BULETINUL INSTITUTULUI POLITEHNIC DIN IAŞI

BULETINUL 

INSTITUTULUI 

POLITEHNIC 

DIN IAŞI 

Tomul LVIII (LXII) 

Fasc. 4 

MATEMATICĂ. MECANICĂ TEORETICĂ. FIZICĂ 

2012 Editura POLITEHNIUM

BULETINUL INSTITUTULUI POLITEHNIC DIN IAŞI 

PUBLISHED BY 

“GHEORGHE ASACHI” TECHNICAL UNIVERSITY OF IAŞI 

Editorial Office: Bd. D. Mangeron 63, 700050, Iaşi, ROMANIA 

Tel. 40-232-278683; Fax: 40-232-237666; e-mail: polytech@mail.tuiasi.ro 

Editorial Board 

President: Prof. dr. eng. Ion Giurma, Member of the Academy of Agricultural 

Sciences and Forest, Rector of the “Gheorghe Asachi” Technical University of Iaşi 

Editor-in-Chief: Prof. dr. eng. Carmen Teodosiu, Vice-Rector of the 

“Gheorghe Asachi” Technical University of Iaşi 

Honorary Editors of the Bulletin: Prof. dr. eng. Alfred Braier, 

Prof. dr. eng. Hugo Rosman, 

Prof. dr. eng. Mihail Voicu Corresponding Member of the Romanian Academy, 

President of the “Gheorghe Asachi” Technical University of Iaşi 

Editors in Chief of the MATHEMATICS. THEORETICHAL MECHANICS. 

PHYSICS Section 

Prof. dr. phys. Maricel Agop, Prof. dr. math. Narcisa Apreutesei-Dumitriu, 

Prof. dr. eng. Radu Ibănescu 

Honorary Editors: Prof. dr. eng. Ioan Bogdan, Prof. dr. eng. Gheorghe Nagîţ 

Associated Editor: Associate Prof. dr. phys. Petru Edward Nica 

Prof.dr.math. Sergiu Aizicovici, University „Ohio”, 

U.S.A. 

Assoc. prof. mat. Constantin Băcuţă, Unversity 

“Delaware”, Newark, Delaware, U.S.A. 

Prof.dr.phys. Masud Caichian, University of Helsinki, 

Finland 

Prof.dr.eng. Daniel Condurache, “Gheorghe Asachi” 

Technical University of Iaşi 

Assoc.prof.dr.phys. Dorin Condurache, “Gheorghe 

Asachi” Technical University of Iaşi 

Prof.dr.math. Adrian Cordunenu, “Gheorghe Asachi” 


Prof.em.dr.math. Constantin Corduneanu, University of 

Texas, Arlington, USA. 

Prof.dr.math. Piergiulio Corsini, University of Udine, 

Italy 

Prof.dr.math. Sever Dragomir, University „Victoria”, of 

Melbourne, Australia 

Prof.dr.math. Constantin Fetecău, “Gheorghe Asachi” 


Assoc.prof.dr.phys. Cristi Focşa, University of Lille, 

France 

Acad.prof.dr.math. Tasawar Hayat, University “Quaid-i- 

Azam” of Islamabad, Pakistan 

Prof.dr.phys. Pavlos Ioannou, University of Athens, 

Greece 

Prof.dr.eng. Nicolae Irimiciuc, “Gheorghe Asachi” 


Assoc.prof.dr.math. Bogdan Kazmierczak, Inst. of 

Fundamental Research, Warshaw, Poland 

Editorial Advisory Board 

Assoc.prof.dr.phys. Liviu Leontie, “Al. I. Cuza” 

University, Iaşi 

Prof.dr.mat. Rodica Luca-Tudorache, “Gheorghe 


Acad.prof.dr.math. Radu Miron, “Al. I. Cuza” 

University of Iaşi 

Prof.dr.phys. Viorel-Puiu Păun, University 

„Politehnica” of Bucureşti 

Assoc.prof.dr.mat. Lucia Pletea, “Gheorghe Asachi” 


Assoc.prof.dr.mat.Constantin Popovici,“Gheorghe 


Prof.dr.phys.Themistocles Rassias, University of 

Athens, Greece 

Prof.dr.mat. Behzad Djafari Rouhani, University of 

Texas at El Paso, USA 

Assoc.prof.dr. Phys. Cristina Stan, University 

„Politehnica” of Bucureşti 

Prof.dr.mat. Wenchang Tan, University „Peking” 

Beijing, China 

Acad.prof.dr.eng. Petre P. Teodorescu, University of 

Bucureşti 

Prof.dr.mat. Anca Tureanu, University of Helsinki, 

Finland 

Prof.dr.phys. Dodu Ursu, “Gheorghe Asachi” 


Dr.mat. Vitaly Volpert, CNRS, University „Claude 

Bernard”, Lyon, France 

Prof.dr.phys. Gheorghe Zet, “Gheorghe Asachi” 

Technical University of Iaşi


BULLETIN OF THE POLYTECHNIC INSTITUTE OF IAŞI 

Tomul LVIII (LXII), Fasc. 4 2012 


S U M A R 

Pag. 

IRINEL CASIAN BOTEZ, Despre derivabilitate şi rezoluţie în fizică (engl., 

rez rom.). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 

ALEXANDRU-MIHNEA SPIRIDONICĂ, Abordarea fuzzy bazată pe 

controlul statistic al proceselor cu aplicaţie în operaţiile de eşantionare 

(engl., rez. rom.). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 

MUGUR B. RĂUŢ, O formă generală pentru ecuaţia liniilor de câmp electric 

corespunzătoare unei distribuţii continue şi cu simetrie axială de sarcină 

(engl., rez. rom.). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 

DAN-GHEORGHE DIMITRIU, MAGDALENA AFLORI, LILIANA- 

MIHAELA IVAN, EMILIA POLL şi MARICEL AGOP, Tranziţii spre 

haos prin bifurcaţii subarmonice în plasmă. (I) Partea experimentală 

(engl., rez. rom.). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 

MARICEL AGOP, EMILIA POLL, DAN-GHEORGHE DIMITRIU, 

MAGDALENA AFLORI şi LILIANA-MIHAELA IVAN, Tranziţii 

spre haos prin bifurcaţii subarmonice în plasmă. (II) Hidrodinamica 

fractală (engl., rez. rom.). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 

EMILIA POLL, MARICEL AGOP, DAN-GHEORGHE DIMITRIU, 

LILIANA-MIHAELA IVAN şi MAGDALENA AFLORI, Tranziţii spre haos 

prin bifunrcaţii subarmonice în plasmă. (III) Model teoretic (engl., rez. 

rom.).. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 

ADRIAN OLARU şi DORU CĂLĂRAŞU, Contribuţii privind realizarea 

unui nou model de comandă electroferofluidică (entgl., rez. rom.). . . . . 71 

VICTOR BILICHENKO (Ucraina), Formularea scopului şi realizarea 

scopului în modelele privind ciclul de viaţă pentru strategiile de 

dezvoltare a întreprinderilor de transport auto (engl., rez. rom.). . . . . . . 77 

VICTOR BILICHENKO (Ucraina) şi SVITLANA ROMANTUK (Ucraina), 

Proiecte privind dezvoltarea bazei tehnico-productive a unei 

întreprinderi de transport auto (engl., rez. rom.). . . . . . . . . . . . . . . . . . . 85 

ELENA RALUCA BACIU, IRINA GRĂDINARU, MARIA BACIU şi 

NORINA CONSUELA FORNA, Consideraţii privind structura de 

echilibru a aliajelor Co-Cr-Mo destinate protezei parţiale mobilizabile 

scheletate (engl., rez. rom.). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93


BULLETIN OF THE POLYTECHNIC INSTITUTE OF IAŞI 

Tomul LVIII (LXI), Fasc. 4 2012 

MATHEMATICS. THEORETICAL MECHANICS. PHYSICS 

C O N T E N T S 

Pp. 

IRINEL CASIAN BOTEZ, About Derivability and Resolutions in Physics 

(English, Romanian summary). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 

ALEXANDRU-MIHNEA SPIRIDONICĂ, A Fuzzy Approach Based on 

Statistical Processes Control with Application in Sampling Operations 


MUGUR B. RĂUŢ, A General Form for the Electric Field Lines Equation 

Concerning an Axially Symmetric Continuous Charge Distribution 


DAN-GHEORGHE DIMITRIU, MAGDALENA AFLORI, LILIANA-MIHA- 

ELA IVAN, EMILIA POLL and MARICEL AGOP, Transition to Chaos 

through Sub-Harmonic Bifurcation in Plasma. (I) Experiment (English, 

Romanian summary). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 

MARICEL AGOP, EMILIA POLL, DAN-GHEORGHE DIMITRIU, 

MAGDALENA AFLORI and LILIANA-MIHAELA IVAN, Transition 

to Chaos through Sub-Harmonic Bifurcation in Plasma. (II) Fractal 

Hydrodynamics (English, Romanian summary) . . . . . . . . . . . . . . . . . . . . . 37 

EMILIA POLL, MARICEL AGOP, DAN-GHEORGHE DIMITRIU, 

LILIANA-MIHAELA IVAN and MAGDALENA AFLORI, Transition 

to Chaos through Sub-Harmonic Bifurcation in Plasma. (III) Theoretical 

Modeling (English, Romanian summary) . . . . . . . . . . . . . . . . . . . . . . . . . 57 

ADRIAN OLARU and DORU CĂLĂRAŞU, Contributions to the Development 

of a New Model of Electromagnetic Ferrofluid Command (English, 

Romanian summary) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 

VICTOR BILICHENKO (Ukraine), Goal Setting and Goal Attainment in the 

Models of Life Cycle of Development Strategies in Automobile 

Transport Manufacturing Systems (English, Romanian summary) . . . . . 77 

VICTOR BILICHENKO (Ukraine) and SVITLANA ROMANTUK (Ukraine), 

Projects of Production-Technical Base Development of a Motor 

Transport Enterprise (English, Romanian summary) . . . . . . . . . . . . . . . . . 85 

ELENA RALUCA BACIU, IRINA GRĂDINARU, MARIA BACIU and 

NORINA CONSUELA FORNA, Considerations Regarding the Balance 

Structure of Co-Cr-Mo Alloys for Removable Partial Denture (English, 

Romanian summary). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93


Publicat de 

Universitatea Tehnică „Gheorghe Asachi” din Iaşi 

Tomul LVIII (LXII), Fasc. 4, 2012 

Secţia 


ABOUT DERIVABILITY AND RESOLUTIONS IN PHYSICS 

BY 

IRINEL CASIAN BOTEZ 

“ Gh. Asachi” Technical University of Iasi 

Faculty of Electronics and Telecomunication 

Received: September 28, 2012 

Accepted for publication: November 20, 2012 

Abstract. In this article we discuss the problem of resolution in physics and 

its relationship with differentiability. We propose using hyper-real numbers to 

represent the resolution and redefining derivative. 

Key words: derivability, resolutions, modern physics. 

1. Introduction 

Traditionally, the derivative is defined using the notion of limit 

f( xd x) f( x) 

f( x ) lim , 

(1) 

dx0 

dx 

However, in light of modern physics, this mathematical definition is not aligned 

with the physical interpretation. For example, if we consider, based on the (1), 

the speed of a material point 

xt ( t) xt ( ) 

vt () lim . 

(2) 

t0 

t 

In order to evaluate the physical validity of such a definition, one could 

imagine a mental experiment, similar to the ones used by Einstein, and which 

e-mail: irinel_casian@yahoo.com

2 Irinel Casian Botez 

would serve to implement this definition. Thus, if speed were defined in this 

way, this assumes we would be able to measure the position of the material 

1 

point at different time intervals (for example 1s, then 10 2 

s, 10 s,…, 10 15 s, 

etc). These intervals would form an infinite set. Even if one was able to go past 

this challenge and, moreover, even assuming that in the future one would be 

capable of performing measurements at increasingly small resolutions of space, 

x , and of time, t , it would still be impossible to perform measurements at 

x 0 or t 

0 . 

Moreover, the 20 th century physics added a fundamental barrier to using 

this mathematical definition in physics. Even if one could measure positions 

and time moments with increasingly small resolutions, one would soon reach 

scales governed by the quantum laws. In this case, speed becomes worse and 

worse defined as scale decreases, which is expressed in Heisenberg’s relation: 

1 

v m x 

. 

(3) 

This behavior of the speed at quantum scale determined Heisenberg to give up 

using position and speed and to build quantum mechanics without them. 

Nevertheless, we believe that the key hypothesis, which seems to be 

exaggerated, is not the very existence of space (or of the space-time 

respectively, in the case of the relativistic motion) but the hypothesis of its 

differentiability. Moreover, the question then arises: should we give up 

differentiability altogether or should we define the derivate other that by using 

the notion of limit? 

2. What Came First: The Derivative or the Differential? 

The answer to this question is: the differential. Thus, let f () t be a 

continuous function defined in an interval I of real numbers. Let t0 

I be a 

random point and let t be an increase in the argument. For this increase in 

argument corresponds a variation of the function f defined as follows: 

f f( t t) f ( t. ) 

(4) 

0 

This variation of function f depends, in turn, both on t 

0 

and on 

t : 

f ( t,t 

0 

). In the case where keep t 

0 

fixed, we obtain a function f ( t) 

. If 

there is a linear form in t , A t (A = constant), such that the difference 

f t and A t is a small infinite of superior order relative to t 

between 

 

f t At O t . 

(5)

Bul. Inst. Polit. Iaşi, t. LVIII (LXII), f. 4, 2012 3 

Then the function f(t) is called differentiable in point and the 

expression At is called the differential of first degree of the function f, 

noted d( f t 

0) 

(or in short df ). Thus 

d f ( t ) A 

t, 

(6) 

0 

O( t) 

is a notation expressing the fact that the difference ( f A 

t) 

is a small 

infinite of a greater order than t . So, what is a small infinite? We will leave 

the answer to this question to the next section. We will note for now the 

following: 

i) the differential is a linear (homogenous) function given the increase 

t 

of the argument; 

ii) the differential differs from the increase of the function by a quantity 

which is a small infinite; 

iii) the constant A is called the first order derivative of the functions f in 

1 

the point t 0 

, noted f ( t 

0) 

or f ( t0) 

. 

3. Can We Define the Derivative without Using the Llmit? 

The answer to this question is: YES, this is possible using infinitesimal 

calculation introduced by Leibniz. The arguments with which he defended, both 

in the mathematical and in the philosophical plane, infinitesimal calculus has 

guided his successors to formulate an infinitesimal analysis whose rigour 

respects present day criteria (Klein, 1932). 

3.1. Indivisible and Infinitesimal 

The term ‘’infinitesimal’’ was used by Leibniz in 1673. On the other 

hand, Archimede’s infinitesimal method used the notion of indivisible. Thus, 

Archimede considered that the indivisibles that made up a line are the dots and 

that the indivisibles that make up a volume are the planes. Leibniz’s 

infinitesimals are not Archimede’s indivisibles because they are entities of the 

same dimension as the entities they are part of. Leibniz treats curves as being 

composed of infinitesimal lines rather than infinitesimal points. In order to 

avoid Zeno’s paradox (if the indivisibles do not have a magnitude, then no 

figure they compose will have a magnitude either, and if the indivisibles have a 

finite magnitude, than the figure they compose will have an infinite magnitude) 

one must consider that a quantity is composed of infinitesimals only when 

infinitesimals and the original quantity have the same dimension. Otherwise, the 

term indivisibles must be used. 

t 0


3.2. Leibniz’s Methodology 

Bos(1974) identifies two approaches to justifying infinitesimal calculus in 

Leibniz’s work. One approach is related to the classical method of proof by 

“exhaustion” and the other is related to a law of continuity. The first approach 

is called methodology A, from Archimede’s methodology of “exhaustion” and 

the second approach is called methodology B, hinting to Johan Bernoulli’s 

learning of the infinitesimal method from Leibniz. Leibniz’s infinitesimals have 

an ideal ontological status, similar to complex numbers or irrational exponents. 

We also point out that Leibniz’s law of continuity is not a mathematical law but 

a methodology generally applicable in mathematics, physics, metaphysics and 

other sciences. 

In what follows, we will look at methodology B. It is based on two laws 

introduced by Leibniz: 

i) The law of continuity mentioned above, introduced by Leibniz around 

year 1701 in his work Cum Prodiisset ((Child, 1920), translation); 

ii) The transcendental law of homogeneity, introduced by Leibniz around 

year 1676 in his work Quadratura Arithmetica (Parmantier, 2004) 

3.2.1. The Law of Continuity. Leibniz states the law of continuity thus: 

“In any transition assumed to be continuous, which ends in a certain “terminus” 

it is possible to institute a general reasoning in which the terminus is also 

included”. “The terminus” is the end of the transition. This law enables the 

transition from quantities to which one can attribute values, assignables 

quantities, to those to which we cannot attribute such values, inassignable 

quantities, 

assignables inassignable 

 

L.C. . 

quantities quantities 

(7) 

This law can be rephrased as follows: “the laws from the finite world 

remain valid in the infinite world and vice-versa”. We cite as an example one of 

Leibniz’s own examples. Invocating the idea that the term ‘equality’ can only 

refer to an equality with an infinitesimal error, Liebniz writes: “… when a 

straight line is equal to another, we say in fact that they are not equal. But that 

the difference between them is infinitely small…”(Gerhardt, 1846, p. 40). In 

Cumm Prodiisset, Leibniz introduces the notion of “transition state”, which is 

the state where exact equality is not attained, but where the difference is smaller 

than any assignable quantity. The example above can be illustrated using a 

positive finite quantity that Liebniz notes as (d)x . This is an assignable quantity 

which evolves by means of infinitesimals dx towards zero. Thus, the 

infinitesimal dx is the transition state or the “terminus”. Zero is only the 

“shadow” of the infinitesimal.


This law says that it is possible to consider that infinitesimals (or small 

infinites, as they are called today) to be these small in-assignable quantities. 

Indeed, if ε is an element (different from the null element) of an ordered field 

K and has the following property 

r 

r, 

(8) 

for any positive real number r, then it is by definition a small infinite. It is 

essential here, in order to understand our reasoning, to point out that small 

infinites are not real numbers. Thus, if we assume that ε is a real number 

satisfying the relation 

0 ε r, 

(9) 

for any positive real number r, then this relation must also be satisfied for 

r ε 2 . But ε 2 is smaller than ε , as a result we arrived at a contradiction 

which means that the hypothesis we initially made is false, and that a small 

infinite cannot be a real number. 

Such situations have occurred in the past in mathematics, in respect of 

irrational number or complex numbers. 

As a result, any element k from the ordered field K, which is an extension 

of the real number set , is an infinite (which means that k > r for any r 

or k r for any r ) or it is a finite (which means that it lies between two 

real numbers ak b). One can easily prove that a finite element k can be 

expressed as c ε , where c is a real number and ε is either zero or a small 

infinite. The real number c is called the standard part of the finite element k in 

the field K 

 

c st k . (10) 

We can visualize small infinites using a rational function defined over the field 

K. Thus, let us consider the application m:K , given by the expression: 

x c 

mx . 

(11) 

ε 

This maps c in zero and c ε in one, thus separating the images of c and of 

c ε . Using this representation and considering each time the standard part of 

the image, st[ mx ( )]. We obtain a projection of field K on the real axis . This 

representation in the set of real number allows us to distinguish the images of 

points c and c ε , whereas these points themselves cannot be distinguished. 

Relation (11) raises the problem of extention of each real function f over 

the field K. The answer to this problem is given in non-standard analysis 

(Robinson, 1966). This analysis defines the derivative of a function using small 

infinites and the standard function, st,


 

 

( 1 ) 

f x f x 

f 

x st , 

 

 

 

 

( 1 ) 

f x f x 

f 

x st . 

 

(12) 

For example, if 

f ( x) 

3 

x then Eq. (12) becomes f ( x) 

. 

2 2 2 

st[3x 3 xεε ] 3x 

This extension of field to field K is done precisely 

by respecting the law of continuity formulated by Leibniz. In the field K, the 

infinites (be they large or small) are Liebniz’s inassignable quantities and the 

real numbers are the assignable quantities. Large infinites are the reverse of 

small infinites relative to the multiplicative law of field K. The elements of field 

K were named by Robinson hyper-reals. In 1908, Felix Klein introduced the 

idea of two continuums: continuum A (from Archimede); continuum B (from 

Bernoulli). 

All values of continuum A are (in theory) possible results of 

measurements. Continuum B has values, such as x dx 

, which can never be 

the result of measurements. Another relationship between the two continuums is 

that the values from continuum A are mathematically represented through real 

numbers, while the values from continuum B are represented through hyper-real 

numbers. The connection between the two continuums is made by the “standard 

part” function, st, which transforms continuum B in continuum A. As a result, 

the derivative is the standard part of the ratio y x instead of the actual ratio 

y 

x . 

At the beginning of the 20 th century it was suggested to build continuum 

B through a refinement of Cantor’s construction for real numbers. In Cantor’s 

construction a real number r is represented by a Cauchy sequence ( q n 

) of 

rational numbers. Cantor’s transition from the sequence ( q n 

) to r is done by 

sacrificing part of the information. As a result, we can retain a little more 

information, such as the speed of convergence, which represent the 

“refinement” mentioned above. In this refinement, two Cauchy real-number 

sequences, ( rn 

) and r 

n , converge toward the same r with different speeds. In 

this case, we say that the two equivalence classes represented by ( r n 

) and r 

n 

, 

are different. If r 0 and ( r n 

) converges toward zero faster than r 

n we say 

that the small infinite represented by ( r n 

) is smaller than the small infinite 

represented by r 

n . If r 0 then the equivalence classes represented by ( rn 

) 

and r 

represent two different hyper-reals, infinitely close to r. 

n


3.2.2. The Transcendental Law of Homogeneity. The transcendental law 

of homogeneity was formulated by Leibniz thus: ”a quantity that is infinitely 

small relative to another quantity can be neglected if it is compared with this 

latter quantity”. Thus, all the terms of an equation, except those with the largest 

order of infinity or the lowest order of infinite smallness, can be eliminated. 

For example 

a 

dx 

a, 

dx ddx 

dx 

Thus, the resulting equation satisfy the homogeneity requirement. 

3.2.3. The Arithmetic of Infinites. In his work “De Quadrtura 

Arithmetica”, Leibniz introduces a distinction between “the precise equality” 

and “approximate equality”. Thus, he considers that two quantities are equal if 

the difference between them can be made arbitrarily small (in modern theory 

this is called infinitesimal). 

In this same paper, Leibniz establishes a dozen rules which constitute the 

arithmetic of infinites. Among these, rule 12 states that:”…(x + infinitely small) 

divided by (y + infinitely small)…” can be replaced by “… x divided by y…”. 

This rule is essential to Leibniz’s conception of a differential ratio dy 

dx . Thus, 

2 

in order to calculate dy dx when, for example, y x , we start with the 

infinitesimal x and form the infinitesimal ratio y x . We then simplify the 

infinitesimal ratio using the law of continuity, according to which the algebraic 

manipulations valid for real numbers are also valid for infinitesimals. We thus 

obtain the quantity 2x x . Then, in order to get to the answer known today, 

2x, we apply the transcendental law of homogeneity to the infinitesimal ratio in 

order to eliminate the infinitesimal portion x. 

4. Conclusions 

1. In the above we consider that we need not give up the notion of 

derivative in general, but that we must reevaluate this notion. 

2. This reevaluation must be made from the following perspectives: the 

derivative is a notion subjacent to that of differential and the definition of the 

derivative using the limit must be reconsidered using small infinites and the 

standard function. 

3. Moreover, the introduction of small infinites as hyper-real numbers – 

which expands (in the Cantor sense) the real number set – enables us to redefine 

physical quantities through functions that depend explicitly on resolutions. 

These resolutions are best expressed by hyper-real numbers, which are the small 

infinites.


REFERENCES 

Bos H.J.M., Differentials, Higher-order Differentials and the Derivative in the 

Leibnizian Calculus. Arch. History Exact Sci., 14, 55 (1974). 

Child J.M., Cum Prodiisset. Chicago-London, 1920. 

Gerhardt C. I, Historia et Origo calculi differentialis a G. G. Leibnitio conscripta. 

Hannover, 1846. 

Klein F., Elementary Mathematics from an Advanced Standpoint. Macmillan, New 

York, 1932. 

Leibniz G.W., Samtliche Schriften und Briefe. Reihe VII, Mathematische Schriften, 

Infinitesimalmathematik, 4 (1670-1673). 

Parmantier M., Leibniz, G. W.: Quadrature arithm étique du cercle, de l’ellipse et de 

l’hyperbole. J. Vrin, Paris, 2004. 

Robinson A., Non-standard Analysis. North-Holland Publishing, Amsterdam, 1966. 

DESPRE DERIVABILITATE ŞI REZOLUŢIE ÎN FIZICĂ 

(Rezumat) 

Se discută problema rezoluţiei în fizică şi a legăturii ei cu ipoteza diferenţiabilităţii. 

Propunem utilizarea numerelor hiperreale pentru reprezentarea resoluţiilor şi redefinirea 

derivatei.


Publicat de 



Secţia 


A FUZZY APPROACH BASED ON STATISTICAL PROCESSES 

CONTROL WITH APPLICATION IN SAMPLING OPERATIONS 

BY 

ALEXANDRU-MIHNEA SPIRIDONICĂ 

”Gheorghe Asachi” Technical University of Iaşi, 

Faculty of Automatic Control and Computer Engineering 


Accepted for publication: October 31, 2012 

Abstract. In the technical area the scientific area is so high that the 

advanced numerical methods are used instead of experimental determinations 

and the use of advanced software techniques in the design and production 

activities is a require in most larger enterprises and the efficient management 

ensures a good market policy choice. The fuzzy applications have known a great 

development in the last decade in many scientific areas, especially in the various 

kinds of industry. The current trend in the systems control is to “humanize” it by 

making automated machines that are equipped with an information processing 

model specific to human thinking. In this paper was developed a fuzzy modeling 

based on an application of statistical process control in order to realize a more 

correctly sampling operations. 

Keywords: fuzzy approach, statistical process control, variations, critical 

lines, acceptance quality level. 


The main goal of most organizations, no matter of their nature, object or 

size, is to be competitive as possible on the market, a crucial factor in ensuring a 

long operating duration. Managing and providing a better view competitiveness 

can not be given unless we use some statistic models. Nowadays most of the 

companies dropped the old and rudimentary methods of quality management 

e-mail: alex_mihnea@yahoo.com

10 Alexandru-Mihnea Spiridonică 

process in favor of high finesse mathematical and statistical models, which give 

a more complete and deeper image about the level of their quality processes. In 

the scientific area, a process is viewed as a transformation of a set of inputs, 

which may include materials, actions, methods and operations, in desired output 

results, results that take the form of products, information or services. In any 

field or function of an organization there may be many processes and is 

compulsory that any process to be analyzed by a careful examination of inputs 

or outputs in order to determine the actions needed to improve the quality of 

this. The output of a process is “something” that is transferred to someone – the 

client. To produce an output to meet customer requirements, it is necessary to 

define, monitor and control the process inputs and also the process may have 

provided outputs from a previous process. 

This paper is based on an application referred on achievement of 

statistical indicators regarding industrial processes control (Spiridonică, 2011). 

The data for this application were received from an important pharmaceutical 

company from Romania and, based on these data, I have illustrated the 

functioning of this application. This application also can be also used in physics 

measurement process of the quantum mechanics where the observer – measure 

device interaction is dominant. 

2. The Initial Application 

A fundamental role in the development of statistical process control has 

the Shewhart diagrams (Oakland, 2003). The basic statistical measure around 

the mean diagrams is the standard deviation. So, if a process is stable, we expect 

most of the individual results belong to interval [ X 3 σ, X 3σ]. The Fig.1 

(Oakland, 2003; Zalila, 1998) is very representative in this case. 

Fig. 1 – The values of a process in a normal distribution.


In the first part of the application we approached the control charts and 

we set the three areas of importance from a control chart: the stable zone (the 

area that meets the central line), the warning zone (where there are the upper 

and the lower warning limits) and the action zone (where there are the upper 

and the lower action limits). Referring back to the mean chart, it is important to 

say that if the sampling take place in a stable process, then most of the sample 

means belong to the interval X 3SE , where SE represents the standard error 

(Oakland, 2003). 

The application (Spiridonică et al., 2010; Spiridonică, 2011) was made in 

ATLAB programming and contains a total number of 9 forms, but only two of 

them being important in actual paper: 

Fig.2 – The statistic values of the process samples. 

Fig. 3 – The mean chart of the process and the warning and action limits. 

In Fig. 3 are represented the process mean, the warning lines, the action 

lines and also the sample means.


Another important notion is represented by the three indices of capability 

that are not in relationship with the process values (mean, warning and action 

lines). A process capability index represents a measure referred to the actual 

performance of a process, the processes being generally considered a 

combination of several factors, like the equipment, the design and 

implementation method, technical team, materials and the environment. The 

compulsory minimum requirement is that these three standard deviation 

belonging on each side of the mean to be contained within specification limits. 

When a process is under statistical control, a process capability index may be 

calculated. The process capability indices represent a method to indicate the 

process variability relative to product specification tolerance. We analyzed the 

three indices: relative precision index, Cp index and Cpk index. We not explain 

here these three indices because they are explained in previous work dedicated 

to this application ([Spiridonică et al., 2010; Spiridonică, 2011). 

The data used for this application was received from an important 

pharmaceutical company from Iaşi, Romania. We received the data regarding 

the amplicillin bottles production process of 1000, 500 and 250 mg. in the 

application we analyzed the process of the ampicillin bottles production of 1000 

mg, within an hour of production. The quality control is made at the end of each 

day and wasted is accounted and destroyed, also at the end of each working day. 

The permitted tolerance is classically set at 5% 

, i.e. the dose from the bottle 

must be contained in the interval [950, 1050] mg, if the machine is set to 1000 

mg. The average number of bottled produced in every hour of work is 8000, 

which means that in one day of 10 effective working hours are made 80000 

bottles. 

3. Fuzzy Modelling in Statistical Process Control 

In recent years the fuzzy modelling has known an important growth in 

various areas of industry and also in statistical process control (SPC). In the 

following lines are mentioned several important papers in this direction. Zalila 

(Zalila, 1998) proposes a fuzzy supervision system because the theoretical and 

practical limitations of SPC techniques. Filey (2004) describes the techniques of 

statistical process control monitoring the concept of rule-based fuzzy modelling 

in order to present the set of steady state input-output relationships when the 

process variations are due to process noise. Spiridonică et al. (2010) developed 

a fuzzy modelling in order to optimize the SPC techniques through Shewhart 

control charts in order to reduce the variability of a process, the principal cause 

that determines a big number of scraps. Tervaskanto et al. (2002) developed an 

application regarding to control the quality of the pulp and the functional 

condition of the refiners using SPC and fuzzy techniques. El-Shal and Morris 

(2000) describe an investigation into the use of fuzzy modelling to modify the 

statistical process control rules in order to reduce the generation of false alarms 

and to improve the detection-speed of real faults.


4. The Proposed System 

Every fuzzy system is based on if…then rules. It will not insist on the 

fuzzy logic theory because this is not the topic of the paper, but on few notions 

that will be used in our system. Every fuzzy system is developed by using the 

knowledge base. Although every knowledge base category has own rules and 

uses various inputs, all of these categories contain the following components: 

i) a normalization module; 

ii) a fuzzification module; 

iii) an inference engine; 

iv) a defuzzification module. 

Each basic indicator is noted by c. To this indicator it is assigned a norm, 

i.e. a minimum value c and a maximum value c . The norm represents a 

singular value or an interval t c , T c that is a series of value acceptable for the 

indicator c (Zadeh, 1994; Zadeh, 1993). Let be x c the value of the system 

indicator want to be evaluated. So, y c , the normalized value is computed in the 

following manner: 

xc 

c 

, c 

xc 

tc, 

tc 

c 

 

 

y ( ) 1 c xc Tc, 

c xc 

 

 

 

c xc 

, Tc 

xc 

c. 

c 

Tc 

(1) 

The normalized value of the indicator c, i.e. , is transformed by the 

fuzzification module in a linguistic variable (LV) (Zadeh, 1994). All of the 

values of this LV are represented by the words and phrases and a LV is 

represented by a fuzzy set using a membership function LV (y) . The 

membership function associates to each value of the normalized indicator a 

number μ ( ) LV 

y from the interval [0, 1] that represent the membership grade of 

the yc 

to the linguistic variable. 

As has been said above, every fuzzy system is based on if…then rules. 

This is also true for each category of basic knowledge. A composed indicator is 

realized from these components, i.e. from the basic indicators, that are 

expressed as fuzzy indicators. The composed indicator is noted by s. The input 

indicators for the composed indicator s are 1, 2,…,c,…It is assumed that s is 

y c


represented by linguistic values, 

functions 

μα, μβ,..., μν,... 

LV ,LV ,...,LV ,... 

α β ν 

with the membership 

Similarly for the input indicators the linguistic values 

are noted with LV 1 , LV2 ,..., LV k ,... with the membership functions 

1, 2 ,..., k ,... (Philis & Andriantiatsohiliaina, 2001). For each input indicator 

are necessary the following data: y c – the normalized value of the c, where c = 

1,2,… and μ ( y ) - the membership grade of y c in every linguistic variable, 

k 

c 

LV k 

, where k = 1,2,…and c = 1,2,… 

The typical form a rule r from the rules base is the following: if 

“indicator 1 is in LV ” and (or) “indicator 2 is in LV ”…and (or) “indicator c 

is in LV k 

”,…, then “indicator s is in L V v 

”. Or is expressed by the operator 

max and and is expressed by the operator min. Based on these, a composed 

phrase has the following form: premise =”indicator 1 is L ” and “indicator 2 

is LV j 

“ and “indicator c is L 

i 

j 

r 

V i 

V k 

”… then μ 

PREMISE 

min{ μi 

y 1 

μ 

j 

y 2 

R 

( ), ( ),..., 

μk( yc ),...}. 

The truth values of individual phrases are μi( y1), μ 

j( y2),... 

In most of the cases of fuzzy modeling a basic rule can contain subsets of 

the same linguistic value LV v 

, which belong to the indicator s. Also, a basic rule 

can contain more rules and, in order to combine these rules in a single truth 

value, it is necessary to use the union of the individual values through the max 

operator. If R s , v represents the rules set of the linguistic value LV v 

of the s 

indicator, the truth value of the conclusion “s indicator is L ” is expressed by 

In the particular case in that 

become 

f 

f 

max 

(2) 

s, v 

PREMISEE 

rR 

r 

sv , 

R s , v 

V v 

contain a single rule r, then the relation 

 

(3) 

s, v PREMISEE 

r 

The inference motor produces a single fuzzy set, LV s , v 

for every 

linguistic value V v . Then, the membership function LV s , v 

for each numeric 

value z [0,1] of the s indicator designates an achievement grade for the 

conditions s,v ( z) 

min{ v ( z), 

f s ,v} 

, where v (z) 

is the membership function 

of the initial linguistic value LV 

v 

. The maximum value is registered in f s , v , that 

also is the height of the fuzzy set LV s , v 

. The height collections f s , v and 

membership functions s, v ( z) 

of fuzzy sets LV s , v , where v , represents 

the output of the inference motor (Pîslaru & Trandabăţ, 2006).


The defuzzification represents the final operation of assigning a numerical 

value belong to interval [0, 1] to the composed indicator. The input for the 

defuzzification process is a fuzzy set, namely the aggregated output fuzzy set. 

The most popular method of defuzzification is the centroid method because 

returns the center of the surface. 

The first fuzzy approach is dedicated to the statistical indicators. The 

statistical indicators used in this application are: mean, amplitude, standard 

deviation (or variance), standard error. The standard deviation was calculated as 

the ratio between the amplitude and Hartley constant [Oakland, 2003] and the 

standard error was calculated as the ratio between standard deviation with 

Hartley constant and root of sample volume. Two approaches are of interest: in 

the first approach it considered the area between central line (CL) and UWL or 

CL and LWL as the normal area, the area between UWL and UAL or the area 

between LAL and LWL as the warning area and the area over UAL or over 

LAL as the action area. The second approach, but that is not so consistent like 

the first is to increase the normal area and the normal area become the area 

between CL and UAL or CL and LAL. In this case they are only the normal 

area and the action area, thus eliminating the warning area. In paper it used the 

first approach due to completeness. So, the input variable is chose to be mean of 

process or simply mean, with a number of five membership functions: lower 

action area, lower warning area, normal area, upper warning area and upper 

action area. The output may be considered the sampling, with the following 

membership functions: bad, acceptable or good. Based on this information, it 

can be design the five elementary rules: 

i) if mean is in lower action area, then sampling is bad (1); 

ii) if mean is in lower warning area, then sampling is acceptable(2); 

iii) if mean is normal area, then sampling is good (3); 

iv) if mean is in upper warning area, then sampling is acceptable (4); 

v) if mean is in upper action area, then sampling is bad (5). 

The membership function acceptable of the output variable, sample is 

quite interpretable. If they are two consecutive samples with the mean in a 

warning area (lower or upper) then the sampling operation is not good and the 

second sample must be reconsidered or removed or the process must be 

reconfigured. Also, if a single sample is outside the action area, then the sample 

must also be removed. 

Another problem is represented by the setting of sampling interval. From 

SPC theoretical studies it known that a small sample size and a long sampling 

interval produce shifts in a process because in this case there are a big tolerance 

and therefore a non-qualitative production. If we have a medium sampling 

interval and also a small sample size, the production not really increase in 

quality. But, a medium sampling interval and a medium-large or simply large


sample size provided small shifts and the production knows a spectacular 

growth in quality. Very important is the next sampling so if we have a sample 

value that is in normal area, then it is used a small sample size and a short 

sampling interval. If a sample value is in lower warning area or in upper 

warning area, then it is necessary to use a large sample size and a medium 

sampling interval for the next sampling. Finally, if a sample value is in lower or 

upper action area, then for the next sampling operation it is necessary a medium 

sample size and a short sampling interval. 

The second fuzzy approach is dedicated to process capability indices. A 

capability index can be defined as a measure referred to actual performance of a 

process. They are three main capability indices: relative precision index, Cp 

index and Cpk index. The relative precision index is the oldest index from the 

SPC theory and, in order to remove waste production, the specification width 

must be greater than the process variation 

2T > 6σ. (4) 

Knowing that 

value therefore 

R / d , where R = average range and = Hartley 

n 

2T 

6 , 

R d 

(5) 

n 

2 TR / is relative precision index (RPI), and the value of 6/ d n is the minimum 

of RPI to avoid the generation of products outside the specification limits. If the 

RPI of the process is lower than the minimum required, the final product is a 

reject. Clearly, for the avoided of rejects, specification limits must be increased, 

but no so much that the production quality to suffer. 

In order to realize products within a specification, the difference between 

upper specification limit and lower specification limit must be lower than the 

entire process variation. So, a comparison of the 6 

with (USL LSL ) or 2T 

offers an process capability index, known as Cp 

USL -LSL 

Cp 

6σ 

Every value lower than 1 means that the process variation is greater than 

the manufacturer’s specified tolerance band so that the process is not in 

statistical control. In the case of increasing the values of Cp the process 

becomes more powerful. The Cp and RPI indices nothing says about the 

centering of the process, but represents a simple comparison of the total 

variation with the tolerances. 

Often it is possible to consider a relatively large tolerance band with a 

small process variation, but a significant proportion of the process is outside the 

tolerance band, as in the Fig. 4. 

d n 

(6)


Fig.4 – A process that is not centered. 

This fact does not invalidate the use of Cp index for the measure of 

potential capability of a process when this is centered, but suggests the need of 

another index that takes into account both the process variation and the 

centering. Such an index is Cpk that is largely accepted as a means of 

communicating process capability. 

In the case of upper and lower specification limits, they are two values of 

Cpk 

USL X 

Cpku 

(7) 

3σ 

X LSL 

Cpkl 

(8) 

3σ 

Fig.5 – 

Cpk u 

index. 

The general value of Cpk index is the lower value of Cpk u 

and Cp k l 

. If 

Cpk index takes the value of 1 or lower means that the process variation and its 

centering is such that at least one of the tolerance limits will be exceeded and 

the process is not capable (Oakland, 2003; Zalila, 1998; Filey, 2004). As in the 

case of Cp index, an increase of Cpk index leads to an increase of the capability. 

A solution for increasing of Cpk index value is the centering of the process so 

that its mean value and the target or specification value coincide. There will be 

no difference between Cp and Cpk indices if the process is centered on the


target value. Cpk index can be used where there is only one specification limit, 

either upper or lower – a one-sided specification. This situation appears 

frequently and the Cp index cannot be used in this case. 

So, for the fuzzy system for the case of process capability it was 

considered three input variables: specification width, Cp index and Cpk index. 

The membership functions for specification width are: lower than process 

variation and higher than process variation, for Cp index are: lower than 1 and 

higher than 1 and for Cpk index are: lower than 1, 1 and higher than 1. The 

output variable is process capability with two simple membership functions: 

existent in doubt and not existent .Based on this information, it can be design a 

various number of rules: 

if specification width is lower than process variation or Cp is lower than 

1 or Cpk is lower than 1 then process capability is not existent (1). 

if specification width is lower than process variation and Cp is higher 

than 1 and Cpk is 1 then process capability is not existent (2). 

if specification width is higher than process variation and Cp is higher 

than 1 and Cpk is 1 then process capability is existent (3). 

if specification width is higher than process variation and Cp is higher 

than 1 and Cpk is higher than 1 then process capability is existent (4). 

if specification width is higher than process variation and Cp is lower 

than 1 and Cpk is higher than 1 then process capability is not existent (5). 

if specification width is lower than process variation or Cp is higher than 

1 and Cpk is higher than 1 then process capability is in doubt (6). 

The membership function in doubt can produces a relatively ambiguity. 

This fact is because when it is used the or fuzzy operand and at least one input 

variable does not meet the requirement that the process to be capable it is 

necessary to analyze the next input variables. 

If it wanted a more sensitive analysis of process capability then it must 

take into account the following values of Cpk index that represents, in a more 

pertinent way, the confident level of process capability: 

Cpk < 1: manufacturer is not capable and therefore appears rejects; 

Cpk = 1: manufacturer is not really capable and any change within the 

process will result in undetected non-conforming output; 

Cpk = 1.33: manufacturer is relatively far from performance and the nonconformance 

is generally not detected by the control charts; 

Cpk = 1.5: non-conformance can also appear and the chances of 

detecting it are also not good enough; 

Cpk = 1.67: finally it can be said that the manufacturer is competitive 

and the non-conformance can be relatively easily to be detected;


Cpk = 2: the manufacturer has a high level of confidence and the control 

charts are in regular use. 


1. In this paper we tried to make a fuzzy approach based on the area of 

statistical process control and particularly based on process capability. 

2. Process capability is a very big and, in the same time, a sensitive 

problem for all manufacturers that want to obtain benefits in a relatively short 

and medium time. 

3. After the implementation of capability indices of the process we 

consider than an important think is the design of a set of fuzzy rules, because 

these rules are closed to the human thinking and many manufacturers are not 

specialists in areas like statistics, mathematics or quality control. 

4. But the first fuzzy system, the system dedicated to statistical control is 

strong related with the second fuzzy system proposed, the system dedicated to 

the capability process. 

5. This thing is because a process is necessary firstly to be in statistical 

control and only if is in statistical control it can be made an analysis of his 

capability. 

6. If the process is not in statistical control the capability process analysis 

and therefore the fuzzy system dedicated to this are not relevant. 

7. The Cpk index analysis was made in a relatively rough way, because I 

considered only the value of 1, that is the criterion that separates the capability 

from non-capability. 

8. Another important index for analyzing a process is acceptance quality 

level, that refers at the ratio or at percentage of products that are accepted 

outside the tolerance band. 

9. The characteristics of a Gaussian distribution can be used for 

determination of maximum acceptance standard deviation, when the target 

value mean and acceptance quality level are specified. 

10. As a future direction we want to concentrate on the following topics: 

i) realize a complete fuzzy system referring to statistical process control 

and to capability indices; 

ii) based on various practical applications, we will try to make fuzzy 

systems for a large number of industries; 

iii) refine this system by take into account also all the values of Cpk 

index; 

iv) introduce the acceptance quality level in the fuzzy system because this 

indicator is a really important criterion both for manufacturer and for buyer;


v) an implementation in a programming language of the complete fuzzy 

system along with the statistical application made in previous work; 

vi) if it extrapolate the present formalism to the measurement processes in 

quantum mechanics it can be seen that such a fuzzy approach of sampling 

allows the selection, from the most probable values, of that has the closest value 

to the real value. 

11. Moreover, it can be obtain equivalent relationships with Heisenberg 

uncertainty relationships in the form of restrictions in the values selection used 

in measurement process. 

REFERENCES 

Spiridonică A-M. , Teză de doctorat, Cap. 3, 2011, pp. 90-97. 

Spiridonică A-M., Pruteanu A., Ursan G-A., Elges A., A Matlab Application for 

Managing the Variation of Industrial Processes Using the Process Capability 

Indices. Proceedings of the 6 th international Conference on Electrical & Power 

Engineering, Vol. I, Iaşi, 2010, pp.337-341. 

Spiridonică A-M., The Use of Statistical Process Control in Pharmaceuticals Industry. 

Proceedings of the 54 th international scientific conference, Durable Agriculture – 

Development Strategies, Iaşi, 2011. 

Oakland J.S., Statistical Process Control, 5 th Ed., University of Leeds Business School, 

2003. 

Zalila Z., Fuzzy Supervision in Statistical Process Control. Systems, Man and 

Cybernetics. IEEE International Conference, 1998. 

Filey D.P., Fuzzy Modelling Within the Statistical Process Control Framework. Fuzzy 

Systems, IEEE International Conference, 2004. 

Spiridonică A-M., Pîslaru M., Ciobanu R., A Fuzzy Approach Regarding the 

Optimization of Statistical Process Control Through Shewhart Control Charts. 

Proceedings of 10 th International Conference on Development and Application 

Systems, Suceava, 2010. 

Tervaskanto M., Hietanen T., Kortela U., The Process Control Using SPC and Fuzzy 

Modelling Techniques. 15 th Triennial World Congress, Barcelona, Spain, 2002. 

El-Shal S.M., Morris A.S., A Fuzzy Rule-Based Algorithm to Improve the Performance 

of Statistical Process Control in Quality Systems. Journal of Intelligent & Fuzzy 

Systems: Applications in Engineering and Technology, 9, 3,4, 207-223 (2000). 

Zadeh L., Soft Computing and Fuzzy Logic. IEEE Software, 1994. 

Zadeh L., Present Situation in Fuzzy Logic and Neural Networks. EUFIT 93, Aachen, 

Germany, 1993. 

Philis Y.A., Andriantiatsohiliaina L.A., Sustainability: An Ill-Defined Concept and Its 

Assesment Using Fuzzy Logic. Ecol.Econ., 37, 435-465 (2001). 

Pîslaru M., Trandabăţ A., Fuzzy Model for Environmental Sustainability Assurances. 

Proceedings of International Conference of Computational Methods in Sciences 

and Engineering (ICCMSE 2006), Chania, Greece, 2006, pp.718-723. 

Young M., The Technical Writer’s Handbook Mill Valley. CA University Science, 

1989.


ABORDARE FUZZY BAZATĂ PE CONTROLUL STATISTIC AL PROCESELOR 

CU APLICAŢIE ÎN OPERAŢIILE DE EŞANTIONARE 

(Rezumat) 

Utilizarea procesării statistice de semnal a devenit o importantă zonă de cercetare 

în cele mai multe din domeniile industriale, indiferent de tipul semnalelor, electrice sau 

non-electrice. Lucrarea de faţă are la bază o aplicaţie unitară din domeniul controlului 

statistic al proceselor, aplicaţie ce a reprezentat subiectul unor lucrări anterioare. Pe 

baza aplicaţiei ce a calculat mai mulţi indicatori statistici, atât indicatori elementari cât 

şi indicatori complecşi, am realizat o abordare fuzzy privind operaţiunea de eşantionare, 

în scopul unei cât mai corecte alegeri a valorilor sistemului. Se realizează două sisteme 

fuzzy, primul fiind destinat indicatorilor de bază ai controlului statistic al proceselor, iar 

al doilea fiind destinat capabilităţii procesului, o problemă majoră în orice tip de 

industrie, deoarece un proces ce nu suferă de variaţii nu este neapărat capabil. Ca o 

concluzie la formalismul fuzzy prezentat aici se poate spune că este relativ uşor de 

înţeles şi aplicat pentru toţi cei ce lucrează în domeniul asigurării calităţii industriale şi 

poate fi aplicat cu succes pentru mai multe tipuri de industrii, de la operaţiunea de 

proiectare până la produsul final.


Publicat de 



Secţia 


A GENERAL FORM FOR THE ELECTRIC FIELD LINES 

EQUATION CONCERNING AN AXIALLY SYMMETRIC 

CONTINUOUS CHARGE DISTRIBUTION 

BY 

MUGUR B. RĂUŢ 

“Al. I. Cuza” University of Iaşi, 

Faculty of Physics 


Accepted for publication: October 31, 2012 

Abstract.. By using an unexpected approach it results a general form for the 

electric field lines equation. It is a general formula, a derivative-integral equation 

structured as a multi-pole expansion series. By solving this equation we can find 

the electric field lines expressions for any type of an axially symmetric multipole 

continuous electric charge distributions we interested in, without the need to 

take again the calculus from the beginning for each case particularly, for instance 

as in discrete charge distribution case. 

Key words: electric field lines equation, multi-pole expansion series, 

axially symmetric continuous electric charge distribution. 


From an axially symmetric magnetic multi-pole of arbitrary degree n, 

(Jackson, 1975), we can derive the exact equation for the field lines, (Jeffreys, 

1988). The method presented in (Jeffreys, 1988) deals with spherical harmonics 

in the most general way. Consequently the equation for the field lines is the 

expression of a general case. Another two exact equations for the field lines are 

given in (Willis & Gardiner, 1988). The equations are for two special magnetic 

e-mail: m_b_raut@yahoo.com

24 Mugur B. Răuţ 

multi-poles of arbitrary degree with no axial symmetry. These cases may be 

classified as either symmetric or anti-symmetric sectorial multi-poles. 

By using the above considerations the aim of this paper is to find a 

general form for an exact equation for the field lines of an electric multi-pole 

with axial symmetry. 

2. Theory 

Let us consider now a continuous electrostatic charge distribution within a 

spatial volume. We must evaluate the electric potential in a point P outside the 

distribution, as we can see in Fig. 1. 

z 

Charge 

element 

d 

r θ R 

y 

x 

V 

Fig. 1 

The electric field lines equation is the well known expression 

E 

dl 

0. (1) 

By assuming that we have a charge distribution with an axial symmetry 

with respect to z axis, we can explicit the length element and the electric field as 

and 

dldRu Rdθu 

R 

θ 

, (2) 

E V 

1 V 

V 

 

R 

 

 

R 

u R 

u . (3) 

The cross product (1) leads after an elementary calculus to the well 

known field lines equation written in polar coordinates 

dR V V 

Rdθ 0. (4) 

R θ R 

For a continuous charge distribution the electric potential V can be 

expanded as a Legendre series, according to (Eyges, 1980) 

 

1 1 

m 3 

VR, θ P cos 

d 

4 

m 1 m θ ρ r r r 

πε 

 

R 

 

. 

0 m0

and 


Consequently the potential derivatives from Eq. (3) can be written as 

 

V 

1 m1 

m 3 

 

P cos 

d 

4 

m 2 m r r 

R 

 

 

0 m0 

R 

 

r 

 

V 

1 1 

 

θ 4πε R θ 

m 3 

P cos 

d 

m 1 m θρr r r 

 

. 

0 m0 

By introducing these results within Eq. (3) and considering the property 

 

θ 

m 3 

m 3 

Pmcosθρrr d r Pmcosθρrr d r 

θ 

 

the electric field lines equation can be expressed as 

dR 

 

1 

 

m1 

R 

Pmcosθ 

Rdθ Pmcosθ 

0 . (5) 

m1 m 2 

m 0 θ 

 

R 

m0R 

This is a general expression for the electric field lines equation under 

continuous charge distribution hypothesis. At first sight it exhibits a complicate 

form which requires for solving a derivative-integral equation method. Despite 

this appearance the solutions can be obtained in a simple and direct manner, as 

it is show in the following examples. 

It is useful for our calculations to consider the Rodrigues representation 

of Legendre polynomials 

m 

1 d 

m 

2 

Pm cosθ 

cos θ 1 

m m 

. (6) 

2 m!dcos 

θ 

Under these circumstances Eq. (5) became more explicit and simple. The 

derivative with respect to θ of expression (6) 

 

 

m 

 

Pm 

cosθ m d 

m1 

2 

 

 

cos θ 1 2cosθsinθ 

θ m m 

2 m!dcos 

θ 

 

 

 

, (7) 

leads to an important observation that we can make the derivatives with respect 

to cosine before we make the integration, and thus the Eq. (5) became only an 

integral equation, more simpler to solve. 

It is obvious that the case m=0 does not exist because the derivatives (7) 

do not exist. More interesting is the dipole case 

m 1. 

By taking into account the Eqs. (6) and (7), the Eq. (5) can be written as

26 Mugur B. Răuţ 

 

dR 

1 1 d 

0 

2 

 

cos θ 1 2cosθsin 

θ 

R 

2 

R 2dcosθ 

 

 

 

 

2 1 d 2 

Rdθ 

cos θ1 0. 

3 

R 2dcosθ 

 

After trivial simplification and obvious derivatives we obtain the equation 

which can be directly integrated as: 

dR 

sin θ 2cos θ d θ 0 , 

R 

2 

R 

Csin 

θ 

 

 

, (8) 

and it is the well-known expression, in polar coordinates, of the field lines for 

an electric dipole. 

The mathematical treatment of the case m 2 is the same as the 

previous case. We obtain the equation 

dR 

1 2 d 

 

R R 22dcos 

2 

3 2 2 

2 

2 1 

[(cos θ1) 2cosθsin θ] 

 

θ 

3 1 d 2 2 

Rd 

(cos θ 1) 

0, 

4 2 2 

R 22dcos θ 

from which is deduced the most simplest form 

2 

dR 

3cos θ 1 d θ . (9) 

R 2sinθcosθ 

Finally, after integrating Eq. (9), we are obtaining the following relation 

2 2 

R ksin 

θcosθ 

, ( 10) 

whi ch is the well-known expression of the field lines for an electric 4-pole. 

Eq. (5) is the direct consequence of the Eq. (3). If the electric field could 

not be an expression of a scalar potential, then all the above mathematical 

statement has no basis. The magnetic analog for V does not support sources. 

Subsequently the magnetic analog for Eq. (3) can be written only with the 

vector potential A. The vector potential is defined in terms of current density. 

Under axial symmetry and continuous distribution of current density 

hypothesis, A can also be expanded in Legendre series. But compared with the 

electric field this is the only similarity. The magnetic field lines equation 

appears in a double cross-product form. The solutions of this equation are more 

complicate than Eq. (5), (Jeffreys, 1988).



1. The aim of this paper is to deduce a new form for the electric field lines 

equation. 

2. We obtain a general formula, a derivative-integral equation structured 

as a multi-pole expansion series. 

3. The equation has exact solutions corresponding to an axially symmetric 

electric multi-pole continuous charge distribution, without the need to consider 

special assumptions for m 0 . 

4. Eq. (5) can be the starting point of the entire Sec. 2, because is valid in 

the mentioned approximations, without the need to deduce it from Eq. (1) for 

each case from the beginning, for instance as in discrete charge distribution 

case. 

REFERENCES 

E yges L., The Classical Electromagnetic Field. Addison-Wesley, Mass. 1972, reprinted 

by Dover 1980. 

Jackson J.D., Classical Electrodynamics. Wiley, New York, 137, 1975. 

Jeffreys B., Derivation of the Equation for the Field Lines of an Axis Symmetric Multipole. 

Geophy. J. International, 92(2), 355-356 (1988). 

Willis D.M., Gardiner A.R., Equations for the Field Lines of a Sectorial Multi-pole. 

Geophy. J. International, 95(3), 625-632 (1988). 

O FORMĂ GENERALĂ PENTRU ECUAŢIA LINIILOR DE CÂMP ELECTRIC 

CORESPUNZĂTOARE UNEI DISTRIBUŢII CONTINUE ŞI CU SIMETRIE 

AXIALĂ DE SARCINĂ 

(Rezumat) 

Folosind o abordare neobişnuită rezultă o formă generală pentru ecuaţia liniilor 

de câmp electric. Este o formulă generală, o ecuaţie integro-diferenţială, structurată ca o 

dezvoltare în serie. Rezolvând această ecuaţie putem afla reprezentările liniilor de câmp 

electric pentru orice distribuţie continuă şi cu simetrie axială de sarcină, fără a mai fi 

nevoiţi să reluăm calculul de la început pentru fiecare caz în parte, ca, de exemplu, în 

cazul distribuţiei discrete de sarcină.


Publicat de 



Secţia 


TRANSITION TO CHAOS THROUGH SUB-HARMONIC 

BIFURCATIONS IN PLASMA 

I. EXPERIMENT 

BY 

DAN-GHEORGHE DIMITRIU 1 , MAGDALENA AFLORI 2 , 

LILIANA-MIHAELA IVAN 1 , EMILIA POLL 1 and MARICEL AGOP 3 

1 ”Al. I. Cuza” University of Iaşi, 


2 Petru Poni Institute of Macromolecular Chemistry of Iaşi, 

3 “Gheorghe Asachi” Technical University of Iaşi, 

Department of Physics 



Abstract. A scenario of transition to chaos through cascade of spatiotemporal 

sub-harmonic bifurcations was experimentally recorded in lowtemperature 

diffusion dc discharge plasma in connection with the appearance 

and dynamics of multiple double layer structures. The phenomenon was 

evidenced by increasing the potential applied on a supplementary electrode 

immersed into plasma. The fast Fourier transforms of the current collected by 

this electrode show the appearance of sub-harmonics of the fundamental 

frequency simultaneously with the emergences of new double layers as part of 

the multiple double layer structure in dynamic state. 

Key words: chaos, period-doubling bifurcation, multiple double layer. 


Multiple double layers are complex nonlinear potential structures in 

plasmas, consisting of two or more concentric double layers attached to the 

Corresponding author: e-mail: dimitriu@uaic.ro

30 Dan-Gheorghe Dimitriu et al. 

anode of a dc glow discharge (Chan & Hershkowitz, 1982; Intrator et al., 1993; 

Conde & Leon, 1994; Nerushev et al., 1998; Strat et al., 2003) or to a positively 

biased electrode immersed into plasma (Ioniţă et al., 2004; Aflori et al., 2005; 

Novopashin et al., 2008). It appears as several bright and concentric shells 

attached to the electrode. The successive double layers are located precisely at 

the abrupt changes of luminosity between two adjacent plasma shells. The 

number of shells depends on the background gas, its pressure, the electrode 

voltage and the discharge current (Aflori et al., 2005; Novopashin et al., 2008). 

The axial profile of the plasma potential has a stair step shape, with potential 

jumps close to the ionization potential of the used gas (Conde & Leon, 1994; 

Ioniţă et al., 2004 Dimitriu et al., 2007). At high values of the voltage applied to 

the electrode the multiple double layers structure evolves into a dynamic state, 

consisting of periodic disruptions and re-aggregations of the constituent double 

layers (Chiriac et al., 2006). The experimental investigations proved the 

important role of the elementary processes such as the electron-neutral impact 

excitations and ionizations in the formation and dynamics of the multiple 

double layers (Ioniţă et al., 2004; Chiriac et al., 2006). Different models were 

proposed for such structures, in which the appearance of the double layers was 

explained as a bifurcation (Conde 2006), or the double layers were assimilated 

with the Turing-type structures (Popescu, 2006). 

Here we report experimental results showing that a plasma conductor 

passes into a chaotic state through a cascade of spatio-temporal sub-harmonic 

bifurcations when the applied constraint in form of the voltage applied on a 

supplementary electrode immersed into plasma increases. The fast Fourier 

transforms (FFTs) of the current oscillations collected by the electrode show the 

appearance of new sub-harmonics of the fundamental frequency simultaneously 

with every appearance of a new double layer structure (as part of a multiple 

double layers structure) in dynamic state. 

2. Experimental Results 

The experiments were performed in a plasma diode, schematically 

presented in Fig. 1. Plasma is created by an electrical discharge between the 

hot filament (marked by F in Fig. 1) as cathode and the grounded tube (made 

of non-magnetic stainless steel) as anode. The plasma was pulled away from 

equilibrium by gradually increasing the voltage applied to a tantalum disk 

electrode (marked by E in Fig. 1) with 1 cm diameter, under the following 

experimental conditions: argon pressure p = 10 –2 mbar, plasma density n pl 10 9 

cm –3 and the electron temperature kT e 2 eV. 

When the voltage on the electrode reaches V E 55 V, a double layers 

structure appears in front of the electrode (see Fig. 2a). Because of the


experimental conditions, this structure appears directly in dynamic state. The 

oscillations of the current collected by the electrode and their FFT’s are shown 

in the Figs. 3a and 3b, respectively. By a further increase of the voltage on the 

electrode, new double layers develops in front of the electrode, giving rise to a 

multiple double layers structure in dynamic state (see photos in Figs. 2b-2e). 

U 1 U 2 

R 2 

A 

F 

E 

R 1 

PS 

Fig. 1 – Schematic of the experimental setup (F – filament, A – anode, E – 

supplementary electrode, P – cylindrical probe, U 1 – power supply for heating the 

filament, U 2 – power supply for discharge, PS – power supply for supplementary 

electrode bias, R 1 , R 2 – load resistors). 

a b c 

d 

e 

Fig. 2 – Photos of the multiple double layers structure in different stages (at different 

potentials applied on the electrode) of its formation.


Simultaneously with every new double layer formation, a new subharmonic 

appears in the FFT spectrum of the oscillations of the current 

collected by the electrode (Fig. 3, second column). Thus, we recorded, in fact, 

spatio-temporal bifurcations in the plasma system (sudden changes in the spatial 

symmetry and the temporal dynamics of the plasma system). At high values of 

the applied potential, the plasma system passes into a chaotic state, 

characterized by uncorrelated and intermittent oscillations (Figs. 3n-3o, 

respectively). 

The stability of a double layer is assured by the balance between the 

production of electrons and positive ions through electron-neutral impact 

ionizations and excitations and the particle losses by recombination and 

diffusion. At high values of the current through the structure, this equilibrium is 

lost and the double layer passes into a dynamic state. When the double layer 

disrupts, the initially trapped particles (electrons and positive ions) are released 

and move towards the electrodes as bunches of particles. In the case of a 

multiple double layer, the free particles have to pass through the others double 

layers and can affect their dynamics. Then, the dynamics of the multiple double 

layer structure becomes more complex and spatio-temporal bifurcation appears 

(sub-harmonics in the current oscillation spectrum and a spatial bifurcation of 

the plasma in front of the electrode are observed). 

a 

b 

c 

d 

Fig. 3 – Oscillations of the current (first column) and their FFT’s (second column), at 

different increasing value of the voltage applied to the electrode.


e 

f 

g 

h 

j 

k 

l 

m 

n 

o 

Fig. 3 (continued) – Oscillations of the current (first column) and their FFT’s (second 

column), at different increasing value of the voltage applied to the electrod.


This looks like the well-known Feigenbaum scenario of transition to 

chaos by cascades of period doubling bifurcations, but is not the case because in 

our experiment we do not record period-doubling bifurcations, except the first 

one, but sub-harmonic bifurcations. The final plasma system state is a chaotic 

one, consisting of uncorrelated and intermittent oscillations with a broad 

spectrum and many peaks being present, which correspond to the unstable 

periodic dynamics of the multiple double layers structure. A similar behavior 

was already reported in connection to gas lasers, Rayleigh-Benard instability or 

ionization waves in plasma. 

The complexity of the phenomenon and the fractal spatial structure of the 

multiple double layers suggest a theoretical modeling in the frame of the scale 

relativity theory. In fact, such an approach has given very good results in the 

theoretical modeling of the appearance and dynamics of simple double layers 

(fireballs). 

4. Conclusion 

A transition to chaos through spatio-temporal sub-harmonic bifurcations 

was experimentally evidenced in plasma by analyzing the oscillations of the 

current collected by a supplementary electrode, related to the nonlinear 

dynamics of multiple double layer structures. 

Acknowledgments. This work was supported by Romanian National Authority for 

Scientific Research, CNCS-UEFISCDI, project number PN-II-ID-PCE-2011-3-0650. 

REFERENCES 

Aflori M., Amarandei G., Ivan L.M., Dimitriu D.-G., Sanduloviciu M., Experimental 

Observation of Multiple Double Layers Structures in Plasma – Part I, Concentric 

Multiple Double Layers. IEEE Trans. Plasma Sci., 33, 2, 542-543 (2005). 

Arecchi F.T., Meucci R., Puccioni G., Tredicce J., Experimental Evidence of 

Subharmonic Bifurcations, Multistability, and Turbulence in a Q-Switched Gas 

Laser. Phys. Rev. Lett., 49, 17, 1217-1220,(1982). 

Atipo A., Bonhomme G., Pierre T., Ionization Waves: From Stability to Chaos and 

Turbulence. Eur. Phys. J. D, 19, 79-87 (2002). 

Chan C., Hershkowitz N., Transition from Single to Multiple Double Layers. Phys. 

Fluids, 25, 12, 2135-2137 (1982). 

Chiriac S., Aflori M., Dimitriu D.-G., Investigation of the Bistable Behaviour of 

Multiple Anodic Structures in DC Discharge Plasma. J. Optoelectron. Adv. 

Mater., 8, 1, 135-138 (2006). 

Conde L., Ferro Fontán C., Lambás J., The Transition from an Ionizing Electron 

Collecting Plasma Sheath into an Anodic Double Layer as a Bifurcation. Phys. 

Plasmas, 13, 11, 113504 1-6, (2006). 

Conde L., Leon L., Multiple Double Layers in a Glow Discharge. Phys. Plasmas, 1, 6, 

2441-2447 (1994).


Dimitriu D.-G., Aflori M., Ivan L.M., Ioniţă C., Schrittwieser R.W., Common Physical 

Mechanism for Concentric and Non-Concentric Multiple Double Layers in 

Plasma. Plasma Phys. Control. Fusion, 49, 3, 237-248 (2007). 

Dimitriu D. G., Physical Processes Related to the Onset of Low-Frequency Instabilities 

in Magnetized Plasmas. Czech. J. Phys., 54, Suppl. C, C468-C474 (2004). 

Dubois M., Rubio M.A., Berge P., Experimental Evidence of Intermittencies Associated 

with a Subharmonic Bifurcation. Phys. Rev. Lett., 51, 16, 1446-1449 (1983). 

Feigenbaum M.J., Universal Behavior in Nonlinear Systems. Los Alamos Science, 1, 1, 

4-27 (1980). 

Intrator T., Menard J., Hershkowitz N., Multiple Magnetized Double Layers in the 

Laboratory. Phys. Fluids B, 5, 3, 806-811 (1993). 

Ioniţă C., Dimitriu D.-G., Schrittwieser R.W., Elementary Processes at the Origin of the 

Generation and Dynamics of Multiple Double Layers in DP Machine Plasma. 

Int. J. Mass Spectrom., 233, 343-354 (2004). 

Nerushev O.A., Novopashin S.A., Radchenko V.V., Sukhinin G.I., Spherical 

Stratification of a Glow Discharge. Phys. Rev. E, 58, 4, 4897-4902 (1998). 

Niculescu O., Dimitriu D.-G., Păun V.P., Mătăsaru P.D., Scurtu D., Agop M., 

Experimental and Theoretical Investigations of a Plasma Fireball Dynamics. 

Phys. Plasmas, 17, 4, 042305 1-10 (2010). 

Nottale L., Scale Relativity and Fractal Space-Time: A New Approach to Unifying 

Relativity and Quantum Mechanics. World Scientific, Singapore, 2011. 

Novopashin S. A., Radchenko V. V., Sakhapov S. Z., Three-Dimensional Striations of a 

Glow Discharge. IEEE Trans. Plasma Sci., 36, 4, 998-999 (2008). 

Popescu S., Turing Structures in DC Gas Discharges. Europhys. Lett., 73, 2, 190-196 

(2006). 

Strat M., Strat G., Gurlui S., Ordered Plasma Structures in the Interspace of Two 

Independently Working Discharges. Phys. Plasmas, 10, 9, 3592-3600 (2003). 

TRANZIŢIE SPRE HAOS PRIN BIFURCAŢII 

SUBARMONICE ÎN PLASMĂ 

I. Partea experimentală 

Se obţine experimental unscenariu de tranziţie spre haos prin bifurcaţii pe 

subarmonice spaţio-temporale în conexiune cu dinamicile straturilor duble multiple 

dintr-o plasmă de descărcare.


Publicat de 



Secţia 




II. FRACTAL HYDRODYNAMICS 

BY 

MARICEL AGOP 1 , EMILIA POLL 2 , DAN-GHEORGHE DIMITRIU 2 , 

MAGDALENA AFLORI 3 and LILIANA-MIHAELA IVAN 2 





3 Petru Poni Institute of Macromolecular Chemistry, Iaşi 



Abstract. By considering that the particles movement in plasma takes place 

on fractal curves, a fractal hydrodynamic model is developed, based on scale 

relativity theory. This model successfully predicts the self-structuring of the 

fractal fluid, the obtained structures being very similar to the multiple double 

layers in plasma. 

Key words: fractal, hydrodynamics, scale relativity theory. 


In many systems where deterministic chaos arises, spatial and temporal 

structures were also experimentally observed. For time scales large with respect 

to the inverse of maximum Lyapunov exponent, deterministic trajectories can 

be replaced by families of potential trajectories and the concept of definite 

positions by that of probability density. This allows the description of the chaos 

effect in a stochastic way by a diffusion process (Lichtenberg & Lieberman, 

Corresponding author: e-mail: m.agop@yahoo.com

38 Maricel Agop et al. 

1983). By considering that the particles movement takes place on continuous 

but non-differentiable curves, i.e. on fractal curves, the scale relativity theory 

approaches the chaotic effects in the same way as in (Lichtenberg & Lieberman, 

1983), but the diffusion becomes a spatio-temporal scale dependent process 

(Notalle, 1989; Nottale, 1993; Nottale, 2011). 

The complex dynamical systems (and particularly the plasma), which 

display chaotic behavior, are recognized to acquire self-similarity and manifest 

strong fluctuations at all possible scales (Lichtenberg & Lieberman, 1983; 

Notalle, 1989; Nottale, 1993; Nottale, 2011; Feynman & Hibbs, 1965; Popescu, 

2006; Dimitriu, 2004; Dimitriu et al., 2003). Since the fractality appears as a 

universal property of these systems, it is necessary to construct a fractal physics 

(Notalle, 1989; Nottale, 1993; Nottale, 2011). In such conjecture, by considering 

that the complexity of the physical processes is replaced by fractality, it is no 

longer necessary to use the whole classical “arsenal” of quantities from the 

standard physics (differentiable physics). The physical systems will behave as a 

special interaction-less “fluid” by means of geodesics in a fractal space. The 

theory which treats the interactions in the previously mentioned manner is the 

Scale Relativity (SR). 

SR applies the principle of relativity to scale transformations. The 

principle of SR requires that the fundamental laws of nature apply whatever the 

state of scale of the coordinate system. In particular, a particle path in quantum 

mechanics may be described as a continuous and non-differentiable curve, i.e. a 

fractal curve. In order to include the non-differentiable fractal quantum motion 

into those described by a theory of relativity, the quantum space-time is 

considered relative and fractal, i.e. divergent with decreasing scale. In this 

theoretical framework, it is not necessary to endow a point particle with mass, 

energy, momentum or velocity. The particle may be reduced to and identified 

with its own trajectory. 

2. Consequences of Non-differentiability 

Let us suppose that the particles movements (electrons, ions and neutrals) 

take place on continuous but non-differentiable curves (fractal curves). The 

non-differentiability implies the followings: 

i) A continuous and a non-differentiable curve (or almost nowhere 

differentiable) is explicitly scale dependent, and its length tends to infinity, 

when the scale interval tends to zero. In other words, a continuous and nondifferentiable 

space is fractal, in the general meaning given by Mandelbrot to 

this concept (Mandelbrot, 1983). 

ii) There is an infinity of fractals curves (geodesics) relating any couple 

of its points (or starting from any point), and this is valid for all scales. 

iii) The breaking of local differential time reflection invariance. The 

time-derivative of an arbitrary field Q (speed, concentration, etc.) can be written 

two-fold


dQ 

 

dt 

dQ 

 

dt 

t 0 

t 0 

 

Qt t Qt 

lim , 

t 

 

 

Qt Qt t 

lim . 

t 

Both definitions are equivalent in the differentiable case. In the nondifferentiable 

situation these definitions fail, since the limits are no longer 

defined. In the framework of fractal theory, the physics is related to the 

behavior of the function during the “zoom” operation on the time resolution δt, 

here identified with the differential element dt (“substitution principle”), which 

is considered as an independent variable. The standard arbitrary field Q(t) is 

therefore replaced by a fractal arbitrary field Q(t,dt), explicitly dependent on the 

time resolution interval, whose derivative is undefined only at the unobservable 

limit dt 0 (from mathematic point of view, these fields are described by 

fractal functions, for details (definition, properties, etc.) (Nottale, 1993). As a 

consequence, this leads us to define the two derivatives of the fractal arbitrary 

field as explicit functions of the two variables t and dt, 

dQ 

 

dt 

dQ 

 

dt 

t 0 

t 0 

, , 

Qt t t Qt t 

lim , 

t 

Qt , tQt t, 

t 

lim . 

t 

The sign “+” corresponds to the forward process and “–“ to the backward 

process, respectively. 

i) The differential of the coordinates, d X t,dt 

, can be decomposed as 

follows 

d X t,dt d x t d t,d t , 

(3) 

 

d xt 

is the “classical part” and d ξ t,dt 

where 

is the “fractal part”. 

i 

ii) The differential of the “fractal part” components ξ t,dt, 

i 1,3 , 

satisfies the relation (the fractal equation) 

i 

i 

i 

 

1/ 

D F 

(1) 

(2) 

d ξ λ d t , 

(4) 

 

where λ are some constant coefficients, and DF is a constant fractal 

dimension. We note that for the fractal dimension we can use any definition 

(Kolmogorov, Hausdorff , etc.). 

iii) The local differential time reflection invariance is recovered by 

combining the two derivatives, d dt and d 

dt , in the complex operator


dˆ 1d d i d d 

 

dt 2 

 

dt 

2 

 

dt 

 

By applying this operator to the “position vector”, a complex speed yields 

 

. 

 

(5) 

ˆ 

ˆ dX 

1dX dX i dX dX V V V V 

V i i 

dt 2 

 

dt 

2 

 

dt 

 

V U (6) 

2 2 

with 

V 

V 

 

V 

U 

 

V 

2 

V 

2 

The real part, V, of the complex speed, Vˆ , represents the standard 

classical speed, which is independent of resolution, while the imaginary part, U, 

is a new quantity arising from fractality, which is resolution-dependent. 

iv) The average values of the quantities must be considered in the sense 

of a generalized statistical fluid like description. Particularly, the average of 

d X is 

with 

d 

 

 

 

, 

. 

(7) 

X d x , 

(8) 

 

d ξ 0 

(9) 

. 

In such an interpretation, the “particles” are identified with the geodesics 

themselves. As a consequence, any measurement is interpreted as a sorting out 

(or selection) of the geodesics by the measuring devices. 

3. Covariant Total Derivative 

Let us now assume that the curves describing the particles movements 

(continuous but non-differentiable) is immersed in a 3-dimensional space, and 

i 

X i 1,3 is the position vector of a point on the curve. 

that X of components 

Let us also consider the fractal arbitrary field Q , t 

differential up to the second order 

2 

X and expand its total 

Q 

1 Q i j 

d Q 

 

dtQdX d X d X . 

t 2 

i j 

(10) 

X X


The relation (10) are valid in any point of the space manifold and also for the 

points X on the fractal curve which we have selected in Eq. (10). From here, the 

forward and backward average values of this relation take the form 

Q 

1 Q i j 

d Q 

 

dt QdX 

d X d X 

t 2 

i j . (11) 

X X 

We make the following stipulation: the mean value of Q and its 

i 

derivatives coincide with themselves, and the differentials d X and dt are 

independent, therefore the average of their products coincide with the product of 

averages. Thus, the Eqs. (11) become 

Q 

1 Q i j 

d Q 

dtQ d d X d X 

t 

X 

 

2 

i j 

X X 

 

 

, (12) 

or more, by using Eqs. (3) with the property (9), 

2 

Q 

1 Q i j i j 

d Q 

 

dtQd d x d x d ξ d ξ . 

t 

x 

2 

i j 

X X 

 

 

(13) 

i 

ξ 

Even the average value of d is null (see Eqs. (9)), for the higher order of 

these average coordinates the situation can be different. First, let us focus on the 

mean dξ 

i d ξ 

j . If i j, this average is zero because of the independence of 

d 

i 

ξ 

and d 

j 

ξ 

. So, by using Eqs. (4), we can write 

i j i j 

 

 

2 

2 

D F 

2 1 

d ξ d ξ λ λ dt 

dt . 

(14) 

Then, Eqs. (13) may be written under the form 

2 2 

Q 1 Q i j 1 Q i j 

 

i j 

i j 

 

D F 

2 1 

d Q dt d x Q d x d x λ λ dt d t. 

(15) 

t 2 X X 2 X X 

If we divide by dt and neglect the terms which contain differential factors (see 

method from (Agop et al., 2008)), the Eqs. (15) are reduced to 

2 

dQ Q 1 Q i j 

2 D 

d 

F 1 V Q λλ t 

dt 

t 2 

i j . (16) 

X 

X 

These relations also allow us to define the operator 

2 

d 

1 i j 

2 D 1 

dt 

F 

dt t 2 

i j 

X X 

V 

 

(17)


Under these circumstances, let us calculate ˆd dt . By taking into account Eqs. 

(17), (5) and (6) we obtain 

dˆ Q 1dQ dQ dQ dQ 

i 

dt 2 dt dt 

dt dt 

 

 

 

1Q 

1 1 Q i j 2 D F 1 

VQ λλ dt 

2 t 2 4 

i j 

X X 

1Q 

1 1 Q i j 2 D F 1 

VQ λλ dt 

2 t 2 4 

i j 

X X 

2 

2 

2 

i Q 

i i Q i j 2 D F 1 

VQ λλ d t 

(18) 

2 t 2 4 

i j 

X X 

2 

i j 

V λλ 

j 2 D F 1 

dt 

VV VV 

i 

2 2 

Q 

2 

Q i j i j i j i j 

i j 

X X 

 

 

2 

ˆ 1 Q i j i j 

V Q λλ λλ 

4 

i j 

X X 

 

i j i j 

2 D F 1 

λλ λλ 

dt 

i Q 

i i Q 

Q 

 

2 t 

2 4 

i 

X X 

Q 

 

 

t 

 

D 

d 

F 

λλ λλ i λλ λλ t 

1 

2 1 

 

4 

Q 

 

 

t 

 

i 

This relation also allows us to define the fractal operator: 

 

2 

ˆd ˆ 1 i j i j i j i j 2 D 1 

i d F 

λλ λλ λλ λλ 

t 

dt t V 4 

 

i j 

X X 

. (19) 

Particularly, by choosing 

 

the Eq. (14) becomes 

i j i j ij 

 

λλ λλ 2Dδ 

i 

j 

D 

t 

F 

2 1 

(20) 

dξ 

d 

2D d dt. (21) 

We note the followings: 

i) The fractal processes given by Eq. (21) with DF 

2 are known as 

“anomalous diffusion” (sub-diffusion for D F < 2 and super-diffusion for D F >2). 

Usually, the “Fokker-Planck equations” for anomalous diffusion do not have the


form of the ordinary diffusion equation. Indeed, it is well-known that the 

“Fokker-Planck equations” for anomalous diffusion have the form of the 

fractional derivative equations, and the equations are called fractional Fokker- 

Planck equations (Gouyet, 1992; El Naschie, 1995; Weibel et al., 2005). 

ii) The Nottale’s theory is formulated in the fractal dimension D F = 2, i.e. 

for movements on Peano curves, and for Wiener’s stochastic processes. In these 

conditions, the fractal operator (19) takes the simple form: 

ˆd ˆ 2 D 

i dt 

1 F 

V D . 

(22) 

dt 

t 

We now apply the principle of scale covariance, and postulate that the passage 

from classical (differentiable) physics to the fractal physics can be implemented 

by replacing the standard time derivative operator, ddt , by the complex 

operator ˆd dt . As a consequence, we are now able to write the equation of the 

field flow in its covariant form 

ˆd Q Q 

ˆ i d 2 1 0 

dt 

D F 

V Q D t Q 

. (23) 

t 

This means that at any point of a fractal path, the local temporal term, t Q , the 

non-linearly (convective) term, V ˆ Q 

and the dissipative one, Q , make 

their balance. Moreover, the behavior of a fractal fluid is of viscoelastic or of 

hysteretic type, i.e. the fractal fluid has memory. Such a result is in agreement 

with the opinion given in (Agop et al., 2008; Niculescu et al., 2010; Gurlui et 

al., 2006; Chiroiu et al., 2005, Ferry & Goodniks, 1997): the fractal fluid can be 

described by Kelvin-Voight or Maxwell rheological model with the aid of 

complex quantities e.g. the complex field, Q, the complex structure coefficients, 

2 

D 

t 

 

F 

iD 

d 

1 

, etc. 

4. Geodesics. Fractal Hdrodynamics 

We are now able to write the equation of geodesics (a generalization of the 

first Newton’s principle) in the form 

dˆ Vˆ 

Vˆ 

Vˆ Vˆ ηV ˆ. 

(24) 

dt 

t 

Formally, at the global scale (with its differentiable and fractal components), the 

Eq. (24) is a Navier-Stokes type equation with the imaginary “viscosity 

coefficient” 

D 

2 1 

F 

η iD dt . 

(25)


This result evidences the rheological properties of the fractal fluid. If the 

motions of the fractal fluid are irrotational, i.e. V ˆ 0 

, we can choose Vˆ of 

the form 

D 

t 

F 

ˆ 2 

2i d 1 ln , 

V D ψ 

(26) 

with ψ being the scalar potential of the complex speed. Then, by substituting 

(26) in (24) and using the method from (Niculescu et al., 2010; Gurlui et al., 

2006) it results: 

dVˆ 

2 D 1 ln 

2i d 2i d 2 

F 

ψ 

DF 

1 ψ 

 

D t 

D t 

0. (27) 

dt t ψ 

Eq. (27) can be integrated in a universal way, which yields 

Lˆ 0, 

D 

F 

D 

F 

ˆ 2 4 2 2 D 

4 d 2i d 

1 

D 

L t t 

t 

(28) 

up to an arbitrary phase factor which may be set to zero by a suitable choice of 

the phase of ψ. Eq. (28a), where ˆL is the differential operator (28), is of 

Schrödinger type. 

iS 

For ψ ρe , with ρ the amplitude and S the phase of ψ, and by using 

(26) the complex speed field (6) takes the explicit form 

D 

F 

DF 

D 

2 D F 1 

t S 

2 D F 1 

dt lnρ. 

ˆ 

2 1 2 1 

2 d i d ln 

V D t S t ρ, 

V 2D 

d , 

U D 

(29) 

By substituting (29) in (24) and separating the real and the imaginary parts, up 

to an arbitrary phase factor which may be set at zero by a suitable choice of the 

phase of ψ, we obtain 

V 

 

m 

t 

 

 

ρ 

ρV 

0, 

t 

0 V V Q, 

(30) 

with Q the fractal potential 

2 

2 4 D 2 0 

2 

F ρ m U 

DF 

1 

0 0 

Q2m D dt m D dt 

U. (31) 

ρ 2


Eq. (30) are the law of momentum conservation and the law of density 

conservation and m 0 is the rest mass of the fractal fluid particle. Together, these 

two equations define the fractal hydrodynamics. 

5. Predictability through Factality. Fractal Fluid Self-structuring 

In the one-dimensional case, Eqs. (30) and (31) with the initial conditions 

 

V x, t 0 

and the boundary ones 

 

α 

c, ρxt , 0 e 

 

ρ x 

2 

x 

 

1 

0 , (32) 

πα 

, , , , 

V x ct t c 

ρ x t ρ x t 0, (33) 

implies the solutions (for details see the method described in [18]) 

 

 

 

2 

1 

x 

ct 

ρxt 

, 

exp 

, 

2 

2 

 

2 2D 2 

2 2D 

 

2 

π α t 

α t 

 

α α 

 

 

 

cα 

V 

α 

2 

2D 

 

 

α 

2 

xt 

2 

2 2D 

2 

 

 

α 

 

 

t 

(34) 

where c is a constant speed, α is the distribution parameter and 

Then, it results the complex speed 

D 

t 

F 

2 1 

D D d . 

(35) 

2 2D 

 

cα xt 

ˆ α 

x 

ct 

V V iU 

 

2Di 

2 

2 2 

2 2D 2 2 2D 

2 

α t α t 

α α 

, (36) 

the fractal potential and force


2 

2 x 

ct 

2mD 

0 

0 

2 

2 2 

 

2 2 

2 

2 2D 

2 

Q2 m D 

, 

D 

α 

t α t 

 

α 

α 

 

 

 

 

Q 2 xct 

F 4 m0D 

. 

x 

2 

2 

2 2D 

 

2 

α 

t 

α 

 

 

 

In non-dimensional coordinates 

(37) 

x 

τ ωt , ξ , 

(38) 

λ 

where ω is a specific frequency of the fractal fluid and λ is a characteristic 

length of the fractal fluid, and with the substitutions 

the Eqs. (36) and (37) become 

2 

α 

 

μ 

λ 

, 2D 

ν , (39) 

αωλ 

2 2 

ˆ 

V i 

2 2 2 2 2 2 

 

Q 

 

 

 

 

2 

2 2 2 

 

 

1 

 

2 

 

2 2 2 

2 2 2 

 

 

2 

The complex current density field is also obtained 

(40) 

(41) 

 

F 

. (42) 

2 2 2 2 

ˆ μ ν τ 

( ξ τ) ξ τ ( ξ τ) 

 

J exp i exp . 

2 2 3 2 

2 2 2 2 2 2 3 2 

 

2 2 2 

(43) 

( μ ντ) ( μ ντ) 

 

μ ντ 

 

Figs. 1a-g show the dependences: (a) ρξ,τ,μ ( ν 1) 

, (b) e V 

ˆ 

( ξ,τ,μν 1) , 

(c) m V 

ˆ 

( ξ,τ,μν 1) , (d) Q( ξ,τ,μ ν 1) 

, (e) F( ξ,τ,μ ν 1) 

, (f) 

e J 

ˆ 

( ξ,τ,μν 1) and (g) m J 

ˆ 

( ξ,τ,μν 1) . It results:


i) The force field induces fractal characteristics to the quantities which 

define the system dynamics. Consequently, they become dependent on the 

spatio-temporal coordinates. 

ii) The observable in the form of the rectilinear and uniform motion, V=c, 

is obtained by annulling the force field. The fractal forces on the semi-spaces 

x x and x x , with x the mean position, compensate each 

other 

m 

dV 

x 

 

m 

0 0 

dt 

x 

dV 

dt 

x 

This means that a fluid particle on free motion locally “polarizes” the fractal 

fluid behind itself, x ct , and ahead of itself, x ct , in such a way that the 

resulting fractal forces are symmetrically distributed with respect to a plane 

through the observable particle position x ct at any time t (see the 

symmetry of the curves from Figs. 1a-g). In this case, the quantities become 

independent on the spatio-temporal coordinates. The presence of an external 

perturbation induces an asymmetry in the distribution of the fractal force field in 

respect to the plane where the particle is, having as a result the “excitation” of a 

specific mode of fractal fluid self-structuring. 

Therefore, in the particular case of plasma, the collisions induced by the 

interactions of their particles can be “substituted” by the fractal field (we will 

return later in the article to this subject), while the presence of an external 

constraint (as, for example, a voltage) “excites” a specific mode of plasma selfstructuring, 

that could lead to the generation of an electric double layer, or 

multiple double layer. 

By using the normalized variables (38) and 

 

N , 

 

0 

V 

V 2 k 

, 

BT 

u , 

u m 

0 

 

x 

2 

2 2D 

1 

0 3 

. 

u 

, (44) 

where ρ 0 is the equilibrium density, u is a specific propagation speed of a 

perturbation in the fractal fluid, k B is the Boltzmann constant and T is the 

“temperature” of the fluid particle (in this model, the fluid particles are 

identified with the geodesics of the fractal space and their distribution satisfies a 

certain statistics. We associate the temperature T to such a statistical ensemble.), 

the Eqs. (30) become 

N 

NV 

 

0, 

 

 

2 

V 

V 

2 1 

V 

 

0 2 N 

 

 

. 

 

N 

 

 

(45)


1 

10 

 

0.75 

0.5 

0.25 

20 

10 

ReVˆ 

5 

0 

-5 

20 

10 

0 

-20 

-10 

0 

τ 

-10 

-20 

-10 

0 

τ 

ξ 

0 

10 

20 -20 -10 

(a) 

ξ 

0 

10 

20 -20 -10 

(b) 

20 

0 

10 

Im Vˆ 

0 

-10 

10 

20 

Q 

-100 

-200 

-300 

10 

20 

-20 

-20 

-10 

0 

τ 

-20 

-10 

0 

τ 

ξ 

0 

10 

20 -20 -10 

(c) 

ξ 

0 

10 

20 -20 -10 

(d) 

20 

1 

F 

10 

0 

-10 

20 

10 

Re Ĵ 

0.75 

0.5 

0.25 

20 

10 

-20 

-20 

-10 

0 

τ 

0 

-20 

-10 

0 

τ 

ξ 

0 

10 

20 -20 -10 

(e) 

ξ 

0 

10 

20 -20 -10 

(f) 

Im Ĵ 

0.4 

0.2 

0 

-0.2 

-0.4 

-20 

-10 

0 

τ 

20 

10 

Fig. 1 – The dependence on normalized spatial coordinate and time of: (a) normalized density 

( , , 1) , (b) normalized differential speed 

ˆ 

e V ( , , 1) , (c) normalized fractal speed 

ˆ 

m V ( , , 1) 

ξ 

0 

, (d) normalized fractal potential Q( , , 

1) , (e) normalized fractal force 

F( , , 1) , (f) normalized differential current density 

ˆ 

e J ( , , 1) , (g) normalized fractal 

ˆ 

-10 

10 

20 -20 

(g) 

current density m J ( , , 1) , respectively.


For localized stationary solution, let us choose a transformed coordinate q 

in the moving frame, such that 

q ξ Mτ , (46) 

where M is the equivalent of the Mach number for the fractal fluid 

V 0 

M , (47) 

u 

and V 0 is the speed of a perturbation moving together with the frame. After 

integration, from the continuity equation, it results 

1 

and from the momentum equation it result 

V 

 

M 1 

N 

 

 

, (48) 

2 2 

V 

2 1 d 

VM 

0 N 

, 

(49) 

2 

2 

N 

dq 

where we have used the restrictions 

q , , d N 

V 0 N 1, 

2 

0 

dq 

d N 

, 0 (50) 

2 

dq 

Now, by substituting (48) in (49) and taking into account the restrictions (50), 

we successively obtain 

 

 

M 

2 

0 

2 2 

2 4 2 

Z 

Z dq 

0 

Z N 

, 

2 

M 1 2 1dZ 

 

Z 

, 

4ν 

2 dq 

 

 

q 

2 2 

Z 

0 

d 

 

1 

1 1 d Z 

1 

, 

 

 

2 

Z M 

q q 

2 

2 ν 

Z 

0 

 

const., 

0 

 

2 

, 

(51) 

which implies the solution


M 

N sh 

qq0 

 

, (52) 

 

ν0 

 

with q 0 an integration constant. In these conditions, the current density is 

the fractal potential is 

M 

J NV M N 1M sh q q0 

M 

, (53) 

 

ν0 

 

M 2 2 

1 

1 M 

cth 

M 

Q q 

q0 

 

2 

 

2 

N 

2 

ν0 

 

and the “voltage-current characteristics” is 

, (54) 

2 

2 

M 

 

J 

Q 1 1 

2 

 

M 

. (55) 

 

 

 

 

Now, if the structured fractal fluid is equivalent with a “circuit element”, such, 

for example, it is happens with the electric double layer in plasma, since for 

J M 1, Q JM and for J M 1 , Q M 2 2 , it results that it 

behaves as a “nonlinear element of circuit”. Moreover, the restriction 

dQ dJ M1J M 3 

0 marks the beginning of the self-structuring 

mechanism, for example the generation of a double layer in the case of plasma. 

We note that if ω is the ion plasma frequency, λ is the Debye length and u is 

ion-acoustic speed, the previous results can describe the dynamics of plasma. 

6. Synchronous Movements in the Fractal Systems. Types of Dynamics in 

Plasma at Differential Scale (Macroscopic Scale) 

Let us allow the next assumptions: 

i) the movements at the two scales, differential (through V) and fractal 

(through U), are synchronous, which implies 

d 

D F 

t 

2 1 

V U D lnn, 

n ; 

(56) 

ii) the movements take place on Peano curves, i.e. with the fractal 

dimension D F = 2. In this situation (35) takes the form 

D D . (57) 

For a ionized gas with only one type o f charge carriers, the total current 

density is null


j ρV enμE eDn 

0 

, (58) 

where μ is the mobility of the charge carriers, e is t he elementary electrical 

charge, D is the diffusion coefficient and E is the electric field. In this case, the 

field current, enE is totally compensated by the diffusion current eD n , and 

the vectors E and n are parallel. By limiting to the case of a non-degenerate 

gas, the partial pressure of the charge carriers is related to their density and 

temperature through the relation 

so that, from Eq. (58), it results 

p nk T , (59) 

B 

μ n 

p 

enE en e 

, (60) 

D nE pE pE p k T 

where we used the relation dp 

eEndl 

between the two expressions of the 

force on a layer of dl thickness and unit surface, normal on the pressure 

gradient. From Eqs. (57) and ( 60) it results first 

μkBT 

D D , 

e 

and then, through (30), (56) and (61), the diffusion equation 

n 

D n 

t . 

B 

(61) 

(62) 

Let us consider an ionized gas with the electron density n e and ion density n p . 

Usually , the gradients of the electron and ion densities, n e and np 

, as well 

as the electric field E, are different from zero, so that the total current density 

V j j (63) 

e 

have a diffusion component, as well as a field compone nt ( Chen, 1984; Popa & 

Sirghi, 2000) 

p 

j 

e 

j 

e( μ n E D n 

), 

p 

e e e e 

e( μ n E D n 

), 

p p p p 

(64) 

The field current appears even if no external electric field is applied. Indeed, 

b ecause De Dp, the electrons radially diffuse outside of plasma, leaving 

behind an excess of positive charge. This leads to the appearance of a radial


electric field, E r , which retards the electrons and accelerates the positive ions, 

the total radial current being null in the stationary regime. By taking into 

account the quasineutrality of plasma 

n n n, (65) 

e 

p 

from (63)-(65) it results 

E 

r 

De 

Dp 

1 n 

 

, (66) 

μ μ n r 

e 

p 

as well as the density of the particles current 

with 

j j 

e 

p n 

Ge Gp Da 

e e r 

, (67) 

D 

a 

μ D 

 

μ 

e p p e 

e 

μ D 

μ 

as ambipolar diffusion coefficient. It results first 

p 

, 

(68) 

D D D a , (69) 

and then, from Eqs. (67) and (30), the ambipolar diffusion equation 

n 

Da 

n. 

t 

(70) 

The presence of collisions dramatically changes the expression of the 

diffusion coefficient. For example, for weak ioniz ed plasma and in the 

approximation of small density gradients, the electron free diffusion coefficient 

is (Chen, 1984; Popa &Sirghi, 2000) 

kT B e 

D De 

, 

(71) 

m ν 

where ν en is the frequency of the elastic electron-neutral collisions, m e is the 

electron mass and T e is the electrons temperature. 

In the general case, we can associate to every type of collision (elastic, 

inelastic) trajectories described by continuous and non-differentiable curves 

(fractal curves). Since the fractality through the Eq. (21) suppose a certain 

statistics, it results that the type of collision have to be described by a certain 

random process. For example, in the case of elastic collisions, the dynamics of 

the plasma particles can be described by Brownian-type movements. Then, the 

e en


trajectories of the plasma particles are not fractals, but can be approximate with 

fractals. Indeed, between two successive elastic collisions the particle trajectory 

is a straight line, while the trajectory becomes non-differentiable in the impact 

point (there are left and right derivatives in this point). Now, by considering that 

all the elastic collisions impact points compose an innumerable set of points, it 

results that the trajectories become continuous and non-differentiable, i.e. fractal 

curves. The random process that can describe the Brownian motion could be, 

for example, the Wiener process. In this case, the mean square distance covered 

by a particle in the mean time τ, can be assimilated to a diffusion coefficient (up 

to a numeric factor) 

2 

x 

D D . (72) 

τ 

Moreover, by taking into account the statistic meaning of the collision crosssection, 

σ, a correspondence with the diffusion coefficient can be established in 

the form (Chen, 1984; Popa &Sirghi, 2000) 

c 

D D , (73) 

n σ 

where n 0 is an equilibrium density and c is a specific propagation speed of a 

perturbation in plasma. Because in the general case σ is a function of the charge 

carrier energy, the scale dependence (35) can be replaced by the normalized 

energy ε dependence 

0 

2 D 1 

() F 

D D ε , 

(74) 

having in mind the fractal characteristics of a relation of type σ σE 

(Mandelbrot, 1983; Gouyet, 1992). 

7. Conclusion 

By considering that the particles movements in a dc gas discharge plasma 

take place on fractal curves, a fractal hydrodynamic model was developed in 

order to describe its dynamics. Thus: 

i) the scale relativity model was more detailed presented compared with 

those presented by Nottale in (1989, 1993, 2011) (consequences of nondifferentiability, 

covariant total derivative, geodesics via Schrödinger-type 

equation or fractal hydrodynamic model); 

ii) through fractal hydrodynamic model we shown that the predictability 

is imposed by fractality and the conditions in which a fractal fluid can selfstructure 

were specified;


iii) in the frame of the fractal hydrodynamic model, the synchronous 

fractal movements were analyzed and some types of plasma dynamics were 

presented, which satisfies such a condition (ionized gas with only one type of 

charge carriers, ionized gas with two types of charge carriers – ambipolar 

diffusion); 

iv) the correspondence between the collisions and fractality was 

established, as well as the way in which this correspondence function in plasma; 

Acknowledgments. This work was supported by Romanian National Authority 

for Scientific Research, CNCS-UEFISCDI, project number PN-II-ID-PCE-2011-3- 

0650. 

R E F E R E N C E S 

Agop M., Forna N., Casian Botez I., Bejenariu I. C., New Theoretical Approach of the 

Physical Processes in Nanostructures. J. Comput. Theor. Nanosci., 5, 4, 483- 

489, (2008). 

Chen F. F., Introduction to Plasma Physics. 2 nd Ed., Plenum Press, New York, 1984. 

Chiroiu V., Ştiucă P., Munteanu L., Dănescu S., Introduction in Nanomechanics. 

Romanian Academy Publishing House, Bucharest, 2005. 

Dimitriu D.-G., Ignătescu V., Ioniţă C., Lozneanu E., Sanduloviciu M., Schrittwieser 

R. 

W., The Influence of Electron Impact Ionisations on Low Frequency Instabilities 

in a Magnetised Plasma. Int. J. Mass Spectrom., 223-224, 141-158 (2003). 

Dimitriu D. G., Physical Processes Related to the Onset of Low-Frequency Instabilities 

in Magnetized Plasmas. Czech. J. Phys., 54, Suppl. C, C468-C474 (2004). 

El Naschie M. S., Rössler O. E., Prigogine I. (Eds.), Quantum Mechanics, Diffusion and 

Chaotic Fractals. Elsevier, Oxford, 1995. 

Ferry D. K., Goodnick S. M., Transport in Nanostructures. Cambridge University 

Press, Cambridge, 1997. 

Feynman R. P., Hibbs A. R., Quantum Mechanics and Path Integrals. MacGraw-Hill, 

New York, 1965. 

Gouyet J. F., Physique et structures fractals. Masson, Paris, 1992. 

Gurlui S., Agop M., Strat M., Strat G., Băcăiţă S., Cerepaniuc A., Some Experimental 

and Theoretical Results on the Anodic Patterns in Plasma Discharge. Phys. 

Plasmas, 13, 6, 063503 1-10, (2006). 

Lichtenberg A. J., Lieberman M. A., Regular and Stochastic Motion. Springer-Verlag, 

New York, 1983. 

Mandelbrot B.B., The Fractal Geometry of Nature. Freeman, San Francisco, 1983. 

Niculescu O., Dimitriu D.-G., Păun V.P., Mătăsaru P.D., Scurtu D., Agop M., 


Phys. Plasmas, 17, 4, 042305 1-10 (2010). 

Nottale L., Fractal Space-Time and Microphysics: Towards a Theory of Scale 

Relativity. World Scientific, Singapore, 1993. 

Nottale L., Fractals and the Quantum Theory of Spacetime. Int. J. Mod. Phys. A, 4, 

5047-5117 (1989).




Popa G., Sirghi L., Fundamentals of Plasma Physics. “Al. I. Cuza” University 

Publishing House, Iasi, 2000. 

Popescu S., Turing Structures in DC Gas Discharge. Europhys. Lett., 73, 190-196 (2006). 

Weibel P., Ord G., Rösler O. E. (Eds.), Space Time Physics and Fractality. Springer, 

Wien – New York, 2005. 

Wilhelm H. E., Hydrodynamic Model of Quantum Mechanics. Phys. Rev. D, 1, 8, 2278- 

2285 (1970). 

T RANZIŢII SPRE HAOS PRIN BIFUNCŢII SUBARMONICE ÎN PLASMĂ 

II. Hidrodinamica fractală 

(Rezumat) 

Propunerea ca particulele unei plasme de descărcare se deplasează pe curbe 

continue şi nediferenţiabile, adică pe curb e fractale, se construieşte o hidrodinamică 

fractală compusă din legea de conservare a impulsului. Prezenţa potenţialului fractal 

“coordonează” atât haoticitatea sistemului cât şi selfstructurarea acestuia (prin straturi 

multiple).


Publicat de 



Secţia 




III. THEORETICAL MODELING 

BY 

EMILIA POLL 1 , MARICEL AGOP 2 , DAN-GHEORGHE DIMITRIU 1 , 

LILIANA-MIHAELA IVAN 1 and MAGDALENA AFLORI 3 





3 Petru Poni Institute of Macromolecular Chemistry, Iaşi 



Abstract. A theoretical model able to explain the transition to chaos 

through a cascade of sub-harmonic bifurcations in connection with the nonlinear 

dynamics of multiple double layers is established based on fractal 

hydrodynamics and scale relativity theory. The obtained results are in very good 

agreement with the experimental ones. 

Key words: chaos, fractal, sub-harmonic bifurcation, fractional revival 

mechanism. 


The scale relativity theory is based both on the fractal space-time concept 

and on a generalization on Einstein’s principle of relativity to scale 

transformations (Nottale, 1989; Nottale, 1993; Nottale, 2011). It is built by 

completing the standard laws of classical physics (motion in space-time) by new 

Corresponding author: e-mail: maflori@icmpp.ro

58 Emilia Poll et al. 

scale laws, the space-time resolutions being used as intrinsic variables, playing 

for the scale transformations the same role as played by velocities for motion 

transformation. 

In the usual theories of plasma physics in which the charged particle 

movement take place on continuous and differentiable curves (Goldstein & 

Rutherford, 1995), it is difficult to determine either the collision terms or source 

terms in connection with the elementary plasma processes (excitations, 

ionizations, recombinations, etc.). A new way to analyze the plasma dynamics 

is to consider that the charged particles movements take place on continuous but 

nondifferentiable curves, i.e. on fractal curves (Nottale, 1989; Mandelbrot, 

1983; Cresson, 2006). Then, the complexity of these dynamics is substituted by 

fractality. Every type of elementary process from plasma induces both 

spatiotemporal scales and the associated fractals. Moreover, the movement 

complexity is directly related to the fractal dimension; the fractal dimension 

increases as the movement becomes more complex. Then, plasma will behave 

as a special collisionless fluid by means of geodesics in a fractal space-time. 

Here, we will use a scale relativity model in the study of the discharge 

plasma dynamics, in order to explain the transition to chaos of the plasma 

system dynamics by cascade of sub-harmonic bifurcations. We demonstrated in 

the second part of this article that such a model can explain the self-structuring 

of plasma in form of multiple double layers. Now, by using the fractional 

revival mechanism (Aronstein & Stronde, 1997), we state a Reynold’s fractional 

criterion of evolution to chaos through a cascade of spatio-temporal subharmonic 

bifurcations, related to the multiple double layer dynamics. A very 

good agreement between the experimental results (described in the first part of 

the article) and those provided by the theoretical model was obtained. 

2. Dynamics in Plasma Induced by the Fractal Potential at Differential 

Scale (Macroscopic Scale) 

The fractal potential (see the second part of the article) comes from the 

non-differentiability and has to be treated as a kinetic term and not as a potential 

term. Moreover, the fractal potential Q can generate a viscosity stress type 

tensor. Indeed, in the form 

2 

2 

2 4 DF 

2 

ρ 1 ρ 

 

Qm0D dt 

, (1) 

ρ 2 ρ 

 

 

the fractal potential induces the symmetric tensor 

4 D 2 

4 2 

F 

D 

F 

iρ 

lρ 

2 2 

 

σil m0D dt ρil lnρm0D d t ilρ 

 

. 

(2) 

 

ρ


The divergence of this tensor is equal to the force density associated with Q 

σ ρ Q. 

(3) 

The quantity can be identified with the viscosity stress type tensor of a 

Navier-Stokes type equation 

dV 

m0 

ρ σ. 

d t 

The momentum flux density type tensor is 

il 0 i l il , 

and it satisfies the momentum-flow type equation 

(4) 

π m ρVV σ 

(5) 

 

m0 ρV π. 

(6) 

t 

In order to complete the analogy to classical fluid mechanics, we 

introduce the kinematical and dynamical types viscosities 

 

 

1 

2 

1 

2 

D 

D F 

t 

0 

2 1 

d , 

m ρD 

D F 

t 

2 1 

d . 

The quantities ν and ν are formal viscosities, both of them being induced by 

the fractal scale. According to the previous paragraph, these viscosities can be 

associated with the collisions dynamics. Then, the tensor σ il takes the usual 

form 

In particular, if σ il is diagonal 

σ 

il 

U 

ν 

xl 

 

il 

i 

il 

U 

 

x 

then we obtain (see also the second part of the article) 

i 

l 

 

. 

 

(7) 

(8) 

, (9) 

V 

σ 

m 

 

t 

 

 

ρ 

ρ 

ρV 

0. 

t 

0 V V , 

(10)


Let us assimilate the tensor (9) with the gas pressure, i.e. σil 

p δil 

, case 

in which Eqs. (10) can define the cl assical hydrodynamics. Then, for the 

normalized variables 

Vk r 

ωt τ , kr ξ , kz η , Vξ 

ω , Vk z 

Vη 

ω , ρ 

N 

ρ , (11) 

0 

and by admitting the adiabatic expansion of the gas, γ = 1.33, Eqs. (10) become 

2 γ1N 

NV ξNV NV V N 

, 

τ ξ ξ η ξ 

1 

ξ ξ ξ η 

1 

η ξ η η 

2 γ1N 

NV ξNV V NV N 

, 

τ ξ ξ η η 

N 

1 

 

τ ξ ξ η 

 

NVξ 

NVη 

0. 

(12) 

In the Eqs. (12) we considered as functional the scaling relation for the unit 

mass, m , 

0 1 

2 

B 0 

2 

kTk 

γ 

ω 

1 . (13) 

If ω is the ion plasma frequency, k is the inverse of the Debye length and T 0 is 

the el ectron temperature, then the Eqs. (12) can describe the dynamics of a 

discharge plasma. Moreover, the 

relation (13) reduces to a usual dispersion 

relation (C h e n, 1984; Popa & Sirghi, 2000). 

For the numerical integration we shall impose the initial conditions 

V 0, ξη , 0 , V 0, ξη , 0 , 0, , 

ξ 

η 

1 

5 

N ξη , 1 ξ 2, 0 η 1, (14) 

as well as the boundary conditions 

V ,1, V ,2, 0, V ,1, V 

,2, 0, 

 

 

 

,1, 

N,2, 

 

 

V , ,0 V , ,1 0, V , ,0 V 

, ,1 0, 

 

N 

1 , 

5 

2 2 

1 

1 5 

3 2 

N 

, ,0 exp exp , 

10 1 5 1 5 

 

N , ,1 

 

 

1 . 

5 

(15) 

Eqs. (12) with the initial conditions (14) and the boundary conditions (15) were 

numerically resolved by using the finite differences (Zienkievicz et al., 2005).


Figs. 1a-l, 2a-l and 3a-l show the two-dimensional contours of the normalized 

density N and the normalized speeds V ξ and V η , respectively, for the normalized 

time values: τ = 1/2 (a), τ = 1/3 (b), τ = 2/3 (c), τ = 1/4 (d), τ = 3/4 (e), τ = 1/5 

(f), τ = 2/5 (g), τ = 3/5 (h), τ = 4/5 (i), τ = 1/8 (j), τ = 3/8 (k) and τ = 5/8 (l). 

1 

1 

1 

0.8 

0.8 

0.8 

0.6 

0.6 

0.6 

0.4 

0.4 

0.4 

0.2 

0.2 

0.2 

0 

1 1.2 1.4 1.6 1.8 2 

1 

0 

1 1.2 1.4 1.6 1.8 2 

0 

1 1.2 1.4 1.6 1.8 2 

(a) τ = 1/2 (b) τ = 1/3 (c) τ = 2/3 

1 

1 

0.8 

0.8 

0.8 

0.6 

0.6 

0.6 

0.4 

0.4 

0.4 

0.2 

0.2 

0.2 

0 

1 1.2 1.4 1.6 1.8 2 

1 

0 

1 1.2 1.4 1.6 1.8 2 

0 

1 1.2 1.4 1.6 1.8 2 

(d) τ = 1/4 (e) τ = 3/4 (f) τ = 1/5 

1 

1 

0.8 

0.8 

0.8 

0.6 

0.6 

0.6 

0.4 

0.4 

0.4 

0.2 

0.2 

0.2 

0 

1 1.2 1.4 1.6 1.8 2 

1 

0 

1 1.2 1.4 1.6 1.8 2 

0 

1 1.2 1.4 1.6 1.8 2 

(g) τ = 2/5 (h) τ = 3/5 (i) τ = 4/5 

1 

1 

0.8 

0.8 

0.8 

0.6 

0.6 

0.6 

0.4 

0.4 

0.4 

0.2 

0.2 

0.2 

0 

1 1.2 1.4 1.6 1.8 2 

0 

1 1.2 1.4 1.6 1.8 2 

0 

1 1.2 1.4 1.6 1.8 2 

(j) τ = 1/8 (k) τ = 3/8 

(l) τ = 5/8 

η 

1/5 N 1 

ξ


(m) legend 

Fig. 1 – Modeled two-dimensional normalized plasma density profiles for different 

values of the normalized time τ. 

1 

1 

1 

0.8 

0.8 

0.8 

0.6 

0.6 

0.6 

0.4 

0.4 

0.4 

0.2 

0.2 

0.2 

0 

1 1.2 1.4 1.6 1.8 2 

1 

0 

1 1.2 1.4 1.6 1.8 2 

0 

1 1.2 1.4 1.6 1.8 2 

(a) τ = 1/2 (b) τ = 1/3 (c) τ = 2/3 

1 

1 

0.8 

0.8 

0.8 

0.6 

0.6 

0.6 

0.4 

0.4 

0.4 

0.2 

0.2 

0.2 

0 

1 1.2 1.4 1.6 1.8 2 

1 

0 

1 1.2 1.4 1.6 1.8 2 

0 

1 1.2 1.4 1.6 1.8 2 

(d) τ = 1/4 (e) τ = 3/4 (f) τ = 1/5 

1 

1 

0.8 

0.8 

0.8 

0.6 

0.6 

0.6 

0.4 

0.4 

0. 4 

0.2 

0.2 

0.2 

0 

1 1.2 1.4 1.6 1.8 2 

1 

0 

1 1.2 1.4 1.6 1.8 2 

0 

1 1.2 1.4 1.6 1.8 2 

(g) τ = 2/5 (h) τ = 3/5 (i) τ = 4/5 

1 

1 

0.8 

0.8 

0.8 

0.6 

0.6 

0.6 

0.4 

0.4 

0.4 

0.2 

0.2 

0.2 

0 

1 1.2 1.4 1.6 1.8 2 

0 

1 1.2 

1.4 1.6 1.8 2 

0 

1 1.2 1.4 1.6 1.8 2 

(j) τ = 1/8 (k) τ = 3/8 

(l) τ = 5/8 

η 

-1 V ξ 1 

(m) legend 

ξ


Fig. 2 – Modeled two-dimensional normalized transversal speed profiles for different 

values of the normalized time τ. 

1 

1 

1 

0.8 

0.8 

0.8 

0.6 

0.6 

0.6 

0.4 

0.4 

0.4 

0.2 

0.2 

0.2 

0 

1 1.2 1.4 1.6 1.8 2 

1 

0 

1 1.2 1.4 1.6 1.8 2 

0 

1 1.2 1.4 1.6 1.8 2 

(a) τ = 1/2 (b) τ = 1/3 (c) τ = 2/3 

1 

1 

0.8 

0.8 

0.8 

0.6 

0.6 

0.6 

0.4 

0.4 

0.4 

0.2 

0.2 

0.2 

0 

1 1.2 1.4 1.6 1.8 2 

1 

0 

1 1.2 1.4 1.6 1.8 2 

0 

1 1.2 1.4 1.6 1.8 2 

(d) τ = 1/4 (e) τ = 3/4 (f) τ = 1/5 

1 

1 

0.8 

0.8 

0.8 

0.6 

0.6 

0.6 

0.4 

0.4 

0.4 

0.2 

0.2 

0.2 

1 

0 

1 1.2 1.4 1.6 1.8 2 

0 

1 1.2 1.4 1.6 1.8 2 

0 

1 1.2 1.4 1.6 1.8 2 

(g) τ = 2/5 (h) τ = 3/5 (i) τ = 4/5 

1 

1 

0.8 

0.8 

0.8 

0.6 

0.6 

0.6 

0.4 

0.4 

0.4 

0.2 

0.2 

0.2 

0 

1 1.2 1.4 1.6 1.8 2 

0 

1 

1.2 1.4 1.6 1.8 2 

0 

1 1.2 1.4 1.6 1.8 2 

(j) τ = 1/8 (k) τ = 3/8 (l) τ = 5/8 

η 

-1 V η 1 

(m) legend 

ξ


Fig. 3 – Modeled two-dimensional normalized axial speed profiles for different values 

of the normalized time τ. 

It results: 

a) generation of multiple structures in plasma (Figs. 1a-l) corresponding 

to the multiple double layers like those described in the first part of this article; 

b) symmetry of the normalized speed field V ξ in respect to the symmetry 

axis of the spatio-temporal Gaussian (Figs. 2a-l); 

c) shock waves and vortices at the structures periphery for the normalized 

speed field V η (Figs. 3a-l). 

All above these induce intermittencies in the dynamics of the plasma 

discharge. Figs. 2a-l and 3a-l show the mechanism of the evolution to chaos 

through sub-harmonic bifurcations of the plasma dynamics, similar to that 

described in the first part of this article. 

3. Fractional Criterion of Evolution to Chaos 

The generation of the double layer implies the phase coherence of the 

plasma particles, i.e. S = const. According to fractal hydrodynamics described in 

the second part of this article, this means that V 0 . It results: 

i) at the macroscopic scale, the specific momentum transfer (for m 0 = 1) 

don’t exist; 

ii) the fractal fluid self-structures in electron-ion pairs; 

iii) the fractal fluid behaves as a quantum fluid (superfluid, 

superconductor, etc.); 

iv) we obtain for the law of momentum conservation and for the law of 

density conservation (see the second part of this article) the simple forms 

2 

d ρ 

 

dx 

2 

ρ 

0, 

t 

E 

2 2 

m0D 

 

24 

D 

dt 

F 

ρ 0, 

where E is the energy integration constant. The solution of the Eq. (16) is 

with 

k 

2 

 

 

0 

 

(16) 

ρ Asin kx , 

(17) 

E 2 4 

 

 

2 d D 

t 

F 

, 

(18) 

2m D 

0 

A and 0 being two integration constants. This solution induces the fractal speed


2 D 1 d 2 

d F E 

Ux 

D t lnρ 

 

dx 

m0 

 

cotankx 0 

 

(19) 

and the fractal potential 

2 

x 

mU 0 

D dU 

d 2 d 

F x 

DF 

Q m t 

0 E m t , k 

2 

D dx 

D 

1 2 

2 1 2 4 2 

2 

0 

const. (20) 

This means that a specific momentum transfer exists at the fractal scale, so that 

the fractal potential determines the energy of the charge carriers; 

v) to the current density 

 

 

jx x ρ xUx x jcsin 2 kx 0 

, 

j 

c 

1 

2 2 

A 2E 

 

 

2 m0 

 

the following current can be associated 

 

 

 

, 

I jc 

sin 

2 

kx0 

 

dxdy 

 

cos 2kx 

sin 2kx 

jc 

yc1x cos2 

 

 

0 sin 2 0 c2 

, 

2kx 

2kx 

 

 

 

(21) 

(22) 

where c 1 and c 2 are two integration constants. Particularly, for c 1 = 0 and c 2 = 

0, and with the notations 

j ξη 

2 = , 2kx = ξ , 2ky = η , I c = j c xy = 

0 

the Eq. (22) becomes 

The dependence , 

 

c 

4k 

2 

2 

, 

I 

i , (23) 

I 

cosξ 

sin ξ 

iφξ 

, cos 

sin . 

(24) 

ξ ξ 

i i ξ is shown in Fig. 4a. For = π/2 the Eq. (24) takes 

the form 

sin ξ 

i π 2, ξ , 

ξ 

(25) 

and this function is shown in Fig. 4b. 

2 2 2 2 

Let us suppose for k the expression k n k 0 induced through the 

generalized coherence ( in the present context, the physica l mean of the 

generalized coherence re fers to the generation of the multip le double layers). 

Then, the relation 

Qn 

2 

2 

Q0n 

, 

4 D 2 2 

0 0 2 0 d F 

Q E m t 

0 

D k , 

(26) 

c


expanded around n either in the form 

or in the form 

n 

2 

2 

0 2 0 0 

Q Q n Q n nn Q n n , (27) 

2 

2 D n n n n 

Qn 

Qn 

4 

m0 

dt 

F 1 

 

D , 

T 

T 

 

 

 

it induces the characteristic times 

T 

T 

 

 

2πm0D 

d 

 

nQ 

4πm0D 

d 

 

Q 

D F 

t 

0 

2 1 

D F 

t 

0 

2 1 

, 

. 

Qn 

2 

Q n (28) 

0 

(29) 

i 

0.2 

0 

-0.2 

15 

20 

-5 

10 

ξ 

i 

/ 2 

0.4 

0 

 

a) 

5 

0 

5 

0.3 

0.2 

0.1 

-0.1 

5 10 15 20 

ξ 

-0.2 

b)


Fig. 4 – a – Non-dimensional current dependence on the non-dimensional phase and 

coordinate ξ; b – non-dimensional current versus non-dimensional coordinate ξ, for 

φ π /2. 

Because T β is independent on n , the expression (29) defines a universal 

time scale. Through (29) and (26), a characteristic frequency can be associated 

f 

0 

1 k0 E0 

 

 

T 

2 

2m0 

 

1 2 

. (30) 

Let us evaluate the expression (30) with respect to the experimental results 

described in the first part of this article. Thus, by identifying L with a 

characteristic length of the double layer, namely the width of the double layer 

1/2 

(Doggett & Lunney, 2009; C harles, 2007), L = (2ε 0V/en 0 ) , the Eq. (30) takes 

the form 

and the fractal potential eigenstate current densities are 

f 

0 

1 2 

1 en0E0 

 

, (31) 

2 m0 

0V 

 

where E 0 is the ion energy, m 0 is the ion mass, n 0 is the ion density in the 

double layer and V is the voltage on the electrode. For the experimental 

conditions 

kT i 0.2 eV, kT e 2 eV, n e n i = n 0 10 9 cm -3 , m i = m 0 40 a.m.u. and V 

100 V, we obtain the width of the double layer L 3.3 mm and the disruption 

frequency f 0 150 kHz. These values are in good agreement with those existing 

in the literature (Charles, 2007; Hershkowitz 2005) as well as those 

experimentally obtained. 

It can be observed that, through the generalized coherence and by using a 

fractional revival formalism (a fractional revival of a physical function occurs 

when a physical function evolves in time to a state th at can be described as a 

collection of spatially distributed physic al sub-functions that each closely 

reproduces the initial physical function shape), the discrete fractal potential 

eigenvalues are 

Q 

n 

Q n 

0 

2 

 

 

(32) 

jn ( x ) A0 sin nk0x 

, (33) 

with A 0 being a constant amplitude. 

In this context, we write the current density at the moment t = 0 as 

 

J xt , 0 Ji 

x . (34)


We expand this current density using the fractal potential eigenstate basis 

with 

 

J x c j x , 

(35) 

i 

 

 

n1 

n n 

 

c j xJ xdx . 

(36) 

n n i 

 

By using the time scale T β , the time evolution in the fractal potential eigenbasis 

is found from a Schrödinger’s type equation (the charge transport takes place on 

fractal curves) 

to be 

2 

4 

2 

2 J 

2 1J 

2 

t 

D 

F 

DF 

t 

t 

D d i d 0, 

x 

D 

(37) 

[ ] n n 

2 

, exp 2 

 

J xt i tT 

n c j x . (38) 

n 

A function F(n) whose domain is restricted to the integers (n ) can be write 

as a finite sum of exponentials if and only if it is r periodic, that is, there is an 

integer r such that F(n) = F(n + r) for all n. Such a finite sum is called the finite 

Fourier series (Apostol, 1976). 

In our case, we identify F(n) = exp[-i2(t/T β )n 2 ]. The necessary and 

sufficient condition for this exponential to be a periodic function of the quantum 

number n is that the time ratio t/T β must be rational, and we write 

p 

Tp, q 

Tβ 

, (39) 

q 

for relatively prime integers p and q ( that is, p/q forms a simplified fraction). In 

terms of frequency, the Eq. (39) takes the form: 

1 q 

f pq , f0 

T p, 

q 

p 

, 1 

f 0 . (40) 

Now, through the fractal expressions 

2 

mV 0 

E 0 

 

 

2 D F 1 2 DF 

1 

p 

m D dt f0 m0D d t f p, 

q, (41) 

2 

q 

we can introduce the Reynolds’s fractional criterion 

VL c c p 

e pq 

, , 

ν 

q 

where we used the substitutions 

T 

(42)


1 

V c = V, Lc 

Vf p, 

q 

, 1 2 D 

d 

1 F 

D t . (43) 

2 

From (42) and (Chen, 1984) it results a critical value for the Reynolds 

number, e c , up to this value the fractal fluid flow becoming turbulent. Because 

from (40) it results sub-harmonics for Re c , according to (Dubois et al, 1983; 

Arecchi et al., 1982; Atipo et al., 2002) a criterion of evolution to chaos through 

cascade of spatio-temporal sub-harmonic bifurcations is stated. 

4. Conclusion 

1. By considering that the particles movements in a dc gas discharge 

plasma take place on fractal curves, a mathematical model was developed in 

order to describe the transition to chaos of the plasma system dynamics through 

cascade of sub-harmonic bifurcations. 

2. By using the fractional revival formalism, a Reynolds’s fractional 

criterion of evolution to chaos through cascade of spatio-temporal sub-harmonic 

bifurcations was stated. We also specified some types of plasma dynamics 

induced by the fractal potential (numerical simulation of the hydrodynamic 

behavior of an ionized gas). 

Acknowledgments. This wo rk was supported by Romanian National Authority 

for Scientific Research, CNCS-UEFISCDI, project number PN-II-ID-PCE-2011-3- 

0650. 

REFERENCES 

Agop M., Mazilu N., Fundamentals of Modern Physics. Junimea Publishing House, 

Iaşi, 1989. 

Apostol T.M., Introduction to Analytic Number Theory. Springer-Verlag, New York, 

1976, pp. 157-160. 

Arecchi F. T., Meucci R., Puccioni G., Tredicce J., Experimental Evidence of 

Subharmonic Bifurcations, Multistability, and Turbulence in a Q-Switched Gas 

Laser. Phys. Rev. Lett., 49, 17, 1217-1220, (1982). 

Aronstein D.L., Stroud C.R., Fractional Wave-Function Revivals in the Infinite Square 

Well. Phys. Rev. A, 55, 4526-4537, (1997). 

Atipo A., Bonhomme G., Pierre T., Ionization Waves: From Stability to Chaos and 

Turbulence. Eur. Phys. J. D, 19, 79-87, (2002). 

Charles C., A Review of Recent Laboratory Double Layer Experiments. Plasma Source 

Sci. Technol., 16, 4, R1-R25, (2007). 

Chen F.F., Introduction to Plasma Physics. 2 nd Ed., Plenum Press, New York, 1984. 

Cresson J., Non-differentiable Deformations of n 

. Int. J. Geom. Methods Mod. Phys., 

3, 7, 1395-1415, (2006). 

Doggett B., Lunney J.G., Langmuir Probe Characterization of Laser Ablation Plasmas. 

J. Appl. Phys., 105, 3, 033306 1-6, (2009). 

Dubois M., Rubio M.A., Berge P., Experimental Evidence of Intermittencies Associated 

with a Subharmonic Bifurcation. Phys. Re v. Lett., 51, 16, 1446-1449, (1983).


Goldsten R.J., Rutherford P.H., Introduction to Plasma Physics. IOP, Bristol, 1995. 

Hershkowitz N., Sheaths: More Complicated than You Think. Phys. Plasmas, 12, 5, 

055502 1-11, (2005). 

Landau L.D., Lifshitz E.M., Fluid Mechanics. 2 

nd Ed., Butterworth-Heinemann, Oxford, 

1987. 

Mandelbrot B.B., The Fractal Geometry of Nature. Freeman, San Francisco, 1983. 

Niculescu O., Dimitriu D.G., Păun V.P., Mătăsaru P.D., Scurtu D., Agop M., 


Phys. Plasmas, 17, 4, 042305 1-10, (2010). 

Nottale L., Fractal Space-Time and Microphysics: Towards a Theory of Scale 

Relativity. World Scientific, Singapore, 1993. 

Nottale L., Fractals and the Quantum Theory of Spacetime. Int. J. Mod. Phys. A, 4, 

5047-5117, (1989). 



Popa G., Sirghi L., Fundamentals of Plasma Physics. “Al. I. Cuza” University 

Publishing House, Iaşi, 2000. 

Stoler D., Equivalence Classes of Minimum Uncertainty Packets (I). Phys. Rev. D, 1, 

12, 3217-3219, (1970). 

Stoler D., Equivalence Classes of Minimum-Uncertainty Packets (II). Phys. Rev. D, 4, 

6, 1925-1926, (1971). 

Zienkievicz O.C., Taylor R.L., Zhu J.Z., The Finite Element Method – Its Basis and 

Fundamentals. Elsevier-Butterworth-Heinemann, Oxford, 2005. 

TRANZIŢII SPRE HAOS PRIN BIFURCAŢII SUBARMONICE ÎN PLASMĂ 

III. Model teoretic 

(Rezumat) 

Se propune un model theoretic care explică tranziţia spre haos prin bifurcaţii 

subarmonice în plasmă. Rezultatele teoretice sunt validate de datele experimentale.


Publicat de 

Universitatea Tehnică „Gheorghe Asachi“ din Iaşi, 


Secţia 


CONTRIBUTIONS TO DEVELOPMENT OF A NEW MODEL 

IN ELECTROMAGNETIC FERROFLUID COMMAND 

BY 

ADRIAN OLARU and DORU CĂLĂRAŞU 

“Gheorghe. Asachi” Technical University of Iaşi, 

Department of Fluid Mechanics 


Accepted for publication: September 12, 2012 

Abstract. The conceptual model of a servo-element with electromagnetic 

ferrofluid command is based on the rheological behavior of magnetically 

controllable fluids. 

The magnetorheological fluid type MRHCCS4-B, discussed about in this 

paper, is able to provide high shear stress at small applied magnetic fields. This 

class of magnetorheological fluid leads to major changes in fluid rheology when 

a relatively modest external magnetic field is applied. The producing company 

provides directions for its applications where high shear stresses are required. 

The conceptual model suggests a new application for a magnetically 

controllable fluid, namely the control of linear movement of an electromagnetic 

ferrofluid element, by varying the external applied magnetic field. 

The conceptual model uses a sealed hydraulic system to prevent fluid loss. 

Key words: servo-element, magneto-rheological fluid, electromagnetic 

ferrofluid command. 


From the point of view of workability and construction, the servo-valves 

have been constantly improved in what regards cost reductions and productivity 

increase. 

This paper deals with a new application of a magnetically controllable 

fluid, namely the linear movement control of an electromagnetic ferrofluid 

Corresponding author: e-mail: dorucalarasu@yahoo.com

72 Adrian Olaru and Doru Călăraşu 

element, by varying the applied external magnetic field. Such control could 

replace either the electromechanical converter (torque motor) and flapper nozzle 

amplifier of servo valves, or the control with proportional electromagnet for 

proportional servo-elements. 

2. The Conceptual Model of a Servo-element with Electromagnetic 

Ferrofluid Command 

The conceptual model of a servo-element with electromagnetic ferrofluid 

command is based on the rheological behavior of magnetically controllable 

fluids. 

The thixotropic magnetorheological fluid type MRHCCS4-B, produced 

by Liquids Research Limited (www.liquidsresearch.com ) is able to provide 

high shear stress at applied small magnetic fields. 

This class of magnetorheological fluid leads to major changes in fluid 

rheology when a relatively modest external magnetic field is applied. 

The company mentioned above provides directions for its applications, 

especially where high shear stresses are required, i.e. the vehicle suspension 

systems, suspension seats and exercise equipment. 

Fig. 1 shows the functional scheme of the electromagnetic ferrofluid 

control for controlling linear movement. 

The sealed hydraulic system consisting of: S1 (2) – tubular hydraulic 

resistance RH (4) with diameter d; S2 (5), is filled with magnetorheological 

fluid type MRHCCS4-B produced by Liquids Research Limited. 

The value of the hydraulic resistance RH can be changed by varying the 

external magnetic field of an intensity H. Changing the H intensity of the field 

is obtained by energizing the coil (3) at different voltages. This changes the 

magnetic induction B and, subsequently, the magnetic fluid viscosity η. 

7 6 5 4 3 

U 

2 1 

= 

F2 

Fluid 

MRHCCS4-B 

Q 

Fluid 

MRHCCS4-B 

= 

F1 

F1 

Fig. 1 – The basic scheme of the conceptual model. 

The size of the input pressure p 1 for the hydraulic resistance RH is given 

by the size of the applied force on the bellows S 1 minus its elastic force 

(corresponding to the axial elastic deformation) and friction force. Pressure p 2 is 

given by the resistance force of the load element ES (return spring (6) and 

displacement measuring device spring, namely the comparator (7)). The 

pressure drop Δp=p 1 –p 2 , the speed of the bellows compression S1, v s , (equal to


axial deformation speed of bellows S2) and the magnetic fluid speed through 

hydraulic resistance RH, v RH , become functions of the intensity of the applied 

magnetic field. 

3. The Theoretical Model for the Electromagnetic Ferrofluid Command 

The pressure force which gives pressure p 1 in the bellows chamber S1 is 

obtained with the equation p 1 =(F 1 -F fm -F e.s )/S s . Force F 1 is the external force 

applied to the bellows S1. The Force F fm is the friction in bearings, and F e.s is 

the elastic force (corresponding to the bellows S1). 

On the load circuit, the force F 2 , which creates backpressure p 2 in 

chamber of the bellows S2 is given by the total elastic forces of the return spring 

(6), the comparator (7) and bellows S2 (5). The bellows S1 and S2 are identical. 

The elastic constant of the three series connected springs (k s - bellows S2, 

k s3 - return spring and k s4 - spring comparator) is calculated with the equation 

1 

n 

1 

(1) 

k k 

s 

On the hydraulic resistance RH, the magnetic fluid flow occurs under the 

pressure difference Δp=p 1 –p 2 . The factors being defined above, the pressures 

p 1 , p 2 can be calculated using the equation 

s 

i1 

i 

x 

F1 Ffm 

F 

F 

e. 

s 

1 Ffm 

ks 

p 

2 

1 , 

(2) 

S 

S 

where: F 1 – external force applied to bellows S1; F fm – friction in bearings; F e.s 

– elastic force corresponding to bellows S1; S s – surface of bellows S1; k s – 

elastic constant of bellows; x – displacement; 

p 

2 

s 

x 

F sac . . 

es . Fea . F 

k 

ec . 

2 , 

(3) 

S 

S 

where: F e.a – elastic force of return spring (6); F e.c – elastic force of comparator 

(7); k s.a.c – elastic constant of bellows, return spring and comparator. Therefore 

s 

s 

x x 

F1 Ffm ks ks. 

a. 

c 

p 

2 2. 

(4) 

S S 

To determine the motion parameters is necessary to know the 

characteristics of the magnetorheological fluid. For the magnetorheological 

fluid MRHCCS4-B, the producing company specifies the physical 

characteristics and the characteristics of the variation of induction B(H) and 

s 

s


shear stress τ by shear rate γ at different temperatures (www.liquidsresearch. 

com ). 

The flow regime through the tubular hydraulic resistance RH used, 

having a d diameter, is determined by the value of Reynolds number 

vd 

e 

, (5) 

υ 

where v – the speed of the fluid flow, υ - kinematic viscosity. 

To calculate the average velocity of flow through the tubular hydraulic 

resistance RH, having known the values of the pressure drop, geometric 

parameters and characteristics of magnetorheological fluid, the Hagen- 

Poiseuille equation can be used for the laminar flow regime (Re


The average speed of movement is determined by calculating the 

maximum speed obtained for the initial moment, i.e. displacement x = 0 and the 

minimum speed calculated for x = x max . 

Maximum speed is obtained by using force F 1 for x = 0 

v 

max 

2 

( F1 

Ffm) 

d 

. 

(9) 

32 lS ( B) 

Minimum speed is obtained when the sum of forces is 0, or Δp=0 

v 

min 

s 

( F1 Ffm ksxks. 

a. 

cx) 

d 

. (10) 

32 lS η( B) 

Average speed is determined as semi-sum between maximum and 

minimum speed 

v 

med 

 

s 

 

vmax 

v [2 F1 F ( 

min 

fm x ks ks. a. 

c)] 

d 

. (11) 

2 64lS η B 

s 

 

2 

 

2 

4. Conclusion 

1. The conceptual model suggests a new application for a magnetically 

controllable fluid, namely the control of linear movement of an electromagnetic 

ferrofluid element, by varying the external applied magnetic field. 

2. The magnetorheological fluid type MRHCCS4-B is able to provide 

high shear stress at applied small magnetic fields. 

3. This class of magnetorheological fluid leads to major changes in fluid 

rheology when a relatively modest external magnetic field is applied. 

4. The value of the hydraulic resistance can be changed by varying the 

external magnetic field of intensity H. 

5. Changing the H intensity of the field is obtained by energizing the coil 

at different voltages. This changes the magnetic induction B and, subsequently, 

the magnetic fluid viscosity η. 

REFERENCES 

Călugăru Gh., Cotae C., Lichide magnetice. Ed. Ştiinţifică şi Enciclopedică, Bucureşti, 

1978. 

*** www.liquidsresearch.com


CONTRIBUŢII PRIVIND REALIZAREA UNUI NOU MODEL DE COMANDĂ 

ELECTROFEROFLUIDICĂ 

(Rezumat) 

Lucrarea propune o nouă aplicaţie privind un fluid controlabil magnetic, 

respectiv controlul deplasării liniare a unui element electroferofluidic, prin variaţia 

câmpului magnetic exterior aplicat, care să permită controlul poziţiei sertarului de 

urmărire al unui servoelement hidraulic. Controlul deschiderii distribuitorului permite 

un control al debitului şi, respectiv, controlul vitezei unghiulare a unui motor hidraulic 

rotativ. 

O astfel de comandă ar putea înlocui motorul electric de cuplu şi etajul de 

amplificare de tip ajutaj – paletă al unei servovalve, sau comanda cu electromagnet 

proporţional la servoelementele proporţionale. 

Modelul conceptual de servoelement cu comandă electroferofluidică se bazează 

pe comportarea reologică a fluidelor controlabile magnetic. Se utilizează un sistem 

hidraulic etanş, care să nu permită pierderi de fluid. 

Comportarea reologică a fluidului controlabil magnetic (magnetoreologic) 

depinde de stimuli externi, respectiv de temperatură şi câmp magnetic aplicat, cât şi de 

structura fizică a elementelor prin care are loc curgerea.


Publicat de 



Secţia 


GOAL SETTING AND GOAL ATTAINMENT IN THE MODELS 

OF LIFE CYCLE OF THE DEVELOPMENT STRATEGIES OF 

AUTOMOBILE TRANSPORT MANUFACTURING SYSTEMS 

BY 

VICTOR BILICHENKO 

Vinnitsya National Technical University, 

Ukraine 

Received: November 15, 2012 


Abstract. The paper suggests the determination of the terms “goal setting” 

and “goal attainment”, there had been considered the content of the process of 

goal setting and goal attainment in the projects of the life cycle of the strategies 

for the development of the automobile transport enterprises, developed the 

structure of the model system of control over the strategic development of these 

enterprises. 

Key words: goal setting, goal attainment, strategic development, life cycle, 

automobile transport enterprise, project, strategy. 

1. Urgency of the Issue 

Problems, relating to the strategic development of the automobile 

transport enterprises, became visual in the Ukraine beginning with the second 

half of 90-th. The general educational processes, in particular, the tendencies to 

the globalization and corporation of the world economy acted as the external 

influencing factors. The internal motives appeared in the result of mass 

privatization. The native economy entered the stage, when the absence of the 

scientifically substantiated development strategy becomes a real obstacle on the 

way to the successful operation of an enterprise. 

e-mail: bilichenko_v@mail.ru

78 Victor Bilichenko 

Today more and more attention is paid to the research of the issue of the 

development of automobile transport enterprises. This is dew to the fact that 

the issue of maintenance and development of transport directly influences the 

national economy, external policy, social stability, scientific and technical 

progress as well as allows to strengthen the national interests of the country. 

The urgency of the issue of the strategic development of the transport 

enterprises increases under the crisis conditions since the development of the 

economy of the country depends upon the results of the economic activities of 

each separate enterprise. 

Automobile transport enterprises, as well as any system of other origin 

independent of the form of property, sphere and range of activity, is 

subordinated to the life-sustaining activity laws. The possibilities to modify, to 

transfer to the higher stages of the development, or, vice-versa, to face crisis, 

requires the enterprise to change the goals, strategies and means for their 

realization. Learning and taking into account theoretical, practical processes of 

cycle development of both, manufacturing systems and strategies for their 

development, enable to stipulate for the state of the manufacturing system in 

future and for the substantiated decision making in management. 

The issues concerning the management in the development of the 

enterprise had been considered in the scientific works of the famous native and 

foreign scholars and experts in economy. It should be noted that the significant 

contribution to the solution of the above issue was made by such scholars as D. 

Bell, O.O. Bogdanov, N. Viner, V.M. Geets, V.M. Gryniova, O.A. Yerokhina, 

Dz. Clark, M.D. Kondratiev, Yu. G. Lysenlo, I.R. Prygozhyn and other. The 

issues of the essence and model mechanisms of the life cycle of the strategies 

of the enterprise development had been researched by such scholars as І. 

Adises, S. Bushuev, L.Greiner, O. Kuzmin, N. Stepanenko, О. Melnyk, Zh. 

Lippit, І. Mazyr, N. Olderrogge, V. Shapiro, G. Kozachenko, G. Atamanchuk, 

N. Nizhnik, V. Tsvetkova and other. 

2. The Unsolved Part of the General Issue 

Acknowledging the scientific and practical value of the woks of the 

above authors, it is necessary to emphasize, that some issues of conceptual, 

methodological and methodical character required further researches. Thus, the 

issues of goal setting and goal attainment in the projects of life cycle of the 

development strategies for the automobile transport enterprises need to be 

further researched. 

3. Task Setting 

The control over the strategic development of the automobile transport 

enterprises in the conditions of changing environment is the urgent problem in 

Ukraine, considering the current stage of the development of market economic


relations. The objective of the paper is the system analysis of the goal setting 

and goal attainment for building the model system of the life cycle of the 

organization development strategy. 

4. Solution to this Issue 

Goal setting and goal attainment is the integral component of the 

functions of strategic development control. The uniting properties of goal 

setting are realized in the system of strategic planning and provide for the 

connection of the mission, vision of the strategic goals of the enterprise with the 

goals of the incorporated subsystems, which operate in the subsystems of 

business processes. Consideration of the functions, the goal setting and goal 

attainment in the models of life cycle of the development strategies, it is 

expedient to specify the essence the terms “goal setting” and “goal attainment”. 

In general, goal setting is a practical understanding of the his or her 

activity from the point of view of the formation “setting” goals and their 

realization “attainment” by most efficient means. Goal setting in strategic 

control over the development is a process of goals formation for the enterprise. 

The result of the goal setting process is the unique determination of the goals of 

development and their understanding by managers. 

Goal attainment is the mobilization of resources, energy and means 

to attain the goal. In accordance with (Bazarov, 2002) for the substantive 

formation and task setting, goal setting and goal attainment of the life cycle of 

the development strategies by its decomposition, it may be divided into the 

small life cycles of formation, realization and strategy control. 

In such a case the project of the organization development strategy, on 

the base if the principle regulations of system analysis, may be described by the 

model 

Pe P( Pf , Pr, P c ), 

(1) 

where Pf , Pr, 

P c are corresponding projects models (subprojects) of formation, 

realization and strategy control. 

The structure of the life cycle of the development strategy of the 

organization as a project is presented on Fig. 1. 

The building process of the model system stipulates for two stages: goal 

setting and goal attainment 

On the goal setting stage there has to be formulated the system goal of the 

strategic development of the automobile transport enterprise, which generally 

includes the multitude of local goals, which ensure the attainment of the system 

or global goal of the enterprise, namely 

n n n n 

G G g : g G , n1, 2, ..., N; i1, 2, ..., I; j 1, 2, ..., J; , (2) 

 

s ij ij ij ij


where 

n 

G ij 

– the multitude of local goals of the automobile transport enterprise, 

which must be realized in the projects of the n-th stage of the life cycle of the 

strategy as for the і-th type of the activity on the j-th time interval, which ensure 

the attainment of the system (global) goal of the above enterprise. 

Fig. 1 – Structure of the life cycle of the development of the economic enterprise as the 

project (Х – input parameters, У – environmental influence upon the organization, R – 

result of the activity of the economic enterprise). 

The well formed strategy must answer the SMART principles (S – 

specific, significant, stretching, M – measurable, motivational, manageable, A – 

attainable, achievable, acceptable, ambitious, action-oriented, agreed upon, R – 

realistic, relevant, reasonable, rewarding, result-oriented and T– timely, timebound. 

The analysis of the conditions of development and functioning of the 

automobile transport enterprise allows to determine the conditions of activity 

efficiency of transportation, ensuring the working capacity as well as 

expeditionary servicing as the types (kinds) of the local goals. 

Thus, the target level for the stage of the formation of the strategy 

is possible to write as follows 

1 

G 1 j 

1 1 1 1 1 

ij ij ( 1j , 2j , 3j 

1 

G ij 

G G G G G ), (3) 

where – the multitude of the local goals in the projects of the stage of the 

strategy formation as for the activity concerning the transportation on the j-th 

it

time interval; 

1 

G 2 j 


– the multitude of the local goals in the projects of the stage 

of the strategy formation as for the activity concerning the provision of the 

working efficiency of technical vehicles on the j-th time interval; 

– the 

multitude of the local goals in the projects of the stage of the strategy formation 

as for the activity concerning the expeditionary servicing on the j-th time 

interval. Correspondingly, it is possible to write the target level for the stage of 

2 

G ij 

the realization of the strategy : 

2 2 2 2 2 

ij ij ( 1j , 2j , 3j 

1 

G 3 j 

G G G G G ), (4) 

and the stage of the control and correction of the strategy, 

3 3 3 3 3 

G G ij ( G , G , G ). (5) 

ij 1j 2j 3j 

On the base of the model of goal setting there will be built the model of 

goal attainment as for the following algorithm, which stipulates for the 

determination of: 

n 

1) multitudes of functions, F ij , which must be realized in the projects of 

the n-th stage of the life cycle of the strategy as for the і-th type of activity on 

n 

the j-th time interval for the attainment of the set : 

 

n n n n n 

ij ij ij ij ij 

G F f : f F , n1, 2, ..., N; i1, 2, ..., I; j 1, 2, ..., J ; (6) 

n 

2) multitudes of the tasks, O ij , which must be solved in the projects of 

the n-th stage of the of the life cycle of the strategy as for the і-th type of 

n 

activity on the j-th time interval for the realization of the set F ij , 

 

n n n n n 

Fij Oij Oij : Oij Oij , n1, 

2, ..., N; i1, 2, ..., I; j 1, 2, ..., J ; (7) 

n 

3) multitudes of methods and models, M ij , which must be used in the 

projects of the n-th stage as for the і-th type of activity on the j-th time interval 

for the set 

n 

O ij 

 

n n n n n 


O M m : m M , n1, 2, ..., N; i1, 2, ..., I; j 1, 2, ..., J ; (8) 

4) multitudes of the algorithms, A ij , which must be used in the projects 

of the n-th stage of the life cycle of the strategy as for the і-th type of activity on 

n 

the j-th time interval for the solution of the set : 

n 

O ij 

G ij


 

n n n n n 


M A a : a A , n1, 2, ..., N; i1, 2, ..., I; j 1, 2, ..., J ; 

 

(9) 

Project oriented strategic control over the automobile 

transport enterprises 

System goal - C 

Controlling goals RS of ATF– 

n 

Controling goals TS – 

n 

G 2 

G 1 

Controling goals ES – 

n 

G 3 

Controlling function RS of ATF– 

n 

F 2 

Controlling function TS – 

n 

F 1 

Controlling function ES – 

n 

F 3 

Controlling tasks RS of ATF– n 

O 2 

Controlling tasks TS – n 

O 1 

Controlling tasks ES – n 

O 3 

Methods and models the solution of 

the controlling tasks 

n 

RS of ATF – M 2 

Algorithm for the solution of the 

controlling tasks 

n 

RS of ATF – A 2 

Soft and hardware means for the 

solution of the controlling tasks 

RS of ATF – n 

P 2 

Methods and models the 

Methods and models the 

solution of the controlling 

solution of the controlling 

n 

tasks RS of ATF – M 1 

tasks RS of ATF – n 

M 3 

Algorithm for the solution of Algorithm for the solution of 





A 1 

A 3 

Soft and hardware means for Soft and hardware means for 

the solution of the controlling the solution of the controlling 

n 

n 

tasks RS of ATF – P 1 

tasks RS of ATF – P 3 

Structural developments of the 

n 

control realization RS of ATF – S 2 

Structural developments of the 

control realization 

n 

RS of ATF – S 1 

Structural developments of 

the control realization 

n 

RS of ATF – S 3 

Result of the solution of the 

n 

controlling tasks RS of ATF – R 2 

Result of the solution of the Result of the solution of the 



n 

RS of ATF – R 1 


R 3 

Fig. 2 – Structure of the control model of the transportation system (TS), ensuring the 

working capacity automobile transport facility (ATF) (repair system (RS) of ATF), 

expeditionary services (ES) in the projects of the life circle of the strategy life cycle of 

the development strategy of the automobile transport enterprise. 

5) multitudes of soft- and hardware means, P ij , which must be used in 

the projects of the n-th stage of the life cycle of the strategy as for the і-th type 

n 

of activity for the solution of the set : 

 

n n n n n 


A ij 

A P P : P P , n1, 2, ..., N; i1, 2, ..., I; j 1, 2, ..., J ; (10) 

6) multitudes of structural formations, , which are being realized in 

the projects of the n-th stage of the life cycle of the strategy as for the і-th type 

of activity of the set 

n 

O ij 

n 

S ij 

n


 

n n n n n 

Pij Sij Sij : Sij Sij 

, n1, 2, ..., N; i 1, 2, ..., I; j 1, 2, ..., J ; (11) 

7) multitudes of the results, 

n 

R ij , the solutions in the projects of the n-th 

stage of the life cycle of the strategy as for the і-th type of activity of the set 

 

n n n n n 


S R r : r R , n1, 2, ..., N; i1, 2, ..., I; j 1, 2, ..., J . (12) 

In accordance with the above, the structure of the model system of 

controlling over the strategic development of an automobile transport enterprise 

may be presented as is shown on Fig. 2. 


The structure of the system model for controlling over the strategic 

development, suggested on the base of the system analysis of the processes of 

goal setting and goal attainment in the projects of life cycle in the strategies of 

development the automobile transport enterprises, allows to coordinate the short 

term interests with the goals of attainment of the long term stable advantages on 

the market, which will provide the enterprise with the relative independence on 

the market state in the period of temporary worsening of the market conditions 

and to keep the potential possibilities on the high level. 

REFERENCES 

Bazarov Т.Yu., Staff Management: Textbook. Masterstvo, 2002. 

Blank I.А., Principles of Financial Management. Nika-Tsentr, 1999. 

Radionova N.V., Anti-recessionary Management. Textbook for colleges. M. YuNITI- 

DАNА, 2001. 

Lange О. Introduction to the economic cybernetics. Progress, 1988. 

FORMULAREA SCOPULUI ŞI REALIZAREA SCOPULUI 

ÎN MODELELE PRIVIND CICLUL DE VIAŢĂ PENTRU 

STRATEGIILE DE DEZVOLTARE A ÎNTREPRINDERILOR 

DE TRANSPORT AUTO 

(Rezumat) 

Se studiază structura modelului pentru controlarea strategiei de dezvoltare a 

întreprinderilor de transport auto, care să permită coordonarea intereselor acestora pe 

piaţă pe termen scurt, cu realizarea obiectivelor pe termen lung, astfel încât 

întreprinderile să obţină o independenţă relativă în perioadele de înrăutăţire temporară a 

stării pieţei. 

 

 

n 

O ij


Publicat de 



Secţia 


PROJECTS OF PRODUCTION-TECHNICAL BASE 

DEVELOPMENT OF A MOTOR TRANSPORT ENTERPRISE 

BY 

VICTOR BILICHENKO and SVITLANA ROMANTUK 



Vinnitsya National Technical University, 

Ukraine 

Abstract: The product and the results of development projects of an 

enterprise are examined. The main directions of development of industrialtechnical 

base of a motor transport enterprise are defined. 

Keywords: development project, production and technical base, motor 

transport enterprises. 


The successful development of motor transport enterprises in many 

respects depends on the perception of the adequacy and speed of response to 

changes in internal and external environment. Currently the project is 

considered as the most effective form of implementation of the targeted changes 

at the enterprise level. 

Change management when creating the project of development of 

production system of the motor transport enterprise (MTE) is a purposeful 

influence presented in the planning, organization and control of implementation 

of actions, aimed at creating or refining a project of development of production 

system with account of changes in the external environment and the internal 

environment of the MTE. It is obvious, that the product of the project can be 

adjusted for all phases of the life cycle of the project of development of 

industrial systems, including phase of the operation or failure of the operation of 

the MTE and the elimination of an MTE as an existing business or at the 

Corresponding author: e-mail: bilichenko_v@mail.ru

86 Victor Bilichenko and Svitlana Romantuk 

liquidation value. Development projects for production-technical base, as the 

main material-technical component of the passive and active assets of the 

enterprise is a priority direction of development of the motor transport 

enterprise (ATP). 

2. The Main Part 

Life cycle of a project, as it is known, can be viewed as a set of logically 

related activities, in the process of completion of which one of the main results 

of the project is achieved. In this case, as has been observed in many studies, 

under conditions of the life cycle of the project it is natural to start with the life 

cycle of an object, which is the product of the project (a house, an information 

system, equipment and etc.) (Tsipes & Torb, 2009, p. 205). 

Life cycles of the project are specific not only in respect of the area in 

which the project management is implemented (construction, pharmaceuticals, 

intellectual technologies, etc.), but in relation to the individual organizations. In 

practice the formation of the so-called corporate standard of the project takes 

place Thus, as noted by W. Duncan, in the U.S. many companies consider the 

life cycle of their projects «practically the object of religious worship», which is 

not a subject of review or criticism (Grashina & Duncan, 2006, p. 24). 

Let us consider the conditions of interaction of the life-cycle of a project 

development facility and the life cycle of the actual object, which is the product 

of the project, at an example of projects of construction. 

So, according to P. W. Morris, a typical life cycle of the construction 

project consists of four phases: feasibility study, planning and designing of 

production, and also reception and commissioning (*** 1 , 2000, p.15). 

However, as noted in the work (Tsipes & Torb, 2009, p. 206), the life 

cycle of the object (building) is not limited by the given phases. Proceeding 

from this, a number of contemporary approaches to the management of 

construction project proposes a significantly broader view of the life cycle of a 

construction project, including the last phase of strategic development, as well 

as the following the phase of putting into operation the phases " like operation 

proper, reconstruction, liquidation. 

In work (*** 2 , 2003, p. 91) one of the main differences of the 

construction projects is indicated: «Creation of the project of construction is 

never the end result, after which any of the results of the project do not remain». 

The next step in the logical chain of reasoning is the necessity to reject the 

vision of the project as an activity, which is aimed at the achievement of a 

single goal. In contrast to the «traditional» views to a single project the author 

proposes to include not only the creation of the object, but also its further 

development in the process of exploitation. 

In the project, which is viewed as an evolving, not all of the ultimate 

objectives are defined in advance, their appearance is often determined by 

external circumstances, which may result in re-profiling and / or redevelopment


of the building. And such a project is accomplished only together with the 

completion of the life cycle of an object (Tsipes & Torb, 2009, p. 206). 

This idea finds confirmation in real modern practice of realization of the 

investment-construction activity, which is based on the concept of development, 

when the goal is not just to create the object, but to create the object, which will 

bring big profit and for as long as possible. And if this is so, then the traditional 

construction project is only a particular case of the project development. At the 

same time, projects development, in addition to the above-mentioned phases of 

construction, also include the operational phase and the elimination one (Tsipes 

& Torb, 2009). 

This approach has been implemented in the work (Scharova, 2011, p. 5) 

in which, in particular, the author refers to the difference between the product 

and the result of investment and construction project. As the product of this 

project the author understands the material embodiment of the concept and the 

design and estimate documentation by means of use of the investment funds 

that are invested in the property, and the result of the project is the possibility of 

the technical operation of the latter (Scharova, 2011, p. 5). 

In its turn, the product of a development project is the considered as the 

actual use of the result of the investment-construction project according to the 

development concept. The result of the project the author understands the 

satisfaction from receiving the product of the project of development and 

obtaining the planned project (commercial, economic, financial, social) on the 

stage of operation (Scharova, 2011, p. 5). 

On the basis of the above mentioned considerations the project of 

development of the system of provision of services for the technical preparation 

of vehicles (TPV) of an MTE is the development of the production system of 

the MTE and / or supply of the services from the outside on the principles of 

outsourcing, which is intended, in accordance with the chosen strategy of 

development of MTE to ensure, on a given level of the transport process 

productive exchanges at minimal costs. In this case we proceed from the fact 

that the production system of the MTE includes production-technical base, 

together with repair and service personnel and engineering and technical staff, 

as well as with the elements of the technical organization and production 

management. 

At the same time, production and technical base of MTE is formed by the 

funds, which are intended for technical support of the process of maintaining 

and restoring the workability of the TPV, as well as the maintenance of 

buildings, constructions, communications and other objects in proper condition. 

The structure of the funds, which form production and technical base, can also 

be represented as such, which consist of passive (buildings, constructions) and 

active (technical equipment, tools, appliances) parts. Project of development 

(updating) of the production and technical base of the MTE can be classified 

according to the existing classification of processes of reproduction of the basic


funds and directions of the investment. According to the latest classification 

such processes are (Kanartchiuk & Kurnikov, 1997, pp. 163-164): technical reequipment, 

reconstruction, expansion, new construction. 

Technical reequipment is a renewal of the active part of the production 

assets on the basis of: the introduction of new technology (technical equipment, 

fixtures, equipment for technical service and repair of TPV) and techniques; 

increase of the level of mechanization and automation of processes of technical 

service and repair of TPV; modernization of the existing equipment; 

improvement of production and labour organization methods. 

The peculiarity of technical re-equipment is updating means of labour 

without increasing the production area of the enterprise and compulsory 

reduction of number of workers. In the process of technical re-equipment there 

is a need for partial reconstruction of the production, household and warehouse 

premises, providing or liquidation of communications, improvement of energy 

supply. However, the passive part of fixed assets should not exceed 10…15 %. 

The main indicators of the technological modernisation of the MTE are 

summarizing technical-and-economic indices, which characterize the ultimate 

goal and the results of technical re-equipment; measures of technical reequipment; 

the need for material and technical resources and equipment; 

construction-assembly works; the value of the investment. 

Depending on the forms of updating means of labour small, medium and 

complex technical re-equipment.are distinguished. 

Small technical re-equipment provides the replacement of a small part of 

morally obsolete equipment, as well as modernization and improvement of 

existing instruments of labour. 

For small technical re-equipment of the coefficient of renewal of fixed 

capital (K), as a rule, exceeds the disposal of (K 2 ), that is K>K 2 , and their values 

oscillation within the following limits: 0.1≤K 0 ≤0.3 and 0.1≤K 2 K 2 . Their 

values lie in the range of 0.3≤K o ≤0.5; 0.2≤K 2 ≤ 0.4. 

The complex technical re-equipment, respectively, is characterized by a 

significant updating of the equipment park, increase of mechanization level and 

automation of production processes, introduction of the latest technologies. In 

this case 0.3≤K o ≤0.5, and 0.4≤K 2


structures the restructuring and conversion of areas, shops and sites on a new 

technical basis is carried out. The decision is made as to mechanisation and 

automation of industrial processes, replacement of morally and physically 

obsolete equipment, introduction of the newest technologies, growth of 

production space and installation of auxiliary equipment. Reconstruction means 

defining the scope (quantity) of MTE and the level of concentration of TPV. 

The need for the reconstruction due to the changes taking place in the 

structure of TPV parks , their design and the terms of their operation, the 

requirements to the quality of transport service and technical maintanence, the 

levels of consumption and saving fuel and energy resources, policies for the 

protection of the environment, etc. 

Reconstruction is connected with such objective economic laws, as 

dominating growth of the active production funds and labor productivity, 

reduction the share of living labour and the increase of labour share in the 

process of production intensification. 

Depending on the volume of works in respect of the existing production 

assets the following types of reconstruction are distinguished: 

1. Small (partial), aimed as a rule, at replacement of morally and 

physically obsolete active fixed assets, i.e. K 2 =K o , and numerical values of 

these indicators correspond to the following conditions: 0.1≤K o ≤0.2 and 

0.1≤K 2 ≤0.2. It envisages the implementation of insignificant volume of 

construction works, connected with re-planning of shops, offices and 

installation of new technological equipment. 

2. Middle, which has a purpose, as a rule, of replacement of active and 

passive elements of the basic production assets, complex mechanization and 

automation of production. In this case >K 2 , the numeric value is within the 

following limits: 0,21≤ Ko≤ 0.4 and 0.21≤ K2≤ 0.3. 

3. The complex, which has a purpose, as a rule, of a radical renewal of 

fixed assets, based on introduction of the newest scientific and technical 

achievements. In this case K 0 >K 2 , and numerical values are in the following 

ranges: 0.31≤K 2 ≤0.5 and 0.41≤K o ≤ 0.6. 

Reconstruction and technical re-equipment are aimed at the increase of 

production capacities, increase of labour productivity of maintenance workers, 

as well as the improvement of the values of other technical and economic 

indicators. So with the concept of «reconstruction» the concept of «technical reequipment 

of the existing MTE is inseparably linked. 

Extension presupposes the construction of separate shops, premises, 

production units, communications and other facilities on the territory of the 

existing MTE. 

New construction means the erection of MTE buildings, constructions, 

technical equipment, TPV parks, gas stations, communication, etc. on new sites. 

New construction implies the unity of the processes of creation of active 

and passive parts of the main funds of MTE according to the project, in which


the volume of the works of technical maintenance and repair and technical level 

of the production and technical base are balanced. 

The modern practice concerning the development of the productiontechnical 

base, gives grounds to consider the reconstruction to be the most 

widespread and generalized form of realization of scientific-technological 

process at the MTE. During this reconstruction could cover not only the 

technical re-equipment of production-technical base and its expansion. 

Reconstruction provides the transition from individual technical maintenance 

and repairs in the framework of the closed technological cycle of an individual 

MTE to the development of specialized production and co-operative forms of 

relations between production units and the creation of industrial technology of 

technical service and repair of TPV. 

Thus, one of the main directions of development of the production system 

of the MTE is realization of projects of updating the production and technical 

base of objects of the fixed production assets), the main types of which are 

projects (programmes) of the technical re-equipment, reconstruction, expansion 

and new construction. 

In this case the project of updating of the production and technical base of 

the MTE itself can be considered as the investment activities following in the 

implementation the technological sequence of works to create within the 

established deadlines and budgetary constraints an updated object of the fixed 

production assets of MTE, the availability and use of which are necessary for 

the effective implementation of the strategic objectives of the development of 

an MTE. 

3. Conclusion 

1. The material embodiment of the concept and the design and estimate 

documentation of an updated object of the main production assets is considered 

as the product of a project of updating the production and technical base of a 

motor transport enterprise. 

2. The result of project aimed at the upgrade of the production and 

technical base of a motor transport enterprise is the possibility of the technical 

operation of the product of the aforementioned project. 

REFERENCES 

Tsipes G.P., Torb A.S., Proekti i upravlenie proektami v sovremennoi kompannii. 2AO, 

Plimp-Biznes, Moskva, 2009. 

Grashina M., Duncan W., Osnovi uravnenia proktami. SPb., Pitersburg, 2006. 

*** Kerivnitstvv z pitani proektnogo menedjimenttu (trans. from engl.) (Buscheva S.D., 

Ed.), Vidavnichnii dim. “Delovaia Ukraina”, Kiev, 2000. 

*** Construction Extension to A Guide to the Project Management Body of Knowledge. 

PMBOK Guide 2000 Ed.,– Pennsylvania, 2003.


Zabvodin Iu., Kotkov V., Saruhanov A., Upravlenie neftegazostroitelinimi proektami. 

Ekonomika, Moskva, 2004. 

Scharova O.S., Upravlinnia formuvanniam batchennia produktiv proektu developmentu 

na fazi proktuvannia. Upravlinnia proektami ta programami, Kiev, 2011. 

Kanartchiuk V.E., Kurmikov I.P., Virobnitchi sistemi na transporti. Pidrutchnik. 

Bitschaia sch., 1997. 

PROIECTE PRIVIND DEZVOLTAREA BAZEI TEHNICO-PRODUCTIVE 

A UNEI ÎNTREPRINDERI DE TRANSPORT AUTO 

(Rezumat) 

Se examinează efectele proiectelor de dezvoltare a unei întreprinderi şi se definesc 

direcţiile principale de dezvoltare a bazei tehnico-productive a unei întreprinderi de 

transport auto.


Publicat de 



Secţia 


CONSIDERATIONS REGARDING THE BALANCE STRUCTURE 

OF Co-Cr-Mo ALLOYS FOR REMOVABLE 

PARTIAL DENTURE 

BY 

ELENA RALUCA BACIU 1 , IRINA GRĂDINARU 1 , MARIA BACIU 2 

and NORINA CONSUELA FORNA 3 

“Gr. T. Popa” University of Medicine and Pharmacy of Iaşi, 

1 Department of Dental Materials 

3 Department of EPI Clinic and Therapy 

“Gheorghe Asachi” Technical University of Iaşi, 

2 Department of Material Engineering and Industrial Security 


Accepted for publication: December 5, 2012 

Abstract. Removable partial denture (RPD) is a representative example of 

prosthetic restoration having a very complex structure and manufacture 

technology. Choosing the field of use of any material relies on the knowledge of 

its physical, chemical, technological and usage properties as well as the 

reciprocal influences between them. The analysis of the binary equilibrium 

diagrams of the ternary system Co-Cr-Mo shows the formation of a solid 

solution γ (of alloyed austenite type) rich in cobalt where appear numerous 

carbides with hardening effect resulted from the invariant reactions specific to 

these alloys. 

Keywords: partially movable skeletal prosthesis, Co-Cr-Mo alloys, binary 

equilibrium diagrams 


Prosthetic restorations specific to dentistry have a complex structure and 

numerous metal components may be found in their structure. The construction 

Corresponding author: e-mail: irigrad@yahoo.com

94 Elena Raluca Baciu et al. 

of the partial skeletal prostheses includes the following structural elements: 

artificial dental arches, prosthesis saddles, the main connectors, secondary 

connectors, and the sustaining and stability elements. 

Each constitutive part has its specific functional role, geometry and 

manufacture technology. To fulfil these conditions, one must take into account 

an essential criterion: the choice of material. 

2. Goal 

The study aims at knowing as completely as possible the Co-Cr-Mo 

alloys by mainly following their characterization from the structural viewpoint. 

3. Material and Method 

Taking into consideration the recommendations made in the specialized 

literature and the results of the practical experience from the dental labs, we 

analysed the following to materials belonging to the system of Co-Cr-Mo alloys 

for metal components of the removable partial denture (Table 1). 

Class of 

materials 

Co-Cr-Mo 

alloys 

Table 1 

Non-noble dental alloys subjected to experimental researches 

Material 

Producer Usage recommendations 

trademarks 

Sismo - similar 

Dentaurum, - skeletal prostheses with 

Remanium 6 M 

Germany 

clasps, grooves 

800 + 

Robur 400 

Eisenbacher Dental 

– Waren ED 

GmbH, Germany 

- skeletal prostheses with 

grooves and special 

systems 

4. Results and Discussions 

The chemical combinations taking place between the main components 

Co-Cr-Mo allow us to obtain, by the casting operations, metal alloys whose 

physical, mechanical technological and usage properties may be evaluated 

through the analysis of the characteristic equilibrium diagrams. 

Ternary alloys Co-Cr-Mo may be analysed by means of the binary 

equilibrium diagrams Co-Cr (Fig. 1), Cr-Mo (Fig. 2) and Mo-Co (Fig. 3) or by 

means of the isothermal sections made in the ternary diagram. 

The role of the alloying elements in the formation of structural 

constituents will be represented by the fact that the presence of molybdenum 

associated with chrome will create conditions for the formation of intermetallic 

phases, whereas chrome will provide the resistance to high temperatures of 

these alloys.


Fig. 1 – Equilibrium diagram 

of Co-Cr alloy system (* * * 

Cobalt-Chromium (Co-Cr) 

Phase Diagram. 

www.calphad.com). 

Fig. 2 – Equilibrium 

diagram of Cr-Mo alloy 

System (* * * Cobalt- 

Chromium (Co-Cr) Phase 

Diagram. 


Fig. 3 – Equilibrium 

diagram of Mo-Co alloy 

system (* * * Cobalt- 

Chromium (Co-Cr) 

Phase Diagram. 


In case of Co-Cr alloys (Fig. 1), we notice that the basic metallic mass is 

made up of a solid solution γ (of alloyed austenite type) rich in cobalt (min. 50 

%) and with a CFC crystal network. Carbides appear by precipitation processes 

in solid state having a hardening effect for the austenitic matrix. The main 

carbides are of M 23 C 6 type, but we also identified MC, M 3 C 2 , M 6 C M 7 C 3 

(Metal x C y ) carbides. The precipitation and morphology of carbides are 

determined by their solubility into cobalt (Ghiban & Borţun, 2009). 

M 23 C 6 may also precipitate under the form of very fine particles thus 

obtaining to an exaggerated hardening of the metallic mass and the basis and the 

diminution of its plasticity (Ardelle, 1994). 

Following these secondary transformations, we may notice the formation 

of two intermediary compounds with incongruent melting: sigma σ and ε phases 

(Co), (Gupta, 2005; Meyer & Degrange, 1992). 

Intermediary phases Co 9 Mo 2 (), Co 3 Mo, Co 7 Mo 6 (µ) and sigma-σ phase 

are present in Co-Mo alloys (Fig. 3) (Okamoto, 1991). σ and µ phases result 

from the peritectic reactions 

and 

0 

T 1620 

C 

L ασ, 

0 

T 1510 

C 

L σ 

μ. 

The eutectic reaction of these alloys is 

(1) 

(2) 

0 

T 1355 C 

L μ γ. 

(3) 

And at the end of the three peritectoid reactions, we obtained Co 

9Mo 2,Co3Mo 

carbides and ε phase, according to the invariant reactions


0 

T1200 

C 

μγ 

CO MO , 

9 2 

0 

T1025 

C 

9 2 

μ 

3 

CO MO 

CO MO, 

(4) 

(5) 

3 

0 

700 

γ+ CO MO T C ε. 

(6) 

The two phases and σ sustain the eutectoid disintegrations 

9 

Co Mo 2 

α μ 

0 

T1018 

C 

9 2γ 

 

3 

CO MO 

CO MO, 

(7) 

0 

T1000 

C 

σ . 

(8) 

Since Cr and Mo have isomorphic crystalline networks, they will form an 

alloyed solid solution α (Co, Cr), according to the equilibrium diagram, with 

the reciprocal solubility of components and the formation of a minimum point 

at 12.5 % Mo and 1820 o C (Fig. 2). We may notice the absence of miscibility in 

solid state below 880 o C. The semi-products of non-noble dental alloys under 

analysis were purchased from the manufacturing companies they having 

established values for the chemical composition, physical and mechanical 

properties pursuant to the technical sheet of the product. As for the chemical 

composition, we notice that the two materials have the character of complex 

alloys since they exhibit a large number of alloying elements (Table 2). 

Table 2 

Chemical composition of the dental non-noble alloys under study 

Alloy Chemical composition, [ % ] 

trademark Co Ni Cr Mo Ti Nb Al W Si Mn C 

Sismo 63.3 – 30.0 5.0 – – – – 10 – – 

Robur ~62 – 29.10 5.85 – – – 0.72 0.48 0..57 0..52 

Technical 

norm 

DIN EN 

ISO 

22674:2007 

DIN EN 

ISO 

22674:20 

07 

The properties of the elements from the chemical composition will exert 

their influence on the physical properties of the alloys formed (Table 3). 

In correlation with the chemical composition, it is obvious that the 

selected alloys need high melting and casting temperatures, a fact that makes 

their processing by casting require special vacuum or inert atmosphere melting 

equipment to mainly avoid the oxidization process. 

An important role in appreciating technological processability by 

chipping, plastic deformation, welding etc. and behaviour during use of the 

metal components obtained is held by resistance mechanical properties (R m and 

R p 0,2 ), plasticity (A 5 şi E) and hardness of the alloys used (Table 4).

Alloy 

trademark 


Table 3 

Physical properties of the alloys under study 

Density, 

[g/cm 3 ] 

Melting 

temp., 

[ o C] 

Casting 

temp., 

[ o C] 

Expansion 

coef. α 

(25...600 o 

C), 

[K -1 ] 

Colour 

Sismo 8.2 1240-1410 – – – 

Robur 400 8.3 1350-1390 1450 – alb 

Table 4 

Mechanical properties of the non-noble alloys under study 

Breaking 

Proportionality 

Alloy strength, 

Elongation Microhardness 

limit, Rp 

trademark Rm 

0,2 

[%] HV 

[MPa] 

10 

[MPa] 

Elasticity 

module E, 

[GPa] 

Sismo 960 720 6 370 – 

Robur 

400 

900 – >6 410 230 

Since the metallic components are technologically made by casting 

operations and subsequent mechanical processing, the value of the chemical 

composition of the main physical properties and resistance and plasticity 

mechanical characteristics will exercise their influence on the technological 

properties of the alloys under study. 


1. The chemical combinations occurred between the alloying elements 

during the elaboration-casting processes of alloys may have the character of 

some invariant reactions (eutectic, peritectic, eutectoid, peritectoid), variation 

reactions of reciprocal solubility of the solid components etc, they being 

identifiable on the equilibrium diagrams. 

2. From the metallographic viewpoint, the products of chemical reactions 

appear under the form of phases and constituents in the microstructure of each 

alloy. Consequently, by optical and electronic microscopy, one may highlight 

phases of solid solution type and intermetallic compounds as well as 

constituents of eutectic, peritectic, eutectoid, peritectoid type etc. 

3. The analysis of the binary equilibrium diagrams of the ternary system 

Co-Cr-Mo shows the formation of a solid solution γ (of alloyed austenite type) 

rich in cobalt where appear numerous carbides with hardening effect resulted 

from the invariant reactions specific to these alloys.


REFERENCES 

Ardelle A.J., Metallic Alloys Experimental and Theoretical Perspectives. Kluwer 

Academic Publishers, 1994, p. 93. 

Ghiban B., Borţun C.M., Aliaje dentare de cobalt. Ed. Printech, Bucureşti, 2009, pp. 6- 

7. 

Gupta K.P. The Co-Cr-Mo (Cobalt-Chronium-Molybdenum) System. Journal of Phase 

Equilibria and Diffusion, 26, 1, 87-92 (2005). 

Meyer J.M., Degrange M., Alliages nickel-chrome et alliages cobalt-chrome pour la 

prothèse dentaire. Encyclopédie Médico-Chirurgicale, p. 1992, 23065T10:12. 

Okamoto H. Mo-Ni (Molybdenum-Nickel). Journal of Phase Equilibria, 12, 6, 703 

(1991). 

* * * Cobalt-Chromium (Co-Cr) Phase Diagram. www.calphad.com. 

CONSIDERAŢII PRIVIND STRUCTURA DE ECHILIBRU A ALIAJELOR Co-Cr- 

Mo DESTINATE PROTEZEI PARŢIALE MOBILIZABILE SCHELETATE 

(Rezumat) 

Proteza parţială mobilizabilă scheletată (PPMS) este un exemplu reprezentativ 

de restaurare protetică cu o constituţie şi tehnologie de fabricaţie deosebit de complexe. 

Alegerea domeniului de utilizare a oricărui material este bazată pe cunoaşterea proprietăţilor 

sale fizice, chimice, tehnologice şi de utilizare, precum şi a influenţelor reciproce 

dintre acestea. Analiza diagramelor de echilibru binare ale sistemului ternar Co-Cr-Mo 

indică formarea unei soluţii solide γ (de tip austenită aliată) bogată în cobalt în care sunt 

dispuse numeroase carburi cu efect durificator rezultate în urma reacţiilor invariante 

specifice acestor aliaje.

BULETINUL INSTITUTULUI POLITEHNIC DIN IAŞI

Create successful ePaper yourself

Delete template?

Save as template?