BULETINUL INSTITUTULUI POLITEHNIC DIN IAŞI

BULETINUL
INSTITUTULUI
POLITEHNIC
DIN IAŞI
Tomul LVIII (LXII)
Fasc. 3
MATEMATICĂ. MECANICĂ TEORETICĂ. FIZICĂ
2012 Editura POLITEHNIUM

BULETINUL INSTITUTULUI POLITEHNIC DIN IAŞI
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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”
Technical University of Iaşi
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”
Technical University of Iaşi
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”
Technical University of Iaşi
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
Asachi” Technical University of Iaşi
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”
Technical University of Iaşi
Assoc.prof.dr.mat.Constantin Popovici,“Gheorghe
Asachi” Technical University of Iaşi
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”
Technical University of Iaşi
Dr.mat. Vitaly Volpert, CNRS, University „Claude
Bernard”, Lyon, France
Prof.dr.phys. Gheorghe Zet, “Gheorghe Asachi”
Technical University of Iaşi

BULETINUL INSTITUTULUI POLITEHNIC DIN IAŞI
BULLETIN OF THE POLYTECHNIC INSTITUTE OF IAŞI
Tomul LVIII (LXII), Fasc. 3 2012
MATEMATICĂ. MECANICĂ TEORETICĂ. FIZICĂ
S U M A R
Pag.
NICOLETA NEGOESCU, Rezultate de punct fix pentru operatori şi şiruri de
operatori contractivi pe spaţii pseudocompacte (franc., rez rom.). . . . . . 1
ION CRĂCIUN, Asupra unor probleme axial-simetrice ale teoriei elasticităţii
micropolare (engl., rez. rom.). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
GHEORGHE NEGRU, Model pentru evaluarea impactului fiabilității asupra
analizei costului pe durata de viață aplicabil sistemelor tehnice
complexe(engl., rez. rom.). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
RADU IBĂNESCU şi CĂTĂLIN UNGUREANU, Analiza unei came
circulare excentrice (engl., rez. rom.). . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
PETRONELA PARASCHIV, Metode de corecţie ale modelului biomecanic în
cazul mişcării antebraţului (engl., rez. rom.). . . . . . . . . . . . . . . . . . . . . . 43
IRINA GRĂDINARU, ELENA RALUCA BACIU şi DANIELA CALAMAZ,
Consideraţii generale privind culoarea în practica stomatologică (engl.,
rez. rom.).. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
MARIUS CATANĂ şi DANIELA TARNIŢĂ, Analiza statică a articulaţiei
genunchiului uman folosind MEF (entgl., rez. rom.). . . . . . . . . . . . . . . . 59
BOGDAN HORBANIUC, GHEORGHE DUMITRAŞCU şi MIHAI ADAM,
Studiul printr-o schemăimplicită cu diferenţe finite a unui sistem de
stocare termică în sol cu un singur tub (engl., rez. rom.). . . . . . . . . . . . . 67
GELU COMAN şi VALERIU DAMIAN, Transferul de căldură în pista
patinoarului artificial (engl., rez. rom.). . . . . . . . . . . . . . . . . . . . . . . . . . . 79
FLORIN POPA, EDWARD RAKOŞI şi GHEORGHE MANOLACHE,
Consideraţii asupra unui model comportamental de funcţionare a
sistemelor de propulsie pentru automobile (engl., rez. rom.). . . . . . . . . . 87
IULIAN AGAPE, LIDIA GAIGINSCHI, ADRIAN SACHELARIE şi
VASILE HANTOIU, Utilizarea funcţiilor spline la generalizare a
modelului de deformaţie (engl., rez. rom.). . . . . . . . . . . . . . . . . . . . . . . . 95
VLAD MARŢIAN, MIHAI NAGI şi CIPRIAN FLUIERAŞ, Simulări 1D
asupra schimbului termic al bateriilor vehiculelor electrice folosind
pachetul software OpenModelica (engl., rez. rom.). . . . . . . . . . . . . . . . . 105
PETRU CÂRLESCU, VASILE DOBRE, IOAN ŢENU şi RADU ROŞCA,
Modelarea transferului de căldură în procesul de uscare a malţului
(engl., rez. rom.). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

BULETINUL INSTITUTULUI POLITEHNIC DIN IAŞI
BULLETIN OF THE POLYTECHNIC INSTITUTE OF IAŞI
Tomul LVIII (LXI), Fasc. 3 2012
MATHEMATICS. THEORETICAL MECHANICS. PHYSICS
C O N T E N T S
Pp.
NICOLETA NEGOESCU, Résultats de point fixe pour des opérateurs et suites
d’ opérateurs contractifs dans des espaces pseudocompacts (French,
Romanian summary). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
ION CRĂCIUN, On Some Axial-Symmetric Problems of the Micropolar
Elasticity Theory (English, Romanian summary). . . . . . . . . . . . . . . . . . . . 7
GHEORGHE NEGRU, Model for the Realibility Assessment Impact on the Life
Cycle Cost Analysis Applicable th the Complex Technical Systems
(English, Romanian summary). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
RADU IBĂNESCU and CĂTĂLIN UNGUREANU, Analysis of a Circular
Cam (English, Romanian summary). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
PETRONELA PARASCHIV, Correction Methods of the Biomechanical Model
in the Case of Forearm Movement (English, Romanian summary) . . . . . 43
IRINA GRĂDINARU, ELENA RALUCA BACIU and DANIELA CALAMAZ,
General ConsiderationsConcerning the Color in the Dental Practice
(English, Romanian summary) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
MARIUS CATANĂ and DANIELA TARNIŢĂ, Static Analysis of the Virtual
Human Knee Joint Using FEM (English, Romanian summary) . . . . . . . . 59
BOGDAN HORBANIUC, GHEORGHE DUMITRAŞCU and MIHAI ADAM,
Study of a Single Tube Borehole Thermal Energy Storage by Means of
an Implicit Finite Difference Method (English, Romanian summary) . . . 67
GELU COMAN and VALERIAN DAMIAN, The Heat Transfer Inside an
Artificial Skating Rink (English, Romanian summary) .. . . . . . . . . . . . . . 79
FLORIN POPA, EDWARD RAKOŞI and GHEORGHE MANOLACHE,
Considerations on a Behavioural Model of Car PropulsionSystems
(English, Romanian summary). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
IULIAN AGAPE, LIDIA GAIGINSCHI, ADRIAN SACHELARIE and
VASILE HANTOIU, Usage of Spline Function Method for
Generalization of the Deformation Coefficient Model (English,
Romanian summary). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
VLAD MARŢIAN, MIHAI NAGI and CIPRIAN FLUIERAŞ, Ons for 1
Electrical Vehicle Battery Using Openmodelica Software Package
(English, Romanian summary). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

PETRU CÂRLESCU, VASILE DOBRE, IOAN ŢENU and RADU ROŞCA,
Modeling of Heat Transfer During Malt Drying Process (English,
Romanian summary). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

BULETINUL INSTITUTULUI POLITEHNIC DIN IAŞI
Publicat de
Universitatea Tehnică „Gheorghe Asachi” din Iaşi
Tomul LVIII (LXII), Fasc. 2, 2012
Secţia
MATEMATICĂ. MECANICĂ TEORETICĂ. FIZICĂ
RÉSULTATS DE POINT FIXE POUR DES OPÉRATEURS ET
SUITES D’OPÉRATEURS CONTRACTIFS DANS
DES ESPACES PSEUDOCOMPACTS
Reçue: February 28, 2012
Accepteé pour publication: May 15, 2012
PAR
NICOLETA NEGOESCU ∗
Université Technique “Gheorghe Asachi” de Iaşi,
Départment de Mathématiques
Abstract. One proves some fixed point theorems for operators satisfying
contractive conditions (1) and (1’) in pseudocompact spaces.
Key words: fixed point, contractive operator, pseudocompact spaces.
Dans cette Note nous démontrons quelles théorèmes de point fixe pour
des opérateurs et suites d’opérateurs continues contractifs defines sur des
espaces pseudocompacts.
Les conditions contractives (1) (1’) et (1’’) que nous avons emploées nous
ont été suggérées par celles des théorèmes de point fixe pour des applications
contractives (defines sur des espaces pseudocompacts) obtenus par Jain et Dixit
(1986) et par Nicoleta Negoescu (1998).
Nous començons par rappeller q uelques notions et résultats qui nous sont
nécessaries.
D é f i n i t i o n 1. On dit que l’espace topologique X s’appelle espace de
Tychonoff (ou espace T 1 ) et X est un espace T 1 (l’espace topologique X est un
3 2
espace T 1 si et seulement si pour ∀x , x2∈X, x1
≠ x2
il existe un ensemble,
ouvert D ⊂ X tel que x 1
∈ D et x2
∈ D ) et si pour choqur point x∈ X et
chaque ensemble fermé F ⊂ X, x∉ F,
il existe une function f : X → I=[0,1]
∗ e-mail: napreut@yahoo.com

2 Niooleta Negoescu
continue telle que f(x) = 0 et f ( y) = 1, ∀y∈
F.
Les espaces de Tychonoff s’appellent aussi espaces complètement
réguliers. Il est evident que espace complètement régulier (espace T 1 ) est aussi
3 2
un espace régulier (espace T 3 ) et tout espace régulier est aussi un espace T 2
(séparé Hausdorff).
D é f i n i t i o n 2. Un espace topologique X s’appelle pseudocompact si
X est un espace de Tychonoff et chaque function réelle continue, définie sur X
est bornée.
Observations 1. Tout espace compact est aussi un espace pseudocompact;
l’affirmation réciproque n’est pas vraie, cependant dans un espace métrique les
notions d’espace compact et pseudocompact coincident.
2. Le produit cartesien de deux espaces de Tychonoff est aussi un espace
de Tychonoff, mais le produit cartesien de deux espaces pseudocompacts n’est
pas toujours un espace pseudocompact.
L e m m e (Engelking, 1968). Un espace de Tychonoff est
pseudocompact si et seulement si toute function rélle, continue sur X attaint ses
bornes.
D é f i n i t i o n 3. On appelle pseudométrique sur un ensemble
arbitraire, nonvide X une application μ : X × X →
+
qui a les proprétés
suivantes:
i) μ( xx , ) = 0, ∀x∈
X;
ii) μ( xy , ) = μ( yx , ), ∀xy , ∈ X;
iii) μ( xy , ) ≤ μ( xz , ) + μ( zy , ), ∀xyz , , ∈ X.
Observation 3. La pseudométrique μ est une function continue. Dans ce
qui suit nous présentons trios resultants de point fixe en partant des notes de L.
B. Cirić (1974), S. N. Lal et Mohan Das (1982) et de Nicoleta Negoescu (1998;
1989).
T h é o r è m e 1. Soient X un espace de Tychonoff pseudocompact,
T : X → X un opérateurs continu sur X et μ : X × X → une pseudométrique
qui satisfont à l’inégalité
min{ μ( Tx, Ty), μ( xTx , ), μ( yTy , )} + βmin{ μ( xTy , ), μ( yTx , )} < αμ( xy , )
pour x, y ∈X , x ≠ y, 0< α < 1 et β ∈
+
.
D é m o n s t r a t i o n. Soit l’application φ:
X →
+
définie par
φ( p=μ ) ( p,Tp ) por ∀p∈X.
φ est une application continue car elle est la
(1)

Bul. Inst. Polit. Iaşi, t. LVIII (LXII), f. 2, 2012 3
composition de deux functions continues T et μ . X est un espace de Tychonoff
pseudocompact et alors chaque function réelle continue sur X est fornée et
attaint ses bornes (sur X). Donc il existe un point u∈ X tel que
φ( u) = inf{ φ( p): p∈X}.
Nous montrerons que u est un point fixe de T et pour cela nous supposons
que u n’est pas un point fixe de T; donc Tu ≠ u et nous prenons x = Tu et y = u
dans l’inégalité (1) en obtenant
min{ μ( T x, Tu), μ( Tu,
T
0< α < 1
2 2
u), μ( uTu , )} + βmi n{ μ( Tu,
Tu), μ( uT , 2 u)}
< αμ( uTu , ).
Alors ils existent les deux cas suivants:
2
min{ ( , )
2 2
a) Si μ uTu μ( Tu, T u)} = μ( Tu,
T u ), on a μ( Tu) < αμ( uTu , ) où
donc
μ TuTu
2
( , ) μ( uTu , )
< et φ(
Tu) < φ( u ), une contradiction.
2 2
b) Si min{ μ( uTu , ) μ( Tu, T u)} = μ(
Tu, T u ), alors μ( uTu , ) 0 u est un fixe unique pour T et pour
β− α<
0 il suit que le point fixe de T n’est pas unique
Maintenant nous nous occuperens des points fixes des suites d’opérateurs
continues d’un certain type de contractivité Tn
: X → X.
T h é o r è m e 2. Soient X un espace de Tychonoff pseudocompact,
μ : X × X →
+
une pseudométrique et ( Tn),
T
n: X→X , n∈ , une suite
d’opérateurs continues sur X qui est convergente (simplement) à une function
continue T: X×
X.
Si chaque opérateurs T n satisfait à l’inégalité (1) pour
n∈ , ∀x, y∈X,
x ≠ y, 0< α < 1, β ∈
+
, β > 0, alors la suite (u n ) des point
fixes des opérateurs T n est convergente au point fixe u ∈ X unique de l’ opérateurs
T.
D é m o n s t r a t i o n. En procédant de la meme manière qul celle du
Théorème 1, il suit qu’il existe un point fixe u n de chaque T n pour β > α .
Alors il nous reste a montrer que μ( u , ) n
u → 0 pour n →∞. Si u n = u
pour une infinite de n, le résultat est demontré.
Si u n = u pour une infinite de n, nous obtenons

4 Niooleta Negoescu
min{ μ( Tu , Tu), μ( u , T u ), μ( uTu , )} + βmin{ μ( u , T u), μ( uTu , ) < α μ( u , u)}
n n n n n n n n n n n
ou
α
min{ μ( un, Tnu), μ( uu ,
n)} < μ( un, u)
β
et il suit
α
μ( Tu
n n, Tu
n
) < μ( un, u) ≤ μ( TuTu
n
, ) + μ( Tu
n n,
Tu)
n
,
β
ou
α
μ( Tu
n n, Tu) < μ( TuTu
n
,
n) + μ( Tu
n n,
Tu),
β
c’est-à-dire
ou
⎛ α ⎞
⎜1 − ⎟μ( Tu
n n, Tu) < μ( TuTu)
n
, ,
⎝ β ⎠
⎛ α ⎞
⎜1 − ⎟μ( un, u) < μ( TuTu).
n
,
⎝ β ⎠
Mais μ( Tu , Tu) → 0 pour n→∞
T →T
par l’hypothèse) et il suit que
n
n
( n
μ ( un,
u) → 0 pour ∀n
∈ , donc un
→ u.
Il faut montrer encore que l’opérteurs T a un point fixe uniqe pour β > α;
ce que signifié qu’il est suffisant de démontrer que T satisfait à l’inégalité (1).
Pour tous les x, y∈X, x≠ y,
nous avons
min{ μ{ Tx, Ty), μ( xTx , ), μ( yTy , )} + βmin{ μ( xTy , ), μ( yTx , )} < αμ( xy , ).
Alons T a un point fixe unique pour β > α (conformément au théorème 1).
Si la suite d’opérateurs (T n ) ne satisfait pas à l’inégalité (1) mais
l’opérateur T satisfait à l’inégalité (1) pour β > α nous pouvons démontrer le
suivant:
T h é o r è m e 2. Soient X un espace de Tychonoff pseudocompact,
μ : X × X →
+
une pseudométrique, Tn
: X→X,
n∈ , une suite d’opérateurs
continues sur X qui ont les points fixes {u n } et la suite (T n ) est convergente
uniormément à une function continue T : X → X.
Si T satisfait à l’inégalité (1)
pour β > α, l’opérateur T a un point fixe unique u et u → u.
n

Bul. Inst. Polit. Iaşi, t. LVIII (LXII), f. 2, 2012 5
D é m o n s t r a t i o n. Il faut seulement montrer que la suite (u n ) est
convergente a u. De la convergence uniforme de la suite (T n ) sur l’ensemble
{: un∈ N}
il suit que pour ∀ ε > 0 il existe un nombre entier N( ε)
yel que
n≥ N( ε ) implique
ε
1
μ( TuTu
n
,
n) < pour tous un, où M= .
M
α
1−
β
(2)
En procédant de la manière du théorème precedent nous obtenons de (2):
α
μ( Tun, Tu) < μ( un,
u)
et donc
β
α
⎛ α ⎞
μ( un, u) = μ( Tu
n n, Tu) ≤ μ( Tu
n n, Tun) < μ( un, u)ou μ( un, u) ⎜1 − ⎟μ( Tu
n n, Tu n
).
β
⎝ β ⎠
μ( Tn, un)
Donc μ( un, u) < , c’est-à-dire μ( un, u)
< ε pour tous n≥
N()
ε
α
1−
β
(de (2)). Alors μ( u , u) → 0 pour n →∞ et donc un
→ u.
n
Observation 4. Pour β = − 1 dans le Théorème 1, nous obtenons un
analogue du résultat de L. B. Cirić (1972) dans un espace de Tychonoff
pseudocompact.
Observation 5. Si dans les conditions du Théorème 1 nous remplaçons
l’inégalité (1) par l’une des inégalités suivantes
2
min{[ μ(
Tx, Ty), μ( xTx , )] , μ( yTy , )} +
+ β μ xTy μ yTy < αμxy μ yTy
2
min{ ( , ), ( , )} [ ( , ), ( , )] .
(1’)
2 1/3 1/2
[ μ( Tx, Ty), μ( xTx , )] , μ( yTy , )] βμxTy [ ( , ), μ( yTy , )] αμ( xy , ),
+ < (1’’)
p our tous xy , ∈X, x≠ y, 0< α< 1, β ∈
+
, nous obtenons des Théorèmes
analogues du Théorème 1. On peut aussi obtenir des analogues des Théorèmes 2
et 3 pour les memes inégalités.
Observation 6. La méthode est employée dans les démonstrations des
Cirić ( 1972), Lal et Das (1982) et par Nicoleta Negoescu (1998; 1989).
Observation
7. Les Théorèmes 1, 2, 3 et toutes leurs consequences sont
aussi vrais si X est un espace métrique compact.

6 Niooleta Negoescu
Observation 8. Si μ ne satisfant pas à l’inégalité (2) de la Definition 3
( μ n’est pas symétrique) alors μ est une pseudo-quasimétrique (Hicks, 1988).
Les Theorems 1, 2, 3 sont
vrais aussi dans ce cas, mais nous ne pouvons plus
démontrer l’unicité du point fixe.
BIBLIOGRAPHIE
Cirić L. B., On Some Maps with Non-unique Fixed Point. Publ. Inst. Math., 17(34), 52-
58 (1974).
Engelking R., Outline of General Topology. Amsterdam–Warszawa, 1998.
Fisher B., Fixed Points and Constant Mappings on Metric Spaces. Rend. Acad. Lincei,
61, 129-332 (1970).
Hicks T. L., Fixed Point Theorems for Quasi-metric Spaces. Math. Japonica, 33, 233-
236 (1988).
Jain R. K., Dixit S. P., Some Fixed Point Theorems for Mappings in Pseudocompact
Tihonov Spaces. Bull. Math., Debrecen, 33, 195-197 (1986).
Lab S. N., Das M., Mappings with Common Invariant Points in 2-Metric Spaces (I and
II), Math. Sem. Notes, Kobe, 8, 83-90 (1980) and 10, 691-695 (1982)..
Negoescu Nicoleta, Extensions des théorèmes de R. K. Jain et S. P. Dixit. Vol. Itinerant
Seminar of Functional Equations, Approximation and Convexity, Univ. Cluj-
Napoca, 1988, pp. 243-248.
Negoescu Nicoleta, Un théorème de point fixe pour deux applications commutatives
d’un certain type de contractivité. Studii şi cercetări ştiinţifice, Univ. Bacău, Ser.
Matematica, 2, 85-86 (1992).
Taniguchi T., Common Fixed Point Theorems on Extension Type Mappings on
Complete Metric Spaces. Math. Japonica, 34, 139-142 (1989).
REZULTATE DE PUNCT FIX PENTRU OPERATORI ŞI ŞIRURI DE
OPERATORI CONTRACTIVI PE SPAŢII PSEUDOCOMPACTE
(Rezumat)
Se demonstrează trei teoreme de punct fix pentru operatori şi şiruri de operatori
care satisfac inegalităţi contractive (1), (1’), (1’’) definiţi pe spaţii metrice
pseudocompacte.

BULETINUL INSTITUTULUI POLITEHNIC DIN IAŞI
Publicat de
Universitatea Tehnică „Gheorghe Asachi” din Iaşi
Tomul LVIII (LXII), Fasc. 3, 2012
Secţia
MATEMATICĂ. MECANICĂ TEORETICĂ. FIZICĂ
ON SOME AXIAL-SYMMETRIC PROBLEMS OF THE
MICROPOLAR ELASTICITY THEORY
BY
ION CRĂCIUN ∗
“Gheorghe Asachi” Technical University of Iaşi,
Department of Mathematics and Informatics
Received: May 28, 2012
Accepted for publication: July 20, 2012
Abstract. The steady vibrations of the linear theory of micropolar elasticity
of an isotropic, homogeneous, and centrosymmetric elastic body occupying the
three dimensional region B are considered. A generalized Galerkin representation
of a regular solution of basic differential equations in B of this theory is given. In
the case of cylindrical coordinates (, r θ, z)
under the assumption of axially
symmetry (causes and effects do not depend of the variable θ), from governing
equations in the amplitudes of displacement and rotation vectors we obtain two
independently systems, each one containing three partial differential equations.
The unknowns of the second system are the second component of the amplitude
of displacement vector u, and the first and the third components of the amplitude
of rotation vector φ , all depending only of variables r and z in B, for each θ.
The aim of this paper is the study of such obtained system. Thus, the Galerkin
representation of a regular solution by stress functions is obtained, the actions
both of body and couple-body loadings are analysed, and the action of a
concentrated couple-body is considered.
Key words: micropolar elasticity; axially-symmetric problems; stress
functions; concentrated loadings.
1. Introduction
The classical theory of elasticity is inadequate to represent the behaviour
of certain materials such as polycrystalline materials, granular bodies, bodies
∗ e-mail: ialcraciun@yahoo.com

8 Ion Crăciun
with large molecules like polymers, because they have an internal structure
demanding that each point of them to enjoy of six degree of freedom. In a
remarkable monograph, the brothers E. and F. Cosserat (1909) gave a
systematic development of the mechanics of continuous media in which each
point has six degree of freedom of a rigid body. The orientation of a given
particle of such a medium can be represented mathematically by the values of
three mutually perpendicular unit vectors, called directors by Ericksen and
Truesdell (1958). Thus, the theory of oriented media has been appeared.
An exposition of the kinematics of oriented bodies, together with
references to other contributions on the subject up to 1960, is given in the
monograph by Truesdell and Toupin (1960)
Micropolar elasticity termed by Eringen (1966), or asymmetric elasticity,
after Nowacki (1986), as well as Cosserat elasticity theory are used to describe
deformation of elastic media with oriented particles. For engineering
applications it can model composites with rigid chopped fibres, elastic solids
with rigid granular inclusions, and other industrial materials.
However, in spite of its novelty, the theory of Cosserat brothers was not
appreciated for a long time. Only in the sixties the research development in the
area of the general theory of continuum helped the Cosserat theory to attract the
attention of researchers and it becames the starting point for a number of related
theories in mechanics and physics of oriented media.
Many papers written over the years in the area of mechanics have been
devoted to the media of Cosserat type. Due to its cognitive values, possible
complete experimental verification as well as technological applications, the
Cosserat theory is being developed in a great number of research centers in the
world (Poland, USA, Russia, Georgia, Germany, Great Britain, Sweden,
Romania, Canada, India, Italy, Japan, Holand). The first well-known and widely
appreciated monographs concerning the asymmetric elasticity theory have been
written in Poland. The most up-to-date monograph investigation devoted to
linear asymmetric elasticity theory is the book of W. Nowacki (1986). The work
of Nowacki contains a long list of references. A more extensive reference list
can be found in the recent monographs of Dyszlewicz (2004), and Ieşan (2004).
Thus, it has been created a new model of a continuum known today as the
Eringen-Nowacki model.
There are situations in practice when the causes acting on a micropolar
i t
elastic body depend on the time t by the function e − ω . A such dependence is
expressed in the form
i t
f% ( x
ω
, t) =Re[ f( x )e − ],
(1)
where ω is called the frequency of vibration, f % is a scalar or a vector field,
x=x1e1+ x2e2+
x3e3is
the position vector of a point of the body B referred to

the Cartesian coordinate system
Bul. Inst. Polit. Iaşi, t. LVIII (LXII), f. 3, 2012 9
Ox1x 2x3
whose the unit vectors of axes are
e1, e2 ande 3, i = − 1 is the imaginary unit, R e is the real part of the complex
function between brackets, and f ( x)
is called the amplitude of that field.
Consequently, the effects also depend on time by the rule (1).
If both the causes and effects are of the form (1), then the derivatives of a
function f %( x , t)
into respect to the time variable t became the product between
2
−iω
and f ( x ), in the case of first derivative, and the product between − ω
and f ( x),
when a second derivative is applied. In this case, any equation of the
−iωt
governing differential system has the factor e and by its elimination, through
i
a division by e − ωt
, we arrive at a similar equation in which only the amplitudes
appear. Therefore, in the place of temporary variable t we will have the
frequency of vibrations ω and all unknowns of the field equations governing an
elasticity theory in the case of steady vibrations will be amplitudes.
A lot of papers and books are devoted to the steady vibrations of Cosserat
elasticity theory. Many results on steady vibrations in this field were established
by Georgian research workers, some of them being published in the well-known
monograph by Kupradze and others (1986).
2. The Eringen-Nowacki Model in the Case of Steady State Vibrations
2.1. Notation and Mathematical Preliminaries
Physical quantities are mathematically represented by tensors of various
orders. The equations describing physical laws are tensor equations. Quantities
that are not associated with any special direction and are measured by a single
number are represented by scalars, or tensors of order zero. Tensors of order
one are vectors, which represent quantities that are characterized by a direction
as well as a magnitude. More complicated physical quantities are represented by
tensors of order greater than one.
Throughout this paper light-faced Roman or Greek letters stand for
scalars, and Roman letters in boldface denote vectors.
A system of fixed rectangular Cartesian coordinates Ox1x 2x3
is sufficient
for the presentation of a mathematical theory in the three-dimensional space
3 . In an index notation, the coordinate axes may be denoted by x
j
and the
unit base vectors of the frame by e
j
, where j = 1,2,3. We mention that a
repeated index implies the Einstein summation convention over values 1, 2,3
3
unless explicitly otherwise specified. The position vector of a point in is
x =xe
j j.
Particularly significant in vector calculus is the Hamilton's vector
operator or the nabla operator denoted by ∇ , where

10 Ion Crăciun
∂ ∂ ∂ ∂
∇ = e1 + e2 + e3
= ej
.
∂x1 ∂x2 ∂x3
∂x j
When applied to the scalar field f( x1, x2, x3) = f( x ), the vector operator
∇ yields a vector field or a tensor of rank one, who is known as the gradient of
that scalar field. Thus,
f = ∇ f = f e ,
grad
, i i
A comma in the front of an index i denotes partial differentiation into
respect the x
i
variable.
In a vector field, denoted for example by ux ( ), the components of the
vector are functions of spatial coordinates x , x , x denoted by
1 2
ui
( x).
Assuming that functions ui
( x1,
x2, x3)
are differentiable, the nine partial
∂ui
derivatives can be written in an index notation as u i , j
. It can be shown that
∂x
u i , j
j
are the components of a second-rank tensor.
3
u ( x , x , x )
i
1 2 3
When the vector operator ∇ operates on a vector ux ( ) = uie i
in a way
analogous to scalar multiplication, the result is a scalar field termed the
divergence of that vector field ux ( ) having the expression
div u= ∇ ⋅ u= u1,1 + u2,2 + u3,3 = u ii ,
.
By taking the cross product of the operator ∇ to the vector field
ux ( ) = uiei
we obtain a vector field termed the curl of ux ( ) and denoted by
curl u or ∇ × u, whose analytical expression is
where
ε ijk
∇ × u=
curl u= ε u ,
ijk
k , j
is a component of the Ricci's alternating tensor.
2
The Laplace operator ∇ is obtained by taking the divergence of a
gradient. The Laplace operator of a twice differentiable scalar field f is the
following scalar field
2
div grad f = ∇∇ f =∇ f = f ii ,
.
The operator ∇ can be applied to the divergence of an amplitude vector
ux ( ) = uiei
and the result can be written as
2
grad div u= ∇∇ ⋅ u=∇ u=u j , ji
e i
.
2
The Laplace operator ∇ of the vector field ux ( ) = uie i
is the vector
2
∇ u= ∇⋅ ∇u= u k , jj
e k
= ∇∇⋅u− ∇× ( ∇ × u ).
or

Bul. Inst. Polit. Iaşi, t. LVIII (LXII), f. 3, 2012 11
2.2. Basic Equations of the Eringen-Nowacki Model in the Case of
Steady Vibrations
We consider an isotropic, homogeneous, and centrosymmetric micropolar
3
elastic solid occupying a region B ⊂ with piecewise smooth boundary ∂ B.
The closed region B is the union of the sets B and ∂ B,
that is B = BU ∂B.
Let
n be the components of the unit outward normal vector n to ∂ B.
j
We refer our considerations to the micropolar elastic medium
B( λμαβγερJ
, , , , , , , )
described by the constants of micropolar elasticity theory λ,μ,α,β,γ,ε, the
density ρ, and the rotational inertia J. All the physical fields defining a
micropolar elastic state of the medium B are real valued functions of the
position vector x called amplitudes.
The fact that the function f is defined in the set B will be denoted by
n
f : B → . A function f belongs to the class C in the set B , and we write
n
this in the form f ∈ C ( B),
if f and all partial derivatives up through n order
are continuous in the set B , where n is a natural number. Generally, throughout
the present paper we assume that the considered functions are sufficiently
regular to make the applied procedures meaningful.
The basic equations of the Eringen-Nowacki model, in the case of
steady vibrations, can be divided into the following groups:
i) the equations of motion in B
where
σ μ C B)
1
ji
,
ji
∈ (
2
⎧
⎪
σ
ji,
j
+ ρω ui + Xi
= 0,
⎨
2
⎪⎩ εijkσ jk
+ μji,
j
+ Jω φi + Yi
= 0,
(2)
are the amplitudes of the physical components of
asymmetric stress force-stress tensor and asymmetric couple-stress tensor,
respectively, X , Y ∈C 0 ( B ) are the fields of the amplitudes of body loadings
i
i
2
and body moments, respectively, and ui, φi∈ C ( B)
are the amplitudes of the
components of displacement and rotation vectors, respectively;
ii) the geometric relations in B
⎧⎪
γ
ji
= ui,
j
−εkjiφk
,
⎨
⎪⎩ κji
= φi , j;
iii) the constitutive relations in B
(3)

12 Ion Crăciun
⎧σ ji
= ( μ+ α) γji + ( μ− α) γij + λγkkδji,
⎪
⎨
⎪⎩ μ = ( γ+ ε) κ + ( γ− ε)
γ κ + βκ δ .
ji ji ij ij kk ji
(4)
Substituting the constitutive relations (4) into equations of motion (2)
and using the geometrical relations (3) we obtain the equations of motion in
terms of the amplitudes of displacements u and rotations ϕ
i
i
⎧
⎪
⎨
⎪⎩
u + ( λ+ μ− α) u + 2αε φ + X = 0,
2 i j, ji ijk k,
j i
ϕ + ( β+ γ− ε) φ + 2αε u + Y = 0,
4 i j, ji ijk k , j i
(5)
where the differential operators
2 and 4
are given by
= ( μ + α) ∇ + ρω , = ( γ+ ε) ∇ + Jω − 4 α.
(6)
2 2 2 2
2 4
The principal aim of the theory is to determine a regular solution
u φ ∈C B ∩ C B) (7)
2 1
i, i
( ) (
of the equations of motion in the amplitudes of displacements and rotations (5)
satisfying the following boundary conditions:
⎧⎪
σ
jinj = pi , μ
jinj = mi ,on ∂Bσ
,
⎨
⎪⎩ ui = fi, φi = gi,on ∂Bu,
where p , m : ∂B → , f , g : ∂B
→ are given functions defined on subsets
i i σ
i i u
of the set ∂ B with ∂Bσ
∩∂ Bu
=∅, ∂Bσ
∪∂ Bu
=∂ B.
The functions p i
are
called surface tractions, are the components of the surface couple-stress
vector,
form
f and
i
g i
m i
are the surface displacements and rotations, respectively.
The fundamental differential equations (5) can be written in the vector
where the new vectors
⎧⎪ 2u+ ( λ+ μ−α) ∇∇ · u+ 2α∇× φ+ X=
0,
⎨
⎪⎩ 2 α∇× u +
4φ+ ( β+ γ− ε) ∇∇·
φ+ Y=
0,
u 2
and
2φ are equal to
2 2 i i 4 4 i i
(8)
(9)
u= ( u ) e ; ϕ = ( ϕ )e .
(10)
The vector form of the boundary conditions (8) is
⎧σ = p, μ = m,on∂Bσ
,
⎨
⎩u = f, φ = g,on ∂ Bu
,
(11)

Bul. Inst. Polit. Iaşi, t. LVIII (LXII), f. 3, 2012 13
where σ = σ
jin
je i
is the stress (traction) vector, μ = μ
jin
jei
is the couple-stress
vector, and ( pmf, , , g) = ( pi, mi, fi, gi) ei.
Thus, the main initial boundary-value problem of this theory is to
determine a regular solution
2 1
u, φ ∈C ( B) ∩C ( B)
(12)
satisfying the motion vector equation (9) and the boundary conditions in vector
form (11).
By using modern methods of mathematics, many authors Nowacki
(1966), Nowacki (1986), Dyszlewicz (2004), Ieşan (2004), Hetnarski (1987),
Ignaczack (1971), Teodorescu (1975, Crăciun (1977), Crăciun (1978), and
others, have been obtained various results on boundary value problems both of
the linear theory of micropolar elasticity and micropolar thermoelasticity in the
case of steady vibrations.
2.3. Generalized Galerkin Representation of a Regular Solution
Let us consider the set of functions
where
2,
4
are given in (6),
( )
⎧
⎪ u =
1 4−
∇∇ · Γ Φ − 2α
∇ × Ψ 3
,
⎨
⎪
⎩ ϕ= (
2 3−∇∇· Θ)
Ψ−2α∇×
1Φ,
(13)
and
2 2
⎧
1= ( λ + 2 μ) ∇ + ρω ,
⎪
2 2
⎪ 3= ( β + 2 γ) ∇ + Jω
−4α,
⎨
2
⎪Γ= ( λ+ μ −α) 4
−4 α,
⎪
2
⎩Θ= ( β + γ −ε) 2
−4 α ,
(14)
3 3
Φ: B→
, Ψ : B→
(15)
8 6
are functions of classes C ( B)and C ( B),
respectively.
To obtain the generalized Galerkin representation (11), the following
relations between the operators (6) and (14) have been used
2 2 2 2
1 4 2 3 2 4
4 α .
−∇ Γ = −∇ Θ = + ∇ = Ω . (16)
The functions Φ: B→
3
and Ψ:
B→
3
are called stress functions or Galerkin
vectors.
In solving problems of asymmetric elasticity in the case of steady
vibrations the stress functions are of considerable importance.

14 Ion Crăciun
If we introduce the notations
⎧ ω 2 2
2 4α
2 4α
⎪σi
= , σ3 = σ3 − , σ4 = σ4
− ,
⎪
ci
β+ 2γ γ+
ε
⎪ 1 ρ 1 ρ 1 J 1 J
⎨ = , = , = , = ,
2 2 2 2
⎪c1 λ+ 2μ c2
μ+ α c3
β+ 2γ c4
γ+
ε
⎪
⎪ 2α
2α
p= , s=
,
⎪
⎩ γ+ ε μ+
α
then the operators , , , can be written as
where
1 2 3 4
2 2 2 2
⎧
1= ( λ+ 2 μ)( ∇ + σ1), 2= ( μ+ α)( ∇ + σ ),
⎪
2
⎨
2
2
2 2
⎪ ⎩ 3= ( β+ 2 γ)( ∇ + σ3 ),
4= ( γ+ ε)( ∇ + σ4
).
Another forms of the operators Ω and Θ are given by
2 2 2 2
⎧Ω=
⎪
( μ+ α)( ∇ + k1 )( ∇ + k2),
⎨
2 2 2 2 2 2
⎪Θ= ⎩ ( μ + α)( β+ 2 γ)( ∇ + σ ˆ
2
+ σ3) − ( μ+ α)( γ+ ε)( ∇ + k1 + k2),
2
k 1
and are the quadratic roots of the equation
(17)
(18)
(19)
k − ( σ + σˆ
+ ps)
k + σσˆ
=0. (20)
4 2 2 2 2 2
2 4 2 4
T h e o r e m 1 (Nowacki, 1986). If the functions (13) satisfy the
differential system
⎧ 1Ω Φ + X=
0,
⎨ (21)
⎩ 3 Ω Ψ + Y = 0,
then the set of functions (11) represents a solution of the basic differential
system (9).
Eqs. (11) and (13) are useful in the determination of the fundamental
solutions for the system of differential equations (9) in the unbounded elastic
space, and they have been obtained by using different methods by Nowacki
(1986), Kupradze (1986), Stefaniak (1968), Şandru (1966), and others.
3. The Basic Equations in Cylindrical Coordinates
For the analysis of specific problems of micropolar elasticity,
orthogonal curvilinear coordinates often lead to simplification of the
mathematical treatment.
In the study of some axially-symmetric problems of the micropolar
elasticity it is convenient to use the cylindrical coordinate system ( r, θ , z).
In
3
every point of the vector base of this coordinate system is formed by the
unit vectors er
, e
θ
, ez.
The first unit vector, orthogonal both to e
θ
and e z
, is

Bul. Inst. Polit. Iaşi, t. LVIII (LXII), f. 3, 2012 15
chosen in a such way that the set { er, eθ
, e z
} to be a right-oriented system of
vectors. In this case, the relationships between the Cartesian coordinates
x , x , x and the cylindrical coordinates ( r, θ , z)
are
1 2 3
⎧x = rcos θ,
⎪
⎨
⎪
⎩
⎧ 2 2
r = x1 + x2,
⎪
⎪ x
cosθ
=
1
1 2 2
⎪ x1 + x2
x2
= rsin θ,
⇔ ⎨
x2
x3
= z; ⎪
⎪
sinθ
= ,
2 2
x1 + x2
⎪
⎪
⎩z
= x3.
,
(22)
These equations give the meaning of each cylindrical coordinate. By
using Achenbach (1973), the operators in the previous sections, in cylindrical
coordinates, have the expressions:
∂ 1 ∂ ∂
∇ = er
+ eθ
+ ez
,
(23)
∂r r ∂θ
∂z
f 1 f f
∇f
= grad f = ∂ er + ∂ e
∂
θ
+ e
z
,
∂r r ∂θ ∂z
(24)
1 ∂ur
1∂uθ
∂uz
∇ · u= divu= ur
+ + +
r ∂r r ∂θ ∂z ,
(25)
er reθ ez
1 ∂ ∂ ∂
∇ × u= curl u=
,
(26)
r ∂r ∂θ ∂z
u ru u
r θ z
where the vector u, written in cylindrical coordinates, has the analytical
expression
u= ure .
r
+ uθeθ + uze
z
The scalar Laplace operator of a vector u is more complicated, but we will
write its expression by using the identity
u ∇∇ ( · u) ∇ ∇ u grad (div u) curl curl u .
(27)
2
∇ = − × × = −
By employing the expressions for the gradient (22), the divergence (23), and the
curl (24), from (25) we obtain
2 ⎛ 2 2 ∂uθ
⎞ ⎛ 2 2 ∂uθ
⎞ 2
∇ u = ⎜∇0ur −
2 r
+ ∇
0uθ + ,
2
θ
+∇
z z
r ∂θ ⎟ ⎜
r ∂θ
⎟
⎝ ⎠ e ⎝ ⎠
e u e (28)

16 Ion Crăciun
where the scalar Laplace operator of a scalar f , denoted by
and
2 2
2 1 f 1 f f
f
∂ ⎛
r
∂ ⎞
∇ = + ∂ +
∂
2 2 2
⎜ ⎟
r ∂r⎝
∂r ⎠ r ∂θ ∂z
,
2
∇ f , is
(29)
2 2 1
∇
0
=∇ −
2 .
(30)
r
By starting with (9), and by taking into account (26)-(28) we arrive to the
differential system of steady vibrations of micropolar elasticity in cylindrical
coordinates
⎧ ⎛ 0 2 ∂uθ
⎞ ∂( ∇· u) ⎛1∂φz
∂φθ
⎞
⎪( μ+ α) ⎜ 2
ur
− ( λ μ α) 2 X 0,
2
r
r θ
⎟+ + − + α
r
⎜ −
r θ z
⎟+ =
⎪ ⎝ ∂ ⎠ ∂ ⎝ ∂ ∂ ⎠
⎪
⎪ ⎛ 0 2 ∂ur
⎞ 1 ∂( ∇· u)
⎛∂φr
∂φz
⎞
( μ+ α) 2
uθ
+ + ( λ+ μ− α) + 2α
− + X 0,
2
θ
=
⎪
⎜
r θ
⎟
r θ
⎜
z r
⎟
⎝ ∂ ⎠ ∂ ⎝ ∂ ∂ ⎠
⎪
⎪
∂( ∇· u) 2α
⎛ ∂ ∂φr
⎞
⎪( μ+ α) 2uz
+ ( λ+ μ− α) + ⎜ ( rφθ
) − ⎟+ X
z
= 0,
⎪
∂z r ⎝∂r ∂θ
⎠
⎨
⎪ ⎛ 0 2 ∂φθ
⎞ ∂( ∇ϕ · ) ⎛1∂uz
∂uθ
⎞
⎪
( γ+
ε)
⎜ 4
φr
− + ( β+ γ− ε) + 2α − + Y 0,
2
r
=
r ∂θ ⎟
∂r ⎜
r ∂θ
∂z
⎟
⎪
⎝ ⎠ ⎝ ⎠
⎪
⎛ 0 2 ∂φur
⎞ 1 ∂( ∇· φ)
⎛∂ur
∂uz
⎪
⎞
( γ+ ε) ⎜ 4
φθ
+ ( β γ ε) 2α Y 0,
2 ⎟+ + − + ⎜ − ⎟+ θ
=
⎪ ⎝ r ∂θ ⎠ r ∂θ ⎝ ∂z ∂r
⎠
⎪
⎪
∂( ∇· φ) 2α
⎛ ∂ ∂ur
⎞
⎪( γ+ ε) 4φz
+ ( β+ γ− ε) + ⎜ ( ruθ
) − ⎟+ Yz
= 0,
⎩
∂z r ⎝∂r ∂θ
⎠
(31)
where u= urer + uθeθ + uzez,
φ = φrer + φθeθ + φze
z.
are the amplitudes of the
displacement vector and rotation vector, and X = X
rer + Xθeθ + Xze z
and
Y= Ye
+ Y e + Ye
are the amplitudes of body and couple-body loadings,
r r θ θ z z .
respectively, while the operators
0 0
2,
4
are determined by
4. Axially-symmetric Problems
2 2 1
∇
0
=∇ −
2 .
r
Consider now the case of axially-symmetric deformations of the body B
assuming that all the causes and effects are independent of the variable θ .
Consequently, we find that system (28) splits into two independent systems.
The first of them contains the displacement u= urer + uzezand the rotation
φ = φ θ
e θ
, while in the second system there appears the displacement u= uθeθ
and the rotation φ = φre+ φze
z.
The first system is

Bul. Inst. Polit. Iaşi, t. LVIII (LXII), f. 3, 2012 17
0
⎧
2
ur + ( λ+ μ−α) ∂re −2α∂ zφθ + X
r
= 0,
⎪
−1
⎨ 2uz + ( λ+ μ−α) ∂
ze + 2 αr ∂
r( rφθ ) + X
z
= 0, (32)
⎪
0
⎪⎩ 4
φθ + 2 α( ∂zur −∂
ruz) + Yθ
= 0.
Here, we used the symbols ∂r,
∂
z
to denote partial derivatives with respect to
the position variable r, z , respectively. The dilatation e = ∇ · u in cylindrical
coordinates and in the absence of variable θ , is given by
−1
e= r ∂
r( rur)
+∂
zuz.
0
The symbols X
r, Yθ , Xz
denote body loadings and the operators
i
are defined
through the operator by
2
∇ 0
0 2 2 0 2
2 0 2 4 0
= ( μ + α)( ∇ + σ ); = ( γ + ε)( ∇ + σ 4
).
The role of the position variable x is played by the coordinates ( rz , ).
The form of the second system in B is
0
⎧
4
ϕr + ( β + γ −ε) ∂rκ −2α∂ zuθ
+ Yr
= 0,
⎪
−1
⎨ 4ϕz + ( β + γ −ε) ∂
zκ + 2 αr ∂ r( ruθ
) + Yz
= 0,
(33)
⎪ 0
⎩ 2
uθ
+ 2 α( ∂zϕr −∂
rϕz) + Xθ
= 0,
−1
where κ = ∇ · ϕ = r ∂
r( rϕr) +∂
zϕz, ϕ = ( ϕr, 0,
ϕz),
and Yr, Xθ
, Yz
0 are body
loadings.
5. The Second Axially-symmetric Problem of Micropolar Elasticity
in the Case of Steady Vibrations
This important problem of micropolar elasticity in the case of steady
vibrations has as object of study the system (30). There are many references
about the second axially-symmetric problem of micropolar elasticity in the case
of steady vibrations. We quoute here Nowacki (1986), Kupradze (1986), and
Dyszlewich (2004). The uncoupled form of system (30) may be found in
Dyszlewicz (2004)
2
0 0
⎧Ω uθ
= 2 α( ∂zYr −∂rYz) −
4
Xθ
,
⎪
0 0 0 0 0 0
⎨ 3Ω φr =∂r( Θ Y ) −2 α∂z 3
Xθ −
2 3Yr
,
⎪
⎩ Ω φ =∂ ( Θ Y ) + 2 α [ r ∂ ( rX )] − Y
0
3 z z 3 −1
r θ 2 3 z
,
(34)
where
−1
Y = ∇ · Y= r ∂ ( rY ) +∂ Y , Y=
( Y , 0,
Y ),
r r z z r z

18 Ion Crăciun
and
0 0 0 2 2
⎧⎪ Ω =
2 4+ 4 α ∇0,
⎨
0 0
⎪⎩ Θ = ( β + γ−ε) 2
−4α 2 .
Writing the amplitudes of displacement and rotations appearing in equations
(30) in terms of reduced Galerkin vectors
Φ= Φθ
(,) rzeθ , Ψ = Ψr(,) rzer + Ψz(, rz) ez,
we have
0 0
⎧uθ
=
4Φθ
−2 α
3( ∂zΨr −∂rΨz),
⎪
0 0 0 2
⎨ϕr = 2 α∂ zΦθ
+
2 3Ψr − [( β+ γ−ε) 2−4 α ] ∂rΨ
,
⎪
⎪⎩
ϕ =− 2 α( r +∂ ) Φ + Ψ − [( β+ γ−ε) −4α
−1 2
z r θ 2 3 z
2
] ∂ zΨ ,
(35)
where
−1
⎧ div Ψ = ∇· Ψ = r ∂
r( rΨr) +∂zΨz
= Ψ,
⎪
⎨rot Ψ = ∇× Ψ = ( ∂zΨr −∂rΨz) eθ
,
⎪
⎩grad Ψ= ∇ Ψ= ∂
rΨer +∂zΨez.
Equations for the stress functions Φ
θ
, Ψ
r
and Ψ z
are obtained from
Eqs. (13) by taking into account both the cylindrical coordinates and the form
of fields in the case of the second axially-symmetric problem. We have
0
⎧Ω Φθ
+ X
θ
= 0,
⎪
0 0
⎨ 3
Ω Ψ
r
+ Yr
= 0,
(36)
⎪
3Ω Ψ
z
+ Yz
= 0.
⎩
Therefore, if the set of functions { Φ
θ
, Ψr
, Ψz}
is a solution in B of the system
(36), then (35) is a solution in B of the second axially-symmetric problem (33).
The strain fields in B are given by the matrices
⎛ 0 γrθ 0 ⎞ ⎛κrr 0 κ
rz ⎞
⎜ ⎟ ⎜
⎟
γ = γθr 0 γθz , κ =
0 κθθ
0 ,
⎜ 0 γzθ 0 ⎟ ⎜
⎝ ⎠ ⎝κzr 0 κ ⎟
zz ⎠
whose elements are given by the geometric relations in B
γ = γ =∂ u − φ γ = φ −r u
−1
rθ r θ z
,
θr z θ
,
γ =∂ u + φ , γ =−φ
,
zθ z θ r θz r
κ =∂ φ , κ =∂ φ ,
rz r z zr z r
−1
( κrr , κθθ , κzz ) = ( ∂rφr , r φr , ∂zφz
).
The stress fields in B are defined by the matrices
(37)
(38)

Bul. Inst. Polit. Iaşi, t. LVIII (LXII), f. 3, 2012 19
⎛ 0 σrθ 0 ⎞ ⎛μrr 0 μrz
⎞
⎜ ⎟ ⎜
⎟
σ = σθr 0 σθz , μ =
0 μθθ
0 .
⎜ 0 σ 0 ⎟ ⎜
⎝ zθ ⎠ ⎝μzr 0 μ ⎟
zz ⎠
(39)
Knowing the amplitudes of the rotations
in (32), we determine the stresses
the constitutive relations in B
σij
φ , φ and the displacement u θ
given
r
z
and the couple-stresses
⎧( σrθ, σθr) = ( μ+ α)( γr θ, γθr) + ( μ−α)( γθr, γr
θ),
⎪( σθz, σzθ ) = ( μ+ α)( γθz, γz θ) + ( μ−α)( γz θr, γθz),
⎨
⎪( μrz , μzr ) = ( γ+ ε)( κrz , κzr ) + ( γ−ε)( κzr , κrz
),
⎪
⎩( μrr , μθθ ) = (2 γrr + βκ, γθθ + βκ), μzz = γzz
+ βκ,
where κ = κ + κ + κ = ∇·.
φ
rr θθ zz
6. Action of Body and Couple-body Loadings
Let us consider the special case of body loadings given by
μ
ij
by means of
(40)
X= (0,0, X ), Y = (0,0, Y ).
(41)
z
The loading X
z
is connected with the triple ( ur , φθ , uz
) which appears in the
first axially-symmetric problem characterized by equations (32), so that only the
body moment Y z
enters in the second axially-symmetric problem. This situation
may imply that only the component Ψ
z
does not vanish identically in relation
(36). Consequently, we obtain the following representation for the triple
( uθ , φr, φ
z)
in B :
z
where the stress function
0
⎧ uθ = 2 α
3
∂rΨz,
⎪
0 2 2
⎨φr
= − [( β+ γ−ε) 2
−4 α ] ∂rzΨz, .
⎪ 2 2
⎩φz
= {
2 3− [( β+ γ−ε) 2−4 α ] ∂z
z} Ψz,
Ψ z
satisfies
(42)
3
Ψ
z
Yz
0.
Ω + = (43)
Introducing another version of the stress function of that kind is connected with
body loadings of the form
X= ( X ,0,0), Y = ( Y ,0,0).
(44)
r
r

20 Ion Crăciun
Ψ r
Having assumed that in (30) only does not vanishes, we obtain the following
representation for a regular solution in B of the governing Eqs. (30) of the
second axially-symmetric problem
0
⎧ uθ = −2 α
3
∂zΨr,
⎪
0 0 0 2 −1
⎨φr
= {
2 3− [( β+ γ−ε) 2−4 α ] ∂
r( r +∂r)
} Ψ
r,
⎪
2 −1
⎪⎩ φz
= − {[( β+ γ−ε) 2
−4 α ] ∂
z( r +∂r)
} Ψ
r,
where the stress function
Ψ z
satisfies the differential equation
(45)
0 0
3
Ω Ψ + Y = 0.
(46)
r
r
For Y = Y = 0, the relationship between the stress functions Ψ and Ψ takes
r
z
0
the form
3
( ∂
rΨz +∂
zΨr)
= 0.
Finally, let us consider that the body loadings are
X= (0, X ,0), Y = (0, Y ,0).
(47)
The body force
Y θ
θ
θ
is connected to the displacements and rotation fields of the
first axially-symmetric problem. Assuming that only Φ
θ
does not vanishes
identically in Eqs. (36), we obtain the following representation for u θ
, ϕ
r
, and
ϕ
z
in B :
⎧
0
uθ
=
4Φθ,
⎪
⎨φr = 2 α∂zΦθ
,
(48)
⎪
−1
⎪⎩ φz = − 2 α( r +∂r)
Φθ
,
where Φ
θ
is a solution in B of the equation
0
Ω + =
Φ r
X θ
7. Concentrated Loadings
Supposing that Yr
= X θ
= 0, the corresponding solutions of the second
axially-symmetric problem can be obtained by means of differential equation
(43) which can be write in the form
( μ + α)( β+ 2 γ)( ∇ + k )( ∇ + k )( ∇ + σˆ
) Ψ = Y .
(49)
2 2 2 2 2 2
1 2 3
Having applied Fourier transformations, direct and inverse,
⎧
1 +∞
f%
(, r η) = f(,)exp(i r z ηz)d,
z
⎪
2π
∫−∞
⎨
1 +∞
⎪ f(,) r z = f%
(, r η)exp(i − ηz)dη
⎪⎩
2π
∫−∞
over z and Hankel transformations
z
0.
z
r
z
(50)

Bul. Inst. Polit. Iaşi, t. LVIII (LXII), f. 3, 2012 21
∞
⎧
⎪
fν( ξ, z) = ∫ rf( r, z)J ( ξr)d r,
0
ν
⎨
∞
⎪ f(, r z) = ξ f (, z)J( ξr)dξ ,
0
ν
ξ
ν
⎩ ∫
(51)
over r , to Eq. (49), with Yr
= 0,
X θ
= 0 being taken into account, one can
derived integral representations of the amplitude fields. Subsequently, having
taken the concentrated loading
M
0
Yz
= δ() r δ(), z Yr
= Xθ
= 0,
(52)
2π
r
from (49), and (52) we obtain the stress function Ψ
z.
The expression of Ψ
z
follows to be introduced in (42) to find the expressions of amplitudes of
displacement u θ
and rotations ϕ r
and ϕ
z.
The results are similarly to those
obtained by Dyszlewicz (2004), p. 175, when, in the place of (52), it was
considered that in the dynamical equations of the second axially-symmetric
problem the concentrated loadings are of the form
−iωt
M
0e
Yz = δ()(), r δ z Yr = Xθ
= 0.
2πr
REFERENCES
Achenbach J. D., Wave Propagation in Elastic Solids. North-Holland Publ. Company,
Amsterdam - London and American Elsevier Publishing Company, 1973.
Cosserat E., Cosserat F., Théorie des corps déformables. A. Hermann et Fils, Paris
1909.
Crăciun I. A., Half-space Problem in Micropolar Thermoelasticity. Bull. Acad. Polon.
Sci., Sér. Sci. Techn., 26, 251-260 (1978).
Crăciun I. A., Potential Method in Micropolar Thermoelasticity. Rev. Roum. Sci.
Techn., Sér. Mec. Appl., 22, 19-36 (1977).
Crăciun I. A., Radiation Conditions and Uniqueness Theorems in Micropolar Thermoelasticity.
Rev. Roum. Sci. Techn., Sér. Mec. Appl., 23, 165-180 (1978).
Dyszlewicz J., Micropolar Theory of Elasticity. Lecture Notes in Applied and
Computational Mechanics, 15, Springer-Verlag, 2004.
Ericksen J., Truesdell C., Exact Theory of Stress and Strain in Rods and Shells. Arch.
Rational Mech. Anal., 1, 295-323 (1958).
Eringen A. C., Foundations of Micropolar Thermoelasticity. CISM Udine, Springer,
1970.
Eringen A. C., Linear Theory of Micropolar Elasticity. J. Math. Mech., 15, 909-923
(1966).
Hetnarski R. B., (Ed.), Thermal Stresses 2. North-Holland Company, Amsterdam, 1987,
Ch. 5, pp. 269-328,.
Ieşan D., Thermoelastic Models of Continua. (Solid Mechanics and Its Applications).
Kluwer Academic Publ., London, 2004.

22 Ion Crăciun
Ignaczak, J., Tensorial Equations of Motion for Elastic Materials with Microstructure.
Trends in Elasticity and Thermoelasticity, Groningen, 89, 1971.
Kupradze V. D., Three Dimensional Problems of the Mathematical Theory of Elasticity
and Thermoelasticity. North-Holland, Amsterdam – New York – Oxford, 1986.
Nowacki W., Couple Stresses in the Theory of Thermoelasticity. III. Bull. Acad. Polon.
Sci., Sér. Sci. Techn., 14, 8, 505-514 (1966).
Nowacki W., Theory of Asymmetric Elasticity. PWN - Polish Scientific Publ.,
Warszawa and Pergamon Press, 1986.
Şandru N., On Some Problems of the Linear Theory of the Asymmetric Elasticity. Int. J.
Eng. Sci., 4, 1, 81 (1966).
Stefaniak J., Generalization of Galerkin's Functions for Asymmetric Thermoelasticity.
Bull. Acad. Polon. Sci., Sér. Sci. Techn., 16, 8, 391-399 (1968).
Teodorescu P. P., Dynamics of Linear Elastic Bodies. Ed. Acad. Române, Bucureşti,
1975.
Truesdell C., Toupin R., The Classical Field Theories. In Vol. III/1 of the Handbuch
der Physik (S. Flügge, Ed.), Berlin-Heidelberg, New York, Springer-Verlag,
1960.
ASUPRA UNOR PROBLEME AXIAL-SIMETRICE ALE TEORIEI ELASTICITĂŢII
MICROPOLARE
(Rezumat)
Se consideră sistemul ecuaţiilor diferenţiale fundamentale al vibraţiilor staţionare
din teoria elasticităţii micropolare a unui solid elastic izotrop, omogen şi cu simetrie
centrală care ocupă o regiune tridimensională B. Dacă necunoscutele sistemului sunt
amplitudinile deplasării u şi microrotaţiei φ , atunci expresia acestuia este
⎧⎪ 2u+ ( λ+ μ−α) ∇∇ · u+ 2α∇× φ+ X = 0,
⎨
⎪⎩ 2 α∇× u +
4φ+ ( β+ γ− ε) ∇∇·
φ+ Y = 0.
(1)
Folosind unele rezultate anterioare ale noastre (Crăciun, 1977), precum şi
rezultate ale multor cercecetori de pretudindeni, în deosebi ale celor din şcoala poloneză
fondată de Witold Nowacki (Nowacki, 1986), precum şi ale celor din Georgia,
îndrumaţi de V. D. Kupradze (Kupradze, 1986), se prezintă o reprezentare de tip
Galerkin a soluţiei sistemului (1), după care se scrie forma sistemului (1) în
coordonatele cilindrice (, r θ, z).
Se studiază apoi probleme axial-simetrice, când cauzele şi efectele nu depind
de θ. Sistemul fundamental (1), scris în coordinate cilindrice, se splitează în alte două,
studiul celui de al doilea
0
⎧
4
φr + ( β+ γ−ε) ∂rκ−2α∂ zuθ + Yr
= 0,
⎪
⎨ 4
φ β γ ε κ αr r ru Y
⎪
⎪⎩ u + 2 α( ∂ φ −∂ φ ) + X = 0,
−1
z
+ ( + − ) ∂
z
+ 2 ∂ (
θ
) +
z
= 0,
0
2 θ z r r z θ
(2)

Bul. Inst. Polit. Iaşi, t. LVIII (LXII), f. 3, 2012 23
constituind preocuparea principală a restului articolului. Astfel, folosind reprezentarea
de tip Galerkin a soluţiei sistemului (1), se arată că soluţia generală a sistemului (2) se
exprimă cu ajutorul a trei funcţii Ψ , Ψ şi Φ , soluţii ale sistemului diferenţial
r
z
0
⎧Ω Φθ
+ X
θ
= 0,
⎪ 0 0
⎨ 3
Ω Ψr
+ Yr
= 0,
(3)
⎪
⎪ 3Ω Ψ
z
+ Yz
= 0.
⎩
Se pot găsi soluţiile sistemului (3) în cazul unor încărcări masice particulare, precum şi
al acţiunii unor forţe şi cupluri de forţe concentrate de forma
M
0
Yz = δ()(), r δ z Yr = Xθ
= 0.
2πr
θ

BULETINUL INSTITUTULUI POLITEHNIC DIN IAŞI
Publicat de
Universitatea Tehnică „Gheorghe Asachi” din Iaşi
Tomul LVIII (LXII), Fasc. 3, 2012
Secţia
MATEMATICĂ. MECANICĂ TEORETICĂ. FIZICĂ
MODEL FOR THE RELIABILITY ASSESSMENT
IMPACT ON THE LIFE CYCLE COST ANALYSIS
APPLICABLE TO THE COMPLEX TECHNICAL SYSTEMS
BY
GHEORGHE NEGRU ∗
Military Equipment and Technologies Research Agency,
Bucureşti
Received: February 1, 2012
Accepted for publication: February 24, 2012
Abstract: The paper presents the mathematical apparatus of the Weibull
distribution failure rate model as well as a study case consisting in its application
to the specialized wheeled vehicle systems. The Weibull distribution failure rate
model could be used to asses the achievement the full readiness status of the
specialized wheeled vehicle systems from the inventory. Thus in order to ensure
the operational efficiency of the technical systems this model could contribute at
the problem of solving of an increased number of requirements with reduced
resources.
Key words: Weibull distribution failure rate model, reliability, specialized
wheeled vehicle systems.
1. Introduction
A technical system is "reliable" when it is able to accomplish its function
in a secure and efficient way throughout its life cycle. Thus the key aspect of the
term reliability is related to the operational continuity.
When a technical system begins to be affected by a large quantity of
accidental failure events (low reliability), this scenario causes high costs,
associated mainly with the recovery of the function (direct costs) and with
growing impact in the operational process (penalization costs).
The total costs of non-reliability (Woodhouse, 1993; Barlow et al., 1993)
can be characterized as follows:
∗ e-mail: gnegru@acttm.ro

26 GheorgheNegru
i) costs for penalization regarding the downtime (operational losses,
impact on quality, impact on security and environment).
ii) costs for corrective maintenance:
iii) manpower: direct costs related with the manpower in the event of an
unplanned action;
iv) materials and replacement parts: direct costs related to the consumable
parts and the replacements used in the event of an unplanned action.
In the technical literature (Adolfo, 2007) are presented three basic models
related to the non- reliability cost within the life cycle cost analysis (LCCA):
constant failures rate, deterministic failures rate and Weibull distribution
failures rate. These models include in their evaluation processes the
quantification of the impact that could cause the diverse failure events in the
total costs of a operational asset.
In the followings will be presented the Weibull distribution failure rate
model and its application to the case of the specialized wheeled vehicle systems.
2. Weibull Distribution Failure Rate Model
In terms of cost analysis structure, this model will estimate the nonreliability
cost with failure frequencies calculated from a Weibull distribution
function.
The model proposes the following sequential flow:
1. Identify, for each alternative to evaluate, the main types of failures.
This way for certain equipment there will be f=1… F failure modes.
2. Determine, for each failure mode, the “times to failure” (operational
times). This information will be gathered, based on failure records, databases
and/or experience of the maintenance and operations.
3. Calculate the cost of failures C f (in monetary units/failure). These costs
include replacement of parts, manpower, production loss penalization and
operational impact costs.
4. Determine the frequency of expected failures δ f using the Weibull
distribution function.
Will be used the following notations in Table 1
Variable
ζ
tf
MTBF
Γ
α
β
Table 1
Significance
Frequency of failures
Time between failures
Mean time between failures (inverse of the frequency)
Gamma function
Characteristic life
Shape parameter

Bul. Inst. Polit. Iaşi, t. LVIII (LXII), f. 3, 2012 27
If we assume that the random variable t f is distributed according to a
Weibull function of parameters α>0 and β>0, its density function is
The mean μ is
β−
( ) ( βt )
f t
The variance is given by
σ
−
β
t f
1 α
= e , for t ≥ 0. (1)
f f f
⎛ 1 ⎞
μ= αΓ ⎜1+
⎟
⎝ β
. (2)
⎠
⎛ ⎛ 2 ⎞ ⎛ 2 ⎞
= α
⎜
Γ ⎜1+ ⎟−Γ ⎜1+
⎟
⎝ ⎝ β ⎠ ⎝ β
⎞⎟ . (3)
⎠⎠
2 2 2
The MTBF is the expected value of the random variable t f , which is equal
to the mean μ,
⎛ 1 ⎞
MTBF = μ= αΓ ⎜1+
⎟
⎝ β
. (4)
⎠
The parameters α and β are calculated from the following expressions
⎛
F
F ⎛
⎞ ⎞
⎜
− ln
F ⎜∑
Afi
( tfi
) ⎟ ⎟
⎜
i f
A ⎝ = ⎠ ⎟
fi
⎜
∑
⎟
i=
f
α = exp⎜ ⎟, (5)
F
F
⎜⎛ ⎞ ⎛
⎞
Afi ln ( tfi ) Afi ln ( t
⎟
⎜⎜∑
− ⎟−⎜∑
fi ) ⎟⎟
⎝ i= f ⎠ ⎝ i=
f
⎠
⎜
⎟
⎝
⎠
where
β =
A
fi
F
∑
F
∑
i=
f
1
ln( t )
i=
f fi
A
fi
( 1−
ln( α)
)
, (6)
⎛ ⎛ ⎞⎞
⎜ ⎜ 1 ⎟⎟
ln ⎜ln
⎜
f
⎟⎟
⎜ ⎜1−
⎟
⎟
⎝ F + 1
=
⎝
⎠⎠ . (7)
ln( t )
fi

28 GheorgheNegru
In Eqs. (5)-(7), f is the number of the specific failure event, F is the total
number of evaluated failures and t fi is the time between the failures in issue
δ f
1
= . (8)
MTBF
5. Calculate the total costs per failures per year TCP f , costs generated by
the different failure events, with the following expression
TCP
F
= ∑ δ C . (9)
f f f
f
The equivalent total annual cost represents the value of the money that
will be needed, every year, to pay for the problems caused by failure events
during the expected years of useful life.
6. Calculate the total costs per failure in net present value NPV(TCP f ).
Given a yearly value TCP f this is the amount of money that needs to be saved
(today) to be able to pay this annuity for the expected number of years of useful
life T, and for a discount rate i. The expression used to estimate the NPV(TCP f )
is
T
( 1+ i)
−1
NPV ( TCPf ) = TCPf . (10)
T
i ( 1+
i )
Finally, as in previous models, the rest of the costs evaluated (investment,
planned maintenance, operations, etc.) are added to the cost calculated for nonreliability
and the total cost is obtained in annual equivalent value. The result is
then compared for the different options.
3. Case Study
3.1. Input Data
In the present study case will be applied the above presented method in
order to analyze the costs generated by the low reliability of a specialized
wheeled vehicle system (Fig. 1). The main components of the system are
presented in Table 2.
S2
S4 S5 S1
S3
Fig. 1 – A schematic view of the specialized wheeled vehicle system
The time between failures t f for 27 failures during 10 year time period for
a specialized wheeled vehicle system is defined based on the recorded statistics
data (Table 3).

Bul. Inst. Polit. Iaşi, t. LVIII (LXII), f. 3, 2012 29
Table 2
Main components of the specialized wheeled vehicle system
System
Destination
S1
Vehicle platform
S2
Communication and information system
S3
Electrono-optic system
S4
Software interface system
S5
Vehicle platform control engineering
Table 3
Failures synthesis
t f Failed system
Failure 1 5 S1
Failure 2 7 S2
Failure 3 3 S3
Failure 4 6 S1
Failure 5 2 S4
Failure 6 5 S5
Failure 7 3 S2
Failure 8 3 S3
Failure 9 7 S1
Failure 10 7 S2
Failure 11 2 S4
Failure 12 4 S5
Failure 13 5 S2
Failure 14 4 S3
Failure 15 2 S1
Failure 16 4 S2
Failure 17 3 S4
Failure 18 7 S5
Failure 19 6 S2
Failure 20 4 S2
Failure 21 4 S3
Failure 22 7 S1
Failure 23 4 S4
Failure 24 3 S5
Failure 25 5 S2
Failure 26 3 S3
Failure 27 4 S1
3.2. Output Data
According to those presented in Sec. 2:
i) identify, for each alternative to evaluate, the main failure modes (f)
where f=1…F, for F failure modes. We assume F=1 failure mode;
ii) determine the time between failures t f , see Table 3;

30 GheorgheNegru
iii) calculate the costs of failures C f (in monetary units/failure);
C = 1000.00 monetary units/failure ; (11)
f
iv) determine the expected frequency of failures δ f with the Weibull
distribution, (using Eqs. (4)–(8). The results of the calculus for the expected
frequency of failures are presented in Tables 4 and 5.
Table 4
Calculus results of the expected frequency of failures δ f
t f Ln(tfi) Afi 1/Ln(tfi)
Failure 1 5 1.609 0.748 0.621
Failure 2 7 1.945 0.618 0.514
Failure 3 3 1.098 1.096 0.91
Failure 4 6 1.791 0.672 0.558
Failure 5 2 0.693 1.736 1.443
Failure 6 5 1.609 0.748 0.621
Failure 7 3 1.098 1.096 0.91
Failure 8 3 1.098 1.096 0.91
Failure 9 7 1.945 0.618 0.514
Failure 10 7 1.945 0.618 0.514
Failure 11 2 0.693 1.736 1.443
Failure 12 4 1.386 0.868 0.721
Failure 13 5 1.609 0.748 0.621
Table 5
Calculus results of the expected frequency of failures δ f
t f Ln(tfi) Afi 1/Ln(tfi)
Failure 14 3 1.098 1.096 0.91
Failure 15 4 1.386 0.868 0.721
Failure 16 2 0.693 1.736 1.443
Failure 17 4 1.386 0.868 0.721
Failure 18 3 1.098 1.096 0.91
Failure 19 7 1.945 0.618 0.514
Failure 20 6 1.791 0.672 0.558
Failure 21 4 1.386 0.868 0.721
Failure 22 4 1.386 0.868 0.721
Failure 23 7 1.945 0.618 0.514
Failure 24 4 1.386 0.868 0.721
Failure 25 3 1.098 1.096 0.91
Failure 26 5 1.609 0.748 0.621
Failure 27 3 1.098 1.096 0.91
α = 2.016 , (12)
β = 4.027 , (13)

Bul. Inst. Polit. Iaşi, t. LVIII (LXII), f. 3, 2012 31
⎛ 1 ⎞
MTBF = μ= αΓ ⎜1+ ⎟= 2.016×Γ ( 1.24827)
= 2.016× 0.90675 = 1.82months,
⎝ β ⎠
(14)
where the value Γ(1.24827) =0.90675 is obtained form the tabulated values
chart of the Γ function. Then the frequency of failures will be
δ = 1/1.82 = 0.549 failures/month = 6.59 failures/year
f
Calculate the total costs per failures per year TCP f (using Eq. 9)
TCP = 6593 monetary units/year.
f
(15)
(16)
The equivalent annual total cost of 6593 monetary units represents the
value of money that will be needed every year to pay for the problems caused
by failures, during the 10 years of expected useful life;
Calculate the total costs per failures in present value NPV(TCP f ) (using
Eq. 10), for a period T=10 years and discount rate i=9%:
10
( 1+ 0.09)
−1
× ( + )
NPV ( TCP f
) = 6593 = 42631 monetary units.
10
0.09 1 0.09
4. Conclusions
(17)
1. The Weibull distribution failure rate model could contribute to the
enhancement of the specialized wheeled vehicle systems performances by
diminishing the uncertainty within the process of total life cycle cost estimation.
2. The calculus example emphasized the strong points and weak points of
the presented method as follows:
a) the obtained annual equivalent total cost, represent the mean value of
money that will be needed every year to pay for the problems of failures, during
the 10 years of expected useful life, with a discount factor of 9%. The frequency
of failures varies every year throughout the expected cycle of useful life.
b) in order to conduct an accurate assessment of the process of reliability
it is necessary that the data gathering must be based on a detailed and good
quality information about the more important types of failures and, if possible,
for similar operational conditions.
3. The main limitations of this method are:
a) regardless the use of the Weibull distribution, the model quantifies the
impact of the non-reliability annual costs in a constant way over the years of
expected asset useful life;
b) it restricts the reliability analysis to the exclusive use of the Weibull
distribution, excluding other existent statistical distributions, which could also
be used in the calculation of the MTBF and failure frequencies.

32 GheorgheNegru
4. Further research will develop failures analyses methodologies for the
components of specialized wheeled vehicle systems.
REFERENCES
*** Maintenance Terminology. European Standard. CEN (European Committee for
Standardization), EN 13306:2001, Brussels, 2001
Adolfo C. M., The Maintenance Management Framework. Springer-Verlag, London
Limited, 2007
Alfares H.K., Aircraft Maintenance Workforce Scheduling: A Case Study. J. of Qual. in
Maint. Eng., 5 (1999).
Barlow R. E, Clarotti C. A., Spizzichino F., (Eds.) Reliability and Decision Making.
London, Chapman and Hall, 1993.
Barlow R.E., Hunter L.C., Optimum Preventive Maintenance Policies. Operations
Research, 1960.
Barlow R.E., Hunter L.C., Proschan F., Optimum Checking Procedures. Journal of the
Society for Industrial and Applied Mathematics, 4, 1978-1095 (1963).
Duffuaa S. O. Mathematical Models in Maintenance Planning and Scheduling. in Ben-
Daya M, Duffuaa S.O. Raouf A (Eds.), Maintenance Modeling and Optimization,
Kluwer, Boston, USA, 2000.
Duffuaa S. O., Raouf A., Campbell J. D.,Planning and Control of Maintenance
Systems: Modeling and Analysis. Wiley, NY, 1999.
Ebeling C., Reliability and Maintainability Engineering. McGraw Hill, New York,
1997.
Krause A., Musingwini C., Modelling Open Pit Shovel-truck Systems Using the
Machine Repair Model. The Journal of The Southern African Institute of Mining
and Metallurgy, 107 (2007).
Mohamed B. D., Salih O. D., Abdul R., Jezdimir K., Daoud A. K., Handbook of
Maintenance Management and Engineering. Springer-Verlag, London Limited,
2009.
Woodhouse J., Managing Industrial Risk. Chapman Hill, London, 1993.
Woodward D. G., Life Cycle Costing Theory. Information, Acquisition and Application.
International Journal of Project Management, 15(6), 335-344 (1997).
MODEL PENTRU EVALUAREA IMPACTULUI FIABILITĂŢII
ASUPRA ANALIZEI COSTULUI PE DURATA DE VIAŢĂ
APLICABIL SISTEMELOR TEHNICE COMPLEXE
(Rezumat)
Lucrarea prezintă aparatul matematic al modelului cu rata distribuţiei
defecţiunilor de tip Weibull precum şi un studiu de caz prin care acest model a fost
aplicat unui sistem complex de tip vehicul specializat pe roţi. Modelul cu rata
distribuţiei defecţiunilor de tip Weibull se poate utiliza pentru evaluarea obţinerii unei
operativităţi ridicate pentru sistemele complexe de tip vehicul specializat pe roţi. Astfel

Bul. Inst. Polit. Iaşi, t. LVIII (LXII), f. 3, 2012 33
pentru a se asigura eficienţa operaţională a sistemelor tehnice acest model poate
contribui la rezolvarea unui număr ridicat de cerinţe cu resurse limitate.

BULETINUL INSTITUTULUI POLITEHNIC DIN IAŞI
Publicat de
Universitatea Tehnică „Gheorghe Asachi” din Iaşi
Tomul LVIII (LXII), Fasc. 3, 2012
Secţia
MATEMATICĂ. MECANICĂ TEORETICĂ. FIZICĂ
ANALYSIS OF A CIRCULAR ECCENTRIC CAM
BY
RADU IBĂNESCU ∗1 and CĂTĂLIN UNGUREANU 2
”Gheorghe Asachi” Technical University of Iaşi,
1 Department of Theoretical Mechanics
2 Department of Machine Tools
Received: May 28, 2012
Accepted for publication: July 15, 2012
Abstract. Circular eccentric cam mechanisms are components of various
mechanical devices. The cam follower can get stuck into its guide if the
mechanism dimensions are not properly designed. The paper analyses how this
situation can be avoided by a correct choice of mechanism dimensions based on
solving a series of equilibrium equations. The calculus is performed using the
Mathcad software.
Key words: circular eccentric cam, cam follower stick, proper mechanical
design.
1. Introduction
The circular eccentric mechanism is well known in mechanical
engineering. This kind of mechanism is very simple and does not create difficult
problems in its design (Sclater & Chironis, 2007), (Beer et al. 2007), (Hibbeler,
2007), (Meriam & Kraige, 2007), (Rosculet, 1982). Nevertheless, a troublesome
event can occur, namely the sticking of the cam follower in the cam follower
guide. This problem can be avoided if the length of the cam follower guide is
properly chosen. The choice can be made after a thorough analysis of the forces
acting on the system and solving the equations of equilibrium of the mechanism
parts. On the other hand, even if the cam follower does not stick, it is possible
for a very large moment to be required for the cam to develop its motion. A
∗ Corresponding author: e-mail: ribanesc@yahoo.com

36 Radu Ibănescu and Cătălin Ungureanu
Φ
d
F
B
h
C
A
l
R
M m
e
O
ϕ (t)
G
a
Fig. 1 – Eccentric cam.
h i
diagram is obtained showing the dependence of the cam follower length with
respect to the moment acting on the cam, so that the cam follower does not
stick. The required length of the cam follower guide can be chosen from this
diagram.
2. The Eccentric Cam Mechanism
The eccentric cam mechanism consists of a disk of radius R actuated by a
couple M m (Fig. 1). The disk can rotate about the fix point O located at the
distance e from the centre of the circle. The cam follower is vertical and it has a
horizontal bar at each of its ends. The bar at the bottom end lies on the cam. The
bar at the top end is actuated by a spring having the spring stiffness k e and by a
constant force F collinear to the spring. The disk weight is G and the weight of
the cam follower is assumed to be very small and hence negligible.
The friction coefficient between the cam follower and the disk and the
cam follower and its guide is μ. The friction coefficient in the joint at O is μ 1 .
All the dimensions are known and can be seen in Fig. 1. The problem is to
determine the minimum value of the driving couple M m to maintain the
mechanism in equilibrium at limit state. The disk and the cam follower are

Bul. Inst. Polit. Iaşi, t. LVIII (LXII), f. 3, 2012 37
drawn separately with all the forces (free body diagrams) in order to solve this
problem.
3. The Equations of Equilibrium
The cam follower is represented in Fig. 2. It is assumed that there is a
small clearance between the cam follower and its guide. For this reason, the
contact between the cam follower and its guide is at the points A and B, as it can
be seen in Fig. 2.
y
Φ
d
F r
+
r
F e
B
N r B
h
l 1
N r
C
N r
A
A
F r
fA
F r fB
F r fC
C
q
O
x
Fig. 2 – The cam follower of the mechanism.
The three equilibrium equations of the cam follower are:
− FfC + NA − NB
= 0 , (1)
N − F − F − F − F = 0 , (2)
C
fA
fB
− N q − N l + N l + h)
− ( F + F )( d + Φ)
− F Φ 0 . (3)
C
The length l 1 is
The length q is
A 1 B ( 1
e
fB =
e
l1 = l−R− esin
φ . (4)

38 Radu Ibănescu and Cătălin Ungureanu
q= a− ecos φ.
(5)
The elastic force has the value 10 N when the cam follower is at the
lowest position, when is the angle φ is 3 π . For this reason, the elastic force is
2
NC
F = 10 + k (1 + sin φ)
. (6)
e
e
The normal forces at the points A, B and C are denoted N , N and
A B
F fC
and the friction force at the same points are denoted F , F and .
The disk is represented in Fig. 3.
y
fA
fB
F r fC
V r
N r C
R
H r
M m
M f O
ϕ(t)
G
x
Fig. 3 – The disk.
The reaction forces at the joint
moment at the joint O is
M rμ H V
O are denoted H and V. The friction
2 2
f
=
1
+ , (7)
where r is the radius of the shaft.
The three equilibrium equations for the disk are:
− H + = 0 , (8)
FfC
V − G − N = 0 , (9)
C
M − M −N ecosφ −Gecos φ − F ( R+ esin φ) = 0 .
(10)
m f C fC

Bul. Inst. Polit. Iaşi, t. LVIII (LXII), f. 3, 2012 39
The friction forces are FfA
= μN
A, FfB = μN
B
and FfC = μNC
because the
friction is considered at limit state.
The system of the six equatio ns of equilibrium is then symbolically
solved.
3. The Solution of the System Equilibrium Equations
In order to obtain the symbolic solution of the system of equilibrium
equations ( 1), (2) and (3), the following two matrices are built in Mathcad
(Maxfield, 2009)
⎛ 1 −1 −μ⎞
⎜
⎟
M = −μ −μ
1
⎜− 11 11+ h − μΦ −q
⎟
⎝
⎠
(11)
⎛ 0 ⎞
⎜
⎟
v: = F + Fe
. (12)
⎜( F + Fe
)( d + Φ)
⎟
⎝
⎠
The symbolic solution is obtained by the command
Insolved( M,v) =
2 2 3 2 2
⎡ 2 μ ( Fh+ Feh + FμΦ + FeμΦ + 2Fμd+ 2 Feμd) + ( F+ Fe)( μ h−μ Φ − h+ 2μ 11 + μΦ
+ 2 μq) -( μ − 1)( Fh + Feh+ FμΦ
+ FeμΦ
+ 2Fμd + 2 Feμd
) ⎤
⎢
⎥
2 3 2
2 μμh ( − μ Φ − h + 2μ 11 + μΦ
+2 μq)
Fh + Feh+ FμΦ
+ FeμΦ
+ 2Fμd + 2Feμd
⎢
2 3 2
⎥
⎢
μ h −μ Φ − h+ 2μ 11 + μΦ
+2μq
⎥
⎢
⎥
⎢
⎥
⎣
⎦
⎢
⎢
⎥
⎥
⎢
2 3 2 2
( F + Fe )( μ h − μ Φ−η+ 2μ 11 + μΦ + 2 μq) -( μ − 1)( Fh + Feh+ FμΦ + FeμΦ
+ 2Fμd + 2Fe
μd)
⎥
⎢
−
⎥
2 3 2
⎢
2 μμh ( − μ Φ − h + 2μ 11 + μΦ
+2 μq)
⎥
= ⎢
⎥
⎢
⎢
⎥
⎥
The three elements of the previous matrix are the unknowns N A , N B and N C .
Hence, the normal force is
NC
Fh + F h + FμΦ
+ F μΦ
+ 2Fμd + 2F μd
NC =
μ h −μ Φ −h + 2μ 11+ μΦ
+ 2μq
e e e
.
2 3
2
(13)
The cam follower is blocked if the force NC
is infinite, that is when the
denominator of the fraction is zero. In this case the distance h is
2 11
h =−
2 3
μ − μ + μ +2μ
Φ Φ q
2
μ −1
(14)
In fact, the distance h must be greater then the value given by Eq. (14) in
order for the cam to be able to rotate.

40 Radu Ibănescu and Cătălin Ungureanu
4. Numerical Example
The following values are considered for a numerical example: R=0.05 m,
l=0.2 m, e=0.02 m, a=0.04 m, Φ=0.01 m, d=0.01 m, r=0.005 m, k e =500 N/m,
G=30 N, μ=0.02, μ 1 =0.01, F=200 N and φ=π/2 (for the most inconvenient
situation). In this case, all the unknowns will be functions of the distance h. The
following expressions for the unknowns N A , N B and NC
are obtained by
using these numerical values
⎡
0.184h
+ 0.5759568
⎤
⎢
⎥
⎢
0.039984h
− 0.0000761568
⎥
⎢ − 0.57561606144h
+ 0.001096365477888
⎥
Insolved( M,v)
→ ⎢⎢
.
(0.039984h−0.0000761568)( − 0.9996h+
0.00190392) ⎥
⎥
⎢
9.2h
+ 0.00552
⎥
⎢
⎣
0.039984h
− 0.0000761568
⎥
⎦
(15)
The normal force N C is then
9.2h
+ 0.00552
NC( h) =
.
0.039984h
− 0.0000761568
(16)
The normal force N C is infinity for h=0.00190468 m.
The unknowns H, V and M m are the solution of the very simple system of
equations (8), (9) and (10). The driving torque M m is given by the following
function of the distance h:
2 2 2
Mn( h) = μr μ NC( h) + ( G+ NC( h)) + NC( h)[ ecos φ+μ( R+esin φ)] + Gecosφ.
(17)
30
20
10
Mm( h)
0
− 10
− 20
− 30
110 × − 3 210 × − 3 310 × − 3 410 ×
− 3
h
Fig. 4 – The function M m (h).

Bul. Inst. Polit. Iaşi, t. LVIII (LXII), f. 3, 2012 41
The diagram of the function given by Eq. (16) is represented in Fig. 4.
The diagram in Fig. 4 shows there is a vertical asymptote at the value of h
for which the normal force N C is infinity. At that point the mechanism is
blocked. The driving torque M m is decreasing very quickly after this value. For
example, if h=0.002 m the driving torque is M m is 9.102 Nm, if h=0.003 m the
driving torque M m is 1.098 Nm, for h=0.004 m the driving torque M m is 0.734
Nm and for h=0.025 m the driving torque M m is 0.371 Nm. The value of h is
chosen in accordance with each particular situation. The recommendation for
the ratio h/Φ is usually to be 2.5 according to a ‘rule of thumb’ commonly used
in mechanical design. This study shows that, in certain situations, the ratio h/Φ
can be chosen to be less than 2.5, a case in which a compromise has to be made
between the length of the guide and the required driving torque. This may be the
case when the cam follower guide length is restricted by certain design
constraints.
5. Conclusions
1. The eccentric cam mechanisms can get jammed if the length of the cam
follower guide is not chosen properly.
2. The equations of equilibrium can reveal the minimum value for the
cam follower guide.
3. The diagram of the driving torque with respect to the length of the cam
follower guide is obtained. Based on it, the proper value for the cam follower
guide can be chosen.
4. It is not recommended to choose a value for the length of the cam
follower guide near the value for which the driving torque becomes infinity
because the necessary driving torque for the mechanism movement will be very
large, and possibly unacceptable.
5. The common recommendation for the ratio between the length of the
cam follower guide and the diameter of the cam follower is for it to be around
2.5. This paper emphasises that in certain situations when the length of the cam
follower guide is constrained, a lower value of the ratio can be chosen, but the
required driving torque will increase.
REFERENCES
Beer F. P., Johnston E. Jr., Eisenberg E. R., Vector Mechanics for Engineers. Statics,
Eighth Edition, 2007.
Hibbeler R. C., Engineering Mechanics. Statics, Eleventh Edition, 2007.
Meriam J. L., Kraige L. G., Engineering Mechanics. Statics, Sixth Edition, 2007.
Maxfield B., Essential Mathcad for Engineering. Science and Mathematics Second
Edition, 2009.
Roşculeţ S. V., Gojineţchi N., Andronic C., Şelaru M., Gherghel N., Proiectarea
dispozitivelor. 1982.

42 Radu Ibănescu and Cătălin Ungureanu
Sclater N., Chironis N. P., Mechanisms and Mechanical Devices Sourcebook. Fourth
Edition, 2007.
*** Mathcad Tutorial. 2005.
ANALIZA UNEI CAME CIRCULARE EXCENTRICE
(Rezumat)
Camele circulare excentrice sunt mecanisme componente ale multor dispozitive
mecanice. În anumite situaţii tachetul acestor came se poate bloca în ghidajul acestuia
dacă dimensionarea nu este corectă. Lucrarea analizează modul în care se poate evita
blocarea tachetului printr-o alegere corectă a lungimii ghidajului bazată pe rezolvarea
unor sisteme de ecuaţii de echilibru. Rezolvarea problemelor calculatorii este făcută cu
ajutorul programului Mathcad.

BULETINUL INSTITUTULUI POLITEHNIC DIN IAŞI
Publicat de
Universitatea Tehnică „Gheorghe Asachi” din Iaşi
Tomul LVIII (LXII), Fasc. 3, 2012
Secţia
MATEMATICĂ. MECANICĂ TEORETICĂ. FIZICĂ
CORRECTION METHODS OF THE BIOMECHANICAL MODEL
IN THE CASE OF FOREARM MOVEMENT
BY
PETRONELA PARASCHIV ∗
“Gh. Asachi ” Technical University of Iaşi,
Department of Physical Education and Sports
Received: May 10, 2012
Accepted for publication: June 15, 2012
Abstract. The simplifications that are made in the case of biomechanical
models lead to the necessity of `correcting` the models achieved through
permanent comparison with the experimental registered values. There are
discussed three correction methods of biomechanical model in order to
approach the experimental model to the theoretical one. For the situation in
which there is the need to compare two curves, an experimental one, given by
the relative values and a theoretical one, given by absolute values, the
interpolation can allow finding a convenient function of interpolation, the same
one for both curves to be compared, in which the coefficients of the
mathematical function differ from one curve to the other. The interpolation
function must satisfy the condition that it represents the best approximation for
any other two curves, experimental and theoretical, that express the variation of
the same physical phenomenon.
Key words: biomechanical model.
1. Introduction
At the time being there are no well defined mathematic methods of
correction of an analytical model, but depending on a situation or another there
can be applied diverse “local solution”. The simplifications that are made in the
∗ e-mail:petrouti@yahoo.com

44 Petronela Paraschiv
case of biomechanical models lead to the necessity `correcting` the models
achieved through permanent comparison with the experimental registered
values (Fisher et al., 2007).
Therefore, if through experiment can be achieved numerical values,
respectively, a matrix of real values, then, applying the method of correlation
coefficient between the matrix of real values and the matrix of theoretical
values, it is determined the correlation degree between values or, other way
stated, in which measure the variation of a variable from the analytical model
has as effect a proportional variation of the values achieved through
experimental determinations (Azar & Calandruccio, 2008). Moreover, it can be
applied the method of perturbation functions over the analytical model in such a
way that the values theoretically determined to coincide or to be very closed to
those experimentally determined.
However, if through the experiment cannot be achieved quantitative
information, but only qualitative, then it would be possible the analysis of all
variation charts of experimental sizes, respectively the determination of
interpolation functions that will approximate the best the analysed curves, for
each portion of curve, and then the comparison of these functions with the ones
corresponding to the analytical model and the effectuation of the afferent
changes, if it is imposed, in the theoretical model (Marciuk, 1983).
The comparison of these curves can reveal different speeds of variation,
through curves gradient, of the analyzed size, so that one can intervene in the
theoretical model for approaching it to the experimental one
2. Theoretical Approach on Correction Methods
There are discussed three correction methods of biomechanical model in
order to approach the experimental model to the theoretical one.
2.1. The Correlation Coefficient Method
Correlation is a descriptive static method that shows the existence or not
of a link between two analyzed variables, but does not highlight the causality
relation. Therefore, between the forces of the tendon achieved through
experimental and analytical path can be established the type of link between
them, respectively the linear or non-linear link, proportional or converse
proportional, of weak, medium or strong or lack of strength, i.e. the
independence of the two variables (Antonescu et al., 1986). The determination
of the correlation starts from calculating the note or the standard score, that
shows the deviation of a given value of the analysed variable from the
arithmetic average of all values, this deviation being expressed in standard
deviations. The standard note has the expression
xi
− x
Zi
= , (1)
σ

Bul. Inst. Polit. Iaşi, t. LVIII (LXII), f. 3, 2012 45
where x i is a value of the analyzed variable, x is the arithmetic average of all
values and σ is the standard deviation, calculated with
1 n i
n i = 1
∑ ( ) 2
, (2)
σ = x −x
n being the total number of values of the variable. In this way, there can be
calculated the standard notes for the values of the first and second variable.
These notes can be expressed with Eq. (3)
Z
1
i
1 1
xi
− x
= ,
(3)
1
σ
for the first variable and
2 2
2 xi
− x
Zi
= ,
(4)
2
σ
for the second variable.
The correlation coefficient can be calculated using the Person correlation
formula such as
r =
n
∑
i=
1
Z Z
1 2
i i
n
. (5)
The correlation coefficient can have values between –1 şi +1, with the
following meanings:
r = –1, the link between variables is perfect and converse proportional;
r = 0, between variables there is no link, respectively variables are
independent;
r = +1, the link between variables is perfect and direct proportional;
r ∈ (–1, 1 )\ 0, ithout value 0, the link is not perfect between the two variables,
meaning that for the variation in a variable will correspond a smaller variation
within the other, converse or direct proportional; in this situation, between the
two variables there is a regressive variation, being also possible to establish the
corresponding equation of regression.
For being able to calculate the correlation coefficient there are necessary,
therefore, the values of the two variables that are analysed. These values are,
actually, matrixes with a single dimension, with n values, to which apply the
usual calculations for statistic analysis, for checking the two arrays of data, as
well as achieving the statistical values.
2.2. The Method of Perturbation Functions
In the case of some “converse problems” there can be applied methods of
the perturbations theory with the help of which one can generate algorithms of

46 Petronela Paraschiv
calculation in order to appreciate the influence of diverse factors (parameters)
over a variable, starting from the experimental determined values of that
variable. As “converse problems” can be considered the following types:
i) problems that ask the state determination of a physical process in
anterior moments of time, on the basis of some measurements done over a
variable;
ii) problems that ask for the reconstitution of a physical operator with a
mathematic structure of a known form, but with unknown coefficients, which
can be established on the basis of the information over some functional of the
experimental determined variable.
Therefore, supposing that it was effectuated a set of measurements
(named “functional”) and each functional is associated a function of influence
for the undisturbed problem , i.e. a model in which operator L and its domain of
definition are considered known, there will be solved n problems as such
* *
L φ = p , i = 1, 2, ...., n. (6)
pi
i
*
They are before those n functions of influence φ pi
and it is solved a basic
problem with the operator model “undisturbed L, adjunct of L,
Lφ = q.
(7)
There are built n formulas of the theory of small perturbations as such
*
φ δLφ)= - δJ , i = 1, 2, ...., n, (8)
( pi pi
where δL is the difference between the studied operator L’ and the model one
L, while J pi represents the set of functionals (measurements).
We suppose that the operator L is known, such as
m
L = ∑ [ αuAu + Bu(
βuCu)],
(9)
u−1
where A u , B u and C u are elementary linear operators, for instance differentials or
integral or combination of these ones; αu
( x ) and βu
( x ) are searched
coefficients, usually known with a harsh approximation for the undisturbed
problem (model).
The aim of the mathematic demarche is that of reconstructing the
'
coefficients and β from the expression
α u
'
u
m
'
'
' = [
k k
+
k( k k)
k = 1
L ∑ α A B β C ]. (10)
With the help of the Eqs. (7) and (8) it is obtained
m
δL = ∑ [ δαuAu + Bu( δβuCu)],
(11)
u=
1

Bul. Inst. Polit. Iaşi, t. LVIII (LXII), f. 3, 2012 47
'
'
where: δα = αu -αu, δβ= βu -βu.
If δαu
( x ) and δβu
( x)
are functions, then the solution to the converse
problem may be learnt with the aid of one of the methods of parameterisation,
which consists of essentially, the following aspects: it is supposed that on the
basis of the preliminary analysis of the physical parameters behaviour there are
found some complete orthogonal systems of functions u k,l (x) and v k,l (x) so that
with their aid it is possible a very good approximation of the functions α k
and
β
k
for a small number n(k) it is given by the functions
nk ,
δα ( x) = ∑a u ( x),
u kl kl
l=
1
(12)
nk ,
δβ ( x) = ∑ b v ( x),
(13)
u kl kl
l = 1
where a kl and b kl are the coefficients that need to be determined.
On the basis of some more complex mathematical considerations it can
'
'
be obtained a first approximation for sizes α and β , named through using the
successive approximations method.
The mathematic method of the perturbation functions, described
succinctly before, supposes the knowledge of the values for the analysed
variable, respectively the set of functionals J pi .
Not quantifying these functional, as well as the case of identification of
only qualitative force from the tendon of Achilles, leads to the impossibility to
apply this method in the case of biomechanical analysis effectuated at the level
of the ankle.
2.3. The Interpolation Method
The problem with the interpolation of given sizes on a discrete amount of
points on the entire defining domain of a continuous argument function is
linked to the construction of variation diagrams with differences and to
continuous representation of the solutions to the problems with differences.
The solution to the problem with differences represents, usually, the
approximate solution to the initial problem on a discreet amount of points. The
accuracy order of the data interpolation must be correlated with the accuracy
order of the approximation of the problem with differences and must not be
smaller than this one.
The algorithms of the interpolation of the functions after exact data,
defined on a discrete amount of points, are based, generally, on using the
interpolative polynomials of Lagrange. In this situation it is supposed that the

48 Petronela Paraschiv
interpolate functions φ( x)
have continuous derivates until a random order
(Elias et al., 2004; Tannous et al., 1996; Wagenaar & von Emmerik, 1996).
The interpolation of a discreet set of data [x, y] supposes the
determination of a function f(x) so that f(x) = y, in order to complete the set of
data in any other point x 0 ≠ x 1 . The function f(x) represents the best
approximation of the set of data, in this case not being necessary for the
determined function to pass through all the given points, but it has to be “the
best approximation” after a certain error criterion imposed. Therefore, the
method of the smallest squares gives, for instance, the best approximation in the
sense of minimizing the squre distances from the given points and the
approximation points.
3. Conclusions
1. Applying an interpolation function, linear, spline cubic or polynomial,
for the whole set of data or only for some periods of the values, determines the
interpolation function coefficients so that the interpolation curve to pass through
all given value points.
2. The cubic spline curve is an even curve, defined by a set of
polynomials of the third degree. The curve between each pair of points is a
polynomial of the third degree calculated in such a way as to lead to even
transitions from a polynomial of the third degree to the other.
3. For the situation in which two curves need to be compared, one
experimental given by relative values, and one theoretical given by absolute
values, the interpolation may allow finding a convenient function of
interpolation, the same for both curves to be compared, in which the mathematic
function coefficients differ from ne curve to the other.
4. The interpolation function found needs to satisfy the condition that it
represents the best approximation for any other two curves, experimental and
theoretical, that express the variation of the same physical phenomenon.
REFERENCES
Antonescu D., Buga M., Constantinescu I., Iliescu M., Metode de calcul şi tehnici
experimentale de analiza tensiunilor în biomecanică. Ed. Tehnică, Bucureşti,
1986.
Azar F. M., Calandruccio J. H., Arthroplasty of the Shoulder and Elbow. (Canale, S. T.,
Beaty, J. H., Eds.) Campbell’s Operative Orthopaedics, 11th Ed. Mosby,
Phyladelphia, 2008.
Elias J.J., Wilson D.R., Adamson R., Cosgarea A.J., Evaluation of a Computational
Model Used to Predict the Patellofemoral Contact Pressure Distribution. Journal
of Biomechanics, 37(3), 295-302 (2004).

Bul. Inst. Polit. Iaşi, t. LVIII (LXII), f. 3, 2012 49
Fisher D. M., Borschel G. H., Curtis C. G., Clarke H.M., Evaluation of Elbow Flexion
as a Predictor of Outcome in Obstetrical Brachial Plexus Palsy. Plast. Reconstr.
Surg., 120, 1585-90 (2007).
Marciuk G.I., Metode de analiză numerică, Ed. Academiei, Bucureşti, 1983.
Tannous R.E., Bandak F. A., Toridis T. G., Eppinger R. H., A Three-Dimensional Finite
Element Model of the Human Ankle: Development and Preliminary Application to
Axial Impulsive Loading, SAE 962427, Proceedings of the 40th Stapp Car Crash
Conference, 1996, pp. 219–238.
Wagenaar R.C., von Emmerik R.E.A., Dynamics of Movement Disorders. Human
Movement Science, 15, 161-175 (1996).
METODE DE CORECŢIE ALE MODELULUI BIOMECANIC ÎN CAZUL
MIŞCĂRII ANTEBRAŢULUI
(Rezumat)
Simplificările realizate în cazul modelelor biomecanice conduc la necesitatea
corecţiei modelelor realizate prin compararea permanentă cu valorile înregistrate
experimental. Sunt discutate trei metode de corecţie a modelului biomecanic cu scopul
de a transforma modelul experimental într-unul teoretic. În situaţia în care este necesară
comparaţia a două curbe, una experimentală obţinută prin valori relative şi una teoretică
obţinută prin valori absolute, interpolarea poate permite aflarea unei funcţii de
interpolare convenabile, aceiaşi pentru ambele curbe comparate, în care coeficienţii
funcţiei matematice diferă de la o curbă la alta. Funcţia de interpolare trebuie să
satisfacă condiţia de a reprezenta cea mai bună aproximare între oricare două curbe
(experimentală şi teoretică).

BULETINUL INSTITUTULUI POLITEHNIC DIN IAŞI
Publicat de
Universitatea Tehnică „Gheorghe Asachi” din Iaşi
Tomul LVIII (LXII), Fasc. 3, 2012
Secţia
MATEMATICĂ. MECANICĂ TEORETICĂ. FIZICĂ
GENERAL CONSIDERATIONS CONCERNING THE COLOR
IN THE DENTAL PRACTICE
BY
IRINA GRĂDINARU ∗ , ELENA RALUCA BACIU and DANIELA CALAMAZ
“Gr.T.Popa” University of Medicine and Pharmacy of Iaşi,
Faculty of Dental Medicine
Department of Dental Materials
Received: June 20, 2012
Accepted for publication: July 30, 2012
Abstract. Nowadays, in dentistry, the determination of the color continues
to represent a delicate problem. In the dental office, the doctor can determine
objectively only the identity of the colors or the difference between colors. But
from these findings, to the color characterization and transmission in dental
laboratory is a long and difficult way. This paper presents a synthesis of data
from the literature concerning color which, together with translucency, gloss and
fluorescence, contributes to the aesthetic qualities of dental restorations. For
determining the color can be used: the visual method (subjective), the simplest
and the most common, performed using color key or technical methods
(objectives), more complex and require expensive technical equipment.
Whichever method is used, a more precise determination of color contributes to
the success of the integration of the restoration in the dento- somato- facial
equilibrium.
Key words: dental materials, color, shade matching.
1. Introduction
Nowadays, in dentistry, the determination of the color continues to
represent a delicate problem. In the dental office, the doctor can determine
objectively only the identity of the colors or the difference between colors. But
∗ Corresponding author: e-mail: irigrad@yahoo.com

52 Irina Grădinaru et al.
from these findings, to the color characterization and transmission in dental
laboratory is a long and difficult way .
This paper presents a synthesis of data from the literature concerning
color which, together with translucency, gloss and fluorescence, contributes to
the aesthetic qualities of dental restorations.
The color of a natural dental structure is influenced by the thickness of
the enamel and the color of the subjacent dentin. Similar considerations can be
applied to dental restorations with layered structure (eg porcelain restorations
which are composed of body porcelain over inner opaque porcelain and resin
composites over more opaque resins). In case of these layered structures, diffuse
reflectance and the relationship between the translucency and thickness of the
outer layer and the color and reflectance of the inner layer represent challenges
to esthetic concerns (Fig.1). The outer translucent layer acts as a light-scattering
filter over the inner layer. As the thickness of the outer layer increases, the
effect of the inner layer is diminished. Similar situations exist when the
translucency of the outer layer decreases (Bratu, 1994; O’Brien, 2008).
Fig.1 – Porcelain–fused-to-metal indirect restoration: optical considerations
after O’Brien.
Color is considered a subjective phenomenon which can be regarded
differently by different observers.
Color may be produced in several different ways, including: selective
reflection, selective absorption, diffraction, scattering, interference.
The color of an object or a dental material result from a number of factors
including: the composition of the material, its thickness, the surface
roughness; the nature of illuminating light (McCabe & Walls, 2008).
The translucency of an object represents the amount of the incident light
transmitted and scattered by the respective object. A color which has a high

Bul. Inst. Polit. Iaşi, t. LVIII (LXII), f. 3, 2012 53
translucency appears lighter and reveals more of the backing in the color. It is
important to notice that the translucency decreases with the increasing
scattering.
The opacity represents the opposite of the translucency. Light scattering
in a material is the result of scattering centers (eg air bubbles, opacifiers such as
titanium dioxide, the filler particles in a resin composite matrix) that cause the
incident light to be scattered in all directions. The effect of scattering depends
on different factors as: the size, the shape, the number of the scattering centers.
Scattering is also dependent on the difference in refractive indices between the
scattering centers and the matrix in which the centers are located..
Another optical property, capable to produce a lustrous appearance, is
represented by the surface gloss. Usually, a high surface gloss is associated with
smooth surfaces. For example, in dental composite resin, the surface gloss
decreases with the increasing surface roughness. High gloss reduces the effect
of a color difference, because the color of the reflected light is more prominent.
In a restorative dental material, high gloss also lightens the color appearance.
Fluorescence represents the emission of light by an object at
wavelengths different from those of incident light. After the removal of the
incident light, the emission ceases immediately. Porcelains used in dental
laboratory are fluorescent under ultraviolet light and the quality of the
fluorescence depends on the brand of porcelain (O’Brien, 2008).
At their turn, each of the mentioned factors (color, translucency, gloss,
fluorescence), as perceived by an observer, can be influenced by: the light
source (illuminant), the inherent optical parameters of the materials that dictate
the interaction of the light from the illuminant with the material; the
interpretation of the observer.
In Fig. 2 is represented the interaction between the light source, the object
(dental material) and the observer.
Surroundings in a dental office may influence and modify the actual light
reaching the object. The color of walls, clothing, soft tissues (eg.lips) contribute
to the color of the light incident on teeth, shade guides and restorative materials
(O’Brien, 2008).
In dental medicine are utilized color systems to describe the color
parameters of different dental materials.
The Munsell color system is a three-dimensional system which has as
coordinates : hue (is the attribute of color that makes it appear blue, yellow or
red), value (is the lightness or darkness of a color and it is the most important
color factor in tooth color matching), and chroma (is the intensity of a color;
that is, the amount of hue saturation) (O’Brien, 2008; McCabe & Walls, 2008).
In Fig. 3 is represented the Munsell three-dimensional color system.
Others color systems are the CIE (Commission Internationale de
l’Eclairage) color systems.

54 Irina Grădinaru et al.
The CIE tristimulus values system uses 3 parameters (X,Y,Z), based on
the spectral response functions defined by the CIE observer. Sometimes is used
a CIE chromaticity diagram to define color.
Fig.2 – Schematic representation of the interaction between the light source,
the dental material and the observer after O’Brien.
Fig. 3 – Schematic representation of the Munsell three-dimensional color system
after O’Brien.
The CIE color system (CIE L*a*b*) uses L* (lightness), a* (hue and
chroma on a red-green scale), b* (hue and chroma on a yellow-blue scale)
parameters for defining the color. The values for the three mentioned
parameters can be calculated from the tristimulus X,Y and Z values (Fig. 4).
In the color system CIE L*a*b*, the color difference ΔE* = [(ΔL*) 2
+(Δa*) 2 +(Δb*) 2 ] 1/2

Bul. Inst. Polit. Iaşi, t. LVIII (LXII), f. 3, 2012 55
ΔL*, Δa*, Δb* represents the differences between the CIE L*a*b* color
parameter of 2 samples.
Fig. 4 – Schematic representation of the CIE and Munsell color system
after O’Brien.
In Table 1 is presented the clinical color matching between dental
structure and dental restorations, according to ΔE* values.
Table 1
Color-matching tolerances by the clinical point of view after O’Brien
The color difference ΔE*
The clinical color
matching between dental
structure and dental
restorations
Color match by the
clinical point of view
0 Clinically perfect
0.5…1
Clinically excellent
1…2 Clinically good
2…3.5
Clinically acceptable
> 3.5 Clinically mismatch
The color perception is sometimes a source of disagreement among the
dentists, technicians and patients.
There are different methods for color measurements: visual comparison
using color standards such as Munsell color chips,color measurements made by
using spectrophotometric or colorimetric methods [2].
Spectrophotometric measurements, done with spectrophotometers that
measure the amount of light reflected at each wavelength, are used to evaluate
the color parameters for porcelains, restorative resins, denture teeth, shade
guides, and color modifications in dental materials. A double-beam
spectrophotometer compares the responses from the object with a reference
standard. From the spectral response can be calculated the color parameters for
the object.
Colorimetric measurements are done with colorimeters that measure the
amount of light reflected at selected colors (eg. red, green, blue).

56 Irina Grădinaru et al.
Another modality for tooth shade selection is the use of photography in
combination with a spectrophotometer.
In the last years in dental practice have been introduced different
colorimeters and spectrophotometers that can determine tooth shade by direct
application to the patient tooth (Fig. 5).
Fig. 5 – Spectrophotometer VITA EasyShade (http://www.agd.org).
VITA EasyShade spectrophotometer provides a determination of tooth
hue according to lighting conditions.
Their accuracy is difficult to evaluate and, for some instruments, depends
on the angle at which the instrument is positioned and other operator variables.
The cost of these instruments is very high and limits their use in most dental
offices.
For determining the color of the natural teeth are used shade guides and
so, the artificial substitute restorations will have similar color and esthetics.
The shade guide “VITAPAN classical” is the most widely used
shade system with more than 40 years of world-wide recognition (Fig. 6).
Fig. 6 – Shade guide “VITAPAN classical”
(http://www.globaldentalsolutions.com/shadeguides.html).
The shade guide “VITAPAN 3D-MASTER” represents the results of
many scientific and clinical researches and is considered the consequent further
development of the “VITAPAN classical”. The VITAPAN 3D-MASTER Shade
Guide features a systematic structure in accordance with a colorimetric
classification principle. It incorporates all three dimensions (3D) of color
perception: Value (Lightness Level), Chroma (Color Saturation), and Hue
(Fig.7).
A shade guide must have some characteristics represented by: a logical
arrangement and a adequate distribution in color space matching with natural

Bul. Inst. Polit. Iaşi, t. LVIII (LXII), f. 3, 2012 57
teeth, a inherent consistency among shade guides, and matching between shade
guides and the dental materials (eg. porcelains, resin composites or dentures).
Fig. 7 – Shade guide “VITAPAN 3D-MASTER”
(http://www.globaldentalsolutions.com/shadeguides.html).
Unfortunately, the currently available shade guides do not have all the
mentioned properties and not all shade guides are fabricated from the dental
materials to which they will be matched (O’Brien, 2008).
For the dental office and laboratory, shade matching is a very complex
process (Fig. 8).
Fig. 8 – Schematic representation of the shade-matching transmission after O’Brien.
It is necessary to respect some recommendations:
i) use the manufacturer shade guide for fabricating the dental restoration;
ii) remove the individual shade tab from the guide and hold it close to the
tooth for shade matching;
iii) respect the manufacturer's recommendations for preparing the surface
of the tooth for shade matching;
iv) match the surface texture of the dental restoration to that of the
remaining dentition as closely as possible;
v) choosing a neutral chromatic environment;
vi) the choice of color is always done at the beginning of the treatment
session;
vii) lighting conditions similar to daylight;
viii) remove lipstick and makeup from the face of the patients;
ix) patient mouth should be at the doctor eye level;
x) the choice of color, by comparison, must not exceed 5 seconds, to
avoid fatigue of the cone cells;

58 Irina Grădinaru et al.
xi) between measurements, eyes will be relaxed by looking an blue
object;
xii) samples are changing fast during the determination, by removal;
xii) color is compared in different conditions (eg. dry and wet, lip
retracted and lowered, light sources in different incidents);
xiv) metamerism is tested by evaluating the color with different light
sources (incandescent light, natural light) (Bratu, 1994; O’Brien, 2008).
In conclusion, for determining the color can be used: the visual method
(subjective), the simplest and the most common, performed using color key or
technical methods (objectives), more complex and require expensive technical
equipment.
Whichever method is used, a more precise determination of color
contributes to the success of the integration of the restoration in the dentosomato-facial
equilibrium.
REFERENCES
Bratu D. Materiale dentare: bazele fizico-chimice. Ed. Helicon, Timişoara, 1994.
*** http://www.agd.org
*** http://www.globaldentalsolutions.com/shadeguides.html.
McCabe J. F., Walls A., Applied Dental Materials. Ed. John Wiley and Sons, 2008.
O’Brien W. J., Dental Materials and their Selection. Ed. Quintessence Pub. Co, 2008.
CONSIDERAŢII GENERALE PRIVIND CULOAREA ÎN PRACTICA
STOMATOLOGICĂ
(Rezumat)
Determinarea culorii continuă să reprezinte, şi la ora actuală, o problemă delicată
a medicinii dentare. În practica stomatologică, medicul poate constata obiectiv numai
identitatea culorilor sau diferenţa acestora. Însă, de la aceasta constatare, până la
caracterizarea culorii şi transmiterea datelor la laboratorul de tehnică dentară este o cale
lungă şi dificilă. Lucrarea de faţă prezintă o sinteză a datelor din literatura de
specialitate privind culoarea care, alături de transluciditate, luciu şi fluorescenţă,
contribuie la calităţile estetice ale restaurărilor dentare. Pentru determinarea culorii se
pot utiliza: metoda vizuală (subiectivă), cea mai simplă şi mai răspândită, realizată cu
ajutorul cheii de culori, sau metode tehnice (obiective), mult mai complexe şi care
necesită o dotare tehnică costisitoare. Indiferent de metoda utilizată, o determinare cât
mai precisă a culorii contribuie la asigurarea succesului integrării restaurării în
echilibrul dento-somato-facial.

BULETINUL INSTITUTULUI POLITEHNIC DIN IAŞI
Publicat de
Universitatea Tehnică „Gheorghe Asachi” din Iaşi
Tomul LVIII (LXII), Fasc. 3, 2012
Secţia
MATEMATICĂ. MECANICĂ TEORETICĂ. FIZICĂ
STATIC ANALYSIS OF THE VIRTUAL
HUMAN KNEE JOINT USING FEM
BY
MARIUS CATANĂ and DANIELA TARNIŢĂ ∗
University of Craiova,
Department of Applied Mechanics
Received: May 15, 2012
Accepted for publication: July 10, 2012
Abstract. One of the most important and most complex joint in the human
body is the knee joint. The main objective of this article is to develop a threedimensional
solid finite element model of the knee joint to predict stresses in its
individual components (femur, tibia, peroneus, menisci and cartilages). A nonlinear
analysis was performed. The non-linearities are due to the presence of the
contact elements modelled between components surfaces. This paper presents a
complex human knee joint model with cartilaginous tissue, using ANSYS
Workbench 14.0, which show us multiple contact pairs working together (Nonlinear
analysis) The applied force was equal with 800 N. In this paper the bones
were assumed having isotropic material properties.
Key words: knee joint, FEM, simulation, stresses, displacements.
1. Introduction
One of the most important joint in the human body is the knee joint.
Many papers have been published about it, but most of these works use
simplified models, using 2D or some using 3D approaches but without knee
joint essential parts. While the mathematical models of the knee joint can be
useful to predict forces and stresses in individual structures of the articulation,
as well as to estimate its kinematics, validation of such models is a challenging
task. It is complicated to make a computational model of the knee that
∗ Corresponding author: e-mail: dtarnita@yahoo.com

60 Marius Catană and Daniela Tarniţă
accurately predicts the response and movements of the articulation as it is
resembled by experimental methods. The software packages take anatomical
data from CT and MRI scans and create computer models of anatomical
structures. A user can modify the image by defining various tissue densities for
display (Radu, 2006).
Virtual modeling of human knee joint have been addressed in several
articles on (Bae et al., 2012), (Fening, 2005), (Bahraminasaba et al., 2011),
(Sandholm et al., 2011), (Kazemi et al., 2011), (Kubicek & Zdenek, 2009). In
his articles (Vidal-Lesso et al., 2011) have been considered the components
tibia, femur, cartilage, notwithstanding the menisci and ligaments, while in the
articles (Kubicek & Li, 2009), (Fenning, 2005), was performed complex
modeling, taking into account the ligaments and menisci; (Sandholm et al.,
2011) take in consideration muscles that were simulated as rigid links. In his
work (Harryson et al., 2007) were also taken into account fibula, patella, and
patellar tendon. These geometric patterns were created from images taken by
MRI and CT, dual fluoroscopic images, or using automatic creation programs
(Mimics) for the geometrical model (Bingham et al., 2008). Virtual models
were analyzed with FEM, after performing a finite element model with
tetrahedral type (Vidal et al., 2008), hexaedral, or using automatic meshing
methods. The articles have used an algorithm for meshing with hexahedrons and
bricks in order to analyze with a much better approximation for the tibiofemoral
contact area (Fening, 2005), (Kazemi, 2011), (Kubicek & Zdenek,
2009).
There have been various investigations that present different analyzes in
the human knee. Analyzes were performed the femur, tibia, meniscus, cartilage
and ligaments, (Bingham's it., 2008), (Kubicek & Zdenek, 2009).Very few data
are available for other material properties like transverse and shear moduli
(Weiss & Gardiner, 2001).
2. Method
Components taken into account for this analysis are: the femur, distal
head of the femur (thigh bone cartilage), the two menisci: lateral and medial,
tibia cartilage, tibia, peroneus and fibula (Fig. 1). These components were
placed in a global system XYZ. This is necessary for proper placement locations
of the tasks that apply degrees of freedom for the entire assemble. Nonlinear
structural analysis is static; non-linearity is evident by the appearance of
nonlinear contacts in the cartilage and meniscus.
Changes and proper positioning of the components was done using the
package ANSYS Workbench 14.0, CAE software allows rapid modelling of a
proper record and control of geometric problems arising in the contact zone.
Position requires a real placement of components, a proper distance between
them and correct geometric structure.

Bul. Inst. Polit. Iaşi, t. LVIII (LXII), f. 3, 2012 61
3. Results
Control conditions of the analysis are:
i) the analysis was run in a time of 1 s;
ii) is now nonlinear and large number of nodes and elements is necessary
to implement a system subsets analysis: subset 10, the initial step: 5 step up:
100.
iii) type iterative solver is: PCG level 2.
Fig. 1 – Virtual 3D Model of the complex human knee joint (left) and tibia, meniscus
and cartilage of the proximal head of the tibia in the right image.
Fig. 2 – Force applied and the degrees of freedom assigned
geometrical model for analysis.
Boundary conditions for this analysis are:
i) the red is observed positioning of the femoral bone in the centre of the
sphere, the area where force is applied evenly distribute of 800N, direction-Z;
ii) the yellow picture above to see a drop in the same location, but in that
area is allowed by the Y-axis rotation system, the remaining displacement after
X, Y, respectively rotations x, z are 0;
iii) in the picture below it can be seen through the media type: Remote
Displacement, support which allows movement of the ankle by letting free the
rotation direction Y.
Mesh nodes and elements in the geometric model was made with bolttype
items such as solid type tetrahedral Solid186 and Solid187, both with the
middle nodes; that are required for a better approximation of the results and
their accuracy.

62 Marius Catană and Daniela Tarniţă
Special methods were used to mesh elements bolt type, method "Sweep"
automatically controls the direction of formation of elements from source to
target, complicated areas, narrow and difficult and discretized mesh of
tetrahedra. For cartilage and menisci have used items such as bolt and the
femur, tibia, fibula or used with node tetrahedral elements in the middle.
Fig.3 – Support "Remote Displacement" model that allows rotation
of the head of the tibia and fibula after distal axis Y.
Tabel 1
Mesh nodes and elements on components
Geometric component Number of nodes Number of elements
Femur 185,402 58,604
Cartilage femur 80,036 22,061
Tibia 127,919 38,401
Cartilage tibie 33,787 8,795
Lateral meniscus 8,182 2,596
Medial meniscus 7,965 2,519
Peroneu 39,118 12.876
Cartilage peroneu-tibie 6,678 2.365
LCM 6449 2223
LCL 7876 2747
LIA 1,660 620
LIP 1,885 679
Total 488,092 148,397
For efficient discretization for this analysis we used elements of size 1 or
1.5 mm for areas of greatest interest, the possible transition (transition) from
this to a larger size through options "Smoothing" the environment and
"Transition" on fast, for each component added a dimension element "element
size", obviously a smaller contact areas tibio-femoral, and greater the jamb. For

Bul. Inst. Polit. Iaşi, t. LVIII (LXII), f. 3, 2012 63
areas of less interest used elements of 4mm. Finer mesh or higher but can be
attributed to solid components and parts thereof, that surfaces (faces) or lines.
For contacts between components of this analysis used contact type "Bonded"
and "NoSeparation". Calculation algorithm is used to contact type
"AugmentedLagrange" is used here a detection refer the identical "On Gauss
Points". To better understand these contacts have given option "Pindball
Region" with a radius of 0.1mm, which means that before you come into
contact, the solver to impose a more efficient mathematical representation of
these contact areas.
Fig. 4 – Network of nodes and elements - Mesh geometrical
model wiht nodes and elements.
The linear material properties (Bae, 2012), (Kubicek, 2009) used whitin
this study are shown in the Table 2.
Tabel 2
Material properties for components
Geometriy Young’s Modulus Mpa Poisson’s ratio
Cortical – Femur bone 18600 0.3
Cortical – Tibie bone 12500 0.3
Spongio bone 500 0.3
Cartilage 12 0.475
Meniscus 59 0.49
LCM 10 0.49
LCL 10 0.49
LIP - LIA 1 0.49
For the set of materials values was calculated Von Misses, total and
directional movements and deformations that occur both but the whole
ensemble as well as the individual components.
Due to subjected load 800N, the eqhilibrum of the forces in the knee joint
was achieved for lateral displacement of 1.23 mm (Fig. 5). Because of this

64 Marius Catană and Daniela Tarniţă
eqhilibrum position, the maximum calculated Von Misses stress of 13.8 MPa
occurs near to the femural proximal head (Fig. 6).
The non-linear analysis emphasised the Von Missed stress in the cartilages
of femur (Fig. 7) and of tibia (Fig. 8), the stresses due to contact pressures being
uniformed distributed. The values of the stresses at the cartilages levels are in
the limits reported by others outhors. The maximum strain due to compression
forces in the cartilages (Fig. 10) and meniscus (Fig. 9) is 10%, also this is in the
limits reported by others.
Fig. 5 – a – total displacement;
b – lateral displacement.
Fig. 6 – a – on Misses Stress;
b – femur Stress.
Fig. 7 – stress for femoral cartilage.
Fig. 8 – stress for tibia cartilage.
Fig. 9 – left – stress for the medial
meniscus; right – stress for the
lateral meniscus.
Fig. 10 – The compression strains for
femur and tibia cartilages.

Bul. Inst. Polit. Iaşi, t. LVIII (LXII), f. 3, 2012 65
4. Conclusions
1. It is necessary to developing reliable standard finite element models of
the human knee joint. In developing such models, the first step is to generate
computer models of the bones of the knee joint.
2. A standard finite element model of the knee model will help surgeons
and biomechanical researchers develop improved implants and treatment
method for patients suffering bone loss and diseases.
3. Additionally, the finite element model used in the present study,
characterized by a solid hexahedral element mesh, was able to analyse the stress
in the purpose of validating mechanical and material properties of bone.
4. Finite Element Modelling based on continuum mechanics is a very
powerful instrument in predicting the behaviour of ligaments.
5. However, the construction and validation of models is very difficult
due to the fact that ligaments are nonlinear, anisotropic, viscoelastic, porous
media and inhomogeneous.
6. Ligaments also undergo large deformations when loaded.
7. It is important to determine the effect of various factors such as a
misalignment, overlap, etc., which could cause erosion of the cartilage.
8. It is important to analyze the components of 3D tibia and femur to be
considered deformable bodies, and the main objective to be closely following
the destruction of cartilage and detection of different factors that can cause these
changes.
Acknowledgements. This work was supported by the strategic grant
POSDRU/CPP107/DMI1.5/S/78421, Project ID 78421 (2010), co-financed by the
European Social Fund – Investing in People, Sectoral Operational Programme Human
Resources Development 2007 – 2013.
REFERENCES
Bae Y.J, Kyung S.P., Jong K.S., Dai S.K., Biomechanical Analysis of the Effects of
Medial Meniscectomy on Degenerative Osteoarthritis. Med. Biol. Eng. Comput.
50, 53-60 (2012).
Bahraminasaba M., Saharia B.B., Roshdi H., Arumugamc M., Shamsborhand M., Finite
Element Analysis of the Effect of Shape Memory Alloy on the Stress Distribution
and Contact Pressure in Total Knee Replacement, Trends Biomater Artif.
Organs, 25, 3, 95-100 (2011).
Bingham J.T., Papannagari R., Van de Velde S. K., Gross C., Grill T. J., Felson D. T.,
Rubash H. E., Li G., In Vivo Cartilage Contact Deformation in the Healthy
Human Tibiofemoral Joint. Rheumatology; 47, 1622-1627 (2008).
Fening D.S., The Effects of Meniscal Sizing on the Knee Using Finite Element Methods.
College of Engineering and Technology of Ohio University, 2005.
Harrysson O.L., Yasser A Hosni Y.A., Nayfeh J.F., Custom-designed Orthopaedic
Implants Evaluated Using Finite Element Analysis of Patient-specific Computed
Tomography Data: Femoral-component, Case Study. BMC Musculoskeletal
Disorders, 8, 91 (2007).

66 Marius Catană and Daniela Tarniţă
Kazemi M., Li L.P., Savard P., Buschmann M.D., Creep Behavior of the Intact and
Meniscectomy Knee Joints. Journal of the Mechanical Behavior of Biomedical
Materials, 2011.
Kubicek M., Zdenek F., Stress Strain Analysis of Knee Joint. Engineering Mechanics,
16, 5, 315-322 (2009).
Radu C., Automatic Reconstruction of 3D CAD Models from Tomographic Slices Via
Rapid Prototyping Technology, 10th International Research/Expert Conference
Trends in the Development of Machinery and Associated Technology TMT 2006,
Barcelona-Lloret de Mar, Spain.
Sandholm A., Schwartz C., Pronost N., Zee M., Voigt M., Thalmann D., Evaluation of
a Geometry-based Knee Joint Compared to a Planar Knee Joint. Virtual Reality
Lab., École Polytechnique Fédérale de Lausanne, Lausanne, Switzerland, 2011.
Tarniţǎ D., Popa D., Dumitru N., Tarniţǎ D.N., Marcuşanu V., Berceanu C., Numerical
Simulations of the Human Knee Joint. Springer Publishing House, 2010, pp. 309-
317.
Tarniţǎ, D., Tarniţǎ, D. N., Bizdoacǎ, N., Popa D., Considerations on the Dynamic
Simulation of the Virtual Model of the Human Knee Joint. “Materialwissenschaft
und Werkstofftechnik”, Materials Science and Engineering Technology, Special
Edition Biomaterials, 40, 73-81 (2009).
Vidal-Lesso A., Ledesma O., Lesso A.R, Rodríguez C., Dynamic Response of Femoral
Cartilage in Knees with Unicompartmental Osteoarthritis. Journal of Applied
Research and Technology, 9, 2, 173-187 (2011).
Weiss .J.A., Gardiner J.C., Computational Modeling of Ligament Mechanics. Critical
Reviews in Biomedical Engineering, 29, 4, 1-70 (2001).
ANALIZA STATICǍ A ARTICULAŢIEI GENUNCHIULUI
UMAN FOLOSIND MEF
(Rezumat)
Articolul îşi propune sǎ prezinte o analizǎ staticǎ neliniară pentru modelul
geometric complex al articulaţiei genunchiului uman. Modelul vitrtual al articulaţiei
genunchiului cuprinde urmǎtoarele componente: femurul, tibia, peroneul, cartilajele
oaselor, meniscurile, ligamentul încrucişat anterior (LIA), ligamentul încrucişat
posterior (LIP), ligamentul colateral lateral (LCL) şi ligamentul colateral medial (LCM).
Sunt prezentate condiţiile în care are loc simularea, condiţiile la limitǎ, caracteristicile
de material. Întregul model este solicitat prin aplicarea unei sarcini de 800N (80kg -
poziţie stat într-un picior). Sunt prezentate rezultatele obţinute: tensiunile maxime
echivalente, deformaţiile şi deplasǎrile pe întreg modelul dar şi pe componente
individuale. Modificǎrile geometrice şi poziţionarea corectǎ a componentelor s-a fǎcut
cu ajutorul pachetului Ansys Workbench 14.0, soft CAE ce permite o modelare rapidǎ,
o punere în evidenţǎ şi control al problemelor geometrice ce apar în zona de contact.
Poziţionarea necesitǎ o plasare realǎ a componentelor, o distanţǎ corectǎ între acestea
precum şi o structurǎ geometricǎ corectǎ. Este evidenţiatǎ şi discretizarea modelului
geometric în model de noduri şi elemente. S-au folosit comenzi pentru discretizare
localǎ mai finǎ, necesarǎ în zona de contact şi zona de interes (zona de contact tibiefemur-menisc).
Prin prezenţa contactului neliniar analiza prezentatǎ este static neliniarǎ.

BULETINUL INSTITUTULUI POLITEHNIC DIN IAŞI
Publicat de
Universitatea Tehnică „Gheorghe Asachi” din Iaşi
Tomul LVIII (LXII), Fasc. 3, 2012
Secţia
MATEMATICĂ. MECANICĂ TEORETICĂ. FIZICĂ
STUDY OF A SINGLE TUBE BOREHOLE THERMAL ENERGY
STORAGE BY MEANS OF AN IMPLICIT FINITE DIFFERENCE
METHOD
BY
BOGDAN HORBANIUC ∗ , GHEORGHE DUMITRAŞCU and MIHAI ADAM
”Gheorghe Asachi” Technical University of Iaşi,
Department of Mechanical and Automotive Engineering
Received: May 20, 2012
Accepted for publication: June 30, 2012
Abstract. The paper deals with the problem of storing heat provided by a
solar system for the periods when solar energy is not available. In order to level
the primary energy curve so as to secure a quasi constant energy supply, a
thermal storage is mandatory. One of the multiple options is to store heat in the
ground and extract it when necessary. The solution studied in the paper consists
of transporting heat to and from the heat storage represented by the ground, by
means of vertical tubes. In order to study the dynamics of the heat exchange
process, one needs to know the temperature field in the ground that results from
the thermal interaction with the heat transfer tube as well as the heat transferred
to or extracted from the heat storage. This issue is solved by means of an
implicit finite difference scheme assuming that the tube is isothermal
longitudinally. The temperature is plotted versus the radius for selected moments
during the charge/discharge processes of a daily heat storage system and the
transferred heat is calculated. The model is used for a single tube arrangement in
order to get the primary results necessary to refine the model by taking into
consideration the non-isothermal tube and further a structure consisting of
connected tube arrays.
Key words: sensible heat thermal energy storage, borehole thermal energy
storage, isothermal tube, numerical modelling, finite difference method.
∗ Corresponding author: e-mail: bogdan_horbaniuc@yahoo.com

68 Bogdan Horbaniuc et al
1. Introduction
Thermal energy storage is a technique used to match the energy supply
(such as wind energy, solar energy, etc.) and the consumer’s energy demand
because the former exhibits important fluctuations or cut-offs (periodical or
random).
Thermal energy storage (TES) is of two types: sensible heat TES (heat is
stored as specific heat of the sensible storage material (SSM) – water, rock, soil,
aquifers, etc.), respectively latent heat TES (heat is stored as latent heat of
fusion of a substance that undergoes a phase transition – molten salts, paraffins,
alloys, etc.).
The immediate benefits of TES (Dincer & Rosen, 2011) derive mainly
from the increase of generation capacity by a better match of the supply and
demand energy curves (because very seldom a peak in demand can be
synchronized with primary energy supply), and from the possibility to shift
energy consumption to low-cost periods.
The most important advantages of TES are (Dincer & Rosen, 2001),
(Dincer, 2002): reduced energy costs and consumption, flexibility in operation,
reduced operation costs, increased efficiency of equipment utilization, and
reduced pollutant emissions and thus a reduced ecological impact.
The performance of a TES can be assessed by means of parameters such
as: storage capacity [kWh/m 3 ], storage density [kWh/m 3 , or kWh/kg],
charge/discharge rates [kW], storage efficiency determined by means of heat
losses [kW], and SSM thermal cycling (the capacity of the SSM to preserve its
thermophysical properties during long periods of operation involving a very
large number of charge/discharge cycles). Thermal energy storage in soil has
some additional advantages such as: favourable thermophysical properties of
the soil, availability of the SSM, minimum costs for the storage system set up
(mere drilling and tube positioning costs), and practically unlimited storage
capacity (by increasing the number of boreholes). The main drawback derives
from the TES set up: it cannot be insulated and therefore heat losses are high.
A borehole thermal energy storage (BTES) is a set of vertical tubes
inserted in the soil in order to transfer heat to and to extract heat from the SSM
represented by the soil itself. Basically, such thermal storage systems are best
suited for seasonal (long term) storage, i.e. summer/winter, but short term
storage (on a daily basis) may also be considered.
The charge/discharge rates of the BTES can be assessed by determining
the dynamics of the temperature field within the SSM. Therefore, different
approaches can be considered involving the analytical treatment of the transient
heat conduction or the numerical approach using the finite difference method
that avoids the almost insurmountable mathematical difficulties of the analytical
method.

Bul. Inst. Polit. Iaşi, t. LVIII (LXII), f. 3, 2012 69
The present paper deals with the numerical approach, by using an implicit
finite difference scheme to solve the heat conduction problem.
2. Mathematical Model and Numerical Approach
2.1. Mathematical Model
The schematic of the system is presented in Fig. 1. The tube of inner
radius R 0 , outer radius R W and length L is inserted in the vertical borehole and a
heat transfer fluid circulates along the tube, with k being the convection heat
transfer coefficient. The radius of the SSM domain is R, sufficiently large in
magnitude to make sure that the thermal influence of the tube does not reach
this point in a reasonable time. This way, the depth of the domain can be
considered as infinite. The temperature of the hot heat transfer fluid is t HF and
the temperature of the cold fluid is supposed to be equal to the temperature t ∞ of
the soil, which is supposed to be initially isothermal.
Fig. 1 – Schematic of the system.

70 Bogdan Horbaniuc et al
Hypotheses:
1. The soil is homogenous and isotropic in terms of thermophysical
properties.
2. The soil is dry (the influence of the humidity is negligible).
3. The soil thermophysical properties are constant.
4. The tube is isothermal longitudinally.
5. The heat propagation is one-dimensional (radial).
In order to get the most general form of the results, dimensionless
temperatures will be used, defined by
t−
t∞
θ =
t − t
HF
∞
.
(1)
The equations that describe the heat transfer phenomenon are:
a) the heat conduction equation in cylindrical coordinates
2
∂θ
⎛
i ∂ θi
1 ∂θ ⎞
= ai
+ i
∂τ 2
;
⎜ ∂ r r ∂r
⎟
(2)
⎝
⎠
where a is the thermal diffusivity and subscript i means W for the wall domain,
respectively S for the soil (SSM) region.
b) the heat balance equation at the wall-SSM interface
λ
W
⎛∂θW
⎞ ⎛∂θ
⎞
⎜
∂r
⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
S
= λS
r= R
∂r
W
r=
RW
, (3)
where λ is the thermal conductivity of the material.
The initial condition is
0
[ ]
τ < 0: θ= θ ∀r∈
R , R . (4)
The boundary conditions are:
i) on the inner tube surface (r = R 0 )
W
= : = = 0. (5)
r R0 θW,0
θ R
ii) at the end of the SSM region (r = R)
⎛∂θS
⎞
r = R: ⎜ = 0, τ
r
⎟
⎝ ⎠
∂
r=
R
≥ 0. (6)

Bul. Inst. Polit. Iaşi, t. LVIII (LXII), f. 3, 2012 71
2.2. Numerical Approach
The heat transfer equation (2), transposed in the finite difference form,
becomes (Horbaniuc et al., 2003), (Horbaniuc & Dumitraşcu, 2006)
2
∂ θ 1∂θ 1 ⎡⎛ h ⎞ ⎛ h ⎞
+ ≅ 1 θ 2θ 1 θ
2
r r
2 ⎢⎜ − ⎟ − + ⎜ + ⎟
∂ r ∂ h ⎢⎣⎝ 2rm
⎠ ⎝ 2r
⎠
m− 1 m m+1
m
⎤
⎥, (7)
⎥⎦
where h is the grid step of the domain (wall, or SSM) and subscript m represents
the current node number. After a series of manipulations, Eq (7) becomes
p p p
im , − 1 im , im , m im , + 1 im ,
− θ + σ θ − γ θ = β , (8)
where superscript p accounts for the current time step (time step number p).
The coefficients σ, γ, and β are
σ
i,m
⎡ Ri
⎤
2⎢Ni
+ m⎥
⎡
( ) 2 ⎤
Δ h
i ⎢ i ⎥
=
⎣ ⎦
2
R
⎢
+
i
ai
τ ⎥ , (9)
⎡ ⎤ Δ
2⎢Ni
+ m⎥−1⎢
⎣
⎥
Δ
⎦
⎣ i ⎦
γ
m
⎡ Ri
⎤
2⎢Ni
+ m⎥
+ 1
Δi
=
⎣ ⎦
, (10)
⎡ Ri
⎤
2⎢Ni
+ m⎥
−1
⎣ Δi
⎦
β
m
⎡ Ri
⎤
2⎢Ni
+ m⎥
2
Δi
hi
=
⎣ ⎦
⎡ Ri
⎤ aiΔτ
2⎢Ni
+ m⎥−1
⎣ Δi
⎦
, (11)
where Δ represents the depth of region i which is W or S, h is the grid step (see
also Fig. 1.), Δτ is the time step, N is the total number of nodes of a domain, and
m is the current node for which the equation has been written.
For a time step, all of the finite difference equations of type (8), along
with the heat balance and the boundary conditions equations written in terms of
finite differences, form a set that has to be simultaneously solved. Since the
matrix of this set is tridiagonal (Croft & Lilley, 1977), (LeVeque, 2007), the set

72 Bogdan Horbaniuc et al
of equations will be solved via the Gauss elimination technique which is best
suited for this case, due to the simple structure of the matrix.
3. Results and Discussion
The presented mathematical model and the numerical treatment have
been applied to an example involving a single tube arrangement and the
temperature field as well as the stored/extracted heat have been determined in
order to analyze the process dynamics.
The tube material is steel (λ W = 50 W/mK, a W = 13.8 m 2 /s). The
thermophysical properties of the soil (type: clay soil) are (Arya, 2001):
λ S = 0.25 W/mK, a S = 0.18 m 2 /s, c S = 0.89 kJ/kgK, ρ S = 1600 kg/m 3 ). The tube
geometry: R 0 = 25 mm, R W = 28 mm, L = 10 m. The radius of the SSM domain:
R = 428 mm. Temperatures: t HF = 100°C, t ∞ = 12°C. The convective heat
transfer coefficient: k = 1000 W/m 2 K. Finite difference grids: number of nodes
in the wall: N W = 5; number of nodes in the SSM region: N S = 100. The time
step has been set to 1 second.
The duration of the charge, respectively discharge processes has been set
to 8 hours (28,800 seconds). A single charge/discharge cycle has been
considered. Two computer programs have been written: one for charge and one
for discharge. Exit data from the first program are entry data for the discharge
one. Figs. 2 through 4 refer to the heat storage charging process.
Fig. 2 represents the plot of the temperature field in the SSM versus time.
Fig. 2 – Temperature field evolution during the charging process.

Bul. Inst. Polit. Iaşi, t. LVIII (LXII), f. 3, 2012 73
The maximum x axis value of the plot is only limited to 65 nodes since
the “visible” propagation distance of the temperature perturbation for the
considered charging period is 65 nodes, which corresponds to 260 mm.
A better image of the temperature field evolution is offered by Fig. 3,
which presents it as a 3-D surface, with horizontal plane axes being the distance
from the outer wall surface in terms of node number, and time. This way, the
evolution of the temperature in each of the nodes can be visualized along the
time axis. In this representation, the node number axis range has been extended
to 80, because it provides a more accurate view of the propagation distance of
the thermal perturbation in the radial direction.
Fig. 4. shows the plot versus time of the heat stored in the ground. Except
for a brief time lapse at the start of the process, the amount of heat stored
exhibits an almost linear growth, the total stored heat representing 33,748.83 kJ,
which is 4.7% of the maximum amount of heat that could be stored if the
temperature of the ground were constant and equal to t HF .
Fig. 3 – 3-D surface representation of the temperature field during the charging process.
Figs. 5 through 7 refer to the heat storage discharging process. Fig. 5
represents the plot versus time of the SSM temperature.
Figs. 6 through 8 refer to the heat storage discharging process. Fig. 6
represents the plot versus time of the SSM temperature.
After stopping the charging process, the discharge of the heat storage
begins immediately. As a result, the initial temperature field (the temperature
distribution at the end of the charging phase) starts to be distorted by the
cooling that extracts heat form the SSM layer in contact with the tube outer
surface. As a result, the temperature plots exhibit a maximum that migrates
radially and concurrently becomes more flattened as the process advances. As

74 Bogdan Horbaniuc et al
cooling extracts heat from the region in the vicinity of the tube, the SSM layers
located far from the tube continue to transmit heat to the regions farther away,
until they are reached by the thermal perturbation. This explains why the
temperature plots continue to advance towards the end of the SSM domain,
although a cooling occurs at the other end.
Fig. 4 – The amount of stored heat versus time during charging.
Fig. 5 – The amount of stored heat versus time during charging.

Bul. Inst. Polit. Iaşi, t. LVIII (LXII), f. 3, 2012 75
Fig. 6 – Temperature field evolution during the discharging process.
A 3-D view of the temperature field evolution similar to that of Fig. 4 is
represented in Fig. 7.
In the case of discharging, a plot similar to that from Fig. 5 is less
significant in terms of information offered.
Fig. 7 – 3-D surface representation of the temperature field during discharging.

76 Bogdan Horbaniuc et al
Fig. 8 – The amounts of extracted and of remaining heat in the SSM
versus time during discharging.
For this reason, we have chosen to represent in a bar plot the amounts of
heat that has been extracted from the SSM until the considered moment,
respectively the amount of heat that remains stored in the system at that
moment. At the end of the process, 22,575.07 kJ of heat still remained stored in
the ground, whereas an amount of 11,173.76 kJ of heat had been extracted.
4. Conclusions
1. The single tube borehole configuration of a thermal energy storage
system may represent a serious candidate for a daily-basis solar energy thermal
storage.
2. A mathematical model and the numerical treatment based on the finite
differences approach have been applied to an example and the temperature field
as well as the stored/extracted heat have been determined in order to analyze the
process dynamics.
3. The example studied in this paper shows that the mathematical model
and the finite difference approach that have been developed work well for a
single tube borehole thermal energy storage and thus a more refined model and
numerical treatment can be further developed.
REFERENCES
Arya P., Introduction to Micrometeorology. Academic Press, London, 2001.
Croft D. R., Lilley D. L., Heat Transfer Calculations Using Finite Difference
Equations. Applied Science Publishers Ltd., London, 1977.

Bul. Inst. Polit. Iaşi, t. LVIII (LXII), f. 3, 2012 77
Dincer I., On Thermal Energy Storage Systems and Applications in Buildings. Energy
and Buildings, 21, 1105-1117 (2002).
Dincer I., Rosen M. A., Energetic, Environmental and Economic Aspects of Thermal
Energy Storage Systems for Cooling Capacity. Applied Thermal Engineering, 34,
377-388 (2001).
Dincer I., Rosen M. A., Thermal Energy Storage. Systems and Applications. Sec. Ed., J.
Wiley & Sons Ltd, Chichester, West Sussex, United Kingdom, 2011.
Horbaniuc B., Dumitraşcu Gh., Outward Solidification around Cylindrical Metallic
Walls. Bul. Inst. Polit. Iaşi, LII(LVI), 6C, s. Construcţii de Maşini, 157-164
(2006).
Horbaniuc B., Dumitraşcu Gh., Popescu A., Inward Solidification in Cylindrical Tubes.
ASME Proceedings of the 6 th ICCES Conference, Corfu, Greece (on CD) 2003.
LeVeque R., Finite Difference Methods for Ordinary and Partial Differential
Equations. S.I.A.M., Philadelphia, 2007.
STUDIUL PRINTR-O SCHEMĂ IMPLICITĂ CU DIFERENŢE FINITE, A UNUI
SISTEM DE STOCARE TERMICĂ ÎN SOL, CU UN SINGUR TUB
(Rezumat)
Lucrarea abordează problema stocării termice în sol a energiei solare în scopul
utilizării acesteia ca sursă termică pentru un sistem cu pompă de căldură. Stocarea în sol
este avantajoasă din punct de vedere al simplităţii constructive a sistemului de stocare şi
al capacităţii de stocare practic nelimitate. Un astfel de sistem de stocare termică este
constituit dintr-un tub vertical plasat în sol, prin care curg alternativ agenţii termici: cald
la încărcare, respectiv rece la descărcare. Lucrarea îşi propune să studieze transferul
termic conductiv unidimensional nestaţionar prin care căldura este acumulată în masa
de pământ aflată în contact cu tubul şi apoi extrasă în vederea valorificării într-o pompă
de căldură. Pentru aceasta, se recurge la o schemă implicită cu diferenţe finite cu
ajutorul căreia se determină câmpul de temperatură la fiecare pas de timp în nodurile
reţelei ataşate domeniului substanţei de stocaj termic (solul). Pe baza cunoaşterii
distribuţiei temperaturii şi a evoluţiei în timp a acesteia, se calculează cantitatea de
căldură stocată/destocată. Pe baza modelului matematic şi a metodei numerice de
rezolvare propuse, în lucrare se consideră un exemplu în care tubul metalic este din oţel,
iar tipul de sol adoptat este cel argilos. Sunt trasate curbele de variaţie a temperaturii pe
direcţie radială la diverse momente în timpul unui ciclu încărcare/descărcare cu durata
de 16 ore repartizate egal pentru cele două faze. Pentru o mai sugestivă prezentare
grafică a rezultatelor, evoluţia câmpului de temperatură este reprezentată şi sub forma
unei suprafeţe 3-D, în care coordonatele planului orizontal sunt distanţa în direcţie
radială reprezentată prin numărul de noduri, respectiv timpul scurs de la începerea fazei
ciclului. De asemenea, pentru fiecare din aceste faze este reprezentată variaţia în timp a
căldurii acumulate în sol. În cazul descărcării, reprezentarea este sub formă de diagramă
cu bare, care evidenţiază simultan căldura rămasă în sol şi cea extrasă, suma lor
reprezentând căldura care era acumulată în sol la sfârşitul fazei de încărcarea a ciclului.
Rezultatele prezentate confirmă pe deplin faptul că modelul cu tub singular
funcţionează, ceea ce constituie baza de plecare pentru o rafinare a analizei, prin
considerarea unei reţele de tuburi interconectate şi a transferului termic bidimensional.

BULETINUL INSTITUTULUI POLITEHNIC DIN IAŞI
Publicat de
Universitatea Tehnică „Gheorghe Asachi” din Iaşi
Tomul LVIII (LXII), Fasc. 2, 2012
Secţia
MATEMATICĂ. MECANICĂ TEORETICĂ. FIZICĂ
THE HEAT TRANSFER INSIDE AN ARTIFICIAL
SKATING RINK
BY
GELU COMAN ∗ and VALERIU DAMIAN
“Dunărea de Jos” University of Galaţi,
Department of Thermal Systems & Environmental Engineering
Received: April 23, 2012
Accepted for publication: July 15, 2012
Abstract. The paper presents the modeling of heat transfer and ice
formation around the pipes of a skating rink by the finite element method. In
case of skating rinks equipped with pipe registers, the temperature field during
the ice formation process can be modeled by analytical methods. Thus the paper
advances a method for calculating the temperature and distribution of heat
around the track pipes The heat transfer in the skating rink track is
nonstationary and phase changing.
Key words: solidification, ice, skating rink, heat transfer, phase changing.
1. Introduction
The problem of melting and solidification of substances, regarded in
terms of determining the temperature field in solid and liquid phases and solid
– liquid interface propagation , has aroused considerable (theoretical and
practical) interest , because the conductive transfer processes accompanied by
the phenomenon of phase transition are present in many applications, such as
solidification of ingots, controlled solidification of alloys to obtain certain
metallographic structures, food refrugeration , soil freezing and defreezing, the
ablation phenomenon under aerodynamic heating, thermal storage with phase
change, etc. Mathematical modeling of this phenomenon and especially
∗ Corresponding author: e-mail: gcoman@ugal.ro

80 Gelu Coman and Valeriu Damian
mathematical solution to the problem are made difficult by the fact that general
solutions relate to three-dimensional nonstationary temperature fields in bodies
whose physical properties are temperature dependent.
2. Problem Formulation
Numerical modeling was performed using Fluent 6.3 software for two
cases of skating track construction, namely: track with water immersed pipeline
and track with pipes buried in sand. For symmetry purpose the study was
conducted using a flat discretization network, bordered left and right by two
vertical lines passing through the mid distance between two consecutive tubes, a
straight horizontal line representing the water (or ice) surface and another one
that represents the track foundation board (Fig. 1).
With the water submerged pipe geometry , the calculation range is identically
bounded, specifying that the entire area around the pipe was considered of
liquid type .
The grid with the water submerged pipe geometry contains 13563 nodes
and 12952 quadrilateral elements, and in the second case ,with the pipe buried
in the sand, 11243 nodes and 10896 elements.
The Fluent program has an impressive library of solid and fluid materials
and their properties. Numerical modeling of heat transfer with phase change
made with this software, physically corresponds to the real solidification
process around the cylindrical surfaces. Thus, the thermophysical properties of
materials are no longer constant but vary with temperature and the phase
change is not isothermal
Water
surface
Sand
surface
Pipe
surface
Track
foundation
board
surface
Fig.1 – The surfaces of the analyzed domain.

Bul. Inst. Polit. Iaşi, t. LVIII (LXII), f. 3, 2012 81
The mathematical model chosen is based on pressure based, since heat
transfer modeling is done in incompressible fluids. Formulating the
mathematical model is of default type which makes solver stable and
convergent, although the number of equations and complexity of calculations
increases. Moreover, the default scheme, unlike the explicit one, is
recommended for heat transfer processes with phase change, by providing
freedom of choice of time step within a wide range of values.
The type of solver adopted was Solidification and Mellting. This type of
solver is intended to solve the problems of heat transfer with phase change. For
a better analysis of the thermal field in certain areas of the computing range a
number of surfaces have been introduced to monitor the evolution of
temperature in the course of the process (Fig. 1).
At the same time to follow the surface temperature evolution was
monitored , namely, water surface temperature, track foundation board surface
temperature and pipe surface temperature. Equations underlying the
mathematical model and the initial and limit conditions are
∂ ρ + ∇ =
∂τ
( ρV ) 0,
(1)
( )
∂ ρu ∂p μ
+∇ =− − +∇ ∇
∂τ ∂x k
( ρuV ) u ( μ u),
(2)
( )
∂ ρv ∂p μ
+∇ ( ρvV ) =− − v +∇( μ∇ v) + ( ρm
−ρ)
g, (3)
∂τ ∂y k
( ρh)
( ρβL)
∂
⎛k
⎞ ∂
+∇ ( ρhV ) =∇⎜
∇h ⎟− −∇( ρβLV ),
∂τ ⎝c ⎠ ∂t
(4)
⎛
K = K
⎜
⎝
β
3
( 1−
β)
0 2
⎞
, (5)
⎟
⎠
where: τ – time, ρ – density, p – pressure, V – velocity vector, u, v – velocity
components on x and y axes, μ – dynamic viscosity, β – liquid fraction, L –
latent heat, K – permeability, K 0 – Kozeny-Karman constant and

82 Gelu Coman and Valeriu Damian
h
⎧CT T<
T
s
f
= ⎨
CT
l
T ≥ T
⎬
f
⎩
⎫
, (6)
⎭
h= h + ∫ c dT
, (7)
ref
T
Tref
p
where: T f – solidification temperature, C s ,C l – specific heat for the solid and
liquid phases, h – sensitive enthalpy, T ref – reference temperature.h ref – reference
enthalpy, c p – specific heat at constant pressure
3. Numerical Results
With numerical simulation of ice formation around the pipe submerged in
water, a number of 25 210 iterations were made at a time step τ = 10 seconds.
The total process running time was 7 hours, considering the process conclusion
at the ice free surface temperature of -5,55 0 C. For the pipe is buried in sand a
number of 18.000 iterations at a time step of 10 s were carried out . The time
taken for solidification process was 5 hours, considering the process complete at
an ice free surface temperature of -4,62 0 C.
Initialized parameters in the mathematical model were:
a) water temperature on the skating track t apa = 10 0 C
b) air temperature at the water free surface t aer = 10 0 C
c) temperature of the refrigerating agent inside the pipe t agent = -10 0 C
d) mass flow fate fo the refrugerating agent m = 0.185 kg/s
e) convection coefficient in the pipe α agent = 550 W/m 2 K
f) air Convection coefficient on water surface α aer = 15W/m 2 K
g) heat flow density o the track foundation board q= 10W/m 2 .
Figs. 2 and 3 illustrate the time variation of the water surface
temperature in the two cases analyzed: tube submerged in the water and pipe
buried in sand. When immersed in water (Fig. 2) there is a sharp decrease in
temperature in the nodes above the pipe and a slower decrease in the points
located midway between the pipes. Under isothermal 0 0 C after τ = 300min,
there is an almost constant distribution in nodes.
When the pipe is buried in the sand Fig.3 there is a more rapid decrease
in temperature compated with the pipe submerged in water. Isothermal 0 0 C
reaches the water surface after approximately 90 minutes. Temperature
distribution in the water surface nodes is almost uniform under isothermal 0 0 C,
indicating a high quality of the ice.

Bul. Inst. Polit. Iaşi, t. LVIII (LXII), f. 3, 2012 83
t[ 0 C]
x[m]
Fig. 2 – Temperature variation on the water surface at pipe submerged in water.
t[ 0 C]
x[m]
Fig.3 – Temperature variation on the water surface at pipe buried in the sand.
In Figs.5 and 6 are shown pictures of temperature distribution over time
for the water immersed pipe. It can be noticed the different distribution over the
2 axis, due to different boundary conditions and also due to external influences.

84 Gelu Coman and Valeriu Damian
Thus, for the X and –Y directions we have a swift temperature decrease, and on
Y direction a slower decrease due to the water surface convective heat flow.
Fig. 4 – Temperature distribution at τ = 30min for the water immersed pipe.
Fig. 5 – Temperature distribution at τ = 150 min for the water immersed pipe.
Figs. 6 and 7 show images of temperature distribution for the sand
embedded pipe for some moments in time.In comparison to the water immersed
pipes, it can be noticed a steeper temperature decrease on the domain surfaces
and a more uniform distribution on the air-water contact surface.
Fig. 6 – Temperature distribution at τ = 150 min for the sand embedded pipe.
Fig. 7 shows temperature variation on the three envisaged contours at
some time moments. It can be noticed a different a temperature distribution on
contour surfaces depending on their position within the analyzed domain.

Bul. Inst. Polit. Iaşi, t. LVIII (LXII), f. 3, 2012 85
Fig.7–Temperature distribution at τ=150min for the sand embedded pipe.
Fig. 8 – Temperature variation on the three envisaged contours at τ = 180 min.
4. Conclusions
1. Compared with the submerged pipes, there is a much faster decrease
of surface temperature and a more uniform distribution at air-water contact.At
the end of the process minimum temperatures on the surfaces studied took the
following values: temperature of the inner surface of pipe t=-8.4 0 C,
temperature on the bottom board surface of the track t=- 8.32 0 C, temperature
on sand-water contact surface t= 5.15 0 C, ice surface temperature t=-4.62 0 C.
2. From the data analyzed it follows that in terms of heat transfer and ice
quality, the sand buried pipe design is recommended due to the following
aspects:
i) Complete water solidification time over the range concerned is
reduced.
ii) Water surface temperature distribution is uniform resulting in a high
quality ice.
iii) Solidification front is spreading quickly and evenly.

86 Gelu Coman and Valeriu Damian
iv) More uniform temperature distribution on the inner surface of the
pipe, so there is an increase in the heat transfer from the refrigerant to the
skating rink
REFERENCES
Lewis R.W., Roberts P.M., Finite Element Simulation of Solidification Problems. App.
Scientific Research, 44, 61-92 ( 1987).
Sasaguchi K., Viskanta R., Phase-change Heat Transfer during Melting and
Resolidification of Melt around Cylindrical Heat Source(s)/Sinks. Jl. of Energy
Resources Technology, 111 (1989).
Shih Y., Chou H., Numerical Study of Solidification around Staggered Cylinders in a
Fixed Space. Numerical Heat Transfer, 40, 1343-1354 (1997).
Vick B., Nelson D.J., Yu X., Model of an Ice-on-pipe Brine Thermal Storage
Component. ASHRAE Trans, 102, 45-54 (1996).
Vick B, Nelson D.J., Yu X., Freezing and Melting with Multiple Phase Fronts Along
the Outside of a Tube. J Heat Transfer, 120, 422-429 (1998).
*** Fluent User’s Guide. FLUENT Inc, 2003.
TRANSFERUL DE CĂLDURĂ ÎN PISTA PATINOARULUI ARTIFICIAL
(Rezumat)
Lucrarea prezintă modelarea transferului de căldură şi formarea gheţii în jurul
ţevilor pistei unui patinoar artificial utilizând metoda elementului finit. Modelarea cu
element finit a transferului de căldură în pista patinoarului permite calculul temperaturii,
a vitezei de propagare a frontului de solidificare şi a ratei de solidificare în jurul ţevilor
pistei. În cadrul studiului au fost comparate din punct de vedere al transferului de
căldură două tipuri constructive de piste şi anume pista cu ţevi scufundate în apă şi cu
ţevi îngropate in nisip.

BULETINUL INSTITUTULUI POLITEHNIC DIN IAŞI
Publicat de
Universitatea Tehnică „Gheorghe Asachi” din Iaşi
Tomul LVIII (LXII), Fasc. 3, 2012
Secţia
MATEMATICĂ. MECANICĂ TEORETICĂ. FIZICĂ
CONSIDERATIONS ON A BEHAVIORAL MODEL OF CAR
PROPULSION SYSTEMS
BY
FLORIN POPA, EDWARD RAKOŞI * and GHEORGHE MANOLACHE
“Gheorghe Asachi” Technical University of Iaşi,
Department of Automotive and Mechanical Engineering
Received: 24 March, 2012
Accepted for publication: 6 April, 2012
Abstract. Several of phenomena occur during operation of car propulsion
systems and are influenced by many factors. This paper presents a unified
approach to these problems. This work generates a mathematical model of a
complex pattern study, for further development through a software package.
Key words: behavioral model, car propulsion systems.
1. General Considerations
In the proposed modelling is adopted as a key assumption that the entire
car's propulsion system is a mechanical system consisting of solid bodies,
connecting inner and outer, with a rigid motion, as outlines in Fig. 1.
Fig. 1 – Mechanical equivalent of the entire system of the vehicle propulsion.
* Corresponding author: e-mail: edwardrakosi@yahoo.com

88 Florin Popa et al.
In this figure, I is the reduced moment of inertia of the rotating elements
of the propulsion engine, I' is the drive wheels reduced moment of inertia, and
I'' is the reduced moment of inertia of the gearbox and main gear elements.
To study the movement of rigid, in the initial phase is necessary to
generate equivalent mechanical model of the propulsion system, which involves
calculating in different points the reduced masses of the real system and the
corresponding reduced loads (Cheng et al., 2009). Thus, the mechanical
equivalent of automotive car propulsion systems assembly is replaced by an
equivalent mass, resulting in the model in Fig. 2, whose degree of freedom is
given by the angle of rotation.
Fig. 2 – The mechanical equivalent of automotive powertrain assembly.
2. Mathematical Model
Considering the reduction point at the motor shaft level, the mass
moment of inertia, I, of the reduced mass is calculated according to the
equivalent kinetic energy criterion, by means of the relation
n
2 n
2
r
j
⎛ ω
j ⎞ ⎛ vi
⎞
∑ I
j
⎜
⎟ + mi
⎜ ⎟
j= 1` ω i=
1 ω
I = ∑
⎝ ⎠
where: I j – is mass moment of inertia of the rotating j mass; ω j – angular velocity
of the j mass; m i – mass of the i element moving translational; v i – speed
of the m i mass; n r – number of rotating masses; ω angular velocity of low
mass I (usually equal to the angular velocity of the shaft to which they reduce).
The mathematical model of evolution of the mechanical equivalent of
automotive powertrain assembly is provided as a starting point total energy
conservation of I mass condition, so that:
2
I( φ) ω ( φ)
d = ⎡M m( ωχ ,
1, χ2,..., χp) −M r(
ωψ ,
1, ψ2,..., ψ ⎤dφ
2 ⎣
q ) ⎦
, (2)
and equivalent simplified form is
ω 2 dI dω
+ I = Mm
−M
r, (3)
2 dφ
dt
which is actually a mathematical model of mass movement I reduced its axis.
⎝
⎠
(1)

Bul. Inst. Polit. Iaşi, t. LVIII (LXII), f. 3, 2012 89
Since the mass moment of inertia I variation depending on the angle of
rotation ϕ is small, can introduce approximation,
dI
0
dφ = . (4)
Eq. (3) becomes
dω
I = Mm
−M
r. (5)
dt Analyzing this equation shows that the propulsion system operation may
occur several situations (Heisler, 2002 & Heywood, 1988), depending on the
relationship in which the two moments, M m and M r . Thus, if the M m = M r
resulting from Eq. (5) that the angular acceleration dω/dt = 0, so the motor shaft
angular velocity is constant. In this situation, powertrain operation is performed
in a steady or stable. Equality of torque and moment resistant is done only at
the intersection point a, between the output characteristics of the engine and
transmission input characteristics, as shown in Fig. 3.
Fig. 3 – Coordinates defining the operating point of the propulsion system.
Coordinates of this point, called point of operation (Rakoşi et al., 2006),
is the parameters of the engine required power, represented by the torque M a
and the angular velocity ω a during the stationary regime
Pa
= Maω a
(6)
Operating point stability analysis by analytical mechanical model
involves studying evolution during transient processes. This development is
defined by the solution of differential Eq. (5). Normally, to solve this equation
requires knowledge of the functions that describe the output characteristics of
the engine and the transmission input characteristics. But using the Taylor series
development needs can be assessed in the general case, the performance of
these functions in the vicinity of the operation, obtaining
∂M m( ωa,... χia,... ) ω−ω ∂M m( ωa,... χia,...
) χ
( ,..., ,...) ( ,..., ,...)
a
i−χia
Mm ω χi = Mm ωa χia
+ +
+ ... ,
∂ω
1! ∂χi
1!
respectively, (7)
∂M r( ωa,... Ψia,... ) ωω − ∂M r( ωa,... Ψia,...
) Ψ−Ψ
M ( ,..., ,...) ( ,..., ,...)
a i ia
r
ω Ψi = Mr ωa Ψia
+ + + ...,
∂ω
1! ∂Ψ
1!
i

90 Florin Popa et al.
the notations have the significance of Fig. 3.
Accepting a linear variation of characteristics M m and M r in the vicinity
of the operation a , which involves taking only the terms containing derivatives
of order zero and first order of Taylor development and, second, entering the
following variable transformations
ω= ωa
+ Ω,
χi
= χχia
+ Χ ,
(8)
and notation
ψ = ψ + Ψ ,
i
ia
Δ = tan δ . (9)
kl
Narrowing the difference Δrω −Δ
mω
=Δ, Eq. (5) becomes successively
simplified forms
dΩ I +ΔΩ= ... +Δ
mχ
Χ ...,
i i −Δ
rψΨ
i j + (10)
dt
or still
d
I
Ω + ΔΩ = C , dt
(11)
where C is denoted by the algebraic sum of terms on the right, that is
kl
= ... +Δ Χ −Δ Ψ + ..., (12)
C
χ ψ
m i i r i i
which means, in fact, the system disturbance occurs.
3. System Response Shown by the Mathematical Model
Solving differential Eq. (11) is based on the constant values I, Δ , C. But
since, obviously, equivalent mechanical model is non-zero mass, we have
I ≠ 0 . In this situation, to solve the equation will consider the following cases
further developed.
1. If Δ ≠ 0 and C ≠ 0 , above equation becomes
dΩ dt = , (13)
C−ΔΩ
I
with solution
Δ
C ⎛ −
I
1 e t ⎞
Ω= ⎜ − ⎟. (14)
Δ ⎝ ⎠
2. When Δ = 0 and C ≠ 0 equation takes the form
dΩ
I = C , (15)
dt
with solution

Bul. Inst. Polit. Iaşi, t. LVIII (LXII), f. 3, 2012 91
when we consider the initial condition ( )
C
Ω = t , (16)
I
Ω 0 = 0.
3. If the parameters defined Δ ≠ 0 and C=0, Eq. (11) becomes
d
I
Ω + ΔΩ = 0 . (17)
dt
Solving the characteristic equation is obtained solution
Ω=Ω
Δ
e − t
I
0
where: Ω 0 is the angular velocity value Ω at baseline, considered to t = 0.
If, however, the initial velocity is zero, that is
Ω 0 = 0,
in (18) becomes apparent
Ω = 0,
and
ω= ω a
,
which is understandable because the disturbance is absent, that is C = 0. Where
Ω ≠ 0 , 0
solution of the Eq. (17) equation remains to form (18).
4. A final case under consideration is given to values Δ = 0 , C = 0.
In this case, Eq. (12) is also a simplified form, that is
dΩ I = 0,
(19)
dt
with solution
Ω =Ω , 0 (20)
same question with respect to Ω
0
, as for 3.
With the developed model, based on the solutions (14), (16), (18) and
(20) can be extremely useful analysis of the stability of the vehicle propulsion
system operation. The first case analyzed is the situation defined by
Δ < 0,
that is
tan δ rω
< tan δ mω
.
In this case, the angular velocity decreases or increases indefinitely to a
change in operating point position a; that, in this case the operating point is
unstable.
In the second case, considering the situation
(18)

92 Florin Popa et al.
Δ = 0 and C ≠ 0 ,
it translates into the condition
tan δ rω
= tan δ mω
In this case we also get an unstable operating point.
An interesting case, revealed by this analysis, occurs when Δ < 0, or
Δ = 0 , but in both cases C=0. This case leads to a metastable operating point.
Clearly, this situation should be avoided because any accidental change of the
angular speed leading to the change of operating point.
Finally we discuss the case when
Δ > 0,
or
tan δ rω
> tan δ mω
,
and
C ≠ 0.
Fig. 4 – The relative positions of the characteristics that determine stable operation.
In this case, the angular velocity varies in time, uniquely determined,
between the old and new position of operating point. It follows that in this case
operating point is stable, a situation particularly advantageous in the operation
of vehicle propulsion.
If, in addition provided Δ > 0 or tan δ rω
> tan δ mω
is satisfied the
condition C = 0, results that the operating point returns to the old position,
remaining stable.
Modelling done in this phase of work and analysis on it, leading to a
general conclusion that shows that to achieve stable operation of automotive
propulsion systems is necessary and sufficient for any operating point as the
slope of the input characteristic be greater than the slope of the output
characteristic.

Bul. Inst. Polit. Iaşi, t. LVIII (LXII), f. 3, 2012 93
In this sense, the Fig. 4 shows different mutual positioning of the input
and output characteristics, leading in all cases to a stable operating point
4. Conclusions
1. In this work a model study was performed, which allows to define
more completely stabilized operating modes of propulsion systems and their
dynamic parameters.
2. This provides early-phase design, construction and functional
definition of criteria additional to those present, to ensure stable operation and
economic, in an area as extensive, contributing to an integrated management of
automotive propulsion systems and the environment.
REFERENCES
Heisler H., Advanced Vehicle Technology. Elsevier Science, Reed Educational and
Professional Publishing, 2nd Ed., 2002.
Heywood J. B., Internal Combustion Engine Fundamentals. McGraw-Hill Series in
Mechanical Engineering, Library of Congress Cataloging-in-Publication, 1988.
Rakosi E., Roşca R., Manolache Gh., Sisteme de propulsie pentru automobile. Ed.
“Politehnium” Iaşi, 2006.
Cheng F., Xu H., Sun Y., Study on Optimization for Vehicle Power Train Parameters.
9th International Conference on Electronic Measurement & Instruments, ICEMI
'09, Beijing, 3, 2009, pp. 989-992.
CONSIDERAŢII ASUPRA UNUI MODEL COMPORTAMENTAL DE
FUNCŢIONARE A SISTEMELOR DE PROPULSIE PENTRU AUTOMOBILE
(Rezumat)
Fenomenele care apar în timpul funcţionării sistemelor de propulsie pentru
automobile au o diversitate mare şi sunt influenţate de o multitudine de factori. Aceste
fenomene pot afecta funcţionarea stabilă a sistemelor de propulsie pentru automobile.
Lucrarea prezintă o abordare unitară a acestor probleme, studiile teoretice fiind
concretizate prin elaborarea unui model matematic complex, aplicabil sistemelor de
propulsie cu motor termic.
Studiul modelului matematic realizat permite să se definească mai complet
modurile stabilizate de funcţionare a sistemelor de propulsie precum şi parametrii lor
dinamici. Astfel se oferă încă din faza de proiectare şi definire funcţională o serie de
criterii suplimentare faţă de cele actuale, pentru a se asigura funcţionarea stabilă şi
economică a sistemelor de propulsie auto într-o zonă atât de extinsă.
La elaborarea modelului matematic s-a avut în vedere optimizarea lui astfel încât
acesta să poată fi inclus pe viitor într-un pachet software ce uşurează calculul şi analiza
datelor obţinute.

BULETINUL INSTITUTULUI POLITEHNIC DIN IAŞI
Publicat de
Universitatea Tehnică „Gheorghe Asachi” din Iaşi
Tomul LVIII (LXII), Fasc. 3, 2012
Secţia
MATEMATICĂ. MECANICĂ TEORETICĂ. FIZICĂ
USAGE OF SPLINE FUNCTION METHOD FOR
GENERALIZATION OF THE DEFORMATION
COEFFICIENT MODEL
BY
IULIAN AGAPE 1 , LIDIA GAIGINSCHI *1 , ADRIAN SACHELARIE 1
and VASILE HANTOIU 2
“Gheorghe Asachi” Technical University of Iaşi,
Department of Automotive and Mechanical Engineering
2 ISU Vrancea
Received: 22 March 2012
Accepted for publication: 12 April 2012
Abstract. The utility of deformation coefficient model in kinematic study
of collisions between vehicles has – even in collinear collisions – certain
constraints: the reduced number of the considered points (usually 3, 4 or 6)
negatively affects the accuracy of results and thus the applicability of the model.
The paper proposes a generalization of the model for a finite number of
points; this compensates the disadvantages mentioned above. A deformation
function was determined based on an interpolation function algorithm on exact
data, defined on a discrete lot of points, using spline function method. The
method is advantageous in terms of simplicity, the spline-function used being
III -rd grade polynomial, on portions.
The purpose of the paper was achieved by determining the analytical
expressions of the function - strain energy of deformation and medium
deformation. Based on this calculus algorithm an original calculus program was
developed, that has been calibrated on a numerical example from the literature.
The value of this generalization model consists in the possibilities offered to
improve the accuracy of CRASH3 model.
Key words: spline function, deformation coefficient.
* Corresponding author: e-mail: lidiagaiginschi@yahoo.com

96 Iulian Agape et al
1. Introduction
During the evolution of automotive there were elaborated different
calculus models for the impact moment velocities, differing on the theoretical
bases used to establish the calculus relations.
In most cases, at the scene of impact are found exclusively only later
produced traces, making easier to determine the post – collision speeds for the
involved vehicles. Most times, the only evidence that may be related to pre –
impact kinematic parameters of the vehicles are the remaining deformations
resulting from impact.
The most competitive calculus models, that generated already
acknowledged numerical calculation programs, are based on the application of
the impulse conservation principle with the consideration of the connection
between energy needed to produce deformations, remaining deformation
amplitude, equivalent speed and body rigidity of involved vehicles.
The CRASH-3 model, well – known for the accuracy of the results,
requires knowledge of remaining deformation, stiffness coefficients, and the
main force of impact direction. The procedure involves dividing the
deformation front in 2, 3 or 5 equal intervals of L length, purpose for
establishing on the deformation front 3, 4 or 6 equidistant points, in which the
remaining deformations will be determined.
The evaluation of deformation values in these points is appreciated as
sufficient for the average deformation determination, with an acceptable
accuracy. Evidently, the increase of the number of points where the deformation
is determined leads to the improvement of the method.
2. The Deformation Function
The algorithm requires dividing the impact zone width (L) in n equal,
consecutive intervals, each with the length L/n, n∈∞ * , n finite number. For
defining the deformation function it will be considered a frontal deformed
vehicle and an orthogonal coordinate system will be jointly attached to this
vehicle. The Ox axis of this system will be normal to the longitudinal axis of the
vehicle, and the Oy axis of deformations will contain the first point of the
vehicle front, starting from left (advancing way), in which the deformation is
measurable.
There is the possibility that, based on deformation values determination,
ξ i in a number of (n+1) equidistant points distributed on the L impact zone
⎛ L ⎞
length, ξi
= ξi⎜( i −1)
n
⎟, 1, ( 1)
⎝ ⎠ i=
n+
, n∈ϒ* , n finite number, to determine an
algebraic function for the deformation, with the form: f: [0, L]→ϒ; f = f(l), so

Bul. Inst. Polit. Iaşi, t. LVIII (LXII), f. 3, 2012 97
that in the network nodes would be fulfilled the relation: f(l i ) = ξ i+1 = ξ i+1 (l i );
L
i= 0, n. It must be mentioned that l i
= i ; i= 0, nand l 0 = 0; l n = L.
n
The classic algorithms for function interpolation upon exact data defined
on a discreet lot of points are based on interpolative polynoms (for example,
Lagrange, Newton, Stirling, Bessel polynoms).
The use of these methods is difficult, due to some particularities in the
field of application, and also due to the interpolation polynoms grade, equal to
the n number of intervals. In such situation, the procedure increase in accuracy
would be with a very complex mathematical device. For the simplicity was
preferred an interpolation method with spline – function, which are – usually –
polynomial function on portions. Most used spline functions are III-rd grade
polynomial.
On the [0, L] segment of the real axis attached to vehicle, a network of n
equal intervals was build, separated by (n+1) equidistant points, in which nodes
are specified the values of the function ξ i (l i ), i= 0, n. In the network nodes the
L
“length” variable takes the values l i
= i , i= 0, n, with l 0 = 0; l n = L.
n
Cubic interpolation problem on portions requires the determination of a
function f:[0, L]→ϒ; f = f(l) which satisfies the conditions:
1) f(l) ∈ C 2 (0, L) that means that it is continuous along with it is II -nd
grade derivatives including, on the definition interval. This condition suits the
explicit interpretation of vehicles deformation.
2) on each of the equal segments of the definition interval, [l i-1 , l i ] ,
i= 1, n, the f(l) function is grade III polynom, with the form
3
i k
f () 1 = fi() 1 = ∑ ak ( li
−l)
, i = 1 , n ; (1)
k = 0
3) in the network nodes the equalities are fulfilled
( ) 1 ,
f li
ξ
i +
4) f // (l) satisfies the limit condition
= i = 1,
n ; (2)
( 0) ( )
= =0, i = 1,
n . (3)
// //
f f L
As the function f(l) second derivative is continuous and linear on each [l i--1 ,
l i ] , i= 1, n interval of the network, it can be written that, for l i-1 ≤ l ≤ l i ( i= 1, n)

98 Iulian Agape et al
// l
i−l l−
li−
1
f () 1 = mi−
1
+ mi
, i = 1,
n ,
L L
n n
where: L/n – the network intervals length; m i – the second derivate of function
f(l) values, in the network nodes
m
( )
//
i f l i
(4)
= i = 0,
n , (5)
Integrating twice the Eq. (4) it will be obtained
( ) ( )
3 3
li −l l−li−
1 li −l l−li−
1
i−1
i i i
f () l = m + m + E + F , i=
1, n
L L L L
6 6
n n n n
,
(6)
where: E i , F i are integration constants, determined from
f(l i-1 ) = ξ i ; f(l i ) = ξ i+1 , i = 1,
n . (7)
Customizing the Eq. (6) for l = l i and l = l i-1 will obtain:
2
⎧mi
⎛ L ⎞ + Fi
=
⎪ ⎜ ⎟
⎪ 6 ⎝ n ⎠
⎨
6
,
i+
1
2
⎪mi
−1
⎛ L ⎞ + Ei
= ξi
⎪
⎩
⎜ ⎟
⎝ n ⎠
ξ
i = 1,
n ,
(8)
and, replacing the constants F i , E i in (6)
( ) ( )
3 3 2
li −l l−l ⎡
i−1 mi−
1 ⎛ L ⎞ ⎤li
− l
f()
l = mi−
1
+ mi + ⎢ξi
−
L L ⎜ ⎟ ⎥ +
6 6 ⎢⎣
6 ⎝ n ⎠ ⎥ L
⎦
n n
n
2
⎡ mi
⎛ L ⎞ ⎤l−
l
+ ⎢ξi+
1
− ⎜ ⎟ ⎥
⎢ 6 ⎝ n ⎠ L
⎣
⎥⎦
n
i−1
The first derivative of the function is
()
f l
, i = 1,
n .
m ( l −l) m ( l−l ) ξ −ξ ( m −m 1)
L
=− + + −
2 L 2 L L 6 n
n n n
2 2
i−1 i i i− 1 i+
1 i i i− ,
(9)
(10)
i = 1,
n .
It allows to determine the lateral limits for the derivative in the network
nodes, for the arguments l 1 , l 2 , …, l n-1 ,

Bul. Inst. Polit. Iaşi, t. LVIII (LXII), f. 3, 2012 99
/ mi L ξi+ 1
− ξi mi − mi−
1
L
limsup
l→l
f ( l)
= + − ,
i
2 n L 6 n
n
(11)
/ mi L ξi+ 2
− ξi+ 1
mi+
1
− mi
L
liminf
l→l
f ( l)
=− + − .
i
2 n L 6 n
n
From the first derivative f / (l) continuity condition in the network nodes,
(n-1) equations will be obtained
L
L
n 2 L n ξi+ 2
−ξi+ 1
ξi+
1
−ξi
mi− 1
+ mi + mi+
1
= − , i= 1, n .
6 3 n 6 L L
(12)
n n
From the limit condition upon the second derivative (condition 4) it will
be obtained
m 0 = m n = 0
(13)
equation that gives a natural cubic interpolative character to the function f(l).
From the Eq. (12) and (13), we obtain a system of linear equation with
the unknown m 1 , m 2 , …,m n
Am = Hξ , (14)
where: the square matrix A (n-1 rows and n-1 columns) and the vectors m and ξ
take the forms
⎛2L
L
⎞
⎜
0 ... 0 0
3n
6n
⎟
⎜
⎟
⎜ L 2L L
⎟
⎜
... 0 0 ⎟ ⎛m1
⎞
⎜6n 3n 6n
⎟ ⎜m
⎟
2
A = ⎜
L 2L
⎟; m = ⎜ ⎟;
⎜0 ... 0 0 ⎟
... m3
⎜ 6n
3n
⎟ ⎜m ⎟
⎝ n − 1 ⎠
⎜.....................................................
⎟
⎜
⎟
L 2L
(15)
⎜0 0 0 ...
⎟
⎝
6n
3n
⎠
ξ
⎛ξ1
⎜
ξ
⎜
2
⎜...
ξ
⎜ξn
⎜
⎝
= 3
⎜ ⎟
ξ
n + 1
⎞
⎟
⎟
⎟.
⎟
⎟
⎠

100 Iulian Agape et al
The rectangular matrix H, with (n-1) lines and n columns has the form
⎛ 1 2 1
⎞
⎜ - ... 0 0
Ln / L/n L/n
⎟
⎜
⎟
⎜
1 2
⎟
⎜0 - ... 0 0
L/n L/n
⎟
⎜
H = ........................................................... ⎟
⎜
⎟. (16)
⎜
1
0 0 0 ... 0 ⎟
⎜
L/n ⎟
⎜
⎟
⎜
2 1 ⎟
⎜
0 0 0 ... -
L/n L/n ⎟
⎝
⎠
The matrix A is symmetrical with simple diagonal dominant. According
to Gerşgorin theory upon the localization of own values, this matrix is positive
defined and multiple. So, the m i coefficients will be determined from the
univocally system (14)
Δ j
m j = , j = 0,
n −1. (17)
det A
According to Cramer rule, where Δ j is the determinant obtained by
replacing the j column in the detA with the column formed by the free terms
⎡
⎤
⎢ξj 1
ξ
j
ξ
j 1⎥
⎢
+ − 2 +
−
L L L
⎥, j = 0,
n . (18)
⎢
⎥
⎣ n n n ⎦
3. Application of the Deformation Function
The CRASH 3 model exploits the linear dependence – experimentally
established – between the impact force F and the L width for the impact zone,
and also the ξ deformation on the direction of F force
F
A Bξ,
L = + (19)
where: A – the ratio between the maximum impact force that does not cause
deformations and the width L for the contact surface [N/m]; B – the ratio
between the impact force and the product: L width of the contact x the
deformation average amplitude, on the direction of F force [N/m 2 ].
The deformation energy E d of the vehicle, corresponding to the ξ
deformation can be obtained by integrating the Eq. (19),
L ξ
L
2
⎛ Bξ ⎞
Ed
= ( A+ Bξ
)dξdl = ⎜Aξ + + G⎟
l
2
0 0 0⎝
⎠
∫∫ ∫ d , (20)

Bul. Inst. Polit. Iaşi, t. LVIII (LXII), f. 3, 2012 101
where Gdl (G – integration constant) represents the mechanical work needed to
the deformation in the elastic field (ξ) for the elementary front dl. If between the
deformation and the force is kept (in the elastic field) the same linear
dependence like in (19), then the deformation, when achieving the ratio A, is
A/B, so the constant G is
AB / 2
A
G= ∫ Bξξ d = . (21)
2B
0
If the impact zone width L is equally divided in n intervals, and the
deformation is measured in the (n+1) demarcating points of the intervals, then,
according to (20) and substituting the deformation with the function
fi
(), l i= 1, non the intervals, it can be written
L L 2 L
f () l
Ed
= ∫Af ()d l l + ∫B + Gdl
dl
∫
. (22)
0 0 0
2
A
Confronting the Eq. (22) with the form Ed
= K1+ AK2
+ BK 3
and
B
after the integrals processing, the following values for the coefficients
L
K
1
= ,
2
n
2 n
L ⎡
L
K = ⎢ ( ξ + ξ ) − ( m + m
⎣
2 i+ 1 i 2 i i−
2n
i= 1 12n
i=
1
⎤
⎥
⎦
∑ ∑ 1 ) ,
K
3
n
=
3L
n
∑
i=
1
n
∑
i=
1
3 3
( ξi+
1
− ξi
)
.
( m + m )
i
i−1
(23)
The deformation function was integrated in the form (9). The average
deformation was calculated like an integral average on the length L of the
deformed front
and finally
L
i i=
n
1 1
ξ = med
ξ()d
l l ξi
()d l l
L∫
= ∑ L
∫
0
l
li−1
i=
1
,
(24)
n
2
1 ⎡ 1 ⎛ L ⎞ ⎤
ξmed = ∑ ⎢( ξi+ 1
+ ξi) − ( mi + mi−
1
2n
i=
1
12
⎜
n
⎟ )⎥ . (25)
⎢⎣
⎝ ⎠ ⎥⎦
3. The Calibration of the Model
The calculus model will be calibrated on the calculus presented in (***,
1980), considering that for the Ford vehicle (frontal damaged) the deformation

102 Iulian Agape et al
were measured in 11 equidistant points (so n=10) on the length L = 1.65 m for
the damaged front. The taken deformations were: ξ 1 = 0.53 m; ξ 2 = 0.52 m; ξ 3 =
= 0.50 m; ξ 4 = 0.43 m; ξ 5 = 0.33 m; ξ 6 = 0.31 m; ξ 7 = 0.27 m; ξ 8 = 0.19 m;
ξ 9 =0.13 m; ξ 10 = 0.14 m; ξ 11 = 0.12 m.
We specify that ξ 1 , ξ 3 , ξ 5 , ξ 7 , ξ 9 , ξ 11 , have the exactly considered values in
the 6 equidistant points from the application (Gaiginschi, 2008). The other 5
considered values for the deformation, in this example, do not distort the
deformation character. The adopted values keep the increase or decrease
deformation character, on the considered intervals.
Table 1 and 2 centralize the calculus results for the m i coefficients and
K 1 , K 2 , K 3 coefficients, the average deformation and deformation energy. The
calculus considered both adopting cases for stiffness coefficients A, B and G,
respectively the ones recommended by SAE, and also the experimentally ones.
Table 1
ξ m
[m] [1/m]
0.53 0
0.52 -0.02029
0.5 -2.1227
0.43 -2.50818
0.33 5.543869
0.31 -2.03644
0.27 -1.80584
0.19 0.444351
0.13 4.436144
0.14 -2.76193
0.12 0
Table 2
k 1 k 2 k 3
Coeficients
m sqm Cubm
0.825 0.519236 0.178861
G, N A, N/m B, N/sqm
exp 13007.42 102481.6 807422.2
sae 2715.543 36200.93 482595
Ed exp, [J] 208359.3
Ed sae, [J] 107354.4
ξmed, [m] 0.315
4. Conclusions
1. Comparing the results with the ones obtained in the application from
(Gaiginschi, 2008), we notice an increase of about 32…37% for the
deformation energy, depending on the adopted coefficients type. On the other

Bul. Inst. Polit. Iaşi, t. LVIII (LXII), f. 3, 2012 103
hand, the average deformation decreased with 15%. Considering only 6 points
for the deformations measurement in case of a front impact, especially when
this took place on a large dimension zone, leads to under evaluate the
deformation energy, and, by default, the collision speed variation.
2. The proposed method facilitates the accuracy determination of the
impact deformed volume. This can eliminate one of the drawbacks of the
CRASH 3 used procedure, highlighted by the expertise practice, that the
deformed profile is vertically uniform rectilinear.
3. The method developed in this paper does not limit the number of
points in which the direct deformation is evaluated. The construction of an
interpolative deformation – function opens the direction for establishing a
deformation acquisition and deformation energy evaluation procedure, an
automatic procedure of high accuracy and efficiency.
REFERENCES
*** CRASH 3 Technical Manual. National Highway Traffic Safety Administration,
1980.
Danner M., Halm J., Technische Analyse von Verkehrsunfallen. Eurotax (International)
A.G., GH – 8808 Plaffikon, 1994.
Gaiginschi R., Reconstrucţia si expertiza accidentelor rutiere. Ed.Tehnică, Bucureşti,
2008 pp. 289-301
Siddal E., Day T.D., Updating the Vehicle Class Categories. SAE Paper 960897, 1976.
UTILIZAREA FUNCŢIILOR SPLINE LA GENERALIZAREA MODELULUI
COEFICIENTULUI DE DEFORMAŢIE
(Rezumat)
Utilitatea modelului coeficientului de deformaţie la studiul cinematic al
coliziunilor dintre autovehicule are – chiar în cazul coliziunilor coliniare – anumite
limitări: numărul redus al punctelor considerate (uzual în practică de 3, 4 sau 6)
afectează negativ precizia rezultatelor şi implicit aplicabilitea modelului.
Lucrarea propune o generalizare a modelului pentru un număr finit de puncte şi
care poate compensa dezavantajele menţionate mai sus.
Scopul lucrării a fost atins prin determinarea expresiilor analitice ale funcţiei
deformaţie a energiei de deformare şi a deformaţiei medii. Algoritmul de calcul al
acestora stau la baza unui program de calcul propriu care a fost calibrat pe un exemplu
numeric din literatura de specialitate. Valoarea generalizării propuse constă în
posibilităţile oferite modelului CRASH 3 pentru îmbunătăţirea preciziei acestuia.

BULETINUL INSTITUTULUI POLITEHNIC DIN IAŞI
Publicat de
Universitatea Tehnică „Gheorghe Asachi” din Iaşi
Tomul LVIII (LXII), Fasc. 3, 2012
Secţia
MATEMATICĂ. MECANICĂ TEORETICĂ. FIZICĂ
ONS FOR ELECTRICAL VEHICLE BATTERY USING
OPENMODELICA SOFTWARE PACKAGE
BY
VLAD MARŢIAN ∗1,2 , MIHAI NAGI 1 and CIPRIAN FLUIERAŞ 1
Received: April 23. 2012
Accepted for publication: July 15, 2012
1 „Politehnica” University, Timişoara,
Department of Mechanical Engineering
2 RAAL S.A.,
Research and Development
Abstract. Increasing demand for clean energy consumptions, and also the
continuously rising of gasoline prices, forced the car manufacturers to consider
the electric driven cars (EV) and hybrid traction cars (HEV), as a solution to
this problem. The main challenge in this field is to develop new batteries that
have high power and high storage capabilities, but this comes at the price of
increased heat generation in the battery, heat that must be evacuated so the
battery doesn’t suffer any damage. The present article presents the simulation of
1D thermal model of a battery using the OpenModelica software package. The
aim of this simulation is to develop the cooling system for an electric vehicle.
Key words: heat transfer, electrical battery, electrical vehicles, hybrid
vehicles, simulation.
1. General Considerations
Electric energy seems to be the future of the vehicles driving power. Due
to continuously rising prices of petrol witch some forecast place a figure of 300
$/barrel in 2035 (Paier, 2011), and due to the growing need for a cleaner
environment, more and more car manufacturers are beginning to develop
electrical powered (EV) and hybrid (HEV) vehicles. The advantages of this type
of powered vehicle are obvious, and apart from the clean energy consumption
∗ Corresponding author: e-mail: mailto:%20vlad.martian@raal.ro

106 Vlad Marţian et al
there is also the advantage of efficiency which for the electric engine is around
80% -90%.
The main obstacle in producing on a mass scale this type of vehicles is
represented by the storing capacity of the electrical energy i.e. the batteries.
Actually the storing capacity is not enough, so one of the main directions of
research is to improve the storing capacity of the batteries. This increase in
energy density and also the need for drawing high powers form the batteries has
another side effect such as increasing the temperature of the battery. The
working temperature of the battery is a very important parameter, for example
for a Li-ion cell an increase in temperature of 15°C will reduce the life of the
cell by about 50% (Asakura, Shimomura & Shodai, 2003). The temperature has
also another effect on the charge/discharge of the battery and also on the storage
capacity of the battery. These parameters i.e. charge/discharge and storage
capacity is quantified by using the term SOC (State of charge). In the work of
Zheng Popov and others (Zheng, Popov & White, 1997) an optimum
temperature for a battery is around 25°C, even if now there are batteries that can
have a maximum temperature of 85°C (Winston, 2011). The current
discharge/charge rate grows as the temperature approaches the optimum due to
increased ion mobility and also due to modifications of internal resistance of the
battery, but after the optimum the current charge/discharge rate stats to decrease
due to oxidations that happen inside battery. Increasing the temperature over the
functioning domain make the batteries to have a catastrophic failure, and not
only the performance of the battery will be diminished but also irreversible
oxidations occur and the battery becomes useless (Jiangang, et. al., 2006).
For these reasons, toghether with RAAL S.A., we began to investigate the
necesity of a cooling system for batteries equiped in EV and HEV. This paper
presents the first step from many that includes battery modelling, the modelling
of cooling modules, the modelling of an automatic driver, experimental tests of
the cooling modules, and thermal test on the battery pack, etc.
2. Battery Models
The literature has many models which vary in complexity. There are
complex models that use quantum mechanics for describing the battery at
chemical reaction level (Parthasarathy et. al., 2002), (Aron, Girban & Pop,
2010), finite element models that describe the spatial dynamics in the battery
(Sievers, Sievers & Mao, 2010),electrochemical models, electrical equivalent
circuit (Matthias et al., 2005), Dynamic Lumped parameters models, tabulated
battery data models.
To model the battery as close to the reality as possible every model has to
take into account the parameters on which the battery depends on, and these
parameters are a few. One of the most important parameter that the battery has
is the so called state of charge, SOC, or the electrical energy stored in the
battery. This parameter depends on other parameters of the battery as the

Bul. Inst. Polit. Iaşi, t. LVIII (LXII), f. 3, 2012 107
current drawn from the battery, the time that the current has been drawn, and the
capacity of battery, and can be express in mathematical form as
It
SOC = 1 − , . (1)
C
∫
t
I ()d t t
0
(2)
SOC( t) = 1 − ,
C
where I – the current drawn (A), t – time (s), C – the battery capacity (A.s)
Other parameters of the battery include the temperature of the battery, the
internal resistance and the open circuit voltage.
In the remaining paper we will only describe the electrical circuit models
which are the base for the model in this article, for other model types you can
see (Gomadam, Weidner, Dougal, & White, 2002).
2.1. Simple Model
The simplest model used consists of a constant resistance R b in series
with an ideal voltage source E 0 , sketched in Fig. 1.
Fig. 1 – Simplest model.
Even this is very simple form electrical point of view; this model does
not take into account the true internal resistance of the battery, which is highly
related to the state of charge (SOC). In this case the draw of energy is unlimited.
Another drawback of this model is that it does not take into account the thermal
energy generated during discharge.
There are other, improved, electrical models, some of which modify the
internal resistance according to the SOC, and also include other parameters that
take into account the dynamics of the electrical current during discharge. One of
this improved a model that is worth mentioning it is the Thervein model.
2.2. Thervein Model
This is another basic battery model which describes a battery with an
ideal voltage source (E 0 ), internal resistance R and a capacitance C 0 which
represents the actual capacitance of the battery, and also an over-voltage
resistance R 0 (Ziyad & Salameh, 1992). The main disatvantage of this model is

108 Vlad Marţian et al
that all the components are constant, whereas in reality all these characteristics
are dependent of the SOC, and the dicharge current. The circuit diagram can be
seen in the Fig. 2.
Fig. 2 – Thervein model.
2.3. Non linear Dynamic Model
A more realistic model has been created by extending the Thervein
model. This new model takes into account the nonlinearities in the components
of the Thervein model. As we said earlier the internal resistance of the battery
R+R 0 and the open circuit voltage E 0 are dependent on the SOC of the battery,
and also on the temperature T of the battery.
Since we are interested in how the temperature of the battery changes in
time we will use a modification of this later model, which can be seen in a
simplified version in Fig. 3.
In this model different form Thervein model we have included the
internal resistance in the overvoltage resistance, for simplification purpose, and
the internal resistance R and the open circuit voltage E 0 are dependent on some
function of SOC.
Fig. 3 – Non linear dynamic model.
3. Modeling Implementation
If we want to know how the current and the voltage in the battery are
modified in time we have to solve a system of equations that include first order
differential equations and also algebraic equations. Doing it by hand it takes a
long time, and if one of the parameter is changed we will have to do it again.

Bul. Inst. Polit. Iaşi, t. LVIII (LXII), f. 3, 2012 109
There is a faster and error free method anyway doing this with the help of the
computer.
In the following we will present the modeling implementation steps with
the help of the OpenModelica (OpenModelica, 2012) software package.
The first step in modeling the battery was to model the equivalent
electric circuit of tha battery. Since OpenModelica has a diagram development
interface, and because the Modelica language (Modelica, 2012) is an equation
based language, the implementation of the electrical model was straithforward.
In the Fig. 4 can be seen the end result of the model.
Fig. 4 – Battery model.
The battery model is composed from different components, which are
electrical components represented by: V oc that implements a signal voltage
source, R int that implements a variable resistor, the internal rezistor of the
battery, C that implements a capacitor, ISens that implements a measuring
sensor for curent drawn.
To complete the battery model it was necesarry to include non electric
components such as: mCp implements a heat capacitor, Temp implements a
temperature sensor, Soc implements the SOC parametter acording to Eq.
(2), ExpDataVoc,ExpDataR implements experimental functions for open circuit
voltage and respectively for internal rezistor of the battery

110 Vlad Marţian et al
The battery model is linked with the rest of the circuit by three
connectors: a positive (p) and negative (n) electric connectors, and a heat
connector (heatPort). Let us explain the thermal part of the battery:
It is well known that the energy conservation law stipulates that the
energy that is stored in a domain must equal the energy that comes in minus the
energy that goes out plus the energy generated inside the domain. The equation
form per unit time, of this law can take the form
dE
d
st
dEin
dE
E
out g
= − + , (3)
dt dt dt
dt
and in the case of a solid domain as the battery, and where we do not have phase
change the Eq. (3) becomes
dT
mCp R i () t hA[ T T ]
d
2
int
amb
t = − − , (4)
where m – mass of the battery, Cp – specific heat capacity of the battery, T –
battery temperature, R int – internal battery resistance, h – thermal convection
coefficient with the outside medium, A – exchange surface of the battery, and
i(t) – the current intensity.
The Eq. (4) is implemented in the battery model as in Fig. 5, except the
Temp sensor, which it is used for linking the temperature to the other
components
Fig. 5 – Thermal model.
4. Simulations
For simulations we have choose a Winston Li-ion battery (Winston,
2011) with a capacity of 60 Ah. The internal resistances and the open circuit
voltage where determined by fitting the charts form the manufacturers data. The
capacitor value was taken to be 4.047kF (Valerie, Ahmad, & Thomas, 2000).
Because we wanted only to test the model, first we have simulated the model

Bul. Inst. Polit. Iaşi, t. LVIII (LXII), f. 3, 2012 111
without any cooling and with a constant resistor taken to be the load on the
battery. You can see the modeling in the Fig. 6
Fig. 6 – Battery without cooling.
The simulation was done for a time of 1 hour in which the battery has
been drained for almost the entire energy. In the following charts we can see the
most important parameters of the battery function of time.
Battery Temperature [°C]
70
60
50
40
30
20
10
0
0 10 20 30 40 50 60 70
Time [min]
Fig. 7 – Battery temperature.
As it is observed from the Fig. 7 the temperature is rising and in an hour
of using the battery with a constant load the temperature rises with almost 40
o C.

112 Vlad Marţian et al
Current Intensity [A]
80
70
4.5
4
60
3.5
50
3
2.5
40
2
30
1.5
20
1
10
0.5
0
0
0 10 20 30 40 50 60 70
Time [min]
Battery Voltage [V]
Current Intensity Battery Voltage
Fig. 8 – Current intensity and voltage.
Eoc [V]
4
3.5
3
2.5
2
1.5
1
0.5
0
0
0 0.2 0.4 0.6 0.8 1 1.2
0.09
0.08
0.07
0.06
0.05
0.04
Rint [Ω]
0.03
0.02
0.01
SOC
Open circuit voltage Internal Resistance
Fig. 9 – Battery internal parameters.
Chart in Fig. 8 show the current intensity and voltage evolution in time
and Fig. 9 show the battery parameters, Open Voltage E oc and internal
resistance R int function of the battery state SOC.
Another simulation done was with a simple cooling of the battery, and
with a variable load resistor which changes the current drawn over time.
Here we used a convection model to remove the heat from the battery and
an ambient temperature of 20 o C.
In Fig. 12 can be seen that the temperature and the current intensity are
connected but there is a slight shift between the current maximum and the
temperature maximum, this can be explained if we look at Fig. 11 and Fig. 9. In
Fig. 11 can be observed that the maximum temperature is near a SOC of 0 and
from Fig. 9ig. 9 we can see that at SOC near 0 the internal resistance rises so
more heat will be generated.

Bul. Inst. Polit. Iaşi, t. LVIII (LXII), f. 3, 2012 113
Fig. 10 – Second simulation.
Temperature [ o C]
30
25
20
15
10
5
1.2
1
0.8
0.6
0.4
0.2
SOC
0
0
0 10 20 30 40 50 60
Time [min] Temperature
SOC
Fig. 11 – Temperature and SOC.
30
120
Temperature [ o C]
25
20
15
10
100
80
60
40
Current intensity [A]
5
20
0
0 10 20 30 40 50 60
Time [min]
Temperature
Fig. 12 – Temperature and current intensity.
0
Intensity

114 Vlad Marţian et al
Another fact that can be observed is that due to current intensity the
battery drains out more rapidly, which is in concordance with the reality.
5. Conclusions and Future Work
1. This research main objective was to model the battery to include the
heat generated and to extract information from it.
2. As can be seen in the first simulation in Fig. 7 the Li-ion battery will
need a cooling system to maintain its temperature at an optimal value. Without
it the battery’s temperature can raise above the maximum temperature and it
will damage the battery
3. The OpenModelica is a great tool that can help us in creating what if
scenarios and we rapidly can take decisions about the dynamics of any physical
system.
4. Even OpenModelica helped us to see the extent of the heat generation
in the battery we still need to do experiment and to determine if we took all the
parameters in this model, so the next phase will be to determine experimentally
the battery’s coefficients and to validate the model.
Acknowledgements. The authors would like to thank University “Politehnica”
of Timisoara, and also to RAAL S.A. Company for the support in this endeavor.
REFERENCES
*** Modelica. Modelica A., Retrieved May 5, 2012, from Modelica:
www.modelica.org, 2012.
*** OpenModelica. OpenModelica, Retrieved April 5, 2012, from OpenModelica:
www.openmodelica.org, 2012.
Aron A., Girban G., Pop C., About the Solution of a Battery Mathematical Model. Int.
Conf. of Diff. Geom. and Dynamical Systems, Bucharest, Balkan Society of
Geometers, Geometry Balkan Press, 2010, p. 10.
Asakura K., Shimomura M., Shodai T., Study of Life Evaluation Methods for Li-ion
Batteries for Backup Application. Journal of Power Sources, 119-121, 902-905
(2003, June).
Gomadam P. M., Weidner J. W., Dougal R. A., White R. E., Mathematical Modeling of
Lithium-ion and Nikel Battery Systems. Journal of Power Sources , 110, 267-284
(2002).
Jiangang L., Xiangming H., Maosong F., Hunrong W., Changin J., Shichao Z., Capacity
Fading of LiCr0.1Mn1.9O4/MPCF Cells at Elevated Temperature. Ionics, 12,
153-157 (2006).
Matthias D., Andrew C., Sinclair G., McDonald J., Dynamic Model of a Lead Acid
Battery for Use in. Journal of Power Sources , 161 (2), 1400-1411 (2005).
Paier O., The E-Car Challenge. Kuli User Meeting. Steyr, Austria, 2011.
Sievers M., Sievers U., Mao S., Thermal Modelling of New Li-ion Cell Design
Modifications. Forschung im Ingenieurwesen, 74 (4), 215-231 (2010).

Bul. Inst. Polit. Iaşi, t. LVIII (LXII), f. 3, 2012 115
Valerie H. J., Ahmad A. P., Thomas S., Temperature-Dependent Battery Models for.
17th Electric Vehicle Simposium, National Renewable Energy Laboratory,
Montreal,Canada, 2000, p. 15.
Winston. GWL Power. Retrieved May 5, 2012, from GWL Power: http://www.evpower.eu/docs/GWL-LFP-Product-Spec-260AH-7000AH.pdf,
(2011, May 5).
Zheng G., Popov N.B., White R.E., Effect of Temperature on Performance of
LaNi4.76Sn0.24. Journal of Applied Electrochemistry , 12, 1328-1332 (1997)..
Ziyad M., Salameh M.A., A Mathematical Model for Lead-acid Batteries. IEEE Trans.
Energy Convers., 7, 93-97 (1992).
SIMULĂRI 1D ASUPRA SCHIMBULUI TERMIC AL BATERIILOR
VEHICULELOR ELECTRICE FOLOSIND PACHETUL SOFTWARE
OPENMODELICA
(Rezumat)
Datorită creşterii nevoii de energie cu emisii zero, producătorii de vehicule au
fost forţaţi să caute soluţii către zona vehiculelor electrice (EV) şi a vehiculelor hibride
(HEV). Acestea folosesc pentru propulsie energie electrică, energie cu emisii zero. Deşi
acest tip de locomoţie nu este unul nou, încercări de a realiza maşini electrice datând de
la începutul secolului XX, realizarea acestora fiind temperată de dificultăţile stocării
acestei energii. Totuşi pe la mijlocul secolului trecut, datorită nevoii de mobilitate au
fost dezvoltate baterii solide care pot stoca o densitate mai mare de energie, ceea ce a
ajutat şi la dezvoltarea vehiculelor electrice.
Recunoscând importanţa mare a acestui tip de locomţie, RAAL S.A. a iniţiat un
studiu aspura necesităţii răcirii acestor baterii, această lucrare reprezentând un prim pas
in realizarea unor sisteme de răcire pentru bateriile vehiculelor electrice.
În lucrarea de faţă este prezentată o modalitate de realizare a unui model de
baterie, care să includă şi influenţa temperaturii bateriei în performanţele acesteia, cât şi
pentru a vedea necesitatea unui astfel de sistem de răcire. Modelul teoretic este
implementat folosind pachetul software gratuit OpenModelica. Avantajul acestui pachet
software faţă de altele cum ar fi Mathlab si Mathematica, în afara gratuităţii acestuia,
este limbajul de programare, care este un limbaj bazat pe rezolvarea ecuaţiilor ceea ce
ne permite modelarea sistemelor fizice in limbajul ştiinţific, fără ajutor din partea unor
specialişti in programare structurată.
Pentru exemplificarea avantajelor oferite, lucrarea prezintă rezultatele a două
simulări:
Prima simulare este realizată pe o celulă a bateriei folosind o încărcare constantă,
rezistor în Fig. 6, şi fără o răcire a bateriei. După cum se poate observa din rezltatele
acestei simulari Fig. 7, Fig. 8 şi Fig. 9, în funcţie de curentul extras temperatura bateriei
creşte cu 40 o C în timp de 1 oră. Scopul acestei simulări a fost de a derermina
necesitatea de răcire a unei astfel de baterii.
Cel de al doi-lea exemplu este o simulare în care se ia in considerare şi o răcire a
bateriei prin convecţie şi o încărcare variabilă (Fig. 133). Rezultatele acestei simulări
sunt prezentate în Fig. 122 şi Fig. 111, unde se poate observa termostatarea bateriei dar
şi a variaţiei temperaturii în funcţie de puterea extrasă din baterie, putere reprezentată de
curentul extras.

116 Vlad Marţian et al
În concluzie se poate afirma că bateriile solide de tipul Li-ion necesită o răcire,
iar aceasta depinde de puterea extrasă.
Pachetul OpenModelica este un mediu de simulare util, care permite crearea şi
simularea, în diferite condiţii, a modelelor fizice cu uşurinţă şi cu evitarea erorilor de
calcul.
În continuare se va încerca dezvoltarea unor modele de răcire mai complicate şi
care să reflecte cât mai aproape de adevăr realitatea.

BULETINUL INSTITUTULUI POLITEHNIC DIN IAŞI
Publicat de
Universitatea Tehnică „Gheorghe Asachi” din Iaşi
Tomul LVIII (LXII), Fasc. 3, 2012
Secţia
MATEMATICĂ. MECANICĂ TEORETICĂ. FIZICĂ
MODELING ON HEAT TRANSFER DURING MALT DRYING
PROCESS
BY
PETRU CÂRLESCU ∗ , VASILE DOBRE, IOAN ŢENU and RADU ROŞCA
“Ion Ionescu de la Brad”
University of Agricultural Science and Veterinary Medicine of Iasi,
Department of Pedotehnics
Received: June 5, 2012
Accepted for publication: 3 July, 2012
Abstract. Development of computers and software made possible CFD
(Computational Fluid Dynamics) modeling of heat transfer phenomena in drying
of malt. Validation of mathematical model of heat transfer process in a porous
medium was achieved by measuring the temperature of the malt in several points
in the malt layer. Simultaneously with the temperature profile, obtained by CFD
simulation of air velocity profile is obtained in the drying process. Modeling and
simulation enables optimization of the drying process by increasing dry product
quality and lower power consumption.
Key words: numerical simulation, heat transfer, drying process, malt.
1. Introduction
Malt drying operation is important in the process of brewing. This
process involves high energy consumption and management according to the
drying process resulting in the final of malt and beer quality. The malt is
produced from barley and drying operations are carried out by following malt
bed of warm air flow. Hot air flow through the porous layer malt bottom-up
heating products and takes a certain amount of moisture in it and drains with air.
The loss of moisture is lost and a significant amount of heat particularly where
air is not recovered. Knowing the temperature distribution in the malt bed
∗ Corresponding author: e-mail: pcarlescu@yahoo.com

118 Petru Cârlescu et al
indicate overheating or unheating areas and can layer a uniform temperature. By
knowing the temperature profile in the layer of barley can optimize the air flow
and temperature in the bed.
Many mathematical models have been developed to simulate the heat and
the moisture transfer in aerated bulk stored grains. The models were obtained at
relatively low temperatures and low humidity to grain.
The partial differential equation models for wheat storage with aeration
were developed by (Metzger, 1983) and (Wilson, 1988).
The models simulated forced convective heat and moisture transfer in
vertical direction, but the model was not validated. (Chang et al., 1993, 1994)
and (Sinicio et al., 1997) developed a rigorous model to predict the temperature
and moisture content of wheat during storage with aeration, and found that
prediction result is in reasonable agreement with observed data. (Sun&Wood,
1997), (Jia et al., 2001), (Andrade, 2001) and (Devilla, 2002) simulated the
temperature changes in a wheat storage bin respectively, and however, the
moisture changes were not done. (Iguaz et al., 2004) developed a model for the
storage of rough rice during periods with aeration.
Two models of the phenomenon of mass and heat transfer in a bed of
grains was developed and analyzed (Thorpe, 2007). In a subsequent paper
(Thorpe, 2008) is calculated on CFD models to a software that simulates heat
and moisture transfer in the bad grain. Based model and simulation of (Thorpe,
2008), (Wang et al., 2010) developed and validated by experimental
measurements of temperature transducers introduction the theoretical model at
different points in a grain silo. The models proposed by the authors cited were
introduced and air temperature of product less than 30 ° C, and two-dimensional
simulations were preformed. This paper proposes the modeling and simulations
in FLUENT software in the 3D heat transfer in the malt bed temperatures of up
to 95 ° C.
2. Mathematical Modeling of the Physical Phenomenon of Drying
2.1. Transfer Equation
The physical drying phenomenon that occurs in the grain bed obeys the law
of conservation. However, to solve such a diversity of problems the equations
that govern heat and mass transfer are expressed in very general terms and they
do not model heat and mass transfer in the malt bulks during malt drying per se.
As a result they have to be tailored to suit malt drying applications. To date,
making the modifications to the standard CFD software appears to have been a
stumbling block for most grain-dry technologists. This physical phenomenon is
d escribed mathematically by a partial differential equation of general form
∂
( ρ φ )
a
∂t
( ρ v ) ( )
+∇ φ =∇ Γ∇ φ + Sφ
, (1)
a

Bul. Inst. Polit. Iaşi, t. LVIII (LXII), f. 3, 2012 119
where: Φ is the quantity of interest which in this case is the energy or humidity
of the intergranular air, ρ a is the density of air, v is the superficial or Darcian
velocity of the air as opposed to the average velocity of the air flowing between
the grain kernels, Γ is the effective diffusion coefficient of Φ through a bed of
grains, t is time, ∇ is the del operator Eqs. (2), S Φ is a source term.
∂ ∂ ∂
∇ = + + . (2)
∂x
∂y
∂z
Eq. (1) refers to a differentially small region of grain and this implies that the
properties have been averaged over some finite volume, otherwise they would
be discontinuous at the boundaries of the malt kernels and intergranular air.
2.1. Heat Transfer
The variable, Φ, in the generalised transport Eq. (1) can also represent
energy. In the case of porous media, such as a bulk of malt, the enthalpies of the
fluid (air) and solid (malt kernels) phases must be considered. The computer
software used in this work solves an enthalpy balance that ultimately reduces to
⎛
⎛ ∂HW
⎞⎞∂T
⎜( ρεc
a a
+ ρs(1
− ε) ⎜cs + cwW + ca ( ρavT
)
T
⎟⎟
+ ∇ =
⎝
⎝
∂ ⎠⎠
∂t
(3)
2
= k ∇ T + S ,
eff
en
where: c a , c s and c w are the specific heats of air, malt and liquid water,
respectively, ρ s is the density of malt kernels on a dry basis, ε – void fraction of
the bed of malt (we assume that value 0,15) , W – malt moisture content , H W -
is the integral heat of wetting of the malt, T – temperature, k eff is the effective
thermal conductivity of the bulk of malt (0.157 W/m 2 K), S en is the thermal
source term that results from heat being liberated or extracted from the malt
when they adsorb or desorb moisture. (Thorpe, 2007) demonstrates that ∂H w /∂T
is negligibly small for grains compared with the specific heat of moist grain and
it can be ignored. A feature of FLUENT is that the specific heat of the porous
zone must be entered as a constant, but in this application the term equivalent to
specific heat, i.e. c g +c w W+(∂H w /∂T), varies with moisture content. For he
purposes of the simulation presented here we assume that the start value of the
malt moisture content, W, is 0.557, which corresponds to a moisture content of
49% wet basis and the grain has a temperature of 21 ºC. The source term, S en ,
takes the form
W
Sen =−hs(1 −ε) ρ ∂
s
,
∂t
(4)
where: h s is the heat of sorption of water on the malt. (Hunter, 1989)
demonstrates that the ratio of the heat of sorption to the latent heat of
vaporisation, h v , of free water is given by

120 Petru Cârlescu et al
hs
p dT dr
= +
h r dp
dT
v
sat
1 .
(Hunter, 1987) provides the following empirical relationship between the
saturation vapour pressure of water and temperature
25
6 ⋅10 ⎛ 6800 ⎞
psat = exp .
5 ⎜ − ⎟
(6)
( T + 273.15)
⎝ T + 273.15 ⎠
The relative humidity, r, of the intergranular air is defined as in Eq. (7), where p
is the vapour pressure of water and p sat is the saturation vapour pressure of free
water.
p
r = .
p
(7)
sat
where: p – vapour pressure of water vapour may be expressed in terms of the
humidity, w, of the intergranular air by
wpatm p = .
0.622 + w
(8)
where: p atm is atmospheric pressure. Differentiation and inversion of Eq. (6)
results in
dT
⎛T
+ 273.15 ⎞ ⎛ 6800 ⎞
=⎜ ⎟ 5 .
dpsat
p
⎜ −
sat
T + 273.15
⎟
(9)
⎝ ⎠ ⎝
⎠
and
dr
Ar
= exp( −BWe
).
dT (10)
T + C
( ) 2
where: A, B and C – are empirical constants widely used to sorption isoterm
proposed by (Chung & Pfost, 1967) and for this case assume the values of
(921.69 º C, 18.077 and 112.35 º C).
sat
2.2. Momentum Transfer
A bed of malt constitutes a porous medium and FLUENT accounts for
the resistance to air flow by adding a source term, S i , to the standard momentum
transport equation. S i as the pressure gradient along a bed of grain uniformly
aerated in the i-th direction with air with a velocity v i . It takes the form
⎛
3 3
ρ ⎞
a
S =− ⎜ D μv + C v vj⎟
∑ ∑ , i = 1, 2, 3 (11)
i ij j ij
j= 1 j=
1 2
⎝
⎠
where: μ is the viscosity of the intergranular air, v j represent the components
of the velocity in all three dimensions, ρ a is the density of air, v is the superficial
or Darcian velocity of the air as opposed to the average velocity of the air
flowing between the malt kernels, D ij is tensor component what represent the
Darcian or viscous resistance, C ij is tensor component what represent inertial
(5)

Bul. Inst. Polit. Iaşi, t. LVIII (LXII), f. 3, 2012 121
resistence. The empirical coefficients D ij , C ij can be related to those in the
FLUENT - CFD programme Eq. (12). The components of the resistance in
these two directions and resistance terms in the horizontal direction can be
designated as R h and S ho , and those in the vertical direction R v and S v . Two
orthogonal horizontal directions are designated by the indices 1 and 2, and the
vertical direction has the index 3.
This is implemented in FLUENT by introducing temperature of 30 º C the
viscosity and density of air are, respectively (18.37·10 6 Pa·s and 1.191 kg/m 3 ).
Using these values we find that D 11 =D 22 =1.833·10 8 , D 33 =2.037·10 8 and
C 11 =C 22 =18371, C 33 =26767.
⎧Rh
μ; i= j = 1,2,
⎪
D Rv
; i j 3,
ij
= ⎨ μ = =
⎪
⎩0; i≠ j, i= 1,2,3, j = 1,2,3.
(12)
⎧2 ⋅ Sh0
ρa; i= j = 1,2,
⎪
Cij = ⎨2 ⋅ Sv ρa; i= j = 3,
⎪
⎩0; i≠ j, i= 1,2,3, j = 1,2,3.
3. Simulation
The aim of this paper is to illustrate how a standard CFD package may be
modified so it can be used to simulate heat and moisture processes that occur
during malt drying process. The features that apply specifically to bulk dry malt
are accommodated by User-Defined Functions (UDF-s) written in a high-level
computer language such as C and introduce in the computer package FLUENT.
A possible further outcome of the work is that it demonstrates how standard
CFD codes can be modified to account for the behaviour of other drying
systems such as those used for horticultural and other agricultural products.
Pre-processing as a first step in CFD simulation is achieved by creating
the drying zone geometry and meshes it with the GAMBIT software.
Drying area is composed of three regions shown in Fig. 1, hot air input
with a low moisture content, malt box region and exhaust air region with high
humidity. The geometry is discretization with structured mesh and 1200000
elements.
Table 1
Geometric dimensions
Areas Length, m Width, m Height, m Thickness malt bed, m
input 0.380 0.370 0.5 -
malt bed 0.380 0.370 0.05 0.007
output 0.380 0.370 0.3 -

122 Petru Cârlescu et al
air
malt bed
a
b
air inlet
Fig. 1 – Geometry dryer: a – geometry; b – meshing.
Processing the initial stage to impose conditions and calculation is
performed in software FLUENT. The conditions chosen to study the effects of
temperature and air flow on conditions within a bed of malt are presented in
Table 2.
Table 2
Initial conditions
W – initial moisture
content,of the malt, %
T – initial temperature
of the malt, º C
v – velocity of the
air, m/s
49 21 1
The physical properties of the system investigated
Specific heat of air c a – J/kg º C 1017
Specific heat of liquid water c w – J/kg º C 4187
Specific heat of malt c s - J/kg º C 2008
Density of air ρ a – kg/m 3 1.169
Density of malt ρ s – kg/m 3 639.2
Temperature and humidity content in the input region are introduced as
five order polynomial functions of time. Function of temperature is introduced
in UDF code in Kelvin as follows
−20 5 −16 4 −12 3 −8 2
( ) = 10 −6⋅ 10 + 9⋅ 10 + 5⋅10
f T τ τ τ
+ 0.0012τ
+ 303.22,
τ +
(13)
where: τ – drying time between 0...25200 s.
The UDF is introduced as a function of humidity content X kg vapor/kg
dry air depending on time.
−23 5 −18 4 −14 3 −10 2
( ) 210 10 210 10
f X = ⋅ τ − τ + ⋅ τ − τ +
+ ⋅ +
−7
4 10 τ 0.0101,
(14)

Bul. Inst. Polit. Iaşi, t. LVIII (LXII), f. 3, 2012 123
where: τ – drying time between 0...25200 s. Laminar solver is chosen because
the dryer inlet air velocity is small.
Processing is performed on a TYAN graphics station dual processor
Intel Xeon Six Core 3.33 GHz, 32 GB DDR3 RAM and two Fermi Tesla C2050
cards, and a drying period of 7 hours for which they were made functions of
temperature and moisture introduced initially during the calculation of the
simulation lasted 15 minutes.
Post-processing. The solver produces a map of the distribution of all the
variables throughout the domain. This result must be processed so that it can be
easily reported, visualized and analyzed. This is the main purpose of the postprocessing
task, which is essential for comprehensive evaluation of the
simulation. Usual outcomes of post-processing for visualization are temperature
and velocity maps, plots of other scalar variables and animations. The postprocessor
can also give information about the instantaneous value of all
variables at certain positions in the domain, and can perform balances and
numerical calculations. Simulation describes the temperature distribution in the
malt bed. Temperature transducers can be placed at different points in this bed,
but can not be so numerous that to draw a precise temperature profile in both
horizontal and vertical section of the malt bed. However the presences of these
transducers are needed to calibrate temperature simulations for some initial
parameters kept constant in the CFD simulation.
Image of Fig. 2 shows the evolution of air temperature at the inlet of the
dryer for a period of 25200 sec. equivalent of 7 hours.
Fig. 2 – Evolution of air temperature during the drying process.
Fig. 3 shows the temperature distribution and evolution in the first layer
of malt into contact with hot air, and in Fig. 4 the temperature profile on the exit
surface of malt layer at the beginning and the end of drying.
Fig. 5 shows the temperature profile and evolution in cross section dryer
through of malt bed at the beginning and end of drying.

124 Petru Cârlescu et al
a
b
Fig. 3 – Temperature evolution in the first layer of malt:
a – time 120 s; b – time 25080 s.
a
b
Fig. 4 – Temperature evolution in the last layer of malt:
a – time 120 s; b – time 25080 s.
a
Fig. 5 – Temperature evolution in the dryer cross section:
a – time 120 s; b – time 25080 s.
4. Conclusions
1. A mathematical model was developed based on dynamic heat transfer
in a bed of malt, and based on this model was simulated in FLUENT software
temperature distribution in this bed with a certain degree of porosity.
b

Bul. Inst. Polit. Iaşi, t. LVIII (LXII), f. 3, 2012 125
2. Modeling and simulation the malt bed as a porous medium raises a
number of problems in terms of heat flow (hot air) through it, the more so as
malt drying involves both heat transfer and mass transfer.
3. The complexity of the mathematical model is high and the CFD
simulation was not possible, or would have taken more time without the support
graphics station of high performance.
4. The temperature profile in malt bed resulting from the CFD simulation
are presented with a high degree of accuracy since the previous simulations
were performed its calibration data obtained from experiment. The input air
temperature in the dryer were obtained experimental and functions by
temperature and time are considered as initial date in simulation.
Both experimental and simulation were unstady preformed on a 3D geometry
dryer.
5. Temperature distribution in the bed malt is very useful in futher
development of the mathematical model by introducing mass transfer depends
on the time evolution of temperature.
6. The stand designed for drying allows large variations of temperature
and humidity of hot air inlet, fitted with sensors for temperature and humidity as
well as air parameters variation.
7. The proposed model and CFD simulation can be used for drying other
grains.
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MODELAREA TRANSFERULUI DE CĂLDURĂ ÎN PROCESUL DE USCARE A
MALŢULUI
(Rezumat)
Dezvoltarea tehnicii de calcul şi a programelor CFD (Computational Fluid
Dynamics) a făcut posibilă modelarea fenomenelor de transfer termic în procesul de
uscare a malţului. Validarea modelului matematic a procesului de transfer termic într-un
mediu poros a fost realizat prin măsurarea temperaturii malţului în mai multe puncte din
strat. Simultan cu obţinerea profilului de temperatură, prin simularea CFD se obţine şi
profilul vitezei aerului circulat în procesul de uscare. Distribuţia temperaturii în stratul
de malţ este foarte utilă în dezvoltarea ulterioara a modelului prin introducerea
transferului de masă care depinde de evoluţia în timp a temperaturii. Standul proiectat
pentru uscare permite variaţii largi ale temperaturii şi umidităţii de intrare a aerului cald,
fiind prevăzut cu senzori de temperatură şi umiditate precum şi cu posibilitatea variaţiei
parametrilor agentului termic de uscare. Modelul propus şi simularea CFD pot fi
utilizate şi pentru uscarea altor cereale.
Modelarea şi simularea procesului de uscare permite optimizarea uscării prin
creşterea calităţii produsului uscat şi scăderea consumului energetic.