Editorial BoardHonorary and Founding EditorProf. Dr. Nyagol Manolov, Bulgaria

Editor-in-ChiefProf. Dr. Slavi Ivanov, Bulgaria

EditorsAssoc. Prof. Dr. Zh. Kalitchin, BulgariaMag. Eng. M. Boneva, Bulgaria

Associate EditorsProf. Dr. Niculae Napoleon Antonescu, RomaniaProf. Dr. Eng. habil. K.-D. Bouzakis, GreeceProf. Dr. Branko Ivkovič, SerbiaProf. Dr. Mehmet Karamis, TurkeyAssoc. Prof. Dr. V. Gecevska, FYR MacedoniaAssoc. Prof. Dr. Mara Kandeva, Bulgaria

International Editorial BoardProf. Dr. Vladimir Andonovic, FYR MacedoniaProf. Dr. Miroslav Babič, SerbiaProf. Dr. G. Haidemenopoulos, GreeceProf. Dr. H. Kaleli, TurkeyProf. Dr. A. Michailidis, GreeceProf. Dr. Nikolai K. Myshkin, BelarusProf. Dr. S. Mitsi, GreeceProf. Dr. P. St. Petkov, BulgariaProf. Dr. Alexandar Rač, SerbiaProf. Dr. R. G. Ripeanu, RomaniaProf. Dr. Andrei Tudor, RomaniaProf. Dr. G. E. Zaikov, RussiaAssist. Prof. Dr. Afsin Alper Cerit, TurkeyAssoc. Prof. Dr. Friedrich Franek, AustriaAssoc. Prof. Dr. Mikolai Kuzinovski, FYR MacedoniaAssoc. Prof. Dr. Fehmi Nair, TurkeyAssoc. Prof. Dr. V. Pozhidaeva, BulgariaAssoc. Prof. Dr. Burhan Selcuk, Turkey

Vol. 22 No 1 2016

JOURNAL OF THE BALKANTRIBOLOGICAL ASSOCIATION ™

Journal of the Balkan Tribological Association is an International Journal edited by the Balkan Tribological Association for rapid scientific and other information, covering all aspects of the processes included in overall tribology, tribomechanics, tribochemistry and tribology.The Journal is referring in Chem. Abstr. and RJCH (Russia).

Aims and ScopeThe decision for editing and printing of the current journal was taken on Balkantrib’93, Sofia, October, 1993 during the Round Table discussion of the representatives of the Balkan countries: Bulgaria, Greece, Former Yu-goslavian Republic of Macedonia, Romania, Turkey and Yugoslavia. The Journal of the Balkan Tribological As-sociation is dedicated to the fundamental and technological research of the third principle in nature – the contacts.The journal will act as international focus for contacts between the specialists working in fundamental and practi-cal areas of tribology.The main topics and examples of the scientific areas of interest to the Journal are:(a) overall tribology, fundamentals of friction and wear, interdisciplinary aspects of tribology;(b) tribotechnics and tribomechanics; friction, abrasive wear, ad he sion, cavitation, corrosion, computer simu-

lation, design and calculation of tribosystems, vibration phenomena, mechanical contacts in gaseous, liquid and solid phase, technological tribological processes, coating tribology, nano- and microtribology;

(c) tribochemistry – defects in solid bodies, tribochemical emissions, triboluminescence, triboche miluminescence, technological tribochemistry; composite materials, polymeric materials in mechanics and tribology; special ma-terials in military and space technologies, kinetics, thermodynamics and mechanism of tribochemical processes;

(d) sealing tribology;(e) biotribology – biological tribology, tribophysiotherapy, tribological wear, biological tribo tech nology, etc.;(f) lubrication – solid, semi-liquid lubricants, additives for oils and lubricants, surface phenomena, wear in the

presence of lubricants; lubricity of fuels; boundary lubrication;(g) ecological tribology; the role of tribology in the sustainable development of technology; tribology of manu-

facturing processes; of machine elements; in transportation engineering;(h) management and organisation of the production; machinery breakdown; oil monitoring;(i) European legislation in the field of tribotechnics and lubricating oils; tribotesting and tribosystem monitoring;(j) educational problems in tribology, lubricating oils, fuels and contacts;(k) contacts – mechanical, agricultural, chemical, medical, social environments.

The Journal of the Balkan Tribological Association is indexed and abstracted in the Science Citation Index Expanded (SciSearch®) and Journal Citation Reports, Science Edition, Thomson Scientific, and in Elsevier Bibliographic Database (http://www.info.scopus.com/detail/what/publishers/. The impact factor is 0.443 in the ‘Chemistry’ subject category of the Journal Citation Reports® (JCR) for 2014.

Compiled by S. K. Ivanov, Zh. D. Kalitchin, J. P. Ivanova, M. I. Boneva, N. Evtimova and V. Shishkova.

Subscription InformationThe Journal of the Balkan Tribological Association (ISSN 1310-4772) is published in four separate books. Regu-lar subscription price: 399 Euro for Europe and 587 USD for all other countries. Nos 1, 2, 3 and 4 will be issued at 30.03; 30.06; 30.09 and 30.12.2015, respectively. 10% agency discount, plus extra postage charges: for Europe 20 Euro regular surface mail and 40 Euro air mail; for all other countries 40 USD regular surface mail and 80 USD for air mail.

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ISSN 1310-4772SciBulCom Ltd.

Wear resistance of coating surface layersT. PETROV, P. TASHEV, M. KANDEVA. Wear Resistance of Surface Layers Modified with Al2O3 and TiCN Nanopowders Weld Overlaid Using TIG and ITIG Methods .......................................................................................................................................................... 304Tribochemistry – special materials in military and space technologiesJ. ALEXANDER, B. S. M. AUGUSTINE. Influence of Microwave Post Curing on the Mechanical and Thermal Properties of Basalt / Epoxy Composites for Aerospace Applications ............................................................................................................................................. 316Coatings – friction in brake padsB. GUNEY, I. MUTLU, A. GAYRETLI. Investigation of Braking Performance of NiCrBSi Coated Brake Discs by Thermal Spraying and Melting ..................................................................................................................................................................................................... 331Coating tribologyS. MITROVIC, D. ADAMOVIC, F. ZIVIC, D. DZUNIC, M. PANTIC, D. JEVREMOVIC. Dry Sliding Wear Behaviour of PVD TiN Coatings for Cold Forming Tools ................................................................................................................................................................... 347Coatings – wear intensity measurementsJ. NAPIORKOWSKI, P. SZCZYGLAK, K. LIGIER, R. KUCZYNSKI. Testing the Wear Intensity of Thin Coatings by the Ball-crater-ing Method in the Process of Selecting Punching Die Materials .................................................................................................................... 356Coatings – chromium oxide filmsZHOU SIHUA, GUO YANHUA, ZHOU XIAODONG, LI JIN, TIAN CANXIN. Effect of Technical Parameters on Structure and De-position Rate of Chromium Oxide Films Prepared by Magnetron Sputtering ............................................................................................... 363Special materials in mechanics and tribologyV. ERTURUN, S. KUMRU, H. FIDAN. Aging Effects on the Microstructure of a Tungsten Inert Gas Welded AA2024 Alloy ............... 372Composite materialsKAN XINRONG, FENG XIAOXIN, ZHANG DONGMEI, SUN HUI, WANG XIAOYAN, YU XIAODONGA, SUN HUZA. Study on the Alkali-activated Cementitious Materials Prepared with Alkali Slag ........................................................................................................ 384Biotribology – ex-poro-hydrodynamic lubricationM. RADU, T. CICONE. Experimental Determination of the Damping Capacity of Highly Compressible Porous Materials Imbibed with Water ............................................................................................................................................................................................................... 390Biotribology – treatment with medicinal compounds of human jointsVL. BOYADZHIEVA, N. STOILOV, R. STOILOV, G. PETROVA, S. STEFANOV, D. MEDJIDIEVA. Cost of Treatment of Rheu-matoid Arthritis, Ankylosing Spondylitis and Psoriatic Arthritis with Biological Disease-modifying Antirheumatic Drugs in the Period 2013–2014 in Bulgaria .................................................................................................................................................................................... 401Lubrication – effect of additives on friction coefficientsM. SIMSEK, O. NTEZIYAREMYE, E. DURAK. Additive Effects on Wear and Friction Performance of Micro-arc Oxidative Coated Journal Bearings .............................................................................................................................................................................................. 410Lubrication – computer simulationD. MILOSEVIC, N. JANJIC, M. VULOVIC, Z. ADAMOVIC. Optimisation of Preventive Maintenance of Lubrication Subsystem by Reliability Simulation Model of V46-6 Engine .............................................................................................................................................. 419Lubrication – effect of pseudo-plastic lubricants and surface roughnessJ. JAVOROVA, K. STANULOV, A. ALEXANDROV, V. ILIUTA. Journal Bearings Lubrication of Non-Newtonian Lubricants with Surface Roughness Effects ............................................................................................................................................................................. 433Role of tribology in sustainable developmentWANG MIN, CHENG WEN, MIN LIANG, ZHENG JIAN-GANG, FANG YONG. Distribution and Correlation of Nutrients and Par-ticle Size in Surface Sediments of Tang-pu Reservoir ................................................................................................................................... 444Polymeric materials – friction-related behaviour of polymer compounds used in the construction of coaxial sealing systemsA. DEACONESCU, T. DEACONESCU. Low Friction Materials Used in the Construction of Hydraulic Sealing Systems in the Case of Small Velocities .............................................................................................................................................................................................. 454Management and organisation of the production – land sliding and land slippingLEI XU, DALI GUO, XIAOHUI ZENG, XIJUN KE. Productivity Simulation and Fracturing Treatment Parameters Optimisation for Oil Wells in Low Permeability Bottom Water Reservoirs ................................................................................................................................... 464Machinery breakdown – preventive maintenanceD. SPASIC, N. JEVTIC, Z. JANJIC, Z. ADAMOVIC. Electrohydraulic System for Automatic Gage Control (AGC) for Tandem Cold Mill Plant in Steelworks Smederevo ............................................................................................................................................................... 471Ecological tribology – risk assessment of hazardous gas transportationJINRONG ZHANG, WENXUAN WANG, LING WU, PING LI. Research on the Diffusion of Hazardous Gas Leakage in Road Trans-portation and Its Influencing Factors .............................................................................................................................................................. 483Machinery breakdown – semantic survey-based risk managementWEN LU, SHIDONG FAN, JIYIN CAO. Research on Semantic Survey-based Ship Machinery Health Management Method Consider-ing Ship Structure and Rule-based Document ................................................................................................................................................ 493Tribology in construction engineering – effect of deformation in CFG pileFEI HUA, D. TARTAKOVSKY, DAWEI SUN. Effect of Deformation of the CFG (Cement Fly-ash Gravel) Pile Group on the Adjacent Metro Tunnel Based on the Equivalent Material ............................................................................................................................................ 507Contacts - adsorption isothermSANG QIN, LI MIN. Study on Logging Interpretation for Shale Gas Reservoirs ........................................................................................ 516InformationThe 2015 Tribology Gold Medal Awarded to Professor Shizhu Wen ............................................................................................................ 522New Member of the International Editorial Board – Assoc. Prof Dr. Friedrich Franek ................................................................................ 523New Member of the International Editorial Board – Prof. Dr. Nikolai K. Myshkin ...................................................................................... 524

Instruction to Authors ..................................................................................................................................................................................... 527

CONTENTSVol. 22, No 1, 2016

Tribotechnics and tribomechanicsN. STOICA, A. TUDOR. Some Aspects Concerning the Thermoelastic Distortions of the Brake Drum-shoe Contact, with Application to Vehicles ...................................................................................................................................................................................................... 1Tribotechnics and tribomechanics – prolongation of firearm lifetimeD. JOVANOVIĆ, B. STOJANOVIĆ, R. JANJIĆ. Capacity for Improving Tribological Characteristics of Barrel Interior Line on a Gun ..... 17Tribotechnics and tribomechanics – tensile strength of steel-plastic geogridsFAN PENG-FEI. Study on Tensile Property of the Steel-plastic Geogrid ..................................................................................................... 28Тribotechnics and tribomechanics – measurement of resistance in elastohydrodynamic contactsC. SACHANAS, A. MOISIADIS, A. MIHAILIDIS. Electrical Contact Resistance Measurement in Elastohydrodynamic Contacts ......... 32Tribotechnics – dynamic oscillation featuresG. POPA, C.-N. BADEA, A. BADEA, V. ŞTEFAN, E. C. BOJE, G. DUMITRU. Dynamic Oscilations Features of the Br 185 Locomo-tive Series ........................................................................................................................................................................................................ 48Tribotechnics and tribomechanics – memory alloysK. ALDAŞ, İ. ÖZKUL. Determination of the Transformation Temperatures of Aged and Low Manganese Rated Cu-Al-Mn Shape Memory Alloys ............................................................................................................................................................................................... 56Tribotechnics – statistic energy analysisXING PENG, HUA LIN, LU CHIHUA, DENG SONG, SU ZHUOYU. Analysis and Optimising Approach for Inner Noise Control Based on Statistical Energy Analysis ............................................................................................................................................................. 66Tribotechnics – tungsten welding of aluminiumS. AMAL BOSCO JUDE, G. R. JINU, P. ARUL FRANCO, MC. LURET. Microstructural Characterisation and Mechanical Properties on the Effect of Weld Speed in Autogenous TIG (Tungsten Inert Gas) Welded AA7075 Butt Joints .......................................................... 79Tribotechnics – smart electric vehicleS. F. LIU, M. H. LEE. User-oriented Prospective Smart Electric Vehicles Innovating Design Development .............................................. 93Technological tribology – working performance of pumping unitsFENG ZI-MING, FANG XIN, DING HUAN-HUAN. Study on Working Performance of Phased Beam Pumping Unit ........................... 104Regulation of yaw phenomenaJIAN HU, LINWEI CAO, ZHANJIN WANG, GANGYAN LI. Automotive Yaw Rate Coordinated Control in Advance under Urgent Steering Condition Based on Active Front Steering and Differential Braking .............................................................................................. 112Computer simulations – molten steel flow, large eddy simulationLI CHAO, BAI MINGHUA. Large Eddy Simulation (LES) of the Molten Steel Flow of Continuous Jumbo Billet Casting Mold Using k – e Two-equation ......................................................................................................................................................................................... 130Computer simulations – social tribologyHUANG SHAO-BIN, LI YAN-MEI, LI YA, XU LI, SHEN LIN-SHAN. Research of a Kind of XML Matching Mechanism ................. 142Tribotesting and electrical systems load forecastingLIU QI, HUANG ZHENZHEN, LI SHENG. Research on Power Load Forecasting Based on Support Vector Machine ............................ 151Design and calculation of tribosystemsZHONGJIU ZHENG. Optimum Global Sliding Mode Controller for DC Motor Servo System Based on Disturbance Observer ............... 160JIAN HU, WEI FAN, ZIRU NIU, GANGYAN LI. Digital Design Method of Variable Ratio Gear for Automobile Recirculating Ball Variable Ratio Steering Gear Based on Meshing Theory ............................................................................................................................... 171JIAN HU, KAIKAI DONG, ZIRU NIU, GANGYAN LI. Digital Design Method of Variable Ratio Gear Pair for Automobile Mechanical Variable Ratio Steering Gear Based on Enveloping Simulation .................................................................................................................... 184CHUANXI LIU. Optimum Imaging Time Selection for ISAR Targets with Complex-motion .................................................................... 200Design and calculation of tribosystems – calibration of gear pumps LIN HAI-BO, PAN WAN-GUI. A New Method for Calibrating Vision Measuring Systems Based on the Black-White Square Array and Its Application ................................................................................................................................................................................................. 210Vibration phenomena – acoustic emissions for recognition of coal rock stabilityJING LI, JIANHUA YUE , YONG YANG, LI ZHAO. Acoustic Emissions Waveform Analysis for the Recognition of Coal Rock Stabil-ity .................................................................................................................................................................................................................... 220Friction and wear of sintered materials in the presence of nanodiamondsE. E. FELDSHTEIN, L. N. DYACHKOVA. The Use of Nanodiamonds to Improve Tribological Properties of Sintered Antifriction Materials ......................................................................................................................................................................................................... 227Wear properties of aluminum alloyM.OVUNDUR, F. MUHAFFEL, H. CIMENOGLU. Characterisation and Tribological Properties of Hard Anodised and Micro-arc Oxidised 5754 Quality Aluminum Alloy ........................................................................................................................................................ 233Wear and corrosionR. G. RIPEANU, M. MINESCU, C. A. NITA. Wear Behaviour of Antifriction Coatings Applied at Parallel Gate Valves ........................ 240Wear – influence of molybdenum coating on frettingD. BADOIU, G. TOMA. On a Dynamic Optimisation Problem of the Quadrilateral Mechanism ............................................................... 250Wear behaviour of stainless steel E. NALCACIOGLU, D. OZYUREK, K. CETINKAYA. Investigation of the Effect of Molybdenum Amount on Wear Behaviour of 17-4 PH Stainless Steel Produced by Powder Metallurgy .............................................................................................................................. 261Wear – rating life of cylindrical roller bearingsS. CREŢU, M. BENCHEA, A. IOVAN-DRAGOMIR. On Basic Reference Rating Life of Cylindrical Roller Bearings. Part II. Elastic-Plastic Analysis ............................................................................................................................................................................................... 272Friction coefficient in gear pairL. IVANOVIC, T. MACKIC, B. STOJANOVIC. Analysis of the Instantaneous Friction Coefficient of the Trochoidal Gear Pair ............ 281Wear behaviour of deep cryogenic treated steelK. AMINI, M. SAFARI, A. SHAFYEI. Investigation of Hardness and Wear Behaviour of the Deep Cryogenic Treated 1.3255 Tool Steel ................................................................................................................................................................................................................ 294

see b.c.

* For correspondence.

281

Journal of the Balkan Tribological Association Vol. 22, No 1, 281–293 (2016)

Friction coefficient in gear pair

ANALYSIS OF THE INSTANTANEOUS FRICTION COEFFICIENTOF THE TROCHOIDAL GEAR PAIR

L. IVANOVICa*, T. MACKICb, B. STOJANOVICa

a Faculty of Engineering, University of Kragujevac, 6 Sestre Janjic,34000 Kragujevac, SerbiaE-mail: lozica@kg.ac.rs, blaza@kg.ac.rs b Faculty of Mechanical Engineering, University of Banja Luka, Banja Luka, Bosnia and HerzegovinaE-mail: tihomir.mackic@unibl.rs

ABSTRACT

Wear of contact surfaces is one of the most common causes of gear failure. In addition to physical and chemical processes that occur in the contact zone, ki-nematic meshing parameters have a great effect on the wear process. This paper analyses the influence of geometric and kinematic parameters of the gear profile on the instantaneous friction coefficient of the trochoidal gearing in the gerоtor pump. The friction coefficient is not a constant value and it varies during the teeth meshing as a result of a constant change in the lubrication regime. There are many empirical expressions for calculation of the instantaneous friction coefficient, but due to numerous limitations, many of these expressions can be applied only to certain operating conditions and loads. The analysis of the friction coefficient was performed through two contact periods defined by the change in direction of the rolling velocity of the meshing profiles contact point. The obtained results enable identification of dominant factors that influence the instantaneous friction coefficient and show possibilities for further development of analytical models of mechanical losses in the gerоtor pump.

Keywords: gerоtor, trochoid, sliding velocity, contact stress, friction coefficient, wear.

AIMS AND BACKGROUND

There is a wide range of pumps used in modern industrial practice. One of the simplest is a gerоtor pump. Gerоtor (trochoidal) pumps are rotary pumps with in-

282

ternal gearing characterised by compactness, simple construction and possibility of wide application in various areas. The gerоtor pump consists of an internal and an external gear. The external gear has one more tooth than the internal one. Both these gears rotate in the same direction with all the teeth involved in meshing at the same time, which is a significant advantage because it decreases the contact load. During the gear rotation, the suction zone is in the area where the teeth leave the intertooth space, and the discharge zone is in the area where they enter the intertooth space forcing the fluid through the discharge.

Gear pair parameters have a significant influence on the functional charac-teristics of the pump. Resistance to wear has an important role in the performance and working life of the pump. Hydrostatic gerоtor pumps offer many advantages, but fatigue damage has been noticed on the surface of the teeth in the hydro-static gerоtor pumps used in aeroplanes1. Therefore, scientific investigations are becoming more and more focused on friction and wear problems, as well as on destruction of the working surfaces that result in a great number of gear failures.

There has been a significant advance in analysis of friction and wear in evol-vent gears. In that sense, development of elastohydrodynamic (EHD) lubrication theory is of special importance because it studies the minimum oil layer thick-ness between the meshing surfaces in order to define the right lubrication re-gime. Based on this theory, numerous empirical formulas for calculation of the instantaneous friction coefficient of the contact surfaces have been developed2–8. However, there are no data in the available literature on such investigations being conducted for trochoidal gearing, hence this paper makes an original contribution to this scientific field of study.

The aim of this paper is to determine the character of the change of the in-stantaneous friction coefficient in the trochoidal pump using available formulas.

GEAR GEOMETRY DEFINING

The geometric and kinematic gearing model used in this paper is based on the investigation results given in the references9–13.

The internal gear profile of the studied gerоtor pump is an equidistant epi-trochoid, while the external gear profile is a combination of circular arches. The basic geometric relations between the meshing gear profiles in different meshing phases are given in Fig. 1.

The coordinate contact points in the coordinate system of the epitrochoid Otxtyt, are:

( ) ( )( ) ( )cos cos cos

sin sin sin

t

t

x e z z c

y e z z c

= ϕ + l ϕ − ϕ + d = ϕ + l ϕ − ϕ + d

, (1)

283

where z is the number of external gear teeth, c – the equidistant radius coefficient:

crce

= (2)

and d is the leaning angle:

sinarctan

cosc

c

βd =

l − β. (3)

The rest of the parameters used in equation (1) are explained in details in the references14,15.

Since the internal gear profile has the form of an equidistant epitrochoid, the expression for determination of the curvature radius rc can be written in the form14:

( )

32 2

2

1 2 cos.

1 cosc

cc

ezec

z z

+ l − l β r = −+ l − l + β

(4)

The change of the curvature radius sign occurs in the inflection point, where the curvature changes from convex into concave, and vice versa. Therefore, it is easy to calculate the equivalent radius of the meshing profiles based on the fol-lowing expression:

,c ce

c c

r

r

rr =

r ± (5)

where the sign (+) is used for meshing of two convex surfaces, while the sign (–) is used for the meshing of a concave and a convex surface.

Fig. 1. A geometric model of the trochoidal gear pair with fixed axes: a – initial meshing moment, b – the moment when the next tooth enters the meshing

a b

284

KINEMATIC ANALYSIS OF GEAR MESHING

This section analyses the movement of the meshing profiles contact point. The internal gear rotates with an angular velocity ωt, and the external gear rotates with an angular velocity ωa. The transmission ratio is given with the expression:

1

.aat

t

zu

z

w −= =w

(6)

The trochoidal gear tooth starts meshing at the point 1 (Fig. 1a) when the ro-tation angle of the generating coordinate system is βc=0. The meshing continues in the direction of rotation and when the meshing point comes to the position 2 (Fig. 1b), the following tooth enters the meshing. At that moment the value of the angle βc is 2p/z, hence, based on the angular displacement, we can calculate the value of the angle βc for which the contact at characteristic contact points i = 1, 2, 3, ... is realised (Fig. 1a): ( )2

1 .ci iz

pβ = − (7)

If we consider the movement of a tooth meshing point, we can write its coor-dinates in the fixed coordinate system in the form of14:

( ){ }

( ){ }cos 1 cos

sin sin

P c c

P c c

x e z c

y e z c

= l β − − β + d

= l β − β + d. (8)

The equations (8) describe the contact line of the meshing profiles. When gears mesh, the profiles roll and slide against each other. The rolling speed equals the relative velocity of the contact point v→rt , i.e. v→ra. The sliding of the profile at the contact point P is a result of the difference in relative velocities of the points at the internal, i.e. external gear profiles. Based on this and the geometric relations given in Fig. 2, the expression for determination of the sliding velocity of the meshing profiles can be written as:

( )1

2 21 2 cos .s t c

cv e z

z

= w + l − l β −

(9)

In addition to the sliding velocity, a sum of rolling velocities is also important for wear analysis. The sum of rolling velocities is a vector sum of relative veloc-ities v→rt and v→ra at the meshing profiles contact point, and it can be expressed as:

( ) ( )1

2 21 2 cos 1 2 ,r t c

cv e

z

′= w + l − l β − + d

(10)

285

where

( )[ ]2

1 1 cos.

1 2 cosc

c

z − − l β′d =

+ l − l β (11)

The sliding velocity equals zero only when the profiles are in contact at the pitch point, where it changes its direction. With the trochoidal gearing, the contact line does not always go through the pitch point, which depends on the relation of the geometrical parameters of the gear profile16.

ANALYSIS OF CONTACT FORCES AND STRESSESWITH THE TROCHOIDAL PUMP

While the gears are in motion, the total normal load is transferred through the simultaneous meshing of the gear pair. However, the load is not equally shared among the teeth pairs because they have different stiffness, size, shape and posi-tion of the tooth profile contact point. Under the effect of forces, elastic teeth de-formations occur at the instantaneous contact points and the size of teeth changes during the meshing. If we assume that all the profiles have regular geometric forms and that the equation of the mesh stiffness is constant, the further analysis will be based on the fact that the normal force is proportional to deformation of the gear pair in contact. Based on this and the calculations of the torque in Refs 14, 15, the expression that defines the contact force at the instantaneous con-

Fig. 2. Kinematic parameters of the trochoidal gear pair

286

tact point can be written as:

( )

( )

2

sin,

1 sin

n tF O nini q

njj p

MF

e z=

a=

− a∑ (12)

where p and q are the ordinal numbers of the first and the last tooth of the external gear that transfer the load, MFn(Ot) is the total moment of normal forces around the centre of the gear15 that is equal to the sum of moments of normal forces for individual teeth pairs.

In order to define the necessary geometric values, it is necessary to write the expressions for the projections of the line segment CP—

i (Ref. 14):

( ){ }( ){ }

cos cos ,

sin sin

CPi ci ci i

CPi ci ci i

x e z z c

y e z c

= l β − − β + d

= l β − β + d (13)

where

( )2 1

,ci

i

z

p −β = −ψ (14)

and where i is the ordinal number of the analysed gear pair, and y is reference angle.

Now the size of the angle ani (Fig. 2) needed for calculation of the contact force can be determined:

arctan .CPini

CPi

y

xa = (15)

Finally, when the contact force is known, the maximum Hertzian contact pressure Pmax can be calculated according to the following formula:

max ,2

q

e

WEP =

p r (16)

where W is the specific load, and Eq – the equivalent elastic modulus.

EMPIRICAL EXPRESSION FOR CALCULATION OF THE FRICTIONCOEFFICIENT

The friction coefficient is not a constant value and it varies during teeth meshing due to constant changes in the contact surface lubrication regime. EHD lubrica-tion theory that has been developed in the last 50 years has enabled significant advances in understanding of the processes that occur between the contact sur-faces. However, determination of the instantaneous friction coefficient value is still a considerate problem. There are many empirical expressions for calculation

287

of the instantaneous friction coefficient, but due to numerous limitations, many of these expressions can be applied only to certain operating conditions and loads. Most of these empirical expressions are for the boundary or mixed (partial) EHD lubrication regime. Since the trochoidal pump operates at rather low values of Hertz pressure and low circumferential velocities, some of the experimental ex-pressions2,3 are not considered in this paper.

The boundary and partial EHD lubrication regimes are most commonly used in gear meshing17. Shipheng and Cheng18,19 have given a great contribution to de-velopment of the theory of partial EHD lubrication. They suggested that the par-tial EHD lubrication regime should be used for gear meshing because the thick-ness of the elastohydrodynamic layer between the contact surfaces had the same value as roughness parameter. The paper by Xu2 is especially important because it studies the friction coefficient in the EHD lubrication regime, even though his approach has some disadvantages and needs to be further developed. The first disadvantage is that this model requires a lot of calculation, i.e. processing time. He modelled a cylindrical gear with spur teeth and an ideal load distribution, and considered the teeth stiff bodies. Xu thought that empirical formulae could not be used to determine the friction coefficient at gear meshing, but he oversaw the fact that all these expressions were defined for the boundary and partial EHD lubrication regime. Most of the expressions given in the paper2 cannot be applied to the partial EHD regime, and particularly not to the boundary EHD lubrication regime. An additional disadvantage of the Xu expression is that he worked with the surface roughness of Ra = 0.07 mm which is difficult to accomplish in a gear.

Misharin3 proposed an expression applicable to larger loads, where there are some limitations concerning the lowest value of the Hertzian pressure (Pmax≥ 245 MPa) and some limitations concerning the ratio of the sliding velocity to the summary rolling velocity vs / vr from 0.4 to 1.3. These values are much higher than the ones in the trochoidal pump, thus the Misharin expression cannot be applied. In conditions of larger loads, like with the trochoidal pump, it is possible to apply the Drozdov expression4 even though there are some limitations concerning the maximum Hertzian pressure. This expression is used for determination of the instantaneous coefficient for the meshing surfaces that roll and slide against each other and for the boundary lubrication regime.

O’Donoghue and Cameron5 studied friction during the rolling and sliding contacts on the Amsler machine20. Benedict and Kelley6 proposed an empirical formula for determination of the friction coefficient under the mixed lubrication regime based on the experimental tests performed on a tribological machine with discs. Höhn et al.7 calculated the average friction coefficient as a function of the load, surface roughness, viscosity, velocity and geometric properties. A similar expression is given in the ISO TC60 Standard8, where the sliding velocity has not been taken into account.

288

Marjanović21 used pin on disc tests for tribological investigations. During the testing, variable velocities and normal forces with different pins were used in order to investigate the influence of the equivalent radius, sliding velocity and contact pressure. Lubrication was performed using the standard gear lubrication oil (GALAX HIPOL B, SAE 90). The following expression was obtained:

0.584max0.231

0.00234 .s

P

vm =

(17)

ANALYSIS OF THE INSTANTANEOUS FRICTION COEFFICIENT

The analysis was performed using Matlab® and the program procedure that en-abled monitoring of the changes of the studied values. In order to perform a better analysis of the changes of the instantaneous friction coefficient and the effect of different parameters on the lubrication regime, four models of the trochoidal pump with different basic parameters were used14:

Model A: z = 6, l = 1.375, rc = 9.75 mm, c = 2.75;Model B: z = 6, l = 1.575, rc = 14.06 mm, c = 3.95;Model C: z = 6, l = 1.675, rc = 16.20 mm, c = 4.55;Model D: z = 5, l = 1.85, rc = 13.35 mm, c = 3.75.Other initial data are given in Table 1.

The analysis of the friction coefficient was performed through two contact periods or two contact phases. Figure 3 shows the distribution of velocities and the leaning angle for the model B. The first phase refers to the meshing from the beginning to the moment when the rolling velocity of the external gear changes its direction and when the maximum leaning angle δmax is reached.

Table 1. Initial data

Parameter Sign ValueVolumetric capacity (m3/rev) q 14×10–6

Working pressure (bar) p 6Number of revolutions of the pump shaft (rpm) nt 1500Angular velocity of the pump shaft (s–1) wt 50pDynamic viscosity of hydraulic fluid (kg/ms) h 0.02Density of hydraulic fluid (kg/m3) rf 900Gears width (mm) b 16.46Eccentricity of pump (mm) e 3.56Young modulus of gear material (N/m2) E 2×1011

Poisson coefficient of gear material (–) ν 0.3Arithmetic average of the roughness profile (mm) Ra 0.8

289

The second phase starts at the moment when the maximum leaning angle is achieved and lasts till the end of meshing, when the angle is δ=0. The curves of the sliding velocity and the sum of rolling velocities intersect at the point when the angle reaches the maximum value.

For analysis of the instantenous friction coefficient for the second contact period, the formula (17) was used since it is meant for the half-dry and dry friction area. This formula does not take into account the rolling velocity, and it has no influence on the value of the instantenous friction coefficient. The experiments were performed on a pin/disk system21, where the disc performed rotary move-ment, so the meshing was similar to the one in the second contact period.

The rest of the formulae were used for the analysis of the instantaneous fric-tion coefficient for the first contact period, because they are meant for the area

Fig. 3. Distribution of velocities and the leaning angle for the model B

7

6

5

4

3

2

1

0

–1

454035302520151050

0 10 20 30 40 50 60 70 80 90 100 110120 130 140rotation angle βc (°)

150 160 170 180

sliding velocitysummary rolling velocityleaning angle

velo

city

(m/s

)

leaning angle (°)

δmax

1. part 2. part

Fig. 4. Distribution of the instantaneous friction coefficient for the model A

0.15

0.10

0.05

0.000 5 10 15 20 25 30 35 40

rotation angle βc (°)

fric

tion

coef

ficie

nt

DrozdovCameronHöhnBenedictISO TC60average value

290

with a mixed or full EHD lubrication regime. These formulae were obtained based on the experiments conducted on the disc/disc or ball/disc system, where both parts roll into the same direction during the meshing, i.e. where their cir-cumferential velocities had the same direction at the contact point. This kind of meshing corresponds to the first contact period.

Figures 4, 5 and 6 show distributions of the friction coefficient and the av-erage arithmetic value of the coefficient for the models A, B and C, respectively. There are two characteristic positions where the curves have peaks. The first one is when the contact point passes through the pitch point, where the friction coeffi-cient abruptly increases until it reaches its static value. The second characteristic

Fig. 5. Distribution of the instantaneous friction coefficient for the model B

0.15

0.10

0.05

0.000 5 10 15 20 25 30 35 40 45 50

rotation angle βc (°)

fric

tion

coef

ficie

nt

DrozdovCameronHöhnBenedictISO TC60average value

Fig. 6. Distribution of the instantaneous friction coefficient for the model C

0.10

0.05

0.000 5 10 15 20 25 30 35 40 45 50

rotation angle βc (°)

fric

tion

coef

ficie

nt

DrozdovCameronHöhnBenedictISO TC60average value

291

position is when the contact point overlaps with the inflection point of the tro-choidal gear curve, when the curves reach their local minimum.

The curve for the average arithmetic value of the friction coefficient is very similar to the curves obtained using the Drozdov and Cameron expressions (for models A, B and C).

With the model D (Fig. 7), the teeth start meshing at the pitch point, so there is no peak that appears with the other models. Maximum values of the friction coefficient are lower with this model, which leads to a conclusion that the friction coefficient decreases with the decrease in the number of teeth. With models that have the external gear with 6 teeth, the friction coefficient decreases with the increase in the value of l.

In the second meshing period (Fig. 8), the friction coefficient increases up to its maximum value and then it mainly decreases linearly. In this phase, the model D exhibits the lowest friction coefficient.

Fig. 7. Distribution of the instantaneous friction coefficient for the model D

0.10

0.05

0.000 5 10 15 20 25 30 35 40 45 50 55

rotation angle βc (°)

fric

tion

coef

ficie

ntDrozdovCameronHöhnBenedictISO TC60average value

Fig. 8. Distribution of the friction coefficient for the second contact period

0.06

0.05

0.04

0.03

0.02

0.01

044 61 78 95 112 129 146 163 180

rotation angle βc (°)

fric

tion

coef

ficie

nt

Model AModel BModel CModel D

292

CONCLUSIONS

It has been determined that the friction coefficient value varies in the range from 0.02 to 0.12 for the studied trochoidal pump models. The maximum value (with the models A, B and C) occurs when the contact point goes through the pitch point. The next characteristic point of an abrupt change in the value of the friction coefficient is the trochoidal gear inflection point. The lowest value of the friction coefficient during the distribution is noticed in the model D, which shows that the decrease in the number of teeth has a favourable effect on the decrease of friction between the contact surfaces.

If we consider the decrease in the friction coefficient, the model D has given the best results for the first contact period while the model A has the lowest fric-tion coefficient in the second period.

The empirical expressions given by Drozdov and Cameron have shown the least deviations from the average friction coefficient. It is necessary to study some newer investigations in the field of EHD lubrication, so that a more accurate anal-ysis and a more precise lubrication regime for the second contact period can be defined. A special attention should also be paid to pitting in the upper part of the teeth since in this meshing period the gear rolling velocities, i.e. their relative velocities have opposite directions.

ACKNOWLEDGEMENTS

Research presented in this paper was supported by Ministry of Education, Sci-ence and Technological Development of Republic of Serbia, Grant TR 35033.

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Received 1 July 2015Revised 20 September 2015