, Sweden, edited by

P. Alvfors, L. Eidensten, G. Svedberg and J. Yan, 167-173, 1996;

[17] Organ J. A.: Thermodynamics and Gas Dynamics of Stirling Cycle Machine,

Cambridge University Press, Cambridge, 1992;

[18] Traian Florea, Anastase Pruiu, Marian Ristea, Nicu Olaru, George Dogarescu,

Traian Vasile Florea, February, 27, 2012 - Calculus and materials for Stirling Engine`s

Bolter and Regenerator , Published by WSEAS Press , pag. 183-186, Cambridge, UK.,

ISBN:978-1-61804-071-8;

[19] Florea Traian, Dragalina Alexandru, Florea Traian Vasile, Pruiu Anastase, May

2009, The study of the irreversibility of the operational process of the external combustion

engines with heat regenerators and increase of power and output for the internal

combustion engines using the experimental and grapho-analytical methods, Bulletin of the

Transilvania University of Brasov, Vol II, Series I, ISSN 2065-2119, Special Issue No.1

Vol.1 2009,pag. 347- 354, ISBN 978-973-598-521-9;

[20] Florea Traian, Petrescu Stoian, Florea Traian Vasile, Dancu Constantin, Pruiu

Anastase, The study of the irreversibility of the operational process of the external

combustion engines with heat regenerators, National Conference of Thermodynamics with

International Participation NACOT 2013 ―Present and Future in Thermodynamics‖,

Editura AGIR, Constanţa, 2013,ISSN-L 1222-4057, ISSN 2247-1871 Online.


174

GAUSSIAN MODEL

Assistant Professor doctor ADINA TĂTAR,

―Constantin Brâncuşi‖ University of Tg-Jiu

[email protected]

Abstract: In this work programme shall be used in the mathematical modeling of the dynamics of atmospheric

pollutants.This is important in assessing the impact of pollutants on air.

Mathematical relationships used are presented within this model, the model structure, required

parameters in solving problems.

Key words:gaussian model, pollutants, mathematical relation, parameters, mathematical modeling

1.INTRODUCTION To assess the impact of pollutants on air and need to be fitted to a NOx installation of

desulphurisation plants, dusting, and were taken into account more scenarios and the impact

has been made using dispersion modeling in ambient air.

For the mathematical modeling of the dynamics of atmospheric pollutants (SO2, NO2

and PM10), resulting from the work of C.T.E. i used was the software OML- Multi.

OML Multi is a model of dispersion of pollutants from local scale developed by the

National Institute of Environmental Research-NERI (Denmark).

In the 1990s this model became operational, being widely used in Denmark for the

practical applications for predicting air quality in different areas, and can be run both in urban

areas and in rural areas up to a distance of 30 miles.

Throughout the 1990s the model has been improved both in terms of theory, but also

in terms of the presentation and view the results.

OML - Multi is a multisource gaussian-type model, designed for inclusion in his

theory of the main physical phenomena governing the dispersion of pollutants into the

atmosphere from industrial sources or other sources.

The pattern can include point sources and area sources. In this model were pursued

and better behavior of the model in most conditions possible, avoiding atmospheric

discontinuităţilor in describing the phenomenon of dispersion, the possibility of its application

to operational purposes. The final version OML-Multi is the result of a long process.

From its first validation by experiments, many new phenomena that have been

introduced over the years have imposed new and new tests and experimental validation.

2. THE STRUCTURE OF THE PROGRAMME

Structurally, OML- Multi consists of:

- Pre-processor-computational method of meteorological parameters needed for

modeling physical processes of dispersion, from the meteorological measurements.

Pre-processor weather requires as input meteorological measurements hourly and two

vertical temperature profiles carried out daily by radiosondaj;



175

- The dispersion model itself-computational method of estimation of concentrations in

a system of predefined receptors based on physical parameters and other data entry required

(emission data, information about land, etc.).

Ground-level concentration is described by Gaussian form of the pollutant plume

relationship date 1:

2

2

2

2

2exp

2exp0,,

z

d

yzy

hy

u

Qyxc

(1)

-Q is the intensity of the source (the mass);

-hef is the actual height of the plume of pollutants;

- ζz is dispersion parameter vertically.

The reflection of the lower and upper limit of the layer is made by the method of the

source image, but the above relationship, for simplicity, has not included a detailed

description of all terms exponenţiali needed contributions from image sources.

If the vertical dispersion ζz exceeds 1.2 times the height value of the mixture, then the

model generates a uniform concentration distribution in the vertical plane of the mixing layer.

The OML, dispersion parameters are correlated directly with the physical parameters

of the boundary layer, describing the turbulence of the atmosphere, unlike most classical

models in which the dispersion parameters are calculated by method of Pasquill-Gifford-

Turner. Due to the variation of the height of the unrest, to amend the dispersion parameters

can easily approach makes it possible for sources with different heights, in addition, the

dispersion parameters are calculated by means of the composition of all the contributions

coming from the turbulence associated with each physical phenomenon that it generates.

As a general rule, in the case of any ζ (ζz and ζy) it breaks down as follows:

22

int

22

buildingernturb (2)

ζturb-represents the dispersion due to atmospheric turbulence;

-ζintern is the contribution resulting from the phenomenon of mixing up of ambient air

pollutant. This phenomenon is associated with scattering feathers with large carrying

capacity;

-ζbuilding is the turbulence generated by the presence of buildings near the sources of the

money.

On the basis of the same principle of decomposition based on statistical theory of

diffusion due to atmospheric turbulence, you can write:

222

convmechturb (3)

- ζturb andζconvdispersion parameters are assigned to those two processes generating

atmospheric turbulence: mechanical and convective processes.


176

- OML implements separately these 5 processes generating turbulence and

differentiates the feathers of pollutant in published and non-published.

- Preprocesorul requires that the meteorological input data hourly weather

measurements and two vertical temperature profiles carried out daily by radiosondaj.

- Output Data are turbulence parameters: sensible heat flux (H), Monin Obukhov

length (L), speed of friction (uz) and the mixing height (day).

The meteorological OML was thoroughly described in numerous publications

(Berkowitz and Prahm-1982, Olesen and Brown 1992, Olesen and others 1992).

OML includes a procedure for modelling the dispersion of pollutants from sources

non-damp surface, rectangular area sources for which the ratio between length and width may

not exceed 10 (L/l < 10), and the length varies between 10 m and 1000 m.

Procedure of calculation is based on the assimilation of surface source with a finite

number of linear sources and contributions from these sources are integrated.

For the calculation of the concentration on the direction of the wind (downwind)

integral analytically using the function resolves the error.

Integral lateral dispersion resolves numeric using numerical techniques proposed by

Romberg. In the case of placing a receiver inside source, concentration on the propagation

direction of the wind shall take into account only the corresponding segment stretches from

the border's first direction of propagation up to the receiver.

In the case of placing the receiver at great distance from the source, it is treated as a

simple linear source. OML uses two distinct values of wind speed:

-Uhs-wind velocity at emission level, used for the calculation of supraînălţării and of

the effects of building;

-um-mediated vertical wind speed used in the calculation of sigmelor and the estimated

time of transport.

Dependence on height windspeed is given by the theory of similarity (Monin and

Obukhov, 1954):

L

z

L

z

z

zz

k

uzu mm

0

0

0ln (4)

Similarity functions used are those proposed by Businger (1971).

Consider the existence of a gradient OML wind on height between the ground surface

and the time of length L.

Input data are:

-hourly weather data, generated in a specific format in preprocesorului weather

rolling;

-data relating to the physical parameters of the sources: sources (point sources-basket)

or geometric dimensions-length-width-height, if sources surface;

-emission data: mass flow, exhaust temperatures;


177

-stroke variation: factors that describe the variation in time of the emissions for each

type of source entered in the model: punctual or surface;

-data related to the network of receptors: receptor coordinates definition in a spherical

coordinate system or rectangular.

The output fields are represented by nodes in the network's strengths of receptors.

OML multi generates in all nodes of the network, the average hourly concentrations of

receptors, as well as monthly, annual average, percentile and other values important in

statistical assessment of air quality.

Weather data were processed using pre-packaged CPU Model meteorological OML

have a Multi format recording annual hourly and wind measurements at a resolution of one

degree.

3.CONCLUSIONS

Graphical representation of the results of mathematical modeling of dispersion of the

pollutants of interest shall be made on topographic maps using georeferenţiate software

Global Mapper 11.

The calculations shall be carried out in a grid with the specified size for the main

features of pollutants emitted by stationary sources directed the burning C.T.E.

The ranges of mediation used for mathematical modeling are those corresponding to

the period of mediation, to be used for each pollutant according to the order MAPM nr.

592/2002.

The software allows modelling of the dispersion on these intervals of mediation.

You can also achieve dispersion maps.

Weather data were processed using pre-packaged CPU Model meteorological OML

Multi have a format recording annual hourly and wind measurements at a resolution of one

degree.

BIBLIOGRAPHY [1] INSEMEX, Computer modeling of the dispersion of dust from dust and gas stations for

fans of the Jiu Valley mining operations, Petroșani, 2010

[2] Order MAPM nr.592/25.06.2002 to approve the norm on the limit values, threshold values

and evaluation methods of sulfur dioxide, nitrogen dioxide and oxides of nitrogen, particulate

matter PM10 and PM2.5, lead, benzene, carbon monoxide and ozone in ambient air ,

published in Official Monitor no. 765/21.10.2002

[3] SC.EPC CONSULTANŢĂ DE MEDIU SRL, Study on dispersion of pollutants emissions

of sulfur dioxide, nitrogen oxides and particulates (PM10) emissions from stationary sources

of SC Complex Energetic Rovinari SA, Bucharest, 2010

[4] S.C. ISPE SA , Study Environmental Impact Assessment, Bucharest, 2008

[5] STAS 11103/78Air purity. Determination of Suspended Powder

[6] Tatar Adina -Milena, Modelling,Dynamic processes in the atmosphere in 2010, Report no.

3, University of Petrosani, 2010


178

THE DETERMINATION OF THEORETICAL COMBUSTION

TEMPERATURE OF HEAVY FUELS CONSIDERING THE

DISSOCIATION OF WATER VAPOURS

FROM THE BURNING GASES

Ph. D

1. Corneliu MOROIANU, Email address: [email protected]

Ph. D1. Ligia Adriana SPORIȘ, Email address: [email protected]

Ph. D1. Traian FLOREA, Email address: [email protected]

1The Naval Academy ―Mircea cel Bătrân‖ Constanța

Abstract: Some components of the burning gases dissociate at high temperatures in the burning space of the

combustion engines. Due to the dissociation of water vapours, the temperature of fuel components from the

burning gases may have higher values. In contact with the surrounding cold surfaces, a heat exchange takes

place inducing the fall of gas temperature so that, the balances of dissociation reactions move to the right,

leading to the reduction of fuel components produced as a result of this phenomenon. This paper gives us an

analytical method for determination the theoretical combustion temperature of heavy fuels considering the

dissociation of water vapours.

Key words: fuel combustion, analytical modelling, dissociation, water vapours.

INTRODUCTION

Some components of the burning gases dissociate at high temperatures in the burning

space of the combustion engines, beginning with 1500 0C. By ―dissociation‖ we mean the

reversible splitting of substance molecules into simple molecules, atoms, radicals or ions, as a

result of chemical covalent bond breaking under the action of outside heat power.

Chapter 1. We‘ll take into account the dissociation of water vapours and carbon dioxide

according to the following reactions:

2H2 + O2 2H2O (1)

2CO + O2 2CO2 (2)

The chemical processes in the furnace are simultaneously developed in both

directions: the direct reaction, during which the initial substances are changed into final

products and the counter-reaction (back reaction\ feedback), of dissociation, by which the

reaction products are changed again into initial products.

The development rates of the two opposite reactions depend on the concentration of

the participant substances. At the beginning of the process, when the concentration of the

initial substances is much higher than the final ones, the direct reaction is developed much

faster than the counter-reaction.




179

According as the process is developed, the concentration of the initial substances is

reduced but the concentration of the reaction products is increased, having as a result the

decrease of direct reaction rate, of forming the final substances and the acceleration of the

counter-reactions, of restoring the initial substances. It gets to the chemical balance in which

the rates of the two reactions, of forming and splitting (decomposing) of the reaction products,

are equal.

The dissociation degree (coefficient) represents the ratio between the number of the

dissociated kmoles and the initial number of total kmoles of the respective component. So:

- for the water vapours:

OH

OH

OH

2

d2

2 n

na (3)

- for carbon dioxide:

2

d2

2

CO

CO

COn

na (4)

where the number of kmoles ―n‖ is expressed in [kmole\kg].

There are initially considered in mixture OH2n kmoles of water vapours. After dissociation

according to the relation (1) in the balanced state, we find:

- kmole water vapours = OHOH 22a1n (5)

- kmole hydrogen = OHOH 22an (6)

- kmole oxygen = 2

an

OH

OH2

2 (7)

- total kmoles = 2

)a2(n

OH

OH2

2

(8)

In the balanced state, the partial pressures of gas components will be:

RT

V

a1n OH

OH

2O2H

2

(9)

TRV

anp

OHOH

H22

2 (10)

RTV2

an OH

O

2O2H

2 (11)

where V - mixture volume [m3]

R - constant of gases

T - temperature [C].


180

According to the law of mass action, we can define the balance constant of chemical reaction

such as:

V

TR

)a1(

a

p

ppk

2

OH

3

OH

OH

O

2

H

OHp

2

2

2

22

2 (12)

Where (i = H2, O2, H2O) are the partial pressures of components.

The equation of gas state for the analyzed case becomes:

,TR)a2(n2

1pV OHOH 22

(13)

Where is the mixture pressure.

After substitution, we can write:

p

a1a2

ak

2

OHOH

3

OH

OpH

22

2

2

(14)

and

p

a1a2

ak

2

COCO

3

CO

pCO

22

2

2 (15)

We can estimate that the dissociation degree depends on the partial pressure of the

component (inversely proportional) and on the temperature (directly proportional). The

dissociation of the gases develops with heat absorption (input), a part of the physical heat of

gas mixture passing into the chemical heat of fuel components, appearing as a result of

dissociation, H2 and CO2.

The total heat content (enthalpy) – the physical heat plus the chemical heat – remains

constant in both cases, with dissociation and without dissociation. The theoretical combustion

temperature is reduced.

For determining the theoretical temperature, we made the initial special diagram Ig

(,t), for the marine fuel RMF 25, emulsified 5% without considering the dissociation. Then,

we made the special diagram Ig (I ,t) of emulsified fuel considering the dissociation, for a

temperature and a partial pressure given. We made the calculations and the graphical

interpretation by the help of a mathematical interpreter Mathcad 7.

So, for the same value of total heat content (enthalpy) 30933.4 [kJ\kg] we obtain the

temperature 1886 [C] for the 5% emulsified fuel considering the dissociation and 1947.6 [C]

for emulsified fuel without considering the dissociation, the temperature difference being of

= 61.6 [ C], so the theoretical combustion temperature with dissociation is lower than in the

case in which its determination is made without taking into account the dissociation.


181

1800 1850 1900 1950 2000

Ga

s E

nth

alp

y [k

J/kg

]

3*104

3.03*104

3.05*104

3.08*104

3.1*104

3.13*104

3.15*104

3.18*104

3.2*104

Iti

It3i

Ti

Temperature [C] Fig. 7.1 - The calculation of theoretical combustion temperature for marine fuel RMF 25, emulsified 5% with

dissociation (__red_) and without dissociation (_blu__).

[C]

200

150

100

50

0 5 10 15 20Wf [%]

ΔT

Fig. 7.2 – Variation in temperature difference T depending on the water percentage in the emulsion, for the

marine fuel RMF 25.


182

The temperature difference T depends on the value of emulsification coefficient

recording higher values for higher emulsification coefficients (Fig.7.3). This phenomenon

leads to the fall of theoretical combustion temperature which for the processes taking place in

gas phase can have as a direct effect the reduction of nitrogen oxide quantity.

2. CONCLUSION

Due to the dissociation of water vapours, the temperature of fuel components in the

burning gases can have higher values. In contact with the surrounding cold surfaces, a heat

exchange takes place inducing the fall of gas temperature so that, the balances of dissociation

reactions move to the right, leading to the reduction of fuel components produced as a result

of this phenomenon.

REFERENCES:

[1] WILLIAMS A., Combustion of droplets of liquid fuels, review. Comb. and

Flame 21), p. 12, 1973.

[2] WILLIAMS A., The mechanism of combustion of droplets and sprays of liquid fuels,

Oxidation and Combustion Reviews 3, p. 1.1968.

[3] GHIA Victor, Combustion graphology used to improve emulsions of water-in-heavy fuel

oil, IKP Stuttgart, 2001.

[4] MOROIANU Corneliu, Arderea combustibililor lichizi în sistemele de propulsie navale,

Editura Academiei Navale ―Mircea cel Bătrân‖, 2001.


183

MATHEMATICAL MODEL FOR BURNING THE MARINE DIESEL

FUEL DROP IN A HOT OXIDIZING ENVIRONMENT

Ph. D1. Corneliu MOROIANU, Email address: [email protected]

Ph. D1. Traian FLOREA, Email address: [email protected]

Ph. D1. Ligia Adriana SPORIȘ, Email address: [email protected]



Abstract. It was proposed a mathematical model for burning the liquid fuel drop in a hot oxidizing environment.

The model was numerically solved by the finite element method. The droplet ignition periods were calculated as

a function of ambient temperature, oxygen concentration, initial droplet diameter and activation energy.

Key words: droplet, burning, mathematical model

1. INTRODUCTION

The object of this analysis is the liquid fuel droplet gradually subjected to the action of

hot, gaseous and exogenous medium. The fuel vapors at the surface of drop diffuse to the

environment and from the environment to the surface of drop diffuse the oxygen. In each

point of the area surrounding the drop in which the fuel and oxygen vapour concentrations are

non-zero, a chemical reaction is produced at a rate determined by Arrhenius‘ reaction,

generating the heat release, the fuel and oxygen consumption as well as the appearance of

burning products. In a certain point of which position depends on the heat balance of the

system, the temperature of gas increases reaching the ignition point. We propose the

following working assumptions:

- the fuel droplet considered has a sphere symmetry according to the assumption that

there is no relative motion between the droplet and the gas, as well as the free

convection is negligible;

- the heat transfer is made only by conduction;

- the pressure is the same all over the droplet medium and constant in time;

- the temperature of droplet surface is equal to the boiling temperature of the liquid at

the given pressure;

- all the components of the gaseous medium satisfy the equation of perfect gas state;

- the chemical reaction is a second-order reaction with a rate established by

Arrhenius‘law.

2. THE MATHEMATICAL MODEL

The processes developed in a gaseous atmosphere surrounding the droplet are

described by the set of equations for the energy conservation and for the components. This set

is formed of the equation of non-steady heat conduction including the flux term resulted from

the chemical reaction and other four non-steady equations of mass. Taking into account the

heat dissipation and including the flux terms for vapours of fuel, oxygen, nitrogen and

burning products, we can write:




184

RT

EexppCAC

C

Q

r

Tr

rCr

1

t

T21

p

2

p

2 (1)

RT

EexpTCACb

r

TC

T

rD

rr

1

t

C21i

i

2

i

2

i (2)

where: i = 1,2,3,4 – refers to fuel, oxygen, nitrogen and burning gases;

C – concentration of component

D – mass diffusion coefficient (m2/s);

- coefficient of thermal conductibility;

A – pre-exponential constant (m3/kg s deg

½);

- density (kg/m3);

bi – stoichiometric coefficient for component i.

The indices used in the development of mathematical model:

c – liquid; g – gas

k – droplet N–normal conditions

p – fuel; s – droplet surface

w – boiling; z – ignition;

0 – initial parameters; 1 – fuel;

2 – oxygen; 3 – nitrogen;

4 – burning gases; - parameters of a gas element at a long

distance to the droplet.

The existing conditions in the droplet are described by the equation of non-steady heat

conduction:

r

Tr

rCr

1

t

T 2

p2

(3)

The boundary conditions for the three above equations are the following: for t = 0 and for t

0. Because of the non-linearity of the equations (1), (2) and (3) the analytical resolution of the

problem is not possible and it is necessary the numerical resolution.

3. TEST RESULTS FOR MARINE DIESEL DMA 25 MARINE FUEL To estimate the values of interest, namely, the dependence of delay time of ignition on

the initial diameter of droplet, the dependence of delay time of droplet ignition on the ambient

temperature as well as the dependence of delay time of ignition on the activation energy value

and on the pre-exponential constant mainly for the calculation of the values and the

parameters specific to marine Diesel DMA 25 (marine fuel):

c = 890 [kg/m3];


185

c = 1,163 (0,12993 - 0,000025.T) [J/m s K];

Cpc = 4186,8. (-0,02144 + 0,00149. T) [J/kg K];

Q = 4192.107 [J/kg];

Tw = 353,26 [K ];

L = 3,94. 105 [J/kg].

The constants of diffusion have been calculated according to the relation:

Di = D0 (Ti / 273)2, (4)

where the values of diffusion D0 constants ―i‖ at the temperature 273 K are:

- for marine Diesel fuel: D0 = 0.751 . 10-5

[m2/s];

- for oxygen: D0 = 1.78 . 10-5

[m2/s];

- for nitrogen: D0 = 1.8 .10-5

[m2/s].

The diffusion constant for burning gases was calculated as a weighted average between the

diffusion constants of CO2 for which D0 = 1.39 x 10-5

(m2/s) and the water vapours D0 = 2.0 x

10-5

[m2/s] by the relation:

0H00HCO0CO02222sp

D.gD.gD ; (5)

Taking 2COg = 0,83 and OH2

g = 0,17, we have: sp0D = 1,49 10

-5 [m

2/s].

For calculation we take into account the variation of conductivity coefficient g and

the specific heat coefficient Cpg related to the temperature and the gas composition in the

ignition medium. It was supposed that at the droplet surface the temperature is equal with the

liquid boiling temperature and the concentration of fuel vapours is constant and equal with the

concentration of saturated vapours.

4. THE DEPENDENCE OF DELAY TIME OF IGNITION ON THE ACTIVATION

ENERGY VALUE AND ON THE PRE-EXPONENTIAL CONSTANT

The estimated delay time of droplet ignition mainly depends on the activation energy

value E and on the pre-exponential constant A, appearing within the flux that mark the rate of

oxidation reaction. These values are difficult to determine for the most fuels used. In Figure 1

and Figure 2 it is presented the dependence of delay time of ignition the DMA 25 fuel

droplet with its initial diameter d0 = 500 [m] and the initial temperature Tp0 = 300 K placed

in the air with ta = 1000 K on the activation energy and the pre-exponential constant.

The delay time of ignition the DMA 25 light fuel droplet is inversely proportional with

the value of pre-exponential constant, the variation ratios being different depending on the

activation energy value; thus, the three times increment of the pre-exponential constant value ,

for the same characteristically values of environment, droplet and activation energy, led to the

reduction of delay time value of ignition, from 1.6 to 4.5 times.


186

10

20

30

40

50

60

50 75 100 125 150 A

2/1

3

kskg

m

E=35000

E=40000

E=45000

[ms]z

Fig. 1 - The dependence of delay time of ignition the marine DMA 25 fuel droplet on the activation energy for

different values of pre-exponential constants.

As the delay times of ignition experimentally determined, under the same conditions,

have different values (from a few milliseconds to 2.5 seconds) the estimation of real values of

E and A, on practical data, and of the curves in figures1 and 2, is relative. The values of the

activation energy and the pre-exponential constant were estimated in the following way:

- from the experimental data, they were adopted the values of critical diameter of light

fuel droplet docr = 200 m at an ambient temperature of T0 = 1090 0K;

- the value of the activation energy was chosen so that the ignition should be performed

when the droplet evaporation was completed, E = 41850 (kJ/kmol);

- for the other calculations, they were adopted the data obtained at tests, so E = 41350

(kJ/kmol) and A = 100 (m3

/kg s K1/2

).

0,2

0,4

0,6

0,8

1,0

1,2

1,4

1,6

1,8

2,0

0,00,6 0,7 0,8 0,9 1,0 1,1 1,2 1,3

][5000 md

][3000

KTp zlg

0

1000

T K

1

23.0

0,5

1,0

1.020 g

Fig. 2 - The dependence of delay time of ignition the marine DMA 25 fuel droplet on the pre-exponential

constant for different values of the activation energy.


187

5. THE DEPENDENCE OF DELAY TIME OF DROPLET IGNITION ON THE

AMBIENT TEMPERATURE

The calculations were made for a variable ambient temperature T0 , from 800 to 1300

0K, for different initial parameters of the droplet and different shares of oxygen in the

burning medium. The results have been presented in Figures 3 and 4 by the relation:

lg z = f(1/T0) (6)

The curves represented in the figures are almost some straight lines, resulting that the delay

time of fuel droplet ignition is an exponential function of ambient temperature. The curves

represented in the figures 3 and 4 can be used for determining the constants Az and Ez on the

experimental data, with the relation:

Z = AZ exp (-Ez / R.T0) (7)

where:

Az and Ez are the conventional activation energy and the initiation energy of ignition.

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

1.5

0.60,6 0,7 0,8 0,9 1,0 1,1 1,2 1,3

]m[23,0g2o

][3000

KTp zlg

0

1000

T K

1

700

500300

]m[900d0

Fig. 3 - The dependence of delay time of ignition the marine DMA 25 fuel droplet on the ambient temperature

for different initial diameters.


188

0,5

0,7

0,8

0,9

1,0

1,1

1,2

1,3

1,4

1,5

1,6

1,7

35000 40000 45000

0,6

zlg ][10000 KT

][5000 md

][3000

KTp

23.020 g

A=50

75

100

150

125

][Kmol

KJE

Fig. 4 - The dependence of delay time of ignition the marine DMA 25 fuel droplet on the ambient temperature

for different oxygen concentrations.

6. The dependence of burning time of droplet on the initial diameter

In figure 5 it was represented the dependence of burning time of droplet at the ambient

temperatures between 700 – 1100 0K for different initial diameters of droplet. It is obvious

that the burning time is much reduced with the decrease of initial diameter of the droplet.

a 210ms

40

30

20

10

500 1000 1500 2000 md 0

K700

K800

K1000

K1100

Fig. 5 - The dependence of delay time of ignition the marine DMA 25 fuel droplet for different initial diameters

and different ambient temperatures.


189

7. CONCLUSIONS

The calculations made related to the process of ignition a fuel droplet in hot oxygen

medium, allow to draw the following conclusions:

- the delay time of fuel droplet ignition is an exponential function of the ambient

temperature; the activation energy of ignition (conventional ignition energy) hasn‘t a constant

value, it depends on the oxygen concentration in the medium and on the initial diameter of the

droplet;

- the delay time of ignition depends on the initial diameter of the droplet, specially,

for low relative temperatures of the medium and droplets of average initial diameter;

- the delay time of ignition depends on the oxygen concentration, specially, for low

temperatures and low oxygen concentration;

- the life time of the droplet reduces with the increase of the ambient temperature and

the oxygen concentration;

REFERENCES

[1] DRYER F.L., Water Adition to Practical Combustion Sistem-Concepts and Aplication,

Sympozium of Combustion, 1996, p. 34-38.

[2] LAW C.K., Theory of Multicomponent Droplet Vaporization, Combustion and Flame,

1998, No.4, p. 210- 219.

[3] LAW C.K., Asymptotic Theory for Ignition and extinction in Droplet Burning,

Combustion and Flame, 1989, No.6, p.24.

[4] MARTIN G. F., HEDLEY A. B., Combustion of a single droplets and simplified spray

system, J. of the Institute of fuel, 1971, p.38-54.

[5] GHIA V., COLIBABA A., A method for establishing the optimal quality of water-havy

fuel, Sci.Tech. Electrotehn. et Energ., 1996, Tome 41, nr.3, p. 401-412.

[6] MOROIANU C., Arderea combustibililor lichizi în sistemele de propulsie navale,

Editura Academiei Navale ―Mircea cel Bãtrân‖, 2001.p.44-76.


190

ASUPRA UNUI SISTEM KOROVKIN

ÎNTR-UN CON DE FUNCŢII PONDERATE

Lector univ. Dr. Ligia-Adriana SPORIȘ,

Profesor ing. Dr. Traian FLOREA,

Conf. ing. Dr. Corneliu MOROIANU,

Academia Navală ‖Mircea cel Bătrân‖ Constanța, România

Abstract: The aim of this note is to present a construction of a Korovkin system for a cone of weighted

continuous set-valued functions.

Keywords: liniar space, Korovkin, function

1. PRELIMINARII

Scopul acestei lucrări este de a prezenta o construcţie a unui sistem Korovkin pentru

un con de funcţii ponderate.

Contextul în care ne vom situa va fi următorul:

- V,G , con local convex separat şi G , spaţiu liniar (astfel, putem considera că ne

aflăm în cadrul SOLC, deoarece topologia simetrică poate fi privită ca o topologie local

convexă şi local plină Hausdorff );

GAGCConv Ø topologia încompacta ,,GConvA G pe erioarăsup (1)

ce devine un con local convex ca subcon a conului local convex plin V,GDConc , unde

Vv,vvvV ;

- X , spaţiu local compact Hausdorff;

- W , pondere pe X .

Se verifică fără dificultate că V,GCConv este un con local convex dirijat la dreapta

şi M -uniform, v -semilatice, iar toate elementele sale sunt mărginite.

Reamintim că există o scufundare naturală ,GCConvGj** j , unde

GCConvA ,AaasupA .

Atunci,

**GGCConvM (2)

are următoarele proprietăţi:

a) *0 w,vM V,v -compactă.

b) M este strict separantă pentru GCConv ,

(i.e. V,v 1 ,GCConvBA, a.i. 0vM ,vBA a.i.

1BA ).


191

c) n1,i ,Ga ,aVa,...,a,aco ii

n

1in21

.

Acum, vom considera:

XJ ,VvGCConv;XCfGCConv;XC sW Wvf a.i. compact şi

wvf0 pe J\X (3)

unde

Vv ,W

vvvV WWW este sistemul de vecinătăţi abstracte.

Fie

Xx M, ,GCConv;XCM*W

xWX (4)

Se demonstrează cu uşurinţă că GCConv;XCW şi WXM moştenesc aceleaşi proprietăţi ca

(1) şi (2).

2. UN SISTEM KOROVKIN PENTRU GCConv;XCW

Mai întâi, vom considera

G;XC GCConv;XCfGCConv;XF Wi

WW de rang finit, n,1i a.i.

x,...xcoxf X,x n1 (5)

Prop. 1:

GCConv;XFW este un subcon sup-stabil al conului local convex

WW V,GCConv;XC .

Corolar 2:

GCConv;XFW este un con dirijat la dreapta M -uniform şi v -semilatice.

Prop. 3:

Fie GCConv;XCf W .

Atunci, X,x ,G ,Vv * fvGCConv;XFg WW a.i. gf xx .

Demonstraţie prop.3:

Conform definiţiei lui x şi a continuitatii lui xfa ,xf a.i. fa x .

Conform definiţiei lui XY ,CW compact a.i. Wvxf şi

Y\Xx ,vxf0 W .


192

Pentru Yy , alegem yfay . ( în particular, aax ).

Conform continuităţii lui yVUU ,f y

o

y a.i. yw1

vzfa y

şi

yUz ,1ywzw ; de unde:

zw

vzf

1yw

vzfay

, yUz .

Definim:

n

1iyi i

ag , unde n,1ii

este o partiţie a unităţii corespunzătoare

subacoperirii finite n,1iyi

U

a lui X . g verifică condiţiile: Xy ,vyfyg căci

1x1 şi n1,i ,0xi nyy a,...,aCXg

Prop. 4:

Fie V,G , un con local convex, cu toate elementele mărginite şi GC , un subcon.

Atunci, s*C

*C GGSubGSup .

Teorema 5:

GCConv;XFW este un sistem Korovkin (la stâng ) pentru GCConv;XCW .

Demonstraţie teorema 5:

Conform Prop.3 şi a definiţiei anvelopei inf, avem că ffx

şi deci,

WXF

MSubf W .

Cum WXM este strict separantă, W

X*0

W M0\vEx şi cum WF , duM dirijat,

sW*W

FCCSubf W

, conform Prop.4.

REFERENCES

[1]. Altomare, F. and Campiti, M., Korovkin–type Approximation Theory and its

Aplications, de Gruyter Studies in Mathematics. vol.17, cap.3, pag.122-169, cap.6, pag. 314 -

394, 1994;

[2]. Campiti, M., Approximation of continuous set-valued functions in Fréchet spaces,

L Analyse Numérique et la Theory de L Approximation, Tome 20, Nr.1-2, pag.25-38, 1991;

[3]. Keimel, K. and Roth, W., A Korovkin type approximation theorem for set-valued

functions, Proc. Amer. Math. Soc., pag.819-823, 1988.


193

ASUPRA UNOR ASPECTE CALITATIVE ALE CONVERGENŢEI

ÎN SPAŢII LINIARE ORDONATE TOPOLOGICE

Lector univ. Dr. Ligia-Adriana SPORIȘ,

Conf.ing. Dr. Corneliu MOROIANU,

Profesor ing. Dr. Traian FLOREA,

Academia Navală ‖Mircea cel Bătrân‖ Constanța, România

Abstract: The aim of this note is to present some particular subspaces resembling with Korovkin closures.The

terminology used is like in [1].

Keywords: liniar space, Korovkin, function

1. INTRODUCERE

Definiţia 1.1. Fie (G,) un spaţiu uniform separat, E, F două mulţimi şi

(E,G), (E,G).

Un operator liniar FE:T se numeşte (, ) – admisibil dacă

)( T .

Definiţia 1.2. În acelaşi context, fie E (E,G) (înzestrată cu topologia indusă de

)(ES ) şi , E .

a) ,:,)()(:)( FELLExHW iiiU (,) – admisibil astfel încât iiL )(

este relativ compactă în E , FE:S)( (, ) – admisibil; dacă: SL H

i în

(F, (G)), atunci )x(S)x(Llim ii

în (F, (G)), unde F este )(FS compactă,

F este o mulţime închisă şi }.

b) )(:Ex)H(V , ii )()( , relativ compactă în E , dacă:

Hi în (G, ), atunci )()(lim xxi

i în (G, ) }.

c) )(:)( ExHU , )( ; dacă pe H, atunci

)()( xx }.


194

2. PRINCIPALELE REZULTATE

Teorema 2.1. În contextul anterior şi presupunând în plus că este închisă în E ,

rezultă că )()()( HUHVHWU .

Corolar 2.2. Dacă E este spaţiu liniar topologic, H E, , E astfel încât este

slab*- închisă, atunci )()()( HUHVHWU .

Definiţia 2.3.

a) Fie E un spaţiu liniar reticulat topologic, H E şi V(E) conul homeomorfismelor

laticiale continue pe E.

FExHWU )(:)( spaţiu liniar reticulat topologic, V(F) echicontinuă

slab*- închisă; iiL )()( echicontinuă, FELi : operatori liniari şi pozitivi,

FES :)( homeomorfism laticial continuu, dacă SLHi : în topologia η ,

atunci )()(lim xSxLii

în η

b) Fie E un spaţiu reticulat normat, H E şi 1:)(1 VEV .

FExHW MU )(:)(, spaţiu reticulat normat; )(1 FV slab*-închisă cu

,sup MLii

FELi : operatori liniari şi pozitivi; FES :)( operator liniar şi

pozitiv )(,)( 11 FVEV -admisibil; dacă SLHi : în topologia η , atunci

)()(lim xSxLii

în η

c) EMEEBM :)(

Teorema 2.4.

a) Dacă E este spaţiu liniar reticulat topologic şi H E, atunci

)()( HUHWU ,

b) Dacă E este spaţiu reticulat normat şi H E, atunci

)(),(,)( 1, EBEVHUHW M

MU

.

BIBLIOGRAFIE

[1] Altomare, F., Campiti, M., Korovkin-type approximation Theory and its Applications, de

Gruyter Studies in Mathematics, vol. 17, 1994;

[2] Altomare, F., Teoremi di approsimazione di typo Korovkin in spazi di funzioni, Rend.

Mat. (6) 13, nr. 3, 1980;

[3] Altomare, F., On the universal convergence sets, Ann. Mat. Pure Appl. (4), 138, 1984;

[4] Donner, K., Extension of Positive Operators and Korovkin Theorems, Lecture Notes in

Math., 904, Springer-Verlag, 1982.


195

PROGRAM FOR THE CALCULATION OF GEOMETRIC

OPTIMIZATION OF PRIMARY SEALS

Ph. D. Lecturer Monica BÂLDEA, University of Pitesti, Romania,

[email protected]

Ph. D. Lecturer Mihaela ISTRATE, University of Pitesti, Romania,

[email protected]

Abstract: The problem to be solved optimally is to minimize the flow losses.Program for the calculation of

geometric optimizationof primary seals indicates that the optimal geometry related to a optimum width of the

sealing surface

Keywords: program, primary seals, geometric optimization

1.INTRODUCTION

The face seal assembly operates centrifugal pump consists of the drive shaft equipped with

the pump sub-assembly Optimization of condensation is to satisfy two conflicting

requirements :

- Limit the minimum leakage ( loss ) or their absence . This involves minimizing and

reducing to zero the thickness of the lubricant film between the friction surfaces ( sealed fluid

to the lock-in) ;

- Limiting friction and wear , which involves ensuring satisfactory lubricating surfaces

Principal remedies are :

- Achieving a dynamic balance under the primary sealing element seals the fluid to ensure a

minimum separation of surfaces in relative motion or pressure contact asperities low as

possible ;

- Foster tribological contact conditions rough by a good choice of materials and lubricating

fluid .

Achieving these remedies involves knowing the influence of factors : the dynamic

behavior of the rotor surface deformation , thermal effects , physicochemical properties of

surfaces , etc. . It was found that a properly designed seal but made inappropriate material

does not work well . So in the present circumstances when the life of the seals Front has

expanded considerably assume that the materials are appropriate . This was due and

technologies for obtaining materials and thermochemical treatments.

2.PUTING THE OPTIMIZATION PROBLEM

Experimental to use a face seal size manufacturing firm shaft diameter of 45 mm. For

sealing the laboratory samples was achieved after a loss rate of 0.3 ml / h over a period of 300

hours at a pressure of 3 MPa and at a speed of 2870 rev / min.

Schematic diagram of unaligned primary seal is shown in Figure 1.


196

Fig.1

Scheme friction ring surface geometry cell is presented in Figure 2

Fig.2

For sealing function , the question of optimal design front seal small portable flow

hydrodynamics.

Data entering the optimization program are :

a. Functional characteristics :

- Sealed fluid pressure , p1 = 3 MPa ;

- The speed of the rotor , n = 1500 rev / min ;

- The sealed fluid , characterized by ρ = 900 Kg/m3 ; = 0,02 Pa·s; E = 1,4·10

9 N/m

2;


197

b.Geometrical characteristics :

- Inner radius of the sealing surface: Ri = 22.75 mm ;

- The outer radius of the sealing surface : Re = 26,75 mm;

- Maximum rotor radius Rm = 27,5 mm;

- The elastic element 8 mounted in parallel on the spring circumference d0 = 52 mm , the

rigidity k = 5.84 MPa , prestressed by f0 = 8 mm;

- Shaft diameter da = 45 mm;

- Misalignment of the rotor surface rad5

2 105 ;

- Misalignment of the stator surface rad5

1 105,2 ;

- Thickness of the sealing gap center .10 mh

The problem to be solved optimally is to minimize flow losses. I believe that optimal

geometry related to the optimal width of the sealing surface , b = Re - Ri .

3.SETTING RESTRICTIONS

- Dimensional restrictions:

021

1

a

a

R

Rr (1)

021

2

m

e

R

Rr (2)

012

3

elmi R

RRr (3)

024

44

e

i

R

Rr (4)

- Restriction for stability:

08 0

22

1 tWkfRRp am (5)

- Restrictions to avoid closing / opening gap sealing:

To find these restrictions must determine:

a) trh ,,min , trh ,,max ;

b) the conditions under which the 0000 maxmin hlhhhk , 1,00 k , ,..10 l

The restrictions are: 21000 eRhhk ; 00210 hlRh e

4.RESULTS AND CONCLUSIONS We use a program in MATLAB and EXCEL .In Figures 3 and 4 the change in the width

of the seal with the diameter of the pump shaft and the pressure shaft diameter of 37.5 mm.

Figure 5 shows the variation in flow from the pressure loss with the same width of seal to the

rotor.


198

Fig. 3

Objective function to optimize been met by the shaft diameter of 80 mm. The other

restrictions are met in all diameters. The optimization program we considered as a basis of

comparison performed at the same flow diameter of the seal firm front[1],[7] .Burgmann

model exceeds the required flow rate of loss to a smaller diameter (75 mm). For pressures

above 3 MPa results change. It was considered appropriate for the study of change in the

width of the sealing surface of the shaft diameter while keeping the objective function, the

maximum loss rate of 0.3 ml / h depending on the pressure. It was observed that the width of

the sealing surface pressure doubles 15MP (the width of the pressure 10MP)

Fig 4

In Fig.4 shaft diameter is constant.,Pressure varies but the value of the objective

function.,Losses flow remains constant d = 37.5 mm; Q = 0.3 ml / h


199

Fig 5

In Figure 5 shows that the pressure varies, the flow rate varies, but the difference Re-Ri are

the same for each pressure.

Burgmann losses flow relationship:

2221943

10752,710885,1

ln

dDnp

d

D

hQ

(6)

The optimized results were obtained at different pressures (p = 3 bar, p = 5bar, p = 7bar,

p = 10 bar, p = 15bar). Putting optimization problem is the problem to be solved, namely

optimal is to minimize flow losses and so the have established certain restrictions that lead to

specific outcomes that will help with future research on the study of mechanical seals.

BIBLIOGRAPHY

[1]Burgmann,Dipl.ing.Jochen Seeling-Gleitringdichtungen fur Steriprozesse in der

Pharmazie ,in der Biochemie,der Lebensmittelindustrie und Gentechnik,Nr.330,1986

[2]Crudu,I., Etanşări pentru organe de maşini in mişcare.Tribosisteme industriale,

Tribotehnica 80, Bucureşti,

[3]Istrate, M., Studiul etanşărilor primare la etanşările frontale, Editura Larisa, Câmpulung,

2013

[4]Istrate, M., Baldea,M., On predictive control using vibration detection method on

condensation damage to the machine with single stage pump,Scientific

Bulletin.Automotive,year XII,nr.16

[5]Lazăr, D.,ş.a. Influenţa mediului asupra alegerii cuplului de materiale pentru inelele

etanşărilor frontale. Tribotehnica` 87, 24-26 sept., Bucureşti, vol. III, p. 79-84.

[6]Popa, N.,ş.a. Asupra uzurii si tipurilor de uzură din etanşările axiale ale pompelor din

industria petrochimică, Tribotehnica 87, Bucureşti, 24-26 sept. 1987, p. 139-143.

[7]Schunk ‚Tradition et Progres,1990.


200

A NOTE ON (α,β)-CUT IN INTUITIONISTIC

FUZZY SETS THEORY

Iuliana Carmen BĂRBĂCIORU , Lecturer Ph. D.

‖Constantin Brâncuşi‖ University, Tg. Jiu

Abstract.In this article, we give some basic definitions from Intuitionistic fuzzy sets theory andwe introduced

new operation (α,β)-cut in intuitionistic fuzzy sets theory with examples and characterizations.

Keywords: Intuitionistic fuzzy sets, support, (α,β)-cut and core of a intuitionistic fuzzy sets.

1. INTRODUCTION

The concept of intuitionistic fuzzy sets was introduced by K.T. Atanassov [1] as a

generalization of the notion of a fuzzy set

, AA x x x X (1)

Definition 1.[1] An intuitionistic fuzzy set (IFS) A in X is given by

, ,A AA x x x x X (2)

where Aμ :X [0,1] is called degree of membership and A : X [0,1] is called degree of

non-membership, with the condition

0 1A Ax x , x X (3)

Definition 2. [2] We call degrees of indeterminacy of x to A, for each A in X the numbers:

1A A Ax x x , x X (4)

IfA is a fuzzy set (1) then supp( ) / A(x) 0A x X is called fuzzy support of A. If and

if there is x such thatA(x)=1, i.e. / A(x) 1x X , the fuzzy set (1) is called normal. In

the case of fuzzy set theory an α-cutor a setof level α , α[0.1],of fuzzy set A is:

[A]α =

if 0

0 if

1

0

A

A

x X x

x X x

(5)


201

Remark 1.Here B indicates the topological closure of a set B. Therefore this definition

requires that X be a topological space.

2.(α,β) -CUT AS THE INTUITIONISTIC FUZZY SETS

In the case of IFS we have:

Definition 3. [2] The support of A(denoted supp(A)) is the set of all those elements of an

universal set whose membership andnon-membership grades in A are greater than zero.

supp( ) / 0, 0,0 1A A A AA x X x x x x (6)

Example 1.Let and .

Then supp( ) , ,A a b d .

Definition 4. [2]The core of A(denotedcore(A)) is the set of all those elements of an

universal set whose membership grades in A are greater equal to one andnon-membership

grades in A are greater equal to zero.

core( ) / 1, 0,0 1A A A AA x X x x x x (7)

Example 2.Let and . Then

core( )A c .

Definition 5. [4]The maximum value attained by A x is referred to as the height of A is

denoted by ht(A). If ( ) 1Aht x and ( ) 0Aht x , then the IFS A is called normal.

Otherwise A is said to be subnormal.

Example 3.Let and . ThenA

is normal. If A is subnormal. Since

ht(A)=0.6.

Definition 6. We consider α,β[0.1] then, for any IFS set A,


202

[A](α,β)

=

0, ,0 1 if

0, 0,0 1 if

, 1

0, 0

A A A A

A A A A

x X x x x x

x X x x x x

(8)

is called (α,β)-cut as the IFSA.

Definition 7. We consider α,β[0.1] then, for any IFS set A,

[A]+(α,β)

=

0, ,0 1 if

0, 0,0 1 if

, 1

0, 0

A A A A

A A A A

x X x x x x

x X x x x x

(9)

is called strong (α,β) -cut as the IFSA.

Remark 2. We observe that:

[A] (0,0)

= 0, 0,0 1 supp(A)A A A Ax X x x x x (10)

[A]+(0,0)

= 0, 0,0 1 =XA A A Ax X x x x x (11)

[A] (1,1)

= 1, 1,0 1 (A)A A A Ax X x x x x core (12)

[A]+(1,1)

= 1, 1,0 1 =A A A Ax X x x x x (13)

Example 4.Let and

. Then

[A](α,β)

=

if 0 0.2, 0.5 1

b,c if 0.2 0.4, 0.4 0.5

b,c if 0.4 0.6, 0.3 0.4

if 0.6 0.8, 0.2 0.3

if 0.8 1, 0 0.2

X

b


203

[A]+(α,β)

=

, , if 0 0.2, 0.5 1

, if 0.2 0.4, 0.4 0.5

, if 0.4 0.6, 0.3 0.4

if 0.6 1, 0.4 0

b c d

b c

b c

The principle of excluded middle is not true to IFS. Some operations on IFSs have ben

also introduced in [4]:

Definition 8.Given two IFSs A and B over an universe of discourse X, one can define the

following relations:

A ⊂ B iff∀x ∈ X, BA x x and BA x x

A = B iffA ⊂ B andB ⊂ A

as well as the following operations [1]:

, ,A AA x x x x X

A ∩ B = , ,A B A Bx x x x X , where

min ,A B A Bx x and

max ,A B A Bx x

A ∪ B = , ,A B A Bx x x x X , where

max ,A B A Bx x and

min ,A B A Bx x

Theorem 1. Let A and B two IFS and one α,β, γ, δ[0.1]. Then following are true:

1) [A]+(α,β)

[A](α,β)

2) If α ≤ γ and β ≤ δ then [A](γ,δ)

[A](α,β)

3) [AB](α,β)

= [A](α,β) [B]

(α,β) [DA,B ](α,β) [DB,A ]

(α,β)


204

4) [AB](α,β)

= [A](α,β) [B]

(α,β)

5) [AB]+(α,β)

= [A]+(α,β) [B]

+(α,β)

6) [AB]+(α,β)

= [A]+(α,β) [B]

+(α,β)

7) [AC]

(α,β) ≠ ( [A]

+(1-α,1-β) )

C

Where

[DA,B ](α,β)

=

, ,0 1 if

0, 0,0 1 if

0 , 1

0, 0

BA

A B

A

A

B

B

x X x x x x

x X x x x x

[DB,A ](α,β)

=

, ,0 1 if

0, 0,0 1 if

0 , 1

0, 0

B A B A

B A B A

x X x x x x

x X x x x x

Proof: 1) If ( ),

x A

, then ,A Ax x which means

,A Ax x proving that ( ),

x A

. So, [A]+(α,β)

[A](α,β)

.

2) If ( ),

x A

then , A Ax x . Consequently , A Ax x i.e.

( ),

x A

. So, [A](γ,δ)

[A](α,β)

.

3) Suppose ( ),[ ]x A B then ,max BA x x and

min ,A Bx x , AA x x or

, BA x x or ,B Ax x or ,B Bx x . Thus ( ),

x A

or

( ),

,B Ax D

or,

A,B

( )

x D

or ( ),

x B

( ) ( )( ) ( ) , ,, ,

, ,A B B Ax A B D D

. Conversely, assume that

( ) ( )( ) ( ) , ,, ,

, ,A B B Ax A B D D

.Then ( ),

x A

or

( ),

,B Ax D

or,

A,B

( )

x D

or ( ),

x B

, A Ax x or

, BA x x or ,B Ax x or ,B Bx x

,max BA x x and min ,A Bx x ( ),[ ]x A B .

4) Suppose ( ),[ ]x A B then ,min BA x x and


205

max ,A Bx x , AA x x and , BA x x and

,B Ax x and ,B Bx x ( ) ( ), ,

x A B

.

Proof of (5) and (6) are exactly similar to those of (3) and (4). Only difference is

instead of inequality here we will have > inequality.

7) Suppose ( ),

Cx A

then , ,0 1A A A Ax x x x . But

then ( )1 ,1

C

x A

1 , 1 ,0 1A A A Ax x x x . Therefore

[AC]

(α,β) ≠ ( [A]

+(1-α,1-β) )

C .

Remark 2.In any fuzzy [AC]

α =( [A

C]

+(1-α))C

but in IFS [AC]

(α,β) ≠ ( [A]

+(1-α,1-β) )

C .

Example 5.Let and

.

,0.5,0.1 , , 0.2,0.6 , ,0.8,0 , ,0.1,0.7 ,( ,0.6,0.3CA a b c d e . Then

,( )

, ,C a cA e

for α=0.3, β=0.3. 1-α=0.7,1- β=0.7 [A]+(1-α,1-β)

= { } ( [A]+(1-α,1-β)

)C = {a,b,c,d,e}. Hence in IFSs, [A

C]

(α,β) ≠ ( [A]

+(1-α,1-β) )

C .

Theorem 2. Let A and B two IFS and one α, β, γ, δ[0.1]. Then:

1) A B if and only if [A](α,β)

[B](α,β)

2) A B if and only if [A]+(α,β)

[B]+(α,β)

3) A = B if and only if [A](α,β)

= [B](α,β)

Proof: 1) Assume that A Bwe will prove that[A](α,β)

[B](α,β)

for all α, β[0.1].

A Biff∀x ∈ X, BA x x and BA x x . Suppose there is (γ, δ)[0.1]

such that ( ), ,) (

A B

. This means there is an x in ,( )

A

such that ( ),

x B

.

Then ,B AA Bx x x x , a contradiction since AB.

Conversely, assuming that [A](α,β)

[B](α,β)

for all α, β,[0.1] we will prove that

AB. For this we have to prove that BA x x and BA x x for all x.

[A](α,β)

[B](α,β) ,A AB Bx x x x . Otherwise, y X for

which ,A AB By y y y y[A](α,β)

but y[B](α,β)

[A](α,β)

[B](α,β)

.


206

A contradiction to the fact that [A](α,β)

[B](α,β)

for all α, β,[0.1].

Proof of (2) areexactly similar to those of (1). Only difference is instead of inequality here we

will have > inequality.Proof of (3) is trivial.

4. CONCLUSIONS

I have introduced a new definition of (α, β)-cut, by changing conditions in the

definition of [5], obtaining a generalization to α -cut level .

REFERENCES

[1] Atanasov, K.T., Intuitionistic fuzzy sets, Fuzzy Sets Syst., vol. 20,pp. 87–96, August 1986.

[Online]. Available: http://dx.doi.org/10.1016/S0165-0114(86)80034-3

[2] Atanasov, K.T.,Intuitionistic fuzzy sets: past, present and future, inEUSFLAT Conf., M.

Wagenknecht and R. Hampel, Eds. University ofApplied Sciences at Zittau/G¨orlitz,

Germany, 2003, pp. 12–19.

[3] Che, L.,Zadeh, A., Fuzzy sets, Information and Control, vol. 8, no. 3, pp. 338–353, 1965.

[4] Despi, I., Opriş, D., Yalcin, E., Generalized Atanasov Intuitionistic fuzzy sets, The Fifth

International Conference on Information, Process, and Knowledge

Management,eKNOW2013.

[5] Veeramani,V.,Batulan R., Some Characterisations of α-cut in Intuitionistic Fuzzy Set

Theory.

[6] Zeng, W. and Li, H., Correlation coefficient of intuitionistic fuzzy sets, Journal of

Industrial Engineering International, vol. 3, pp. 33–40, July 2007.

[7] Yusoff, B.,Taib, I., Abdullah, L.,and Wahab, A. F., A new similarity measure

on intuitionistic fuzzy sets,World Academy of Science Engineeringand Technology, vol. 78,

pp. 36–40, 2011.


207

LYAPUNOV TYPE OPERATORS ON ORDERED BANACH SPACES

Iuliana Carmen BĂRBĂCIORU , Viorica Mariela UNGUREANU,

Constantin Brancusi University of Targu Jiu, Romania

Abstract. This paper studies the properties of a family of Lyapunov- type operators which includes the

ones associated with the linear quadratic optimal control of linear stochastic differential systems (SDSs)

affected by infinite Markovian jumps (MJs) [8]. We prove that this family of Lyapunov type operators

generates a positive causal evolution operator acting on certain infinite dimensional ordered Banach

spaces.

Keywords: Lyapunov operators, positive operators, nuclear operators.

1. INTRODUCTION

The applications of the optimal control theory in both science and engineering are

various and numerous. As an example, we mention here the optimal control problem of

minimizing the concentration of the polluted water during the waste water treatment [1].

Although this optimization problem is nonlinear, the linear quadratic (LQ) control methods

can be applied even in this case (see [4]]). Lyapunov equations play an important role in LQ

control theory. If the studied process suffers abrupt changes in its evolution, the classical

approach may not be applicable. For example, this happens in the case where the

mathematical model of the process is a linear stochastic differential system affected by

Markovian jumps. In this situation the Lyapunov equations are more complicated because

they are defined by Lyapunov type operators acting on infinite dimensional, ordered Banach

spaces of sequences of operators (see [8]). The aim of this paper is to identify a more general

class of such Lyapunov-type operators which generate positive evolution operators on certain

ordered Banach spaces.

2. NOTATIONS

In this paper H and U are real separable Hilbert spaces and ,.,. . denote the inner

product and the norms of elements and operators. Also UHL , is the real Banach space of

linear and bounded operators from H into .U If HU , we shall use the short notation

HL for HHL , . The Banach subspace of )(HL formed by all self-adjoint operators will

be denoted by HS . An operator )(HLA is called nonnegative and we write 0A , if

HSA and 0, xAx for all Hx . Following [6], we denote by )(1 HS (resp. )(2 HS )

the Banach space of all nuclear operators (resp. Hilbert-Schmidt operators) from HS .

Denoting by .Tr the trace operator and by the adjoint operator, we known that )(1 HS is a

Banach space when endowed with the nuclear norm TTTrT 1

. Also, it is known that

)(2 HS is a Hilbert space with the inner product


208

RTTrTR 2

,

and the Hilbert-Schmidt norm TTTrT 2

(see for e.g. [3]).

Now, let Z be an interval of integers, which may be finite or infinite and let B be a real

Banach space. Then }sup,}{{ iggBiggl iiB ZZZ

Z is a real Banach space

with the norm Z

. ( see [6]). If HSB (or L(H)) then Z

Bl will be denoted by Z

HSl (or Z

HLl ).

An element Z

HSlX is said to be nonnegative and we write 0X if 0iX for all Zi .

The cone HK of all nonnegative elements of Z

HSl induces an order ― ‖ on Z

HSl . We say that

a linear and bounded operator ZZ

USHS llL , is positive if UH KK . If

Z

HULlA , , ,,

Z

UHLlB then AB and A are defined by Z iiBiAiAB , and

Z iiAiA , , respectively. We note that

Z

HULlA , and Z

HLlAB . Now let us

consider the linear subspace },{11

iTTlTi

HSHZ

ZN of

Z

HSl . It is known (see

[6]) that 1

.,HN is a Banach space.

If B is a Banach space we denote by ),( BJCb , RJ or TsJ , , Ts 0 , the

Banach space of all mappings BJG : that are continuous and bounded [5]. Similarly,

),(1 BJCb is the space of all continuously differentiable and bounded mappings on J which

derivatives are also bounded on J . For ),,R( Hb LC NL we consider the forward

differential equation

.

0,

HDsY

tstYtdt

tdY

N

L

It is known that this equation has a unique solution ),,(;, 1

Hb LTsCDstY N and the linear

operator on HN defined by tsDstYDstU c ,;,:,L is a strong evolution operator (in

the sense of Definitions 4.1 and 4.2 from [2], [5]) and is called the causal evolution operator

generated by L .

3. MAIN RESULTS

Let us consider a stochastic differential equation of the form

,0,

,,,

00

1

0

tttx

tdwtxttAdttxttAtdx kk

r

k


209

where Rttwtwtwtw r ,,..,, 21 is a standard r dimensional Wiener process and

Rtt , is a right continuous, homogeneous Markov chain with the state space Z and the

infinitesimal matrix Z

jiij ,

with ,0ij if Zji, , ji and ciiijijj

,Z

,

ZZ

jcijjii

,1,

. Here itAk , is a notation for the i-th component itAk of tAk .

A Lyapunov-type operator associated with the above equation is given by

(1) HXXttXAXtAXt NL

,)( 1 ,

where HIitAitA ii

20 ),(),(

and jXitAiXitAiXt ijijj

kk

r

k

,1

1 ),(),(,Z

, Zi .

We know (see [8]) that it is well defined on HN . Now, let us consider a family R}{ ttLL

of operators defined by (1) with Z

HLb lCA , R , Hb LC N,R(1 and t1 being a

positive operator for all Rt . Arguing as in [7], we shall prove that L generates a positive

causal evolution operator on HN .

From the hypothesis Z

HLb lCA , R , we deduce easily that HLCiA b ,., R . It

follows (see [5]) that the family iA ., generates a causal evolution operator istU ;, on H,

which have the properties:

a) the mapping istUst ;,, is . continuous on }0,,{ tsst , uniformly with

respect to i ,

b) );,(,);,(

istUitAt

istU

, uniformly with respect to i for st ,

c) steistU ;, , where tAA t RR

sup: .

Now, let us prove the following.

Theorem 1. Assume that Z

HLb lCA , R . Then the linear and bounded operator

HXtXAXtAXt NL

,)( , Rt generates a positive causal evolution operator

on HN , namely HXstXUstUXst N

,,,:, .

Proof. Obviously, tL and st, are linear operators on HN and st, is positive. The

properties of nuclear operators (see the appendix of [6]) imply that ),R( Hb LC NL and

st, is 1

. -continuous on HN for all .0 ts Hence, L generates ( see [5]) a unique

uniformly continuous causal evolution operator stV , on N H . We shall prove that stV ,

coincides with st, and, consequently, L generates a positive causal evolution operator on

.HN Let stts 00 ,0, be fixed.


210

We first observe that

1

00

0

0

1

0000

0

0

0

),,)((),(,,

),(,)(,,

:,,,,

istXUstUtAtt

istXUistUstU

itAXstiXsttAtt

iXstiXst

tsXitg

1

00

0

00

10000

,,,,

,)(,,)(

itAXsttt

istUstUiXstU

iXsttAistXUstUtA

.,,,,:,,)(

,,

,,,)(

,,)(,,

01000

0

0

0100

100

0

0

tsXitfiXstUistUtAtt

istUstU

istUistUiXstUtA

istXUistUtAtt

istUstU

From the properties a)-c) of the evolution operators istU ;, it follows that for any 0

there is 1,0 such that, for all 0tt , 10 3,,,, iXetsXitf . We conclude

that 110 33,,,, XeiXetsXitg

ii

ZZ

. Thus, for any 0 there is 1,0

such that

etAststtAtt

stst3,,)(

,,0000

0

0

for all 0tt . Here .

is the norm of HL N . Thus that the mapping st, is norm differentiable in t on HN and

.,

,,,

HIss

sttt

st

N

L

On the other hand, we know that the unique solution of the above equation is stV , and,

therefore, ststV ,, . The conclusion follows.

Lemma 2. The cone HH NK is closed.


211

Proof. Let HHnX NK ,nN, be a convergent sequence in HN . We have to prove that its

limit also belong to HH NK . Let Hnn XX N lim . It follows that iX n is . -

convergent to iX in HL . Since the cone HS of all nonnegative operators from HL

is closed, we deduce that 0iX for all Zi and HX K . Now it is clear that

HHX NK and the conclusion follows.

The next theorem is the main result of this paper.

Theorem 3. If Z

HLb lCA , R and Hb LC N,R(1 is such that HLt N1 is a

positive operator for all Rt , then the mapping HXXtXtX NL ,1 , Rt

generates a positive causal evolution operator on HN .

Proof. Consider the Cauchy problem

HXtX

tXttXtdt

tdX

N

L

00

1 ,

on HN . Since ),R(1 Hb LC NL it follows that 1L generates an evolution operator

0,ttS on HN and the unique solution 00 ,,: XttXtX of the above Cauchy problem is

given by 0000 ,,, XttSXttX . Moreover, a variation of constant formula shows that

tX is also the unique solution of the integral equation

.,, 100

0

dssXsstXtttX

t

t

It is known that the solution of the above integral equation can be obtained as the limit of the

sequence of Voltera approximations N, ktX k defined by

.,,

R,

1001

00

0

dssXsstXtttX

tXtX

k

t

t

k

Lemma 2, implies that 01 ,0,

0

ttdssXsst k

t

t

if HHk sX NK for all

tts ,0 . Thus, it follows inductively that ttssX HHk ,, 0 NK for all Nk . Using

again Lemma 2 we deduce that 0, tttX HH NK because it is the limit in HN of a

sequence of elements of the closed cone HH NK .


212

REFERENCES

[1] E. V. Grigorieva, E. N. Khailov, Optimal Control of waste water cleaning plant, Eighth

Mississippi State - UAB Conference on Differential Equations and Computational

Simulations. Electronic Journal of Differential Equations, Conf. 19 (2010), pp. 161-175.

[2] W. Grecksch, C.Tudor, Stochastic Evolution Equations, A Hilbert Space Approach Math.

Res. Vol 75, Akademic Verlag, 1995.

[3] C. Kubrusly, The Elements of Operator Theory, Second edition, Birkhäuser, 2011..

[4] Magdi S. Mahmoud, Coordinated Control of Waste Water Treatment Process,

Proceedings of the World Congress on Engineering 2010 Vol III..

[5] A. Pazy , Semigroups of linear operators and applications to partial differential

equations, Applied Mathematical Sciences 44,Springer- Verlag, Berlin, New -York, 1983.

[6] V.M. Ungureanu and V. Dragan, Stability of discrete-time positive evolution operators on

ordered Banach spaces and applications, J. Differ. Eqns. Appl., 19(2013), 6, 952-980.

[7] V. M. Ungureanu, V. Dragan, Nonlinear differential equations of Riccati type on ordered

Banach spaces, Electronic Journal of Qualitative Theory of Differential Equations, Proc. 9th

Coll. QTDE 17 (2012): 1-22.

[8] V. M. Ungureanu, Optimal control for infinite dimensional stochastic differential

equations with infinite Markov jumps and multiplicative noise, J. Math. Anal. Appl.,

DOI:10.1016/j.jmaa.2014.03.052, 417.2 (2014): 694-718.


213

RANDOMLY GENERATED SUBGROUPOIDS OF X×Z×X

Mădălina Roxana BUNECI, University Constantin Brâncuşi of Târgu-Jiu

Abstract. The purpose of this paper is to provide Maple procedures for random generation of subgroupoids of

the trivial groupoid X×Z×X, where Z is the group of integer numbers.

Keywords: discrete groupoid; pseudorandom number generator; orbit.

.

1. INTRODUCTION

We shall use the notation and terminology from [1]-[3]. In [3] we proved that a

subgroupoid G of XZX, where Z is the group of integer numbers, is characterized by the

set X and two functions f : XX and k :XZ satisfying the properties

1. f(f(u))=f(u) for all uX.

2. k(f(u)) 0 for all uX.

3. If k(f(u)) ≠ 0, then k(u){0,1,…, k(f(u))-1}.

If the function f and k are given, then the groupoid G can be recover as in [2]:

G ={(u,ku,v+tku,u,v): f(u)=f(v), tZ},

where

ku,u := k(f(u)) for all uX.

ku,v:= ( k(u)+k(f(u)-k(v)) mod k(f(u)), if k(f(u))≠0

k(u) – k(v), if k(f(u)) = 0

for all (u,v)X×X with the property that f(u)=f(v) and u≠v. Conversely, k(u)=ku,f(u) for all

uX.

The purpose of this paper is to provide Maple procedures for random generation of the

values of the functions f:XX and k:XZ and consequently, of the groupoid G. The

motivation for generating such groupoids is to test the visualization procedures defined in [4].

2. SUBGROUPOIDS OF XZX GENERATED USING MERSENNE

TWISTER ALGORITHM

As in [3] for implementation in Maple of the reduction to A={x1,x2,…,xn}X of a

subgroupoid GX×Z×X characterized by functions f (satisfying f(A)A) and k we use a list

gd of three arrays:

gd[1] contains a sequence obtained by sorting A and eliminating the duplicates,

gd[2][i] = the index in gd[1] of f(gd[1][i]), i=1..n,

gd[3][i] = k(gd[1][i]), i=1..n.


214

In order to generate the subgroupoids G we use the RandomTools package (in the

Maple environment) and more precisely, the RandomTools[MersenneTwister] subpackage

that contains functions for creating pseudo-random number generators using the Mersenne

Twister algorithm. The Mersenne Twister algorithm was developed in 1997 by Makoto

Matsumoto and Takuji Nishimura [5] and it is based on a matrix linear recurrence over a

finite binary field F2. It provides long period length of 219937

-1, high order dimensional

equidistribution property, speed and reliability. For a k-bit word length, the Mersenne Twister

generates numbers with an almost uniform distribution in the range [0,2k − 1].

The below procedure random_groupoid(n, kmax, nmax) randomly generates a

subgroupoid G={(u,ku,v+tku,u,v): u~v, tZ} of X×Z×X, where X is a finite set and Z is the

group of integers. The data of G are stored in gd. The parameter n represents the cardinality of

X, kmax is the maximum of the set

{ku,u, uX}={k(f(u)), uX}

and nmax is maximum of the set

{|ku,f(u)|, uX, k(f(u))=0}

> with(RandomTools[MersenneTwister]): > random_groupoid:=proc(n, kmax, nmax)

local gd,i,j,s,norbits,sn,co,represent,a,b; gd:=[array(1..n),array(1..n), array(1..n)]; for i from 1 to n do gd[1][i]:=i end do;

s:=NewGenerator(range = 1 .. n); norbits:=s(); co:=1; represent:=array(1..n);

represent[1]:=1; gd[2][1]:=1;

s:=NewGenerator(range = 0 .. kmax); gd[3][1]:=s(); for i from 2 to n do

s:=NewGenerator(range = 0 .. n-1); sn:=s();

if sn<norbits then

co:=co+1; represent[co]:=i; gd[2][i]:=i;

s:=NewGenerator(range = 0 .. kmax); gd[3][i]:=s() else

s:=NewGenerator(range = 1 .. co); a:=s();

s:=NewGenerator(range = a .. co); b:=s();

s:=NewGenerator(range = a ..b);

gd[2][i]:=represent[s()]; if gd[3][gd[2][i]]=0 then

s:=NewGenerator(range = 0 .. 2*nmax); gd[3][i]:=-nmax+s()

else s:=NewGenerator(range = 0 .. gd[3][gd[2][i]]-1); gd[3][i]:=s()

end if end if

end do; RETURN(gd)

end proc;


215

The NewGenerator(range=integer..integer) command used in the procedure

random_groupoid requires RandomTools[MersenneTwister] subpackage. It outputs a Maple

procedure, a pseudo-random number generator, which when it is called outputs one pseudo-

random integer. The output of the generator belongs to indicated range. >gd:=random_groupoid(10,8,3);

The groupoid encoded by gd is G ={(u,ku,v+tku,u,v): u,vX, f(u)=f(v), tZ}X×Z×X, where

X={1,2,3,…,10} and

ku,u := k(f(u)) for all uX.


k(u) – k(v), if k(f(u)) = 0

f : XX, f(1)= 1, f(2)=1, f(3)= 3, f(4)=3, f(5)= 5, f(6)=6, f(7)= 5, f(8)=1, f(9)= 9, f(10)=1,

k:XZ, k(1)=3, k(2)=1, k(3)=2, k(4)=1, k(5)=1, k(6)=7, k(7)=0, k(8)=0, k(9)=4, k(10)=2.

Hence for instance

2

8G ={: r()=2 and d()=8}={(2,k2,8+tk2,2,8): tZ}

={(2,k(2)-k(8) mod k(1)+k(1)t,8): tZ} ={(2, 1+3t,8): tZ}

The procedure orbits(gd) [3] displays the graph of the equivalence relation (principal groupoid) associated with the groupoid G

encoded by gd. Let us see it result for the preceding groupoid

as well as for another randomly generated groupoid.

>orbits(gd);


216

>orbits(random_groupoid(100,8,3));

The following procedure random_groupoid_op:=proc(n, kmax, nmax) generate a

subgroupoid G ={(u,ku,v+tku,u,v): f(u)=f(v), tZ}X×Z×X, where X={1,2,3,…,n} with the

property that the probabilities pk of {u: f(u)=f(k)} satisfy: p1p2…pn. The parameters n,

kmax, nmax have the same signification as in the case of the procedure random_groupoid.

> with(RandomTools[MersenneTwister]): > random_groupoid_op:=proc(n, kmax, nmax) local gd,i,j,s,norbits,sn,co,represent;

gd:=[array(1..n),array(1..n), array(1..n)]; for i from 1 to n do gd[1][i]:=i end do;

s:=NewGenerator(range = 1 .. n); norbits:=s(); co:=1; represent:=array(1..n);

represent[1]:=1; gd[2][1]:=1;

s:=NewGenerator(range = 0 .. kmax); gd[3][1]:=s(); for i from 2 to n do

s:=NewGenerator(range = 0 .. n-1); sn:=s();

if sn<norbits then

co:=co+1; represent[co]:=i; gd[2][i]:=i;

s:=NewGenerator(range = 0 .. kmax); gd[3][i]:=s() else

s:=NewGenerator(range = 1 .. co);

gd[2][i]:=represent[s()]; if gd[3][gd[2][i]]=0 then

s:=NewGenerator(range = 0 .. 2*nmax); gd[3][i]:=-nmax+s()

else s:=NewGenerator(range = 0 .. gd[3][gd[2][i]]-1); gd[3][i]:=s()

end if end if

end do;


217

RETURN(gd) end proc;

>gd:=random_groupoid_op(100,8,3);

>orbits(gd);

The procedure perm_unit_space(gd,p) permutes with respect to p: XX the unit

space X of the groupoid G ={(u,ku,v+tku,u,v): f(u)=f(v), tZ}X×Z×X encoded by gd. More

precisely, it returns the groupoid H=

u

v

u,v

H , where

u

vH ={(u,kp(u),p(v)+tkp(u),p(u),v): f(p(u))=f(p(v)), tZ}

> perm_unit_space:=proc(gd,p) local n,i, gd2, gd3, j;

n:=op(2,op(2,gd[1])); gd2:=array(1..n);gd3:=array(1..n);

for i from 1 to n do

j:=1;

while(j<i) and gd[2][p[i]]<>gd[2][p[j]] do j:=j+1 end do;

if(j<i)then gd2[i]:=j;else gd2[i]:=i end if;

end do;


if gd2[i]=i then

gd3[i]:=gd[3][gd[2][p[i]]];

else

if gd[3][gd[2][p[i]]]<>0 then

gd3[i]:=irem(gd[3][p[i]]+gd[3][gd[2][p[i]]]-gd[3][p[gd2[i]]],

gd[3][gd[2][p[i]]])

else

gd3[i]:=gd[3][p[i]]-gd[3][p[gd2[i]]]

end if `

end if


218

end do; RETURN([evalm(gd[1]),evalm(gd2),evalm(gd3)])

> end proc;

In the following we use the procedure random_groupoid_op(n, kmax, nmax) to

generate gd that encodes the subgroupoid

G ={(u,ku,v+tku,u,v): f(u)=f(v), tZ}X×Z×X, X={1,2,3,…,n}

with the property that the probabilities pk of {u: f(u)=f(k)} satisfy: p1 =max {pj, j=1..n}. Then

we use the command randperm(n) (belonging to combinat package) to get a random

permutation p, and lastly the procedure perm_unit_space(gd,p) to permute the unit space X

of the groupoid G with respect to p: XX and obtain a new groupoid.

> gd:=random_groupoid_op(10,8,3);

> with(combinat):

> p:=randperm(10);

> perm_unit_space(gd,p);

> orbits(gd);

>orbits(perm_unit_space(gd,p));


219

BIBLIOGRAPHY

[1] M. Buneci, Groupoids and irreversible discrete dynamical systems I, Fiabilitate şi

durabilitate (Fiability & durability), No. 1/2012, 350-355.

[2] M. Buneci, Groupoid reductions associated to discrete dynamical systems, Annals of the

―Constantin Brâncuşi‖ University of Târgu-Jiu. Engineering Series. No. 3(2012), 171-182.

[3] M. Buneci, Using Maple to represent the subgroupoids of trivial groupoid X×Z×X,

Fiabilitate şi durabilitate (Fiability & durability), Supplement No 1 (2013), 446-454.

[4] M. Buneci, Using Maple for visualization of topological subgroupoids of X×Z×X, 7th

Symposium Durability and Reliability of Mechanical Systems SYMECH 2014, Târgu-Jiu,

May 2014.

[5] M. Matsumoto and T. Nishimura, Mersenne twister: A 623-dimensionally equidistributed

uniform pseudorandom number generator, ACM Transactions on Modeling and Computer

Simulations 8 (1998), 3–30.


220

USING MAPLE FOR VISUALIZATION OF TOPOLOGICAL

SUBGROUPOIDS OF X×Z×X

Mădălina Roxana BUNECI, University Constantin Brâncuşi of Târgu-Jiu

Abstract. The purpose of this paper is to present various ways to visualize topological subgroupoids of the

trivial groupoid X×Z×X, where Z is the group of integer numbers endowed with the discrete topology and X a

topological space.

Keywords: topological groupoid; Khalimsky topology; equivalence relation; visualization;

.

1. INTRODUCTION

We shall use the notation and terminology from [1] - [4]. In [3] we proved that a

subgroupoid G of XZX, where Z is the group of integer numbers, is characterized by the

graph R of an equivalence relation on X and a family of integer numbers {ku,v}(u,v)R

satisfying the following conditions:

1. ku,u0 for all uX.

2. If ku,u ≠ 0, then ku,v + kv,w = ku,w (mod ku,u), else ku,v + kv,w = ku,w.

We also proved that if ku,u ≠0 and u≠v, then we may consider

ku,v{0,1,…, ku,v-1}

for all v such that (u,v)R. The groupoid G is represented as G=

u

v

u,v R

G

where

u

vG ={(u,ku,v+tku,u,v): tZ}, (u,v)R

Alternatively, we prove in [4] that G can be completely characterized by X and two functions

f:XX and k :XZ satisfying the properties

4. f(f(u))=f(u) for all uX.

5. k(f(u)) 0 for all uX.

6. If k(f(u)) ≠ 0, then k(u){0,1,…, k(f(u))-1}.

If the function f and k are given, then the relation R is defined by (u,v)R if and only if

f(u)=f(v), ku,u := k(f(u)) for all uX and


k(u) – k(v), if k(f(u)) = 0

for all (u,v)X×X with the property that f(u)=f(v) and u≠v.

The purpose of this paper is to provide various ways to visualize the topological

subgroupoids G of XZX assuming that X is a topological space and Z has the discrete

topology. We use Maple environment to implement the visualizations.


221

2. VISUALIZATION OF SUBGROUPOIDS OF XZX DATA

Let G ={(u,ku,v+tku,u,v): f(u)=f(v), tZ} be a subgroupoid of XZX, where X is a finite

set {x1, x2, …, xn} and Z is the group of integers. As in [4] we shall use the characterization

of G in terms of f:XX and k :XZ in order to implement in Maple the groupoid G, where

k(u)=ku,f(u) for all uX. More precisely, we use a list gd of three arrays:

gd[1] contains the elements of X (gd[1][i]=xi, i=1..n),

gd[2][i] = the index in gd[1] of f(gd[1][i]), i=1..n,

gd[3][i] = k(gd[1][i]), i=1..n.

The procedure visualization(gd) represent each (r,d)-fibre j

i

x

xG as the rectangle with

top left corner (i-1,j) and bottom right corner (i,j-1) filled with a color uniquely determined by

i jx ,xk and i ix ,xk .

visualization:=proc(gd)

local n,no_iso0, maxku, maxkuv0,i,j,elem,m,c,p,q,z,mk;

n:=op(2,op(2,gd[1]));

no_iso0:=0; maxku:=0; maxkuv0:=0;


if gd[3][gd[2][i]]>maxku then maxku:=gd[3][gd[2][i]]

else

if gd[3][gd[2][i]]=0 then

no_iso0:=no_iso0+1;

for j from 1 to i-1 do

if

(gd[2][j]=gd[2][i] and abs(gd[3][i]-gd[3][j])>maxkuv0)

then

maxkuv0:= abs(gd[3][i]-gd[3][j]);

end if

end do

end if

end if

end do;

elem:=array(1..n*n);c:=array(0..n+1,0..n+1);

for i from 0 to n+1 do

for j from 0 to n+1 do

c[i,j]:=-2

end do

end do;

z:= no_iso0/n; mk:= 2*(1-z)/(maxku*(maxku+1));


for j from 1 to n do

if gd[2][i]=gd[2][j] then

if gd[3][gd[2][i]]=0 then

c[i,j]:=frac(1+(gd[3][i]-gd[3][j])/(2*maxkuv0+1)*z);

else

c[i,j]:=z/2+(gd[3][gd[2][i]]*(gd[3][gd[2][i]]-1)/2+1+


222

irem(gd[3][gd[2][i]]+gd[3][i]-gd[3][j],gd[3][gd[2][i]]))*mk;

end if

end if

end do

end do;

m:=0;




m:=m+1;

elem[m]:=rectangle([i-1,j],[i,j-1],color=COLOR(HSV,c[i,j],1.,1.));

end if;

end do;

end do;

RETURN(display(seq(elem[i],i=1..m),axes=none,style=patchnogrid))

end proc;

For each k{1,2,…,kmax}, where kmax=max{ku,u: uX} the procedure visualization

allocates a hue (in HSV model). Thus we can visualize the distribution of {ku,u: uX} among

{1,2,…,kmax}. However if kmax is big and the length of the minimal interval containing the

set {ku,u:uX} is small, then less details are visible. Therefore in this it would be better to use

a version of the visualization procedure that allocates hues (in HSV model) only for each ku,u.

The left image below is obtain using the first version, while the right correspond to the second

version.

Also if n is big less details are visible. In this case it is better to represent only the (r,d)-fibres j

i

x

xG with i=a1..b1 and j=a2...b2 where a1, b1, a2 and b2 are chosen in concordance to the region

of interest. The pictures below illustrated this situation:


223

3. VISUALIZATION OF TOPOLOGY OF SUBGROUPOIDS G XZX

Let us consider that X is endowed with a topology X and Z is endowed with the

discrete topology. Then under the operations

(x, n, y)(y, m, z) = (x, n+m, y)

(x, n, y)-1

= (y, -n, x)

XZX become a topological groupoid. In the following the unit space of the groupoid

XZX, {(x,0,x), xX}, is identified with X. We endow any subgroupoid G XZX with

the subspace topology G coming from XZX.

In [1] we started with a topological groupoid (G, ηG), we introduced a topology ηR(ηG)

on the principal groupoid R associated with G (called transported topology from G) and a new

topology ηGR on G (called the modified topology on G with respect to R). Let us recall that a

basis for the topology ηR(ηG) is given by the family of sets {U(F)}F, where each F is a finite

collection of open subsets of G (i.e. F G) and

U(F) = Ud,rFU

Moreover a basis for the topology ηGR is given by

FU

1Ud,rd,rV

where V runs over all open sets of G and F runs over all finite collections of open subsets of

G.


224

Let us use the notation

(x,y)(u,v)

if and only the sequence ((xi,yi))i converges to (u,v), where xi=u and yi=v for all i. According

to [2] this is equivalent to (xi)iI converges to u with respect to X, (yi)iI converges to v with

respect to X and

1. for ku,u0: kx,x0, kx | ku (kx divides ku) and kx,x | ku,v –kx,y

2. for ku,u=0: (kx,x =0 and ku,v =kx,y) or (kx,x0 and kx,x | ku,v –kx,y)

We want to know if (u+p,x+q)(u,v) for p,q{-1,0,1}. Let us assume that u+p (respectively,

v+q)U for all neighborhood U of u (respectively, v) with respect to X. In this case if p=0 or

q=0, then (u+p,x+q)(u,v) if and only the procedure visualization allocates the same hue (in

HSV model) to u

vG and u p

v qG

. If (u+1,v+1)(u,v) then we draw a rectangle r1 on the right

top corner of the rectangle r associated to u

vG by the procedure visualization. The dimensions

of r1 are ¼ of the dimensions of r and color is that of u p

v qG

. The below procedure

visualization_top(gd) allows us to visualize the topology of G in this way for the case

when X is endowed with the indiscrete topology X={, X}. We present here only the part of

the procedure that differs from visualization(gd), meaning the construction of the array

elem containing the rectangles associated to each u

vG .

visualization_top:=proc(gd)

………….

m:=0;




m:=m+1;

elem[m]:=[rectangle([i-1,j],[i,j-1],color=COLOR(HSV,c[i,j],1.,1.))];

for p from -1 to 1 by 2 do

for q from -1 to 1 by 2 do

if

((c[i+p,j+q]<>-2) and (c[i,j]<>c[i+p,j+q]) and (gd[3][gd[2][i+p]]<>0) and

(irem(gd[3][gd[2][i]],gd[3][gd[2][i+p]])=0) and (irem(gd[3][i+p]-

gd[3][j+q]-gd[3][i]+gd[3][j],gd[3][gd[2][i+p]])=0))

then

elem[m]:=[rectangle([i-1/2+p/4,j-1/2+q/4], [i-1/2+p/2,j-1/2+q/2],

color=COLOR(HSV,c[i+p,j+q],1.,1.)),op(elem[m])]

end if;

end do;

end do;

end if;

end do;

end do;

…….


225

The following two pictures are the representations of a groupoid G XZX, where

X={1,2,…,n} returned by the procedure visualization (left) and visualization_top (right). We

see, for instance that (4,4)(3,3), (4,4)(5,5), (4,6)(5,5), (4,6)(3,5), etc.

The below procedure visualization_topK(gd) allows us to visualize the topology

of G the case when X={x1, x2, …, xn} is endowed with the topology X with basis

{{x2k+1},k=0..[n/2]} {{x2k-1, x2k, x2k+1}X, k=1..[n/2]}

(Khalimsky topology – see [5], [6] and [7]). For an (r,d)- fibre u

vG we modify the color of the

rectangle associated to u

vG by the procedure visualization by decreasing the saturation (in the

HSV model) by 1/8 for each odd index of u, v in X. We present here only the part of the

procedure that differs from visualization(gd), meaning the construction of the array elem

containing the rectangles associated to each u

vG .

visualization_topK:=proc(gd)

.....

m:=0;




m:=m+1;

elem[m]:=[rectangle([i-1,j],[i,j-1],color=COLOR(HSV,c[i,j],1-

irem(i,2)/8-irem(j,2)/8,1.))];

if irem(i,2)=0 and irem(j,2)=0 then

for p from -1 to 1 by 2 do

for q from -1 to 1 by 2 do

if

((c[i+p,j+q]<>-2) and (c[i,j]<>c[i+p,j+q]) and (gd[3][gd[2][i+p]]<>0) and

(irem(gd[3][gd[2][i]],gd[3][gd[2][i+p]])=0) and (irem(gd[3][i+p]-

gd[3][j+q]-gd[3][i]+gd[3][j],gd[3][gd[2][i+p]])=0))

then

elem[m]:=[rectangle([i-1/2+p/4,j-1/2+q/4], [i-1/2+p/2,j-1/2+q/2],

color=COLOR(HSV,c[i+p,j+q], 1-irem(i+p,2)/8-irem(j+q,2)/8,1.)),op(elem[m])]

end if;


226

end do;

end do;

end if;

end if;

end do;

end do;

The following two pictures are the representations of a groupoid G XZX, where

X={1,2,…,n} returned by the procedure visualization_top (left) and visualization_topK

(right). We see, for instance that (2,2)(1,1) if X is endowed with the indiscrete topology

but not with the Khalimsky topology.

BIBLIOGRAPHY

[1] M. Buneci, Topological groupoids with locally compact fibres, Topology Proceedings 37

(2011), 239-258.

[2] M. Buneci, Groupoids and irreversible discrete dynamical systems II, Fiabilitate şi

durabilitate (Fiability & durability), No. 1/2012, 356-359.

[3] M. Buneci, Groupoid reductions associated to discrete dynamical systems, Annals of the

―Constantin Brâncuşi‖ University of Târgu-Jiu. Engineering Series. No. 3(2012), 171-182.

[4] M. Buneci, Using Maple to represent the subgroupoids of trivial groupoid X×Z×X,

Fiabilitate şi durabilitate (Fiability & durability), Supplement No 1 (2013), 446-454.

[5] E.D. Khalimsky, Pattern analysis of n-dimensional digital images, in: Proc. of the IEEE

Internat. Conf. on Systems, Man and Cybernetics, 1986, 1559–1562.

[6] K. Y. Kong, Concepts of digital topology, Topology and its applications, 46 (1992), 219-

262

[7] V. Kovalevsky, Finite Topology as Applied to Image Analysis, Computer Vision, Graphics

and Image Processing, 46 (2) (1989), 141-161.


227

ASPECTS OF A LINEAR PROGRAMMING MODEL DEDICATED TO

THE TRANSPORT SYSTEM

Lect. PhD. eng. Elisabeta Mihaela CIORTEA,

University ‖1 Decembrie 198‖ of Alba Iulia, [email protected]

Lect. PhD. math. Mihaela ALDEA,

University ‖1 Decembrie 198‖ of Alba Iulia, [email protected]

Abstract: In the paper is presented a linear programming model whose theory has benefited from the

contribution of the interdisciplinary approach that allowed deeper analysis of complex systems maximum

efficiency. For modeling system we have used Petri nets because we have in our case discrete event systems. In

use of the non-timed and with auxixliary times Petri model, the transport stream was divided into sections, and

these sections will be analyzed successively. Due to the complexity of the system and the large amount of

calculations required it was not possible to analyze the system as a unitary whole. A first attempt to model the

system as a unitary whole led to the blocking of the model during simulation, because of the large processing

times.

Keywords: Petri nets, transport, processing

LINEAR PROGRAMMING MODEL

Mathematical programming models and especially their subclass, linear programming

models, plays an extremely important role both in theory and in practice Error! Reference

source not found..

The theory benefited from the contribution of the interdisciplinary approach which

allowed: further analyzing the maximal efficiency of complex systems, the discovery of new

concepts of optimum, it has perfected the methods of research and knowledge, and the

practice has been enriched with an extremely useful instrument for analysis and substantiation

decisions.

The structure of general linear programming model is established primarily by the set of

activities {A1, A2, ... An} that compose the analyzed system, the set of used resources {R1, R2,

... Rm} as well as technical relations between these. The relationship between activities and

resources is determined by the manufacturing technology for each activity Aj (j=1,...,n) and

may be characterized numerically by the column vector a(j)

with components (a1j, a2j, ... amj).

The elements {aij, i = 1,...,m; j = 1,...,n} are called technical coefficients or specific

consumption coefficients and show how much of the resource Ri is consumed to produce a

unit of the product Pj (as a result of the activity Aj). All the manufacturing "technologies"

defined by the column vectors a(j)

can be organized in a matrix A with m lines and n columns;

each line refers to a resource Ri (i = 1,...,m) and each column relates to an activity Aj (j =

1,...,n).

Denoting by xj (j = 1,...,n) the result of Aj in a given period and bi (i = 1,...,m) available

quantities of resources Ri (i = 1,...,m), the following technical restrictions can be written

mathematically:


228

mnmn2m21m1

2n2n222121

1n1n212111

bxa...xaxa

bxa...xaxa

bxa...xaxa

or Ax b (1)

where

nnmnmm

n

n

b

b

b

b

x

x

x

x

aaa

aaa

aaa

A...

si ...

;

...

............

...

...

2

1

2

1

21

22221

11211

(2)

Each inequation / restriction incorporates two statements:

- the amount of a resource consumption can not exceed the amount available;

- the total consumption Rij of resource Ri for the activitiy Aj is proportional to the

intensity, i.e. xj, so

jijij xaR (3)

The system of restrictions described above makes the link between resources and

activities through the m linear restrictions.

The linear programming model 0 contains restrictions on the type (1) as well as a

criterion for "performance" to assess the effectiveness of each activity. Depending on the

purpose, we can choose as a criterion of efficiency an indicator that measures the effort, one

that measures the result or an indicator expressed as the ratio between result and effort.

The maximum efficiency means minimizing the effort and maximizing the outcome,

and the concept of optimum is defined as a program that minimizes or maximizes an objective

function and in the same time satisfies all technical restrictions.

Assuming that each component of the vector line c = (c1, c2, ..., cn) measure the

efficiency of a unit of the result of Aj, then we can introduce the linear function:

nn xcxcxcxf ...)( 2211 (4)

assessing the performance of any program x.

Summarizing, we obtain the following linear programming software:

)(xfoptimx

(5)

(6)

(7)

(8)

The relations (5), (6), (7) and (8) together constitute the general model of a linear

programming problem, each having a specific role:

(1)

(2)

(3)

njx

mIIIkbxa

mIIIibxa

j

n

j

kjkj

n

j

ijij

,1 0

,...,2,1 unde

,...,2,1 unde

1

212

1

211


229

- the relation (5), where f(x) =

n

1jjjc x is called the objective function for efficiency of

the problem, evaluates the performance of each program options x;

- the relation (6) by type

n

1jijij bxa represents resources type restrictions;

- the restriction (7) such

n

1jkjkj bxa it refers to the technical restrictions qualitative;

- the relation (8) xj 0 j = 1,...,n called the non-negativity variables condition, ensures a

feasible solution from the point of view of economic logic.

In the case of determining optimal production assortment structure, are known the

available quantities of each raw material bi, the technological coefficients aij, the maximum

quantities jx and the minimum

jx that can be produced from each assortment in the analyzed

period and the unitary profits pj, for each type of product.

The general dynamic model of manufacturing system is given by the matrix vector

equation:

x k

x k

x k

A A A

A A A

A

A A A

x k

x k

x k

B B B

B B B

B

B B B

u k

u k

u k

f x k u

n

n

n

ij

n n nn n

n

n

ij

n n nn n

1

2

11 12 1

21 22 2

1 2

1

2

11 12 1

21 22 2

1 2

1

2

1 11

1

1

.

.

.

. . .

. . .

. . . . . .

. . . . .

. . . . . .

. . .

.

.

.

. . .

. . .

. . . . . .

. . . . .

. . . . . .

. . .

.

.

.

,

1

2 2 2

k

f x k u k

f x k u kn n n

,

.

.

.

,

(9)

and so x k A x k B u k f x k u ka ij a pq a 1 , (10)

1, , , , , , , , , ,n i j n n p q n m i j p q (11)

where x k 1 is the matrix of system conditions at the time of t x k 1 , and u k

are the matrices of states and inputs at time t , Aij is the interaction matrix Sb j with Sbi

influencing state x k 1 of the entire system, Bpq - the interaction matrix between u kq

and x kp in which u k x kq

f

p

pq influencing state x k 1 of the entire system,

f x k u k , (12) – the internal interaction matrix between the states x t and their inputs

u t of the entire system in time t , that influence the state x t 1 of the entire system in

the next moment t 1 .

The order of matrix A n nij l c, , is given by the product of the number of each of the

kinematic couplings Sbc and the number of state parameters of each coupling. The elements

aij are linear and nonlinear expressions.


230

The elements bpq are determined by the rule:

0 then ),( influencesnot )(

0 then ),( influences )())(),((

pqpq

pqpq

qppqbkxku

bkxkukukxf (13)

This rule also applies to the matrix elements (12).

The equations of state are subject to the conditions and restrictions:

- for S continue t

- for S discrete t k t Te

e

Ne

,1

;

if t cte then T N te e ,

- 0x – a initial state known of S , and x xa0 0 for ,

- R x k u k , 0, R expressing a set of restrictions in the matrix (12),

- k q x k , and 0 0 0 q x for .

ANALYSIS AND INTERPRETATION SYSTEM

The sequence of events is reduced to a simple ordering of their occurrence. The

simulation involves the consecutive execution of transitions, according to the enforcement

rule of transitions Error! Reference source not found.0 Error! Reference source not

found..

SECTION II with auxiliary times:

Fig.1: Petri net SECTION II with auxiliary times:


231

By using the generalized stochastic Petri net, it obtains the model of the production

system to be analyzed. The execution durations of transitions are distributed using an

exponential law which allows obtaining a graph of the Petri net marking, and this is an

homogeneous Markov process. In the timed Petri nets it is considered that the viable and

marginal properties are some of the most important in the distributed system under review.

The results obtained contribute to finding and studying a large number of timed Petri nets for

which it can produce an effective analysis.

By adopting a new location are reduced the transportation distances. The ovens are

situated on the line making it easier for the administration of cargo passing through the ovens.

The packaged goods do not have to enter in the box and crowding the existing space. A

summary of the presented simulation results is indicated in the following tables.

The results shown in the tables above are obtained when using the simulation of a

number of 1300 iterations.

The second model is applied to section 2 P-timed, deployed in a time of 12,055,273 sec.

for the existing model, and 11756868 sec. for the proposed model with a number of 1300

events. In the figure are shown only the fields that change its value. At this section the actual

time of transport is 81.82 min. and the transportation time obtained from the application of the

new location is 82.70 min. Model: Section 2 with timed positions.xml

Events:1302

Time:12055273

Place Name Arrival Sum Arrival Rate Arrival Dist. Throughput

Sum

Throughput

Rate

Throughput

Dist.

Waiting

Time

Queue

Length

Vagonet disponibil 68 5.6407e-006 177283.4265 69 5.7236e-006 174714.1014 2799.8261 0.016025

Incarcare Vagonet 69 5.7236e-006 174714.1014 69 5.7236e-006 174714.1014 15 8.5855e-005

Transport Vagonet 69 5.7236e-006 174714.1014 69 5.7236e-006 174714.1014 60 0.00034342

Vagonet neutilizat 69 5.7236e-006 174714.1014 69 5.7236e-006 174714.1014 174654.1014 0.99966

Descarcare in Stocator 1 69 5.7236e-006 174714.1014 69 5.7236e-006 174714.1014 15 8.5855e-005

Stocator 1 69 5.7236e-006 174714.1014 69 5.7236e-006 174714.1014 15 8.5855e-005

Spatiu 1 69 5.7236e-006 174714.1014 69 5.7236e-006 174714.1014 174699.1014 0.99991

Incarcare in Cuptor 69 5.7236e-006 174714.1014 69 5.7236e-006 174714.1014 4320 0.024726

ARDERE BISCUIT 69 5.7236e-006 174714.1014 69 5.7236e-006 174714.1014 86400 0.49452

Ardere neutilizata 69 5.7236e-006 174714.1014 69 5.7236e-006 174714.1014 88314.1014 0.50548

Descarcare din Cuptor 69 5.7236e-006 174714.1014 69 5.7236e-006 174714.1014 30 0.00017171

Transport cu Vagoneti 69 5.7236e-006 174714.1014 69 5.7236e-006 174714.1014 60 0.00034342

Vagoneti neutilizati 69 5.7236e-006 174714.1014 69 5.7236e-006 174714.1014 174654.1014 0.99966

Incarcare pentru Biscuitare 69 5.7236e-006 174714.1014 69 5.7236e-006 174714.1014 30 0.00017171

BISCUITARE 69 5.7236e-006 174714.1014 68 5.6407e-006 177283.4265 81000 0.4569

Biscuitare neutilizata 68 5.6407e-006 177283.4265 69 5.7236e-006 174714.1014 94888.0145 0.5431

Descarcare de la Biscuitare 68 5.6407e-006 177283.4265 68 5.6407e-006 177283.4265 15 8.461e-005

Stocare 2 68 5.6407e-006 177283.4265 68 5.6407e-006 177283.4265 15 8.461e-005

Spatiu 2 68 5.6407e-006 177283.4265 68 5.6407e-006 177283.4265 177268.4265 0.99992

Descarcare pentru transport vagonet

68 5.6407e-006 177283.4265 68 5.6407e-006 177283.4265 15 8.461e-005

Transport vagonet 68 5.6407e-006 177283.4265 68 5.6407e-006 177283.4265 60 0.00033844


232

vagonet neutilizat 68 5.6407e-006 177283.4265 68 5.6407e-006 177283.4265 177223.4265 0.99966

Incarcare dispozitiv

Glazurare 68 5.6407e-006 177283.4265 68 5.6407e-006 177283.4265 15 8.461e-005

GALZURARE 68 5.6407e-006 177283.4265 68 5.6407e-006 177283.4265 720 0.0040613

Glazurare neutilizata 68 5.6407e-006 177283.4265 68 5.6407e-006 177283.4265 176563.4265 0.99594

Descarcare 68 5.6407e-006 177283.4265 68 5.6407e-006 177283.4265 120 0.00067688

Transport vagon 68 5.6407e-006 177283.4265 68 5.6407e-006 177283.4265 200 0.0011281

vag neutilizat 68 5.6407e-006 177283.4265 68 5.6407e-006 177283.4265 177083.4265 0.99887

Fig. 2: Global performance indicators relating to the positions in the SECTION 2 model, considered with

stochastic time delay by type P

Model: Section 2 with timed positions.xml

Events:1302

Time:11756868

Place Name Waiting Time Queue Length

Vagonet disponibil 2799.8261 0.016432

Incarcare Vagonet 15 8.8034e-005

Transport Vagonet 45 0.0002641

Vagonet neutilizat 170344.3913 0.99974

Descarcare in Stocator 1 15 8.8034e-005

Stocator 1 30 0.00017607

Spatiu 1 170359.3913 0.99982

Incarcare in Cuptor 15 8.8034e-005

ARDERE BISCUIT 86400 0.50707

Ardere neutilizata 83989.3913 0.49293

Descarcare din Cuptor 15 8.8034e-005

Transport cu Vagoneti 90 0.0005282

Vagoneti neutilizati 170299.3913 0.99947

Incarcare pentru Biscuitare 15 8.8034e-005

BISCUITARE 81000 0.46849

Biscuitare neutilizata 90563.3043 0.53151

Descarcare de la Biscuitare 15 8.6758e-005

Stocare 2 30 0.00017352

Spatiu 2 172865.1176 0.99983

Descarcare pentru transport

vagonet 15 8.6758e-005

Transport vagonet 45 0.00026027

vagonet neutilizat 172850.1176 0.99974

Incarcare dispozitiv

Glazurare 15 8.6758e-005


233

GALZURARE 720 0.0041644

Glazurare neutilizata 172175.1176 0.99584

Descarcare 120 0.00069406

Transport vagon 180 0.0010411

vag neutilizat 172715.1176 0.99896

Fig. 3: Global performance indicators relating to the positions in the SECTION 2 model, considered with

stochastic time delay by type P

CONCLUSIONS

The simulation of the proposed manufacturing system using timed Petri nets, offers the

possibility of viewing over time the manufacturing process.

Applying product manufacturing times and transportation times obtained by measuring

the spot, it is obtained graphical representations that show the average times of transport

activity, but also the evolution of the average time of processing related to the transport

activity time, using as parameters sets of finite products.

REFERENCES

1. Abrudan Ioan, „Sisteme flexibile de fabricaţie. Concepte de proiectare şi

management‖, Editura DACIA, 1996, ISBN 973-35-0568-4

2. Bauşic Florin, „Dinamica maşinilor de construcţii‖, Editura MatrixRom, Bucureşti,

2001, ISBN 973-685-229-6

3. Bejan Andrei, „Modelarea timpului de orientare în sisteme de aşteptare cu priorităţi‖,

Teza de doctorat, Chişinău, Septembrie 2007, Universitatea de Stat din Moldova,

Facultatea Matematică şi Informatică

4. Camerzan Inga, „Proprietăţi structurale ale reţelelor Petri temporizate‖, Teza de doctor

în informatică, Universitate de Stat din Tiraspol, Chişinău, 2007-12-15

5. Mihăilă N., „Introducere în programare liniară‖, Editura didactică şi pedagogică,

Bucureşti, 1964

6. Păstrăvanu Octavian, Matcovschi Mihaela, Mahulea Cristian, „Aplicaţii ale reţelelor

Petri în studierea Sistemelor cu evenimente discrete‖, Editura Gh. Asachi, 2002, ISBN

973-8292-86-7


234

AN EXTREMAL PROBLEM FOR UNIVALENT FUNCTIONS

Professor dr. Miodrag IOVANOV, ―Constantin Brâncuşi‖ University of Târgu-Jiu,

[email protected]

Abstract: 1. Let K be a compact class of functions of the form

f(z)=z + a2z2 +………. (1)

which are regular and univalent in the unit disc U and let p and a be complex numbers,

p 0; p 1; 0 < |a| < 1.

For f K, p and a fixed, the equation

f(z) = pf(a) (2)

has at most a solution zf in U.

If p = 0 or p = 1, then for any f K the equation (2) has only the solution z = 0 or z = a

respectively. We will exclude the case p = 0 or p = 1 and consider the following extremal problem:

given K, p and a, p 0; p 1; 0 < |a| < 1, determine

m(p; a; K) = min{: = |zf|, f K} (3)

This problem was first considered by p. T. MOCANU [4] who solved it for the whole class

S of normalized univalent functions.

I. KACZMARSKI [3] obtained m(p, a, K) for K = SR, the subclass of S consisting of

functions with real coefficients and K = S, the class of starlike functions. L. BURSHTEIN [1]

solved this problem for K = TR, the class of tipically real functions and for some other classes

represented by Stieltjes integrals [2]. M. READE and E. ZTOTKIEWICZ [7] obtained m(p,a,K)

for K =

S , the class of starlike functions of order , K = S°, the class of convex functions and

also for K = TR, by using a different method.

In this paper we consider the following more general problem: given K, a, 0 < |a| < 1,

and the polynomial pn(z) = cnzn + Cn-1z

n-1 + ……+ c0, where c0 0, c, are complex numbers,

determine

m(c0; c1…..cn; a; K) = min{: = |zf|, f K } (4)

where zf satisfies the equation

f(z) = pn(z)f(a) (5)

We‘ll solve this problem for K =

S , K = Sc and K = TR.

We recall that the function f of the form (1) is starlike of order ; 0 < α < 1, iff

)(

)(Re

'

zf

zzf , z U (6)

The class of such functions shall be denoted by

S …..

0S = S is the class of starlike

functions.

A function f of the form (1) is convex iff

0)(

)(1Re

'

"

zf

zzf , z U (7)


235

The class of such functions shall be denoted by S°.

A function f of the form (1) is tipically — real iff f(z) is real if and only if z (-1; 1).

The class of such functions shall be denoted by TR.

2. If K, z and a are given, the set

Kfzf

zf;: K) z, D(a, (8)

is called the domain of variability of f(z)/f(a) for the class K. If p, a, K are given, then (2) has

a solution if and only if p D(a, z, K), z U.

We need the following lemmas :

Lemma 1. [7] The domain of variability of

112,

azf

zaf for the class

S

is the closed disc

22

1

1

1

1

zz

za

(9)

Lemma 2. [5]. The domain of variability for the class TR is the convex hull of the set

22;: TR) z, D(a,1

1

ttzz

taa (10)

We will find (4) by using the condition pn(z) D(a,z,K) for

S , S*, S° and TR

3. For the class

S we have the following result

THEOREM 1. If c0, c1,..., cn are complex numbers, c0 0; 0 < |a| < 1

and then,121

zap

zaz

zap

zzSacccm

nn

n 1:inf,,,.......,, 10 (11)

Proof. We put the condition pn(z) D[a, z,

S ).

Using Lemma 1 we deduce that the solution z of (5) must satisfy the condition

2211

1

z

az

z

za

z

zapn

(12)

which can be written in the following form

zap

zaz

zap

z

nn

1 (13)

From this we easily get (11).


236

For 12

1 we obtain

zap

zz

zap

zzSacccm

nn

n 1:inf,,,.......,,2

110 (14)

If we take c1 = c2 = …. = cn = 0; c0 = p we get

11

,,2

1

pa

paSapm (15)

The extremal function is f(z) = z(1 + ei·z)

-1; = arg a(p-1). Since f is convex we deduce

11

,,

pa

paSapm c (16)

This result is due to M. READE and E. ZLOTKIEWICZ [7]. If α = 0, from (11) we get

zap

zz

zap

zzSacccm

nn

n 1:inf,,,.......,, 10 (17)

If we take c0 = p, ci =0, i = 1,2, …,n, we deduce

12

11

2

121,,

2222

ap

paa

ap

paaSapm (18)

The extremal function in this case is f(z) = z(1-ei

·z)-2

. This result is due to

KACZMARSKI [3].

4. Let K = TR and consider the equation (5), where pn(z) = cnzn+…+c0, ci are real, c0 0,

pn(a) 1, z is real and -1 < a < 1.

In this case we nave the following result:

THEOREM 2. If ci, i = 0,1,…,n, are real numbers c0 0 and 0 < a < 1, then

,min,,,.......,,

1

110 zTRacccm

zn

(19)

where z and , satisfy the equations

01

12

a

zzp

a

zn (20)

and

01

12

ap

an

(21)

respectively. The extremal functions are f(z) = z(1-z)2 and f() = ·(1 +)

-2 respectively.

Proof. We put the condition pn(z)D(a, z, TR).


237

Using Lemma 2 we deduce

2,2,1

1

ttzz

taazpn (22)

Hence our problem is to find the minimum of |z| where z verifies (22).

If we let A = a + a-1

the equation (22) can be written

012 ztAzptzz n (23)

The solution of this equation will be a function of t, z = z(t); t [-2; 2].

If at a point t (-2; 2), z(t) has an extremal value, then z’(t) = 0. If we take the

derivative in (22) and put the condition z’(t) = 0, we deduce z(t)[pn(z(t))-1] = 0.

Since f is univalent on (-1,1), pn(z)=1 and f(z) = pn(z)f(a) imply z = a. This contradicts

pn(a)1. Hence z(t) = 0. But this holds only for c0 = 0 which contradicts our hypothesis.

Hence the only points where z has an extremal value are t = ±2. This completes the proof of

Theorem 2.

The equation (22) can be written

zpz

Azzpzt

n

n

1

12

(24)

Remark. Consider the case c0 = 1 and n = 2m + 1.

The polynominal qn-1(z) = c1 + c2z + … + cn2n-1

is of even degree 2m.

In [6] T. POPOVICIU has given sufficent conditions for the positivity of a polynomial

of even degree. Using these conditions we get:

THEOREM 3. If

1,...,1,0,0,0 2

2232121 miccc iii (25)

where

3

2cos

1

2

n

, then in the class TR we have

0

1

1:min,,,.......,,

2

110

a

zzp

a

zzTRacccm n

zn (26)

The extremal functions is the Koebe function

21 z

zzf

(27)

We shall apply Theorem 2 to the particular case: c0 = p; c1 = q; c2 = c3 = … = cn = 0.

In this case the equation (5) becomes

f(z)=(p+qz)·f(a) (28)

From Theorem 2 we deduce that the minimum of |z| is obtained for z = ±2.

Equation (24) becomes

qzpz

pzAqpzqzt

1

23

(29)


238

Suppose 0 < a < 1. Then we have

THEOREM 4. If 0 < p ≤ 3

2, 0 < q <

3

1 then in the class TR we have

m(p; q; a; TR) =

z (30)

where

z is the smallesi positive root of the equation

0122 23 pzpAqzqpqz (31)

The extremal functions is the Koebe function 21 z

zzf

Proof. The graph of the function defined by (29) has the following form (fig.1).

We remark that

z (0, 0

z ) where 0

z is the solution of the equation (29) for t = 0 i.e. the

solution of

023 pzAqpzqzzg (32)

Since

g(0) = p > 0, (33)

g(1) = 2(p + q)- A < 0 (34)

and

1,2(010 1 qpaaAgg ), (35)

we deduce 0

z (0;1).

Fig. 1.


239

Example. For p = q = a = 7

1 we get m (

7

1,7

1,7

1,TR) 0,02.

THEOREM 5. If 0< p <1; q > 3

1; p + q <

2

A then in the class TR we have

zzTRapm

z

z,min,,

1

1 (36)

where

z is the smallest positive root of the equation (31) and 0

z is the biggest negative root

of the equation

0122 23 pzpAqzqpqz (37)

The extremul functions are f(z) = z(1 + z)-2

and f() = (1-)-2

respectively.

If p, q, are real numbers, we deduce :

THEOREM 6. If p, q are real numbers, 0 < a < 1, then in the class TR we have

,min,,,

1

1zTRaqpm

z

(38)

where z and , are the Solutions of (31) and (37) respectively.

The extremal functions are f(z) = z(1 + z)-2

and f() = (1-)-2

respectively.

REFERENCES

[1] Burshtein, L., Roots of the equation f(z) = αf(a) for the class of typically — real

functions. Mat. Zametki 10, 41-52 (1971).

[2] Burshtein, L., The solution of the equation f(z) = αf(z) in the class of star — shaped

functions, Izv. Vyss. Ucebn. Zaved. Mat., no., 12 (151), 47-50 (1974).

[3] Kaczmarski, I., Sur l’equation f(z) =pf(c) dans la famille des functions univalentes á

coefficients reels, Bull. Acad. Pol. Sci, vol. XV, 4, 245-251 (1967).

[4] Mocanu, P. T., On the equation f(z) = αf(a) in the class of univalent functions.

Mathematica vol. 6 (29), 1, 63-79 (1974).

[5] Pilat, B., On typically — real functions with Montel’s normalization,. Annales UMCS,

XVIII, 53-72 (1964).

[6] Popoviciu, T., Sur une condition suffisante pour qu’un polynom soit positif, Mathematica,

vol. XI, 247-256 (1935).

[7] Reade, M. O., Zlotkiewicz„ E., On the equation f(z) = pf(a) in certain classes of analytic

functions. Mathematica vol. 13 (36), 2, 281—286 (1971).


240

BOOLEAN NORMED ALGEBRAS

Prof.Drd. Constantin P. BOGDAN

Universitatea din Craiova

As.univ. Olimpia PECINGINA

Universitatea ―Constantin Brancusi‖ ,Tg-Jiu

ABSTRACT: In this section we begin a systematic study of algebras given by algebraic measures.Knowing that

the measure is an essential positive total additive function.The most important lesson is made up of probability

measures.Each algebra with measures of probability can be interpreted as a system of events, with the measure

itself as the probability of this system.The bulk of it in the next chapters allow the translation in the language of

probability theory.

ALGEBRE BOOLEAN NORMATE

1. standardized Algebra 1.1 Definition and its properties of Boolean algebra normed

topological definition. A Boolean algebra (abbreviated normata ABN ) is a pair , ,

where is complete and AB is the measure of . Through the routine of the language,

we'll talk often about a "ABN ". If It is a measure of the probability then sometimes

use the term algebra to probability. As is always the outer measure.Therefore, all statements

for Boolean algebra complement proven with external measures applied to an ABN.In

particular, each ABN , turns into a metric space with metric

: ,x y x y . (1)

Considering that the measure is totally additive, nx involve ,© 0n np x x .

Indeed,

1 ,n k kk n

x x Cx

1 n k k

k n

x x Cx

. (2)

Theorem 1. The Topology the metrical of a on coinciding with the topology ( )o .

This theorem is a special case. It is enough to recall the fact that the metric topology

satisfies the condition ( )os .

Corollary 1. Either si two measures on a AB complete.Then for each 0 real,

there are 0 so that the inequality x involve x .


241

To prove it is sufficient to observe that the relationship 0nx and 0nx , in

virtue of the theorem 1, expressing the same thing: convergence in topology from scratch ( ).o

This convergence can be called "convergence to the extent".It is obvious that each measure is

a function. Corollary 1 looks like this concept of absolute continuity, while important for a set

of functions, it becomes useless in our case. For each AB complete all the steps, admits

metric spaces , are homeomorfe to each other.

Corollary 2.

Because the relationship 0nx keep it off, it is necessary and sufficient that each absolute

increase of indices kn There is a joint 1 2( ...)kin k k so ( )

©ki

o

nx .

In fact, the coincidence of topology ( )o si ( )os It is sufficient to refer to the corollary of

Theorem 1 2 shows us that each sequence kn anulandu-it contains a joint to the extent ( )o

convergence knx . This statement can be made more precise method of choice highlighting

such consequential: a convergence ( )o It is enough that the series 1k

knx converge.

We demonstrate that.

Lemma 1. If 1i

iy then ( )

©o

iy .

Demonstratie.Se Yes m and si k , We have m km k

i ii m

i m

y y

i

i m

y

.

Considering that so far as is ( )o - continue:

i ii m

i m

y y

(3)

The right side of this inequalities disappears: 0 lim inf 0i i i imm i i m

i m

y y y

(4)

The right side of this inequalities disappears: 0 lim inf 0i i i imm i i m

i m

y y y

Considering that the measure is essentially positive, we see that lim ©i iy or

( ) lim ©o y

Lemma is proved. Using this Lemma, we consider the problem of complete metrics of a

Boolean algebra.


242

Teorema 2

A AB complete with metric It is a complete metric space.

Demonstration

Fie 1n nx

a Cauchy sequence which comply with . We can choose a joint

1 1 2 ...kn kx n n

so series 11 ,

k kk n np x x

1

,©k kn np x x

converged.

Therefore, according to Lemma 1,1

( )

k k

o

n nx x x

. So, we have

© limnk

x lim 1

lim ©.n k kk

x n nx x

Therefore, there is an element x so ( )

k

o

nx x . Now, the inequality

,( ) ( , ) ( , )k kn n n nx x x x x x We can easily notice that

( )

,( ) 0o

nx x taking into account that

1n nx

is a Cauchy sequence. It has been demonstrated that ,x is complete.

Observation. Theorem 2 remain valid if is a foreign measure, to demonstrate, it is

sufficient to slightly change the proof Theorem 2.

1.2 its properties of a converged ( )o

Starting from Lemma 1, it is easy to demonstrate another important theorem expressed by m.

Frechet.

Theorem 3 Every ABN is regular. I actually set a fact even more generally: regularity of the algebra.

Thus, each can be considered a ABN space metric. What else can we say about the properties

in this space? For example, will be connected? The answer is negative, in general, since some

lie algebras are normed listed all finite AB, which are clearly disconnected.

Theorem 4 An ABN is a metric space continue connected in the shape of an arc. This theorem implies, in

particular, that the set of values in a measure coincide (in accordance with the terms of the

theorem) with the range[0, 1] . But this has been proven previously: base is a Theorem

Demonstration 4 follow now. So either two elements 0 1,x x x database. We will build a

path between these elements. This is done in a few steps. 1. Show that for all , ,u v u v

There are w so 12

( , ) ( , ) ( , )u w w u v .Suppose u u Cv şi v v Cv .

These elements are disjoint, but not equal to zero at the same time and

( , ) ( , )u v u v u v u v u v (5)

Let's assume that for the determination of 0u si u v . (6)


243

Moreover, we can consider the 0 with all quasimasurile regulated by condition

of ( ) 1. 1 It is clear that 0 0.K These categories (and their subclasses) are listed in

mathematical statistics "statistical structures." We will build some elements u şi v with

the following properties: 1 12 2

, , , .u u u v u u v v

We Place .w u v u v Then 1 12 2

( , ) ( )

( ) ( , ).

w u u u v

u v u v

Similarly 1

2( , ) ( , )w v u v .

2. Now, knowing that 0u x si 1v x as we build the item above 01w x a which distance

between 0x and 1x is 10 12

( , )x x In addition, we define 001x and 011x also, continuing this

process we will apply our crowd of AB has all the numbers 0 1( , )r x x , where [0,1]r

dyadic rational-it. Moreover, such a built function 0 is uniformly

continuous: 0 0, .p t t t t

3. We remain to extend this application continuity through the whole range 0, 1)0, ( .x x

Get some continuous functions that apply in AB such that 0(0) x and

0 1 1( ( , )) .x x x It's a good trajectory between points 0x and 1x .

The demonstration is complete. We will add to this very important Observation. in an ABN

form each lot continues |M x x

It is connected in the shape of an arc, 0, 1 .

The proof of the theorem is clear that: u v a then .w a The Crowd M is

closed.That's why each trajectory What unites 0 1,x x M lies entirely in .M

This observation will allow us later to demonstrate his famous Theorem of Lyapunov vector

measures about. Definition: a complete AB is called normata if it admits a few steps. Each

normata holds many measures AB in general; but are equivalent in some places such as the

corollary of the theorem 1 shows 1: define the same topology, topological order

metric.Theorems 3 and 4 are obviously valid for lie algebras normed. In essence, a normata

is a AB AB analyzed together with complete system of measure . Moreover, it is

considered that the system is not empty.From now on we will recover through ( )M M .

Also consider the set K of all total additive on quasimasurile . Each of these is a continuous

function quasimasuri.If v K then the crowd | 0x vx is an ideal of the form u .

Quasimasurilor restriction v the complementary area Cu It is a measure.Usually we

consider probability measures satisfying the condition 1. 1 We note the probability

measures with ( ).oM Similarly we will put Moreover, we can consider the 0 with all

quasimasurile regulated by condition of ( ) 1. 1 It is clear that 0 0.K These

categories (and their subclasses) are listed in mathematical statistics "statistical structures."


244

1.3 The ABN Izomorfisme

Either , and ,v the two ABN. An application f on poset E (for example, a

subalgebra or area) It is called as constanta v f x x for all .x E First of all we are

interested in homomorfism and isomorphism.Talking about a homomorfism or homorfism of

the , on ,v , We always refer to it as homomorfismul (izomorfismul) keeps his

on . The method is often useful in izomorfismelor.

The following theorem relies on this method.

Theorem 5.

If 0 and 0 There are two subalgebre of everywhere and ( )o - dense

then each measure constant monomorfismului 0 of 0 on 0 extends to a measure of

constant izomorfismului 0 of 0 on 0 . Such an extension is unique.

Demonstration.

Lie Algebras and I'm a regular. Constant-mean that measures homomorfismele 0 si 1

0

satisfying the condition ( )E . Indeed, either: 1

,nn

x x

0., nx x

If: 1 1x x , 2 2 1,x x Cx 3 3 1 2 ,....x x C x x Then

0 0 0

0 .

n n nn n

n

n

n

v x v x v x

x x v x

Hence, 0 01

.nn

x x

The last equality says that that condition ( )E is fulfilled. In our case it involves ( )E

.

Thus, homomorfismul 0 satisfies the condition ( )E

. By analogy, we will check if this

condition is satisfied for the 1

0 .AB and meet the condition; It is easy to see, for

such terms of algebras ( )E and ( )E

are equivalent to ( )E and ( )E

In this case, the extension of monomorfismului 0 is an isomorphism on . Show

that it is able. For each x There is a sequence nx in 0 converging at x .. Passing to

the limit in the relationship 0 ,n n nv x v x x (7)


245

We arrived at the desired equality .v x

The uniqueness of an extension obtained thus is obvious. The demonstration is complete. We

will cite two examples in which the theorem applies to 5. In these examples and There

are two normed algebras. And and v There are two measures of the probability of these

algebras.

I. Be and containing compact and autonomous subalgebre everywhere 0 and that 0

. Let's say these subalgebre include not only of systems alternators, but also (considering

some measures and v ) with the same power.Either 0

E an independent system of

generators for 0 and either 0

E an independent system of generators v for 0 .

Finally, we assume for simplicity that

12

x vy

for all 0

x E and 0

E . In this case, each of which applied to each from 0

E on 0

E

an isomorphism is extensive 0 None of his 0 on 0 . It is easily checked that this

measure is isomorphism. In fact, if

1 2 ... ,kx x x x (8)

where 0 0

1,2,...,ix E CE i k iar i jx x for i j then 1

2kx At the same time

0 1 2 ... ,kx x x x (9)

si de asemenea 0 0

1,2,..., ,i xx E CE i k i jx x pentru .i j Asadar

10 2

.kv x x

We must take into account the fact that each element of 0 It is a finite sum of

elements of disjoint form (1). Thus, by Theorem 5, there is a constant measure of

izomorfismului on .

II. Now let's consider a typical situation when there are two pairs of regular subalgebre

,

and ,y y with the following properties:

1) ( )x x x x

for any x

and x ; in a similar way

( )v y y vyvy for any y si y ;

2) ,

and

,


246

Suppose there are izomorfisme with constant measures and many of his

on

and that of on . We show that in this case there is an isomorphism with the

constant measure None of his on by extension of and .

To this end, we introduce subalgebrele 0 ,

and 0

, .

To this end, we introduce subalgebrele 0 ,

and 0

, . If 0

and 0 , It is sufficient to construct an isomorphism with the constant measure of 0 pe

0

CONCLUSIONS

The main issue is topical because it approaches the fuzzy systems underlying the artificial

intelligence that is implemented in the economic and industrial machines.

REFERENCES

[1] Balbes si Dwinger,The curatours of the University of Missouri,1974

[2] Dumitru Buşneag ,Categories of Algebric Logic, Editura Academiei


247

EXTENSION OF AN ADDITIVE FUNCTIONS NUMARABILE

Prof.Drd. Bogdan P. Constantin


As.univ.Pecingina Olimpia

Universitatea ―Constantin Brancusi‖ ,Tg-jiu

ABSTRACT: All measures to start the construction of theories from user a fundamental theorem of a. Lebesgue

whose contemporary formulations is due to c. Caratheodory.

NORMED ALGEBRAS

Lebesgue Theorem Caratheodory-Demonstration.

Either o AB complete.We consider subalgebra 0 with a quasimasura . We

will now undertake the following tasks: construction of a subalgebras regular

that

includes 0 and a quasimasura additive numarabila

on

so x x

, where it is

possible, x x

for any x 0 . In order to accomplish the task, we will define an

outer quasimasura by associating with each .x the crowd xS that is all most families

numarabile 0 satisfying the condition sup x , and by:

inf xS

.

The Demonstration. Either a AB complete.We consider subalgebra 0 with a

quasimasura . We will now undertake the following tasks: construction of a subalgebre

regular

that includes 0 and a quasimasura additive numarabila

on

so

x x

, where possible, x x

for any x 0 . In order to accomplish the task,

we will define an outer quasimasura by associating with each .x the crowd xS that

is all most families numarabile 0 satisfying the condition sup x , and by:

inf xS

.

We note the main characteristics of this external quasimasuri.

1. If 0 0x x then x 0 .x

This property is obvious.Implies that is finite.The following property is also evidence

2. Outside measurement este monotona: x involve x .

3. If kx x then x .k kx


248

Considering that the measure is totally additive, nx Indeed, involve establishing a

chance 0 , associated with each kx family k xS so that the inequality

2k

k

kx

remains.Clearly, gender equality k k belongs to xS ;

and, that's why

x

kk k

x

.

Because is random, we obtain the inequality. Now we arrange the

with all the elements

x so equality

( ) ( ) ( )xu x u C u (4)

keep for any u . It is clear that always ( ) ( ) ( ),xu x u C u so we no longer

stays than to validate the seeming inequality backside.

Lemma 2.Either 1,2...nZ n

si be the sequence nZ What tends to repeat .z Then

( ) lim ( )nn

u z u z

for any .u

Demonstration. Everything is obvious nz decreases.Suppose that nz z and 1 ©.z Then

complete all the steps, admits metric spaces , are homeomorfe to each other.

1 11 1

( )nn n n z

n n

u z u z z u z C

11

lim .

n nn

nn

u z u z

u z

Lemma is proved.

Lemma 3. The Crowd

is a subalgebra on regular basis.

Demonstration.

In condition of the main (4) items x si xC have equal status, so

contains each element

together with the complementarul or. Is Yes ,x y

and we have z x y cu .u

We will have

( ) ( ) ( )xu u x u C

( ) ( )

( ) ( )

y

x x y

u x y u x C

u C y u C C

( )

( ) ( ) ( )y x x y

u x y

u x C u C y u C C

( ) ( ),zu z u C


249

Since ( ) ( ) ( )x xu x C u C u C C

( ) ( )

.

x x

z

u x C C C C

u C x u C

Thus .z

Note that

is the subalgebra. We remain to show that

is on a regular

basis.Either 1 2, ,..., ,...nx x x a

numarabila and submultime m k

ii m

x y

m km k

i ii m

i m

y y

i

i m

y

.

Considering that so far as is ( )o - continue:

In the previous Lemma,

( ) lim ( ),nn

u x u y

( ) lim ( )nx y

nu C u C

Hence, ( ) ( ) ( ).xu x u C u

Thus, .x

It has been demonstrated that

is a subalgebra on regular basis.

Lemma 4 0

Demonstration. Either x 0. Choose an item u and a correspondent by chance uS .

Families of elements of the form x z z and xC z the corresponding elements

are x u and that .xC u

So ( ) xz z

z z x z C

( ) ( ).

xz z

x

z x z C

x u C u

Taking into account that by chance, we conclude that ( ) ( ) ( )xu x u C u

Thus, .x

Axiona is demonstrated.

Lemma 5. If ( ) 0x then .x

Demonstration.Se Yes u , as a result ( ) ( ) ( )xu x u C u ( )

©ki

o

nx .


250

In fact, the coincidence of topology ( )o and ( )os It is sufficient to refer la Theorem 4

Suppose that ,x y

and x d .y Atunci

x

x y x y

x y x x y C

( ) ( )x y

x

.y

So, see for example aditivitatii.So, for any sequence disjuncta nx in ,

We have

1

nn

x

11

,mm

n nnn

x x

and thus:

11

.m

n nnn

x x

Considering the inequality:

1

nn

x

1 11

,m m

n n nn nn

x x x

which is maintained, we get to equality:

0 lim inf 0i i i imm i i m

i m

y y y

(4)

The right side of this inequalities disappears:

They are studying the structure of subalgebrei

that is defined quasimasura

.

Se da un x

Incidentally, we choose the corresponding n xS 1,2,...n so


251

1( )n

ny

x x y x

1,2,...n to be valid.

Now we will put x sup .n n It is clear that x x and x belongs to subalgebrei 0

and also

1

sup

( ) .n

n

n

x x

y x

So, , 0.x x x x x x

Thus, each element x is represented as ,x x u

where 0 , 0,x u si .x x

Lemma 1, 1

( )

k k

o

n nx x x

. So, we have

Even more so, since the is there a submultime numarabil 0 numarabila such

as: 02sup ,x x

where:

0 0 2sup sup .x x

Now we can consider 0sup a substitute for 0.x Therefore,

0 0 2 2.x x x x x x

Lemma is proved.

Definition: a quasimasura v a subalgebra It is called complete as long as the

conditions , ,x x and 0v involve x (and obviously 0v ).

As well as in Lemma 6, quasimasura

I built it is complete.

Lemma 9. Either an additive numarabila full qasimasura to some subalgebra - that

includes regular 0. It is assumed that x x for any 0.x Then:

1)

considered as a metric space. What else can we say about the properties in this space? For

example, will be connected? The answer is negative, in general, since some lie algebras are

normed

listed all finite AB, which are clearly disconnected.


252

Teorema 4

An ABN is a metric space continue Astfel, .x x Applying this to an element

,x

result .x x

It remains to prove the Lemma enunt first. First we will

show that if 0v then v and 0.v Indeed, as in Lemma 8, There is an

element ,v v satisfying 0.v v

All items belong to him 0 and thus

his . So v and

0 0.v v

Quasimasura It is thus complete; v and 0.x Now it is clear that each x

belong to him , taking into account that x can be written as ,x x u where x It is an

element and 0.u So inclusiveness

and Juliet are demonstrated. We need.

complete the demonstration only point 1).

Lemma 10. If

sup ( )ny

y

In general: the regularity of the algebra. Thus, each can be considered a ABN space metric.

What else can we say about the properties in this space? For example, will be connected?

The answer is negative, in general, since some lie algebras are normed listed all finite AB,

which are clearly disconnected. Indeed, in this case

x x x

for any 0 ,x so the 1 of an external steps leads to the desired

equality.

Teorema 6. Either a AB complete, and 0 a subalgebra of , and a

quasimasura in 0 . Then there is a subalgebra) regular

including 0 and (b)) a

quasimasura additive numarabila

in

with the following properties:

1)

x x for any 0;x

2) for each additive numarabila quasimasura in

and for any 0x Image check

,x x the inequality x x

It is true for any .x

How is numarabila then the additive equality

x

x

It is true for any 0.x in other words

It is an extension of .

Call quasimasura

It is built in the proof theorem 6


253

CONCLUSIONS

The main issue is topical because it approaches the fuzzy systems underlying the artificial

intelligence that is implemented in the economic and industrial machines.

REFERENCES

[1] Balbes si Dwinger,The curatours of the University of Missouri,1974

[2] Dumitru Buşneag ,Categories of Algebric Logic, Editura Academiei


254

ABN AND METRIC STRUCTURES SPACES OF MEASURES

As.univ. Olimpia-Mioara PECINGINA

Universitatea ―Constantin Brancusi‖ ,Tg-Jiu

[email protected]

Prof.Drd. Constantin P. BOGDAN


[email protected]

ABSTRACT Is given a measurable space , , .m This space (ii) corresponds to the "metric structure,

Boolean algebra ̂ the resulting algebra is factorization- According to maiprany negligible.This is a

metric structure complete with AB as m̂ . Taking us a certain freedom, we apply the term to the metric

structure of ABN ˆ ˆ,m too.So, every metric structure associated with a metric space offers an example of

ABN.As a result every ABN can be represented in this form. We refer particularly to the Boolean

izomorfismele; izomorfismele metric structures.However, the theory of measure uses another important concept

of isomorphism between spaces.It happens in a few variations, but in all cases we are dealing with a "point" of a

space on the other.Of the many definitions you choose only those that are really used in this part.

Keywords and phrases: boole algebras

1. THE STRUCTURE OF THE METRIC MEASURE SPACE

Is given a measurable space , , .m This space (ii) corresponds to the "metric structure,

Boolean algebra ̂ the resulting algebra is factorization- According to maiprany

negligible.This is a metric structure complete with AB as m̂ .

Taking us a certain freedom, we apply the term to the metric structure of ABN ˆ ˆ,m too.So,

every metric structure associated with a metric space offers an example of ABN.As a result

every ABN can be represented in this form.Before you formulate a corresponding theorem we

refer to the concept of isomorphism. We refer particularly to the Boolean izomorfismele;

izomorfismele metric structures.However, the theory of measure uses another important

concept of isomorphism between spaces.It happens in a few variations, but in all cases we are

dealing with a "point" of a space on the other.Of the many definitions you choose only those

that are really used in this part. Considering the two categories ? and ? whose

elements are the spaces in the form of measures , , .m Morfismele of the first category

are possible applications 1 2: with the following property:

a) 1

1e for any 2e

b) If 2 0m e then 1

1 0m e

If c) 1

1 2m e m e for any 2e (conservation measure) then homomorfism is called



255

a metric.These homomorfisme are taken as morphisms of category ? .

In fact every object of category ? is defined by the , algebra and negligible ideal;

quasimasura sets m playing a supporting role.

These two categories they correspond to two forms of izomorfisme.Later we will call an

isomorphism between two measures of spaces 1 1 1, ,m and 2 2 2, ,m as part of his ?

an isomorphism of these spaces.Such an isomorphism is an application bijectiva

2 1: having its properties a) and b) with reverse aplicatiia 1. Moreover, if

(question 1 ) has the property c) then we call an isomorphism metric.If there is an

isomorphism between two measures of spaces then say that these spaces are izomorfice.

Measure spaces are called izomorfice mod0 (izomorfice metric mod0 ). If we exclude

some negligible, we get crowds izomorfice spaces. It is easy to notice that an

isomorphism of spaces imply a corresponding Boolean structures metric isomorphism,

izomorfismul and metric involves an isomorphism of Boolean algebras normed.The converse

is not generally available but can be valid if successfully restrict our chathosts from well-

conducted group spaces that we have considered.

Theorem 8. For every ABN , There's a space masurabilbil , ,m so ABN , si

ˆ ˆ,m sunt izomorfice.

In other words, every ABN is izomorfa metric structures. We refer here to an isomorphism

metric (measurement).

Demonstration.

We can use any of the three representations of proved the Loomis-Sikorski.The last two

theorems are applicable, since every algebra is regular normata.According to one of these

theorems, we can find ABN , a space an algebra of poset from , and a -

ideal I of this algebra, so that there is an isomorphism the quotient algebra 1| on .

In this case, there is a measure on ; This makes it possible to define a quasimasuri

additive numarabile m on According to the following conditions: .m e e

Here is a homomorfism He became a Canon of on 1| . With key on positive

, We can set up easily as the ideal I coinciding with negligible maiprany m , and is an

isomorphism necessary ABN 1ˆ| ,m on the ABN , . Primarily the role of the main

space play space Stone ; In the case of 2 and 3 main space is "standard", bound

only by the value of but not of algebra itself.In other words we get R the Cantor of the

value r ; so lead to the production of r elements of the range 0,1 . Among the most

interesting questions is the following: Why masurabilbil is the best place in terms of the

representation of an ABN?


256

2 THE STONE OF AN ABN

We consider the Stone space AB complete . We will identify and 0

.CO Every quasimasura 0 on 0 meet the Lebesgue Theorem hypothesis-

Caratheodory and admits an extension numarabila to a quasimasura additive on algebras

. If quasimasura 0 It's actually a positive numarabila essential additive measure

then this algebra may be described more in detail.We're now in this description. 1. remind you

first of all that each lot matchless belongs to . Be a lot .e rare. Its closure e It is also

rare.We consider the open complementary ? ;G e is represented as a semi-open crowds

reunion

,tt T

G x

tx t T (1)

Taking into account that e then it is; t T 1.tx Our Algebra 0 is normabila (since

measure 0 is available), therefore, meet the requirements of numarabil and the meeting can

be represented can be represented as the smallest common sequences of Hamid Ibrahim:

1

1nt

n

x

, 1,2... .nt T n

We can assume that 1 2

...t tx x Applying the additivity of numarabila 0 This will result in

0 01.nt

x Atunci

e e 0 0 0( ) 1t nn

x tC x 0, 0e , e (2)

2.Algebra 0 , as any normabila, algebra is regular, and thus sets a rare ideal. They note

with I and consider the algebra of maiprany

. (3)

This algebra includes all crowds form 2 0, , .e x q x q I It is clear that 1 we have

(4)

3. Of course a few extensions of numarabile additive 0 exist in . In other words, this is

quasimasura 0 ,oU where U is a homomorfism Birkhoff-Ulam. Of positivity

of his 0 equality 0e is equivalent to the ker .e U I Quasimasura is

complete, since the condition of e e si 0e involve e I and .e

4. Both quasimasurile and numarabile are additive and complete when as 0 is

extended.Taking into account that is an extension of the Lebesgue measure, then,

si | . At the same time, we see that .

5. So we mark the following conclusion.If 0 is an ABN, 0 It is a measure of the 0 , and

is an extension of the Lebesgue

1) his field is algebra includes all crowds form 2 ,x q where 0x and q


257

are rare.

2) negligible sets There are even rare submultimile of compact space .

Therefore, all measures in 0 extend the same algebra with the same crowd negligible.

Application 2x I x x It is in this case izomorfismul that is presented in the

theorem 8, coefficient algebra izomorfismul |I on 0 . As I said above, this is an

isomorphism in the ABN, since is constant measurement:

2 2 0ˆ .x I x q x x (5)

But if does not coincide with 0 then induces an isomorphism between and 0.

Notice that each ABN , is izomorfica metric spaces with structures

, , .

Each isomorphism in the ABN , and , induce a homomorfism between Stone

spaces and an isomorphism between measures of space.

3. LEBESGUE SPACES-ROKHLIN

The Discontinuity Of The Cantor T can be used as a representation of a space ABN

, where .cardT r Indeed, there is a ideal I the algebra of Baire space interrupted

extremely compact and an isomorphism coefficient algebra of 0 |IB on . Let us

remember that the algebra of Baire 0B It is the most insignificant algebra including

algebra semideschisa .TD

If his He is considered a measure then this measure "pass" in quasimasura m in

algebra 0B According to the rule 2me e I . In this case mI I ideal becomes

exactly ideal of negligible Baire maiprany m . The structure of the metric space measures

0, ,B m It is in fact ABN ,

We obtain the same metric structure if you replace 0B with algebra complete

0 ,mmB B I

where mI

includes all related elements of crowds mI ideal. "Measure " m

expands on the mB just like izomorfismul . This representation has the special features as a

result of the exceptional role of algebra TD This algebra has the following properties:

1) Each non-empty crowds centered in TD has the nevida intersection.It is obvious because

TD consists of the compact crowd.

2) Each lot in mB is mod0 with the crowd in algebra generated by TD . This means


258

there is a standard extension to quasimasurii | Tm D .

3) TD separate the original space points. The property is called the compactitudine of

algebra TD . Its properties 2 and 3 by definition means that TD is a basis of , , .mB m

In the case of some measurable space , ,m subalgebra 0 It is also called a

compact if its properties owns the 1,2 and 3 (of course replacing t and tD with

question 0 ).

A space masurabilbil , ,m with a full size m It is called Lebesgue space-Rokhlin has a

compact if numarabila. theory of these places is completely developed by V.A.Rokhlin in

1940; he called "Lebesgue spaces".We note the Lebesgue space-with Rokhlin .

From what I said at the beginning of this chapter, shall be deducted for each ABN separabila

represents the structure of a metric space Lebesgue-Rokhlin.Because the database structure

numarabilitatii of a Lebesgue space metric-Rokhlin will always be separabila.We will not

submit this theory of space in detail

As a consequence m is supposed to be a measure of probability, 1.m

Either , ,i

i im 1 , 2i two Lebesgue spaces-Rokhlin. everyone has meant most

number of atoms.Taking into consideration the crowds zero for maintaining a general lines,

we'll list the crowds in descending order.Either 1 1

1 2

, ,...m m

and 2 2

1 2

, ,...m m

crowd in 1

and that 2 where 1

1 2

... 1,2 .i

m m i It turns out that this crowd is meeting the

invariance image allows us to classify Lebesgue spaces-Rokhlin metric.Namely,

1) If 1 0km for any i and k then izomorfice metric spaces are, In this case izomorfice

metric metric structures are also what is obvious.

2) If 1 2

k km m for any 1,2,...k then the spaces 1

1 1, ,m and 2

2 2, ,m are metric

izomorfice mod0 and their structures are izomorfice metric metric.

3) If the metric izomorfice metric structures are then 1 2

k km m for any 1,2,...k and

izomorfice metric spaces are mod0. The proof of this theorem to "classification" is feasible.For example monografiia of

V.G.Vinokurov, see B.A.Rubshtein and A.L.Fedorov.

Result from this theorem as Lebesgue spaces-izomorfice metric structures with Rokhlin are

izomorfice mod0 and if these structures are continuous ("nonatomice") then there is an

isomorphism between these "pure" Lebesgue spaces-Rokhlin. Izomorfica classification of

Lebesgue spaces-Rokhlin is equivalent to the classification of ABN. A subspatiu of a

measurable space , ,m It is a space masurabilbil , ,e eE m where , 0E mE si

|e em m . If em m then both spaces have the same metric structure and more


259

specifically, there is a canonical isomorphism ˆ ˆ,m on ˆ ˆ,e em generated by the

application e e E .e Note all the fundamental things just now: a subspatiu of a

Lebesgue space-Rokhlin is also Lebesgue space-Rokhlin.Sometimes the concept of subspatiu

is interpreted in a wider sense without it is assumed that E It is measurable.So if the

m E m but E then you can still have E in a measurable space and even more so,

with the same metric structure but in any Lebesgue space-Rokhlin will not be produced. It is

clear that every space , ,m a Lebesgue space isomorphic-Rokhlin is also Lebesgue

space-Rokhlin. If we limit the majority of continuous cardinalitatea will happen the same with

izomorfismele mod0. so the Lebesgue spaces-Rokhlin category is "well built" as a whole.

Lebesgue spaces-Rokhlin comprise a category well built "but are also remarkable in terms of

intrinsic structures.Now we will show the main examples.We'll start with the case of the

Lebesgue spaces continuously-Rokhlin are izomorfe to each other. In fact, one example has

already been displayed; it's the Bernoulli ,,N pq pq . We prefer if 12

p q and

abbreviated writing 1 12 2

and 1 12 2

. The compact numarabila we can take

semideschisa algebra .ND The existence of such databases involves exactly the fact that space

is a Lebesgue space Bernoulli-Rokhlin.Am remember already that points maiprany missing;

Therefore, our example is the Example of a Lebesgue space-Rokhlin., and each of these areas

must be isomorphic Bernoulii.In particular space, all Bernoulli spaces built on N

discontinuous are metric izomorfe. A second example of Lebesgue space-Rokhlin is Lebesgue

space 1, , ,nnI l where nI is the n-dimensional cube.In previous chapters we have

described the : 0,1N by the formula: 21

.n

n

n

x

Similar applications can be constructed in the square 0,1 0,1 , the cube, etc.All are

"almost" bijective and continuous, during the process of "measure" Bernoulli 1 12 2

in

Lebesgue measure: .l e e

Application It is not bijectiva to items N of the form 1, 2,... ,0,0,...nx x x and

1, 2,... ,1,1,...nx x x which corresponds to the rational points of the interval binaries.They form

a lot S numarabila, and its image S includes all rational points of the interval

binaries.Let a bijectie some of S on S (both of them are zealous and

numarabile).Application the date of

,

,

x S

x S

(6)


260

is an isomorphism of Bernoulli's range of Lebesgue space. This Lebesgue interval that is

Lebesgue space-and every Lebesgue space Rokhlin-Rokhlin continuously is isomorphic to

it.The same thing is true for squares, cubes, etc. Lebesgue. gotta call it as the best example for

a Lebesgue space-continuous Rokhlin is simply the square Lebesgue. A Lebesgue space-

Rokhlin differently is represented simply as a numarabila or finite group points to the

component 1 2,...,m m An example of another type of interval ,a b with a group of distinct

points 1 2,...,x x to which extent is entirely supported. Coincidence of corresponding maiprany

elements involved, an isomorphism mod0; but no genuine isomorphism is not possible, since

these areas have different cardinalitati and do not allow any bijectie.

Finally, most general example of Lebesgue space-Rokhlin is the interval or Lebesgue square

(or Bernoulli space) with the most numarabila lot of points associated with it.Every Lebesgue

space-Rokhlin is isomorphic metric mod0 such an "example" of space.To simplify the

exposition, this is often taken as the basis for the definition of the Lebesgue space-Rokhlin.

REFERECES

[1] Balbes si Dwinger,The curatours of the University of Missouri, 1974

[2] Dumitru Buşneag ,Categories of Algebric Logic, Editura Academiei ,1990


261

INCREASING RESISTANCE OF STRUCTURAL ELEMENTS WITH

CFRP REINFORCEMENTS

Lecturer PhD. Cătălina IANĂŞI, “Constantin Brâncuşi” University of Tg-Jiu [email protected]

Abstract: One of the defining features of a new material is to be resistant, easy to process, to

combine the qualities of basic materials, in the current technique, but not to borrow from them

their negative properties. Composite materials are metallic or nonmetallic matrix reinforced by

the dispersion of particles, fiber or gas. Fibers are able to withstand elastic applications but to

resist for all requests. Matrix serves only for ensuring a support for fiber, stabilizing the fibers

against the forces of disruption. In recent years, carbon fiber composites have been increasingly

used in different ways in reinforcing structural elements. Specifically, the use of composite

materials as a reinforcement for wood beams under bending loads requires paying attention to

several aspects of the problem such as the number of the composite layers applied on the wood

beams.

Keywords: wood beams, CFRP composites, epoxy resin, bending test.

1. INTRODUCTION

This paper describes an experimental study which was designed to evaluate the effect of

layers number of composite material on the stiffness of the wood beams [3]. By using

composite materials in constructions is expected growth flexural strength and shearing, and

confinement elements tablets (increased wood resistance) [5]. Also, is done raising ductility

areas of plastic joint and meet the requirements of exploitation (decrease arrows and status

cracking). In civil and industrial constructions, working with elements of resistance as beams,

pillars, floor, masonry portal, is necessary strength when they present phenomena of wear and

fatigue. Consolidation and reinforcement is done with items as sheets and carbon fiber plates

[3,6]. Among the composites used in construction include: glass fiber, cellulose fiber,

KEVLAR fiber, carbon fiber and graphite, etc.

This paper describes an experimental study which was designed to evaluate the effect

of layers number of composite material on the stiffness of the wood beams [3]. The type of

reinforcement used on the beams is the carbon fiber reinforced polymer (CFRP) sheet

SikaWrap 230C with E module of elasticity 230 000 N/mm2 and traction resistance 4100

N/mm2 an epoxy resin for bonding all the elements Sikadur 330. Structural epoxy resins

remain the primary choice of adhesive to form the bond to fiber-reinforced plastics [5,7,8,9]

and are the generally accepted adhesives in bonded CFRP–wood connections. Advantages of

using epoxy resin in comparison to common wood-laminating adhesives are their gap-filling

qualities and the low clamping pressures that are required.


262

Following the experimental tests we developed a numerical procedure, which can

estimate the failure load of the wood beams reinforced with composite materials. The critical

point of the numerical procedure consists in the selection of the most adequate characteristic

values for wood limit stresses. This step was carried out using values found in other studies

and from a comparison of the values obtained from the failure loads of the un-reinforced

beams.

2. EXPERIMENTAL STUDY

The total number of wood specimens manufactured is nine, eight of which are reinforced,

and one is un-reinforced. The wood part of all beams was formed by dry wood beech which

size is equal to 25 by 50 by 500 mm [3]. Three beams were reinforced using one carbon fiber

sheet of thickness equal to 1.5 mm, width equal to 25 mm and the length is equal to 500 cm.

The finished dimension of these beams is equal to 25 by 101.5 by 500 mm because they are

two beams stick together with one carbon fiber sheet of SikaWrap 230C and Sikadur 330

epoxy resin (fig.1) [10].

Fig. 1 Tension failure of a reinforced wood beam with one CFRP sheet (in the middle)

For carbon fiber sheet, once it is placed on the wood beam, with the epoxy resin, all what

is required is to press the carbon fiber sheet with a simple roller and pull out the air [1,2,4,6].

In addition, the carbon fiber sheet has a very high tensile strength (with respect to its weight),

it is available in any length, no joints are required, low thickness, easy to transport, laminate

intersections are simple, economical application no heavy handling and installation

equipment, available in various modules of elasticity, outstanding fatigue resistance, and it

can be coated without preparation. Moreover, the CFRP are compatible with wood with

respect to its mechanical properties. The bending test results for a reinforced wood beam with

one CFRP sheet are shown in table 1.

Table 1. Results for a reinforced wood beam with one CFRP sheet

Force (daN) 0,4 0,8 1,2 1,6 2,0 2,4 2,8 3,2

Deflection f (mm) 1 1,5 3,9 6,1 8,3 11,6 15 18,3


263

Other three beams were reinforced using two carbon fiber sheets of thickness equal to

3 mm, width equal to 25 mm and the length is equal to 500 cm. The finished dimension of

these beams is equal to 25 by 73 by 500 mm because they are two slides of wood up and

down stick by the main wood beam, with two carbon fiber sheets of SikaWrap 230C and

Sikadur 330 epoxy resin (fig.2).

Fig. 2 Tension failure of a reinforced wood beam with two CFRP sheets and two slides of wood (up and down)

The bending test results for a reinforced wood beam with two CFRP sheets and two slides of

wood are shown in table 2.

Table 2. Results for a reinforced wood beam with two CFRP sheets (up and down)

Force (daN) 0,4 0,8 1,2 1,6 2,0 2,4 2,8 3,2

Deflection f (mm)

1 2,2 3,9 4,8 61 8,9 12,7 13,3

The last three beams were reinforced using two carbon fiber sheets of thickness equal to 3

mm, width equal to 25 mm and the length is equal to 500 cm. The finished dimension of these

beams is equal to 25 by 113 by 500 mm because they are two beams stick together with two

carbon fiber sheets of SikaWrap 230C and an epoxy resin Sikadur 330, with a density of 1,31

kg/dm 3 , from Sika-Romania, and another slide wood beam in the middle of the beam (fig. 3).

The bending test results for a reinforced wood beam with two CFRP sheets and a wood slide

in the middle of the beam are shown in table 3.

Fig. 3 Tension failure of a reinforced wood beam with two CFRP sheets (in the middle of the beam) and one

slide of wood


264

Table 3. Results for a reinforced wood beam with two CFRP sheets (in the middle)

Force (daN) 0,4 0,8 1,2 1,6 2,0 2,4 2,8 3,2

Deflection f (mm)

1,3 1,9 2,1 2,7 3,2 4,6 5,8 7,6

The specimens tested were not subjected to lateral instability during loading. The total load on

the beam was applied equally at one point equidistant from the reactions (the half length of

the beam).

We use the bending device of the universal machine for mechanical tests which has the

distance between the rollers l = 460 mm. Standard samples was dry beech wood beam

with a rectangular section of 25x50 x500 mm (bxhxl).

After we study those examples we observe that the reinforced wood beam with two CFRP

sheet and a wood slide in the middle (fig.3) is the most resistant and has a good elasticity

breaking into a 10,5 mm deflection. Also, we can say that the number of composite material

layers influences the stiffness of the wood reinforced beams.

CONCLUSIONS

It is proposed to use an inexpensive and easily processed material that is wood.

As a rigid material with good strength and relatively low cost, we use a composite. A

several un-reinforced and reinforced wood beams were tested in order to find their flexural

capacity. CFRP materials were conditioned in an environment of 65±5% relative humidity

and temperature of 20±2°C as this is the service environment in which CFRP reinforced

beams are expected to be used. The results indicate that the behavior of reinforced beams is

totally different from that of un-reinforced one. The reinforcement has changed the mode of

failure from brittle to ductile and has increased the load-carrying capacity of the beams.

Observations of the experimental load–displacement relationships show that flexural strength

increased and middle vertical displacement decreased for wood beams reinforced with CFRP

composite plates, compared to those without CFRP plates. Also, we can say that the number

of composite material layers of carbon sheet and their positions influence the stiffness of the

wood reinforced beams.

Acknowledgments

The author is grateful to the Building Velmix Ltd, Tg-Jiu, Romania, without whose financial

support the present work could hardly be conducted.

REFERENCES

[1] Abdel-Magid B, Dagher HJ, and Kimball T. The effect of composite reinforcement on

structural wood. In: Proceedings – ASCE 1994 materials engineering conference,

Infrastructure: new materials and methods for repair, San Diego, CA, November 14–16; 1994.

[2] Camille A. Issa, Ziad Kmeid. Advanced wood engineering: glulam beams. Department of

Civil Engineering, Lebanese American University, Byblos, Lebanon. Construction and

Building Materials 19 (2005) 99–106.


265

[3] Ianăşi C. Research on reducing the risk of damage for the resistance elements of wooden

building, 2nd WSEAS International Conference on RISK MANAGEMENT, ASSESSMENT

and MITIGATION (RIMA '13) Brasov, Romania, June 1-3, 2013, pp. 161-164, ISSN 2227-

460X

[4] Yeou-Fong Li, Yao-Ming Xie, Ming-Jer Tsai. Enhancement of the flexural performance

of retrofitted wood beams using CFRP composite sheets. Construction and Building Materials

23 (2009) 411–422.

[5] Plevris N, Triantafillou T. Creep behavior of FRP-reinforced wood members. J Struct.

Eng. – ASCE 1995;121(2):174–86.

[6] Plevris N, Triantafillou T. FRP-reinforced wood as structural material. J Mater Civil Eng

– ASCE 1992;4(3).

[7] Tingley DA, Leichti RJ. Reinforced glulam: improved wood utilization and product

performance. Paper presented at Technical Forum – Globalization of wood: supply, products,

and markets. Portland (Oregon): Forest Products Society; 1993.

[8] Triantafillou T, Deskovic N. Innovative prestressing with FRP sheets: mechanics of short-

term behavior. J Eng Mech – ASCE 1991; 117(7):1652–72.

[9] Triantafillou T, Deskovic N. Prestressed FRP sheets as external reinforcement of wood

members. J Struct Eng – ASCE 1992; 118(5):1270–84.

[10] www.sika.ro

http://www.sika.ro/


266

INDEX AUTHORS

Nr.

crt

Name and Surname, Institution Pagina

1. ADÎR Ancuta

Grigore Cerchez Technological College, Bucharest

[email protected]

Nr.1: 64

2. ADÎR George

University POLITEHNICA of Bucharest,

[email protected]

Nr.1: 64

3. ADÎR Victor


[email protected]

Nr.1: 64

4. ALBULESCU Radu Gabriel

Oil Equipment Company of Ploiesti, Romania

Supliment Nr1:

59

5. ALDEA Mihaela

University ‖1 Decembrie 198‖ of Alba Iulia

[email protected]

Nr.1: 227

6. AMZA Gheorghe

Polytechnic University of Bucharest, Romania,

Department of Materials and Welding Technologies,

[email protected]

Supliment Nr1:

51, 67

7. APOSTOLESCU Zoia



[email protected]

Supliment Nr1:

51, 67

8. BABIS Claudiu Polytechnic University of Bucharest Romania,


e-mail: [email protected]

Supliment Nr1:

84

9. BÂLDEA Monica University of Pitesti, Romania,

[email protected]

Nr.1: 195

Supliment Nr1:

150, 333

10. BĂLTEANU Ancuţa University of Pitesti, Romania

[email protected]

Supliment Nr1:

333

11. BĂRBĂCIORU Iuliana Carmen ‖Constantin Brâncuşi‖ University, Tg. Jiu

Nr.1: 200, 207

12. BOCA Maria Loredana ―1 Decembrie 1918‖ University of Alba Iulia

[email protected]

Nr.1: 78

13. BOGDAN Constantin P.

Universitatea din Craiova Nr.1: 240, 247

14. BRĂNICĂ Diana Nicoleta (MILOSTEANU), University Politehnica of Bucharest

[email protected]

Supliment Nr1:

219




267

15. BULAC Ion

Doctor, University of Pitești

email: [email protected]

Nr.1: 97, 103

16. BUNECI Mădălina Roxana University Constantin Brâncuşi of Târgu-Jiu

Nr.1: 213, 220

17. CHIVU Oana



[email protected]

Supliment Nr1:

76, 84

18. CIOFU Florin University "Constantin Brâncuşi" of Târgu-Jiu,

[email protected]

Supliment Nr1:

46

19. CIORTEA Elisabeta Mihaela

University ‖1 Decembrie 198‖ of Alba Iulia,

[email protected]

Nr.1: 227

20. CÎRŢÎNĂ Daniela

„Constantin Brâncuşi‖ University of Tg-Jiu Supliment Nr1:

313

21. CIRTINA Liviu Marius Constantin Brâncuşi University of Târgu Jiu, Romania,

[email protected]

Supliment Nr1:

294

22. CIZER Laura ―Mircea cel Bătrân‖ Naval Academy

Supliment Nr1:

18, 24

23. DELIU Florențiu ―Mircea cel Bătrân‖ Naval Academy

Supliment Nr1:

18

24. DIMITRESCU Andrei

Universitatea Politehnica Bucuresti, Facultatea IMST,

Departamentul T.M.R.

[email protected]

Supliment Nr1:

5, 12, 106

25. DINU Bogdan ORACLE SA, Bucharest,

[email protected]

Supliment Nr1:

250

26. DINU Marius Bogdan University ―POLITEHNICA‖ of Bucharest,

Faculty of Electronics, Telecommunications and Information Technology,

Department of Electronic Devices, Circuits and Apparatus

Supliment Nr1:

39, 243

27. DOBROTĂ Dan

Constantin Brancusi University of Targu Jiu, Romania,

Department of Systems Engineering and Management Technology,


Supliment Nr1:

59, 84

28. DRAGOMIR Adrian Ioan

Auto Repair Company of Brasov, Romania Supliment Nr1:

51, 59

29. DRAGOMIRESCU Cristian

Univ. Politehnica Bucharest,


Supliment Nr1:

93

30. DUMITRESCU Iosif Department of Industrial Mechanical Engineering and Transport,

University of Petroşani, [email protected]

Nr.1: 110, 120

31. DUMITRIU Mădălina


[email protected]

Nr.1: 129, 137







268

32. FLOREA Traian Academia Navală „Mircea cel Bătrân‖

[email protected]

Nr.1: 84, 91, 165,

178, 183

33. FLOREA Traian Vasile A.P.M. Agigea of Constanta, ROMANIA

Nr.1: 157, 165

34. GHIMIȘI Ștefan

Constantin Brâncuși University of Targu Jiu

[email protected]

Supliment Nr1:

222, 294

35. GRIDAN Horia Constantin Brâncuși University of Targu Jiu

Supliment Nr1:

294

36. IANCU Cătălin

Engineering Faculty,‖C-tin Brâncuşi‖ Univ. of Tg-Jiu,

[email protected]

Supliment Nr1:

229

37. IANĂŞI Cătălina

Engineering Faculty,‖C-tin Brâncuşi‖ Univ. of Tg-Jiu,

[email protected]

Nr.1: 261

38. ILINCIOIU Dan

University of Craiova, Faculty of Mechanics,

Department of Applied Mechanics and Civil Constructions, Calea Bucuresti

Street, no. 107, Craiova,Code 200512, Romania,

[email protected]

Nr.1: 27

39. IONESCU Simona e-mail: [email protected]

Nr.1: 33, 43

40. IONICI Cristina

University ―Constantin Brâncuşi‖ of Tg-Jiu,

[email protected]

Supliment Nr1:

154, 158

41. IOVAN Stefan West University of Timisoara, Computer Science Department,

[email protected]

Railway Informatics SA, Bucharest,

[email protected]

Supliment Nr1:

32, 39, 250, 264

42. IOVANOV Miodrag

―Constantin Brâncuşi‖ University of Târgu-Jiu

[email protected]

Nr.1: 234

43. IOVANOV Valeria Victoria

Technical College No. 2, Târgu-Jiu

[email protected]

Supliment Nr1:

270, 276, 282

44. ISTRATE Mihaela

Faculty of Mechanics and Technology ,University of Pitesti


Nr.1:195

Supliment Nr1:

150

45. ITU Răzvan Bogdan

Department of Industrial Mechanical Engineering and Transport,

University of Petrosani,

[email protected]

Nr.1: 110, 120

46. ITU Vilhelm

Department of Industrial Mechanical Engineering and Transport,

University of Petroşani,

[email protected]

Nr.1: 110, 120

47. IVANUS Cristian TAROM SA, Bucharest

[email protected]

Supliment Nr1:

32, 264









269

48. LUCA Liliana

Constantin Brancusi University of Targu-Jiu

[email protected]

Nr.1: 18

49. MĂRĂȘESCU Daniel Academia Navală „Mircea cel Bătrân‖

[email protected]

Nr.1: 84, 91

50. MARC Gheorghe

―1 Decembrie 1918‖ University of Alba Iulia,

[email protected]

Nr.1: 78

51. Marius STAN

Petroleum - Gas University of Ploiesti

[email protected]

Nr.1: 71

52. MARTINEZ Francico Cavas

Polytehnic University of Cartagena Supliment Nr1:

158

53. MIHUȚ Nicoleta – Maria

Constantin Brancusi University of Targu Jiu

[email protected]

Supliment Nr1:

46, 289

54. MIRIŢOIU Cosmin-Mihai University of Craiova, Faculty of Mechanics

Department of Vehicles, Transports and Industrial Engineering

Calea Bucuresti Street, no. 107, Craiova,Code 200512, Romania,

[email protected]

Nr.1: 27

Supliment Nr1:

112, 119, 213

55. MORARIU Cristin Olimpiu Department of Manufacturing Engineering, Technological Engineering and

Industrial Management Faculty, Transilvania University of Brasov, Romania,

e-mail: c.morariu@unitbv. ro

Nr.1: 57, 178

56. MOROIANU Corneliu

Email address: [email protected] Nr.1: 157, 165, 183

57. NEACSA Marin


[email protected]

Nr.1: 64

58. NICULA Dana

Dunărea de Jos University of Galați Supliment Nr1:

222

59. NIOAȚĂ Alin

Engineering Faculty,

‖Constantin Brâncuși‖ University of Târgu-Jiu,

[email protected]

Supliment Nr1:

46, 126, 133

60. NIŢOI Dan

Universitatea Politehnica Bucuresti, Facultatea IMST,

Departamentul T.M.S.

[email protected]

Supliment Nr1:

12, 106

61. NITOI Dan Florin Polytechnic University of Bucharest, Romania,

Department of Materials and Welding Technologies

[email protected]

Supliment Nr1:

67

62. NOVAC George

―Mircea cel Batran‖ Naval Academy, Constanta

[email protected]

Nr.1: 145, 151

63. PARIS Adrian Stere

Univ. Politehnica Bucharest


Supliment Nr1:

93, 99











270

64. PASARE Minodora Maria

University ―Constantin Brancusi‖ of Tg-Jiu

[email protected]

Supliment Nr1:

138

65. PǍTRAŞCU Mariana (ANTONESCU) e-mail: [email protected]

Nr.1: 33, 43

66. PECINGINĂ Irina Ramona „Constantin Brâncuşi‖ University of Tg-Jiu

irinacornescu yahoo.com

Supliment Nr1:

313

67. PECINGINA Olimpia

Universitatea ―Constantin Brancusi‖ ,Tg-Jiu Nr.1: 240, 247

68. PETRESCU Valentin

Lucian Blaga University of Sibiu, Romania,

Department of Industrial Engineering and Management

[email protected]

Supliment Nr1:

67

69. PIRGHIE Ana-Camelia

Stefan cel Mare University of Suceava,

Department of Mechanics and Technology,

[email protected]

Nr.1: 49

70. PIRGHIE Cristian

Stefan cel Mare University of Suceava,

Department of Mechanics and Technology

[email protected]

Nr.1: 49

71. PLESEA Valeriu

„Constantin Brâncusi‖ Universiy of Tg.-Jiu

e-mail: plesea_valeriu@ yahoo.com

Supliment Nr1:

142

72. POPESCU Constantin

Polytechnic University of Bucharest Supliment Nr1:

162, 173, 182, 195

73. POPESCU George

Constantin Brâncuşi University of Tg-Jiu

Supliment Nr1:

325, 329

74. POPESCU Iulian

University of Craiova

[email protected]

Nr.1: 3,10,18

75. PRUIU Anastase Academia Navală „Mircea cel Bătrân‖

[email protected]

Nr.1: 84, 91, 157,

165

76. RADU Serghei Barklav Company,

[email protected]

Supliment Nr1:

18, 24

77. RĂDULESCU Constanţa

,,Constantin Brancusi‖ University of Tg-Jiu,, România

[email protected]

Supliment Nr1:

46, 294, 301

78. ROSCA FARTAT Gabi


Supliment Nr1:

162, 173, 182, 195

79. ROŞCA Vâlcu

Department of Applied Mechanics and Civil Constructions,


[email protected]

Supliment Nr1:

213






271

80. ROŞU Catălin

University of Craiova,

S.C. DICO Romania S.R.L., Pieleşti, Cîrcea Street, No. 2, Romania,

[email protected]

Supliment Nr1:

206, 213

81. SAMOILESCU Gheorghe ―Mircea cel Bătrân‖ Naval Academy,

[email protected]

Supliment Nr1:

18, 24

82. SASS Ludmila University of Craiova

Faculty of Mechanics, [email protected]

Nr.1: 3,10

83. SPORIȘ Adriana Academia Navală „Mircea cel Bătrân‖

[email protected]

Nr.1: 84, 91, 157,

165, 178, 183

84. STAN Marius

Petroleum - Gas University of Ploiesti

[email protected]

Nr.1: 71

Supliment Nr1:

236

85. STANCIOIU Alin

University Constantin Brancusi of Targu Jiu

[email protected]

Supliment Nr1:

320

86. STĂNESCU Alexandru

Universitatea Politehnica Bucuresti,

Facultatea IMST, Departamentul T.M.R.

Supliment Nr1: 5

87. STǍNESCU Constantin D. Polytechnic University of Bucharest


Nr.1: 33, 43

Supliment Nr1:

162, 173, 182, 195

88. STANIMIR Alexandru


Department of Vehicles, Transports and Industrial Engineering,


[email protected]

Supliment Nr1:

206,, 213

89. SURDU Ionela Ramona

Universitatea Politehnica din Bucuresti

E-mail: [email protected]

Supliment Nr1:

340

90. TǍRǍBUŢǍ Doina (ENE)


Nr.1: 33, 43

91. TĂRÂŢĂ Daniela Florentina


Department of Vehicles, Transports and Industrial Engineering, Calea

Bucuresti Street, no. 107, Craiova,Code 200512, Romania,

[email protected]

Supliment Nr1:

206

92. TÂRCOLEA Constantin Univ. Politehnica Bucharest,


Supliment Nr1:

93

93. TĂTAR Adina

„Constantin Brâncuşi‖ University of Tg-Jiu

[email protected]

Nr.1: 174

Supliment Nr1:

307

94. TĂTARU Ion


Department of Applied Mechanics and Civil Constructions,


[email protected]

Nr.1: 27










272

95. TEMPEA Iosif

University Politehnica of Bucharest

[email protected]

Supliment Nr1:

219

96. TEODORESCU Marius Cornel

Oil Equipment Company of Ploiesti, Romania

Supliment Nr1:

51, 59

97. TOMESCU Cristian INCD INSEMEX Petrosani,

e-mail: critom05@ yahoo.com

Supliment Nr1:

142

98. UNGUREANU Viorica Mariela

Constantin Brancusi University of Targu Jiu, Romania Nr.1: 207

99. VLAICU POPA Marius Eremia SNLO Tg.- Jiu,

e-mail: m.vlaicu @ yahoo.com

Supliment Nr1:

142

100. ZAHARIA Sebastian Marian

Department of Manufacturing Engineering,

Technological Engineering and Industrial Management Faculty,

Transilvania University of Brasov, Romania, e-mail:

[email protected]

Nr.1: 57

Supliment Nr1:

257

101. ZAMFIRACHE Marius


Department of Vehicles, Transports and Industrial Engineering, Calea

Bucuresti Street, no. 107, Craiova,Code 200512, Romania,

[email protected]

Supliment Nr1:

206, 213




273

INDEX KEYWORDS

A

Autodesk Inventor Professional analysis

automation algorithm

aluminium alloy

auto pieces Aluminium alloy

annulus spacer annulus bellows

angular displacement artefact

assessment analysis

analytics accelerated reliability testing

air atom

Avogadro analytical modelling

B

brick elements boundary lubrication

ball bearings, brazing

business analytics, brandjacking

business intelligence bending moment

big data big data

biotechnologies biomass

Boltzmann biodiesel

burning

C

computer simulation censored data

cardan joint cardan transmission

correlating cutting and loading machine

critical speed crankshaft

cloud computing cloud enterprise

cloud governmental cyber attack

corrosion carbon steel

crack testing current density

correlation coal

copper compression

calandria tub Candu reactor,

calandria tube channel closures.

carbon footprint CAD,

complex erosion cloud offline

cloud-based CNC

costs customers,

(α,β)-cut and core of a

intuitionistic fuzzy sets

D

dewaxing dewaxing probe

dynamic study dynamic behaviour

deflections data loss

data theft durability

diffusion dependability

design data compression technique

data capture technique data mining

data warehousing design of experiment


274

discount dissociation

droplet discrete groupoid

E

elastomer Embedded

engine room engine

exhaust efficiency

electric field eigenfrequency

eigenfrequency electric strength

energy end fitting

eruptions environmental

efficiency economicity

electrons effective listening

external environment employees

emissions from oil equivalence relation

F

finite element analysis functional parameters

finire speed filtering

finite element force

fuel channel fuel bundle

feeder coupling feeder coupling

force of electrostatic repulsion fuel combustion

function

G

greenhouse gas gaussian model

geometric optimization

H

hunting heat affected zone

I

isolated defect irreversibility

indenter innovation

integrated management system information sources

internal environment Intuitionistic fuzzy sets

K

kinematic analysis kinematic pair

Korovkin Khalimsky topology

L

life losses

low temperatures linear displacement

lifecycle large volume of data

liniar space Lyapunov operators

M

mechanism with three RRT dyads movement laws

mechanisms for scissors metallic structure

mesh mechanism

molecular dynamics modeling

manufacturing lines mission critical

monitoring marine

measurements marine

multiple role station modulator

materials selection manufacturing systems

mass per unit length microhardness


275

models methane

maintenance manganese

milling tool modeling

mathematics minning activities

managers mathematical relation

mathematical modeling

N

nanotribology NDT

nuclear operators

O

open source open cloud

orbit

P

particular cases pipeline

power pneumatic system

power sensor plating

phishing pipes

parts of the oil industry

positioning system

polypropylene core processing objects

potential difference predictive control

pump pressure tube

pressure tube positioning assembly

processing pallets

product plan

pollution particles

plasma pollutants

parameters program

primary seals positive operators

pseudorandom number generator Petri nets

R

rehabilitation reliability

reliability testing refrigerator compressor

railway vehicle regeneration

rebuilding rotary motor

reliability regenerable

Reduction of environmental pollution road transport

Q

quadrilateral mechanism quality

quality indicator

S

sustainability scraper conveyer

suspension Stirling,

solar and wind energy separator

statistics software

steel fabric support

sandwich bars stiffness

steel sintered sintering process

shortening shield plugs

sawdust SolidWorks

sustainability safe


276

snapshot skills

sedimentable powders subatomic processes

T

trajectories two conductive elements

tetrahedral elements technical deviations

transfer object technological process

topological groupoid transport

U

ultrasound ultrasonic field

ultrasonic motors underground areal waters.

V

ventilations vertical vibrations

valve vibration

visualization

W

wears welding

water vapours working space

working fluid web hosting


277

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