on the influence of a pin type on the friction losses in pin bearings.pdf

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On the influence of a pin type on the friction losses in pin bearings

Zivota Antonic a, Nebojsa Nikolic a,, Dragi Radomirovic b

a Faculty of Technical Sciences, Trg Dositeja Obradovica 6, 21000 Novi Sad, Serbiab Faculty of Agriculture, Trg Dositeja Obradovica 8, 21000 Novi Sad, Serbia

a r t i c l e i n f o a b s t r a c t

Article history:Received 17 July 2010

Received in revised form 27 January 2011

Accepted 13 February 2011

Available online 15 March 2011

The aim of this paper is to analyze lubricated revolute joints from the viewpoint of frictionlosses. The paper deals with the lubricated revolute joints composed of a pin and two more

elements connected by means of the pin. It is considered here how different pin types affect

friction losses in pinbearings. Three existing pintypes areinvestigated: a pinpress-fittedeither

in the first or in the second element or the one free to rotate in both elements. As a measure of

friction losses, a dissipation function is determinedin all three cases. Using the results obtained,

the advantages of using the full-floating pin with respect to two other types are demonstrated

by means of a numerical example.

2011 Elsevier Ltd. All rights reserved.

Keywords:

Full-floating pin

Press-fitted pin

Viscous friction moment

Dissipation function

1. Introduction

The majority of machines and devices comprise moving elements that are in contact with each other. As a result of thisinteraction, the friction between coupled surfaces occurs, which leads to their wear. To reduce the resulting unwanted outcome as

much as possible, a lubricant is supplied into the clearance between coupled elements. Then, instead of dry friction, there is the

inner friction in the lubricant, the coefficient of which can be up to 1000 times smaller than the one corresponding to dry friction

[1]. Consequently, the energy dissipation is significantly reduced and there is higher reliability and durability of the coupled

elements.

Rotating pins in revolute joints are examples of moving elements exposed to this behaviour. What is of interest here is the pin

rotation in revolute joints accompanied with viscous friction, which is frequently found in crank gears of IC engines, piston

compressors and other joint mechanisms. However, this problem has not been investigated widely, perhaps due to the fact that

the pin rotation is difficult to determine. Chun [2] and Ha and Chun [3] considered the pin motion as a two-dimensional problem of

the hydrodynamic lubrication theory. Using this theory, they obtained the coefficient of friction, the oilfilm thickness as well as the

friction moment in pin bearings. Aoki et al. [4] studied the influence of a pin deformation on lubrication conditions at a pin boss

bearing. Livanos and Kyrtatos [5] analyzed the friction in all components of the piston assembly. They also presented a

mathematical model that enables one to predict the oilfilm thickness in pin bearings from the given magnitude of an external load.Spuria et al. [6] presented the model that predicts lubricating conditions around a full-floating piston pin in the connecting rod

small-end and pin boss bearing. The angular velocity of the piston pin and the moments of viscous friction are also determined

therein. Flores et al. [7] analyzed different cases of the clearance of a pin in revolute joints: clearance without friction, clearance

with dry friction and clearance with viscous friction.

Lubricated revolute joints composed of a pin and two more elements, connected by means of the pin, are considered in this

paper. Using the well-known Petroff's equation for the viscous friction moment in hydrodynamic bearings [8,9], it is analyzed here

how the different pin types affect friction losses in pin bearings. Three existing pin types are considered: a) pin press-fitted in the

first element, b) pin press-fitted in the second element and c) pin free to rotate in both elements. The expressions for the

Mechanism and Machine Theory 46 (2011) 975985

Corresponding author. Tel.: +381 21 485 2355; fax: +381 21 6350 592.

E-mail address: [email protected] (N. Nikolic).

0094-114X/$

see front matter 2011 Elsevier Ltd. All rights reserved.doi:10.1016/j.mechmachtheory.2011.02.003

Contents lists available at ScienceDirect

Mechanism and Machine Theory

j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / m e c h m t
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corresponding dissipation functions with the viscous friction coefficient are derived. Using the expressions derived, friction losses

are estimated for the sake of defining the most appropriate pin type in one illustrative example.

2. Determination of a viscous friction coefficient in pin bearings

In lubricated simple revolute joints, a pin of the radius rand the length l performs rotational movement about its own axis in a

bearing of the radius R, as shown in Fig. 1. The pin also performs translation movement with respect to the axis of the bearing, but

this movement will not be considered in this paper. Since there is a lubricant between the pin and the bearing, the moment of

viscous friction Mf opposes the pin rotation. Assuming that the clearance c(c=R-r) between the pin and its bearing is small and

filled with the lubricant of dynamic viscosity , and that there are no misalignments between the axes, the moment of viscous

friction can be determined by using the Petroff's equation [10]. A general form of this equation is

Mf =2lr3

c; 1

where represents the relative angular velocity of the pin with respect to the bearing.Introducing the substitution

k =2lr3

c; 2

Eq. (1) becomes

Mf = k: 3

Fig. 1. Decomposed simple revolute joint.

Nomenclature

l length of a pin [m]

l1= l length of pin bearing in Element 1 [m]l2=(1-)l length of pin bearing in Element 2 [m]L length of Element 1 [m]

m mass of Element 1 [kg]

r radius of a pin [m]c radial clearance [m]

dynamic viscosity [Pas]i angular velocity of the i

th element, i=1,2,3 [s1]

Mijp moment of viscous friction between elements i and j for the pth pin type [Nm],

i = 1; 2; 3; j = 1; 2; 3; ij; p = I; II; III

k viscous friction coefficient (dissipation function coefficient) [Nms]

kp viscous friction coefficient for the pth pin type, p=I, II, III [Nms]

D dissipation function [W]

Dp dissipation function of the system for the pth pin type, p=I, II, III [W]

T kinetic energy [J]

V potential energy [J]

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As a measure of viscous friction losses, a dissipation function is introduced. Physically, it is related to power losses due to

viscous friction. The dissipation function has the form [11]

D = 12

k2: 4

Of interest here is to determine the viscous friction coefficients for different pin types in revolute joints and to compare them

mutually.

2.1. Revolute joint composed of three elements

A schematic review of a revolute joint composed of three elements is shown in Fig. 2. Element 1 is a bar, which has a cylindric

hole at the end. The hole serves as a bearing for the pin (Element 3) and the length of the bearing is l1. The pin has two more

bearings in Element 2, the length of each is l2/2. The sum of the lengths l1 and l2 is equal to the length of the pin, labelled by l,a s it is

seen in Fig. 2b. For the sake of further calculation, the dimensionless coefficient is introduced

= l1 = l: 5

The angular velocities of Elements 1 and 2 are 1 and 2, respectively (Fig. 2a).The pin can be built into the assembly presented in Fig. 2a in several ways, depending on whether the clearance between the

pin and other elements exists or not. Thus, there are three pin types:

- pin press-fitted in Element 2 (which will be considered here as Case I),

- pin press-fitted in Element 1 (Case II) and

- pin free to rotate in both elements (Case III).

All these three cases will be considered separately.

2.1.1. Case I

In Case I, Element 3 (pin) is press-fitted in Element 2, which is illustrated in Fig. 3. Elements 1 and 2 rotate with the angular

velocities1 and 2, respectively, as it is earlier shown in Fig. 2a. The angular velocity of Element 3 is equal to that of Element 2, i.e.3=2, because these two elements behave like a rigid body. The length of the pin bearing in Element 1 is l1 (Fig. 3a). Since thespace between Elements 1 and 3 is filled with the lubricant, the viscous friction moment M13

I opposes the rotation of Element 1.

This moment is in balance with the viscous friction moment M31I acting on Element 3. The cylindrical area on Element 3 where the

viscous friction exists is shown in Fig. 3b as a shaded area.

Using Eq. (4) and taking into account that the relative angular velocity of the pin with respect to the bearing is in this case

(12), the dissipation function is

DI =1

2 kI 1 2 2; 6

Fig. 2. Revolute joint composed of three elements: (a) front view during operation and (b) decomposed assembly with basic geometric parameters.

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where, due to Eq. (2),

kI =2l1r

3

c: 7

Since, according to Eq. (5), l1 = l, Eq. (7) becomes

kI = k; 8

so that the dissipation function can be written as

DI =1

2k 1 2

2: 9

2.1.2. Case II

In this case, Element 3 is press-fitted in Element 1 (Fig. 4) and, therefore, they both rotate with the same angular velocity i.e.

3=1. Element 2 rotates with the angular velocity2, asit is earlier shown in Fig.2a.The bearing ofthepin in Element 2 has twoparts, the length of each is l2/2 (Fig. 4a). The space between Elements 2 and 3 is filled with the lubricant, so the viscous friction

moment M32II opposes the rotation of Elements 1 and 3. The shaded area shown on Element 3 in Fig. 4b depicts the area where the

moment M32II

acts.Taking into account the conditions described and implementing the same procedure as in Case I in section 2.1.1, similar

expressions for the viscous friction coefficient and the dissipation function are obtained

kII = 1 k; 10

DII =1

21 k 1 2

2:

11

2.1.3. Case III

In this case, the pin (Element 3) pivots freely in both elements that it connects and is called the full-floating pin (Fig. 5). The pin

has a bearing of the length l1 in Element 1 and two bearings in Element 2, which have the total length l2 (Fig. 5a). Elements 1 and 2

rotate with the angular velocities 1 and 2, respectively (Fig. 5b).

Fig. 4. Pin press-fitted in element 1: (a) cross-section; (b) viscous friction moments acting on Element 3.

Fig. 3. Pin press-fitted in Element 2: (a) cross-section; (b) viscous friction moments acting on Elements 1 and 3.

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A cylindrical gap between Elements 1 and 3 as well as the gaps between Elements 2 and 3 is filled with the lubricant, so there are

the viscous friction moments acting on each of the elements. Of interest here are the moments acting on Element 3, M31III and M32

III,

caused by rotations of Elements 1 and 2, respectively. A shaded area of the pin denotes the area where M31III acts, while the non-

shaded cylindrical areas show where M32III acts (Fig. 5b). As a result of the viscous friction moments, Element 3 rotates with the

angular velocity 3.According to Eq. (4), the friction losses between Elements 1 and 3 can be described by the following dissipation function DIII

D

0

III =

1

2 kI 1 3 2

: 12

There are also friction losses between Elements 2 and 3, described by the dissipation function DIII,

D00

III =1

2kII 3 2

2: 13

The overall dissipation function is

DIII = D0

III + D00

III: 14

The unknown angular velocity3 can be determined by using the moment equation for Element 3. Taking into account Eqs. (3),(8) and (10), the viscous friction moments acting on Element 3 can be expressed as

MIII31 = k 1 3 15

and

MIII32 = 1 k 3 2 : 16

Neglecting the moment of inertia of the pin with respect to the axis of its rotation, the viscous friction moments M31

III and M32

III

must be in balance, i.e.

k 1 3 = 1 k 3 2 : 17

Solving Eq. (17) for 3, gives:

3 = 1 + 1 2: 18

Substituting Eq. (18) into Eqs. (12) and (13), and then into Eq. (14), one obtains

DIII =1

2 1 k 1 2

2; 19

i.e.:

DIII =1

2kIII 1 2

2; 20

Fig. 5. Full-floating pin assembly: (a) cross-section; (b) decomposed assembly with the viscous friction moments acting on the elements.



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where

kIII = 1 k: 21

2.1.4. Comparison of the viscous friction coefficients

For the sake of the reader, the results obtained above are summarized in Fig. 6 and clearly show three pin types and the

corresponding expressions for the viscous friction coefficient.

Based on the expressions derived, the plots of the viscous friction coefficient versus the parameter are given in Fig. 7 for allthree cases considered. It is seen that kIII has the lowest value. In other words, the coefficient of viscous friction is lower in the case

of the full-floating pin than in the case when the pin is press-fitted in any of the assembly elements. This is completely in

accordance with the fact that the full-floating pin is the most convenient solution in machine design, especially in internal

combustion engines [12,13].

It should be noted that, for practical reasons, the parameter ranges between 0.4 and 0.6, because the values outside theinterval do not provide satisfactory stiffness of the assembly and, at the same time, jeopardize its functionality. The graph in Fig. 7

indicates that the full-floating pin is the most advantageous option compared to both other pin types especially in the region

[0.4, 0.6].

3. Illustrative example

To illustrate different dynamic characteristics of the three pin types analyzed, the example of a rotating rod shown in Fig. 8 is

considered. If the mass of the pin is neglected, the system has one degree of freedom. Therefore, the position of the rod is de fined

by one generalized coordinate, angle . It will be determined here how the revolute joint affects the motion of the rod, causedeither by its own weight or by an external moment Mext for all the three pin types analyzed above. The parameter values used in

this example are given in Table 1.

In order to make use of the dissipation functions derived, the equation of motion will be found starting from the Lagrange

equation

d

dt

T

+T

+Dp

+V

= Q

;p = I; II; III; 22

where the kinetic energy Tand potential energy Vare, respectively:

T =1

6mL

2

2; 23

Fig. 6. Viscous friction coefficient expressions for different pin types.



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V = const: mgL

2cos; 24

where the dots stand for the time derivatives.

The dissipation functions are

Dp =1

2kp

2;p = I; II; III: 25

When motion of the rod is caused by its own weight, then the generalized force Q is

Q

= 0: 26

However, more frequently the external moment Mext acts on the rod. Then, the generalized force Q is

Q

= Mext: 27

Substituting Eqs. (23)(27) into Eq. (22), the non-linear differential equations of free and forced motion of the rod are

obtained, respectively

+ 2p+ 20sin= 0;p = I; II; III; 28

+ 2p+ 20sin=

3 Mextm L2

;p = I; II; III; 29

where

0 =

ffiffiffiffiffiffi3g

2L

r; p =

3kp

2 m L2;p = I; II; III:

Further, Eq. (28) is solvednumerically for the initial conditions given in the Table 1, and the results obtained are plotted in Fig. 9.

Fig. 8. A rotating rod with viscous friction in the revolute joint.

Fig. 7. Viscous friction coefficient versus .

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Fig. 9. Characteristics of the rod motion caused by its own weight: a) angular displacement of the rod versus time; b) angular velocity of the rod versus time;c) viscous friction moment acting on the pin versus time; d) power loss in pin bearings versus time.

Table 1

Values of the system parameters.

Mass of the rod m =3 kg

Length of the rod L = 1 m

Coefficient of dissipation function Case I kI=0.4 Nms

Coefficient of dissipation function Case II kII=0.6 Nms

Coefficient of dissipation function Case III kIII=0.24 Nms

Dimensionless characteristics = 0.4Initial position of the rod

(0)=3rad, ifM

ext= 0

(0)=0rad, if not M

ext= 0

Initial angular velocity of the rod 0 = 0rad/sExternal moment Mext=29.43 Nm, constant

Mext=29.43 sin(t) Nm, variable



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Fig. 9a and b shows that the angular displacement and the angular velocity of the rod have the highest amplitudes in the case of

the full-floating pin, which means that the motion is damped less than in other cases. Unlike the angular displacement and angular

velocity, the situation is not so clear when it comes to the viscous friction moment and the friction loss power, which are presented

in Fig. 9c and d, respectively. Namely, only at the beginning of the period considered (during the first 3 s), the moment and the

power are lower in the case of the full-floating pin (Case III) than in the cases of the press-fitted pin (Cases I and II). Later on, the

opposite is true. This can be explained by the fact that the motion ofthe system with higher values of the viscous friction coefficient

(Cases I and II) is damped faster, so the energy available is almost exhausted after a short time interval.

When the external moment Mextacts on the rod, the advantages of the full-floating pin become more obvious. By solving Eq. (29)

numerically forthe initialconditions given in Table 1, the motion ofthe rod causedby the moment Mextis obtained. By using Eq.(3),the

viscous friction moment acting on the pin is determined. The comparison of the viscous friction moments for different pin types is

shown in Figs. 10 and 11. The graph in Fig. 10 is obtained for Mext= M0=const, and the graph in Fig. 11 for Mext=M0 sin(t).

Fig. 11. Viscous friction moment acting on the pin versus time, Mext=29.43 sin(t) Nm.

Fig. 10. Viscous friction moment acting on the pin versus time, Mext=29.43 Nm.

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Fig. 10 shows that the viscous friction moment is the lowest in the case of the full-floating pin duringfirst 20 s. After this period,

the viscous friction moments for all the three cases become almost equal to the external moment Mext and the steady motion

regime is established.

When the motion is caused by the periodically changing external moment, the full-floating pin is again the one whichgives the

lowest viscous friction resistance. This can be seen in Fig. 11.

It should be noted that the authors have also determined the motion of the rod for parameter values other than those given in

Table 1: m ={2, 1, 0.5}, L={0.8, 0.6, 0.4} and = {0.5, 0.6}. However, the results obtained are not qualitatively different from thosereported in Figs. 9, 10 and 11, and, thus, are not shown here. They imply that the conclusions reached on the pin type influence on

friction losses can be considered as general.

4. Conclusion

In this study, the influence of a pin type in revolute joints composed of three elements on friction losses in pin bearings has

been investigated. The procedure yielding the viscous friction coefficients for different pin types has been developed. The full-

floating pin has been confirmed to be the most convenient solution for revolute joints composed of three elements in terms of

energy losses. It has been shown by means of an illustrative example that for different motion laws of a rotating rod, the viscous

friction moments in the full-floating pin bearings are the lowest.

Acknowledgements

The authors would like to express their gratitude to Prof. Ivana Kovacic from the University of Novi Sad for her very helpfulcooperation and encouragement during the research work.

References

[1] D.D. Fuller, Theory and Practice of Lubrication for Engineers, John Wiley & Sons, Inc., New York, 1984.[2] S.M. Chun, Study on the rotating motion of a piston pin of full floating type, Journal of the Korean Society of Tribologists and Lubrication Engineers 23 (3)

(2007) 95102.[3] D.H. Ha, S.M. Chun, Pin-boss bearing lubrication analysis of a diesel engine piston receiving high combustion pressure, Journal of the Korean Society of

Tribologists and Lubrication Engineers 24 (3) (2008) 133139.[4] Y. Aoki, K. Fujii, M. Takiguchi, Y. Takashima, Effects of piston pin deformation lubrication characteristics at pin boss bearing, Effects of Bend and Elliptical

Deformation, ASME-Publication-ICE 34 (1) (2000) 139146.[5] G.A. Livanos, N.P. Kyrtatos, Friction model of a marine diesel engine piston assembly, Tribology International 40 (2007) 14411453.[6] M. Spuria, D. Bonneau, Y.L. Baratoux, P.G. Molari, A Dynamic Model for an Internal Combustion Engine Full-floating Piston Pin in Lubricated Conditions, XVII

Congresso dell'Associazione Italiana di Meccanica Teorica e Applicata, Firenze, 2005.[7] P. Flores, J. Ambrosio, J.C.P. Claro, H.M. Lankarani, C.S. Koshy, A study on dynamics of mechanical systems including joints with clearance and lubrication,

Mechanism and Machine Theory 41 (2006) 247

261.[8] B. Bhushan, Introduction to Tribology, John Wiley & Sons, Inc., New York, 2002.[9] G.W. Stachowiak, A.W. Batchelor, Engineering Tribology, Butterworth-Heinemann, Woburn, 2001.

[10] A.I. Lurie, Analytical Mechanics (Foundations of Engineering Mechanics), Springer, 2002.[11] N.P. Petroff, Friction in machines and the effect of lubricant, Inzhenernyj Zhurnal 1 (1883) 71140.[12] E. Koehler, R Flierl, Verbrennungsmotoren - Motormechanik, Berechnung und Auslegung des Hubkolbenmotors, 4. AuflageVieweg & Sohn Verlag, GWV

Fachverlage GmbH, Wiesbaden, 2006.[13] D.N. Virubov, S.I. Efimov, N.A. Ivaschenko, M.G. Kruglov, O.B. Leonov, A.S. Orlin, S.G. Roganov, F.F. Simakov, N.D. Chainov, V.K. Chistiakov, Internal Combustion

Engines (in Russian), Maschinostroenie, 4, Edition, Moscow, 1984.

Zivota Antonic was born in 1950 in Brdarica, Serbia. He received his BSc and MSc in Mechanical Engineering from the University of

Novi Sad, Faculty of Technical Sciences. He is currently Teaching and Research Assistant in the field of internal combustion engines at

the Faculty of Technical Sciences. His scientific areas of interest are mechanics and tribology.

Nebojsa Nikolic was born on 27 January 1969 in Loznica, Serbia. He graduated from Mechanical Engineering at the University of Novi

Sad, Faculty of Technical Sciences in 1994. He received his MSc in internal combustion engines from the same faculty, where he

currently holds the position of Teaching and Research Assistant. His scientific areas of interest are dynamics and control of internal

combustion engines. He is also interested in software development and has been involved in several software development projects.

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Dragi Radomirovic was born on 22 July 1957th in Nis, Serbia. He graduated from Mechanical Engineering at the University of Novi

Sad, Faculty of Technical Sciences in 1981, where he also received his MSc and PhD theses in Mechanics. He currently holds the

position of Full Professor at the Faculty of Agriculture, University of Novi Sad. His scientific areas of interest are analytical mechanics,

vibrations, dynamics of machines and engineering design.

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