Kireenkov_article207_Improve (1).pdf

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ENOC 2008, Saint Petersburg, Russia, June, 30-July, 4 2008

COUPLED MODELS OF ROLLING, SLIDING AND WHIRLING FRICTION

Alexey Kireenkov

Institute for Problems in Mechanics

Russian Academy of SciencesRussia

[email protected] u

Abstract

It is proposed the essentially new combined models of

friction of rubbed rigid solids under conditions of combined kinematics when besides the sliding and whirlingthere is a motion of rolling. A correlation between fric-

tion of rolling and sliding is modelling on the base experimental investigations from the tyre and railway in-

dustry. In correspondence with these results, the maininfluence of the rolling on the force state in the area of

contact consists in the asymmetry of the diagram of thedistribution of the normal contact stresses. This asym-

metry is well described by the linear function with onecoefficient that depends on the direction of motion and

velocity of rolling and it leads to appearance of nonzerolateral component of the friction force. Under the pro-

posed model of friction are understudied the interrela-tions between friction force components, torques and

velocities. The model involves the replacement of exactintegral expressions for the net vector and torque of the

dry friction forces, formed with the assumption thatCoulomb's friction law is valid at each point of the con-

tact area, by appropriate Pade approximations. Thisapproach substantially simplifies the combined dry fric-

tion modeling, making the calculation of double inte-grals over the contact area unnecessary. Unlike avail-

able models, the model based on the Pade approxima-tions enables one to account adequately for the relation-

ship between force and kinematical characteristics over the entire range of angular and linear velocities. The

approximate model preserves all properties of the model based on the exact integral expressions and correctly

describes the behaviour of the net vector and torque of the friction forces and their first derivatives at zero and

infinity. Moreover, one does not have even to calculatethe integrals to determine the coefficients of the Pade

approximation. The corresponded coefficients can beidentified from experiments. Consequently, the models

based on Pade approximations may be considered asreological models of combined dry friction.

Key wordsCombined friction of rolling, sliding and whirling.

1. IntroductionThere are many works in the scientific literature de-

voted to the dry friction, classification of that at the dependence on the aims of investigations can be foundat [Zhuravlev, V.Ph., Kireenkov, A.A., 2005]. At the

most of these publications authors are using the Cou-lomb model of dry friction supposed that the friction

force at the point of contact is direct opposite the rela-tive velocities of sliding and it is not depend on the

module of velocity. However, there are many experi-mental facts about the violation of this law at case when

the rubbed bodies are participated simultaneously in thetranslational, whirling and rolling motions. Following

the experimental results from the tyre manufactory inthe work [Svendenius, J. ,2003] was established the

empiric dependence of the distribution of normal con-tact stresses at area of contact from the velocity of roll-

ing. At the corresponded this dependence the influenceof whirling is shifting of the symmetric form of contact

stresses distribution in the direction of rolling. This shiftis good approximated by the liner function with one

coefficient depended on the direction and value of therolling velocity. Asymmetry at distribution of the nor-

mal contact stresses at the case of circle areas of contactcause the appearance of the component of the friction

force directed on normal to the trajectory of motion.

2. Combined model of friction of rolling and slidingConstruction of combined model of friction of rolling

and sliding is performed at he supposition the validities

of the Coulomb law at the differential form for the

small element of area S inside of spot of contact, in

correspondence with the differentials of the net vec-tor and torque

C M of the friction forces relatively

the center of contact circle are defined by the formulas

,

( , )

C f dS dM f dS

v y x

r VVF

V V

V

(1)

where - coefficient of friction, ( , ) x yr - radius vec-

tor of the elementary square inside of spot of con-tact (fig.1), - distribution of normal contact stresses,


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- linear velocity of sliding and - angle velocity of

whirling of contact spot center.

Figure 1.

Asymmetry at the symmetric distribution of normalcontact stresses ( , ) x y , arisen at the non zero velocity

of rolling , in the rectangular coordinate

system { }Oy , axis of which is directed alone the

velocity of sliding (fig.1) is described by the following

dependence:

( , )

( , ) 1 , 1, 0 при 0r r r r

x y x y x y k k k

R

(2)

where - radius of contact circle, - axis of rectan-

gular coordinate system directed perpendicularly to theinstantaneous velocity of rolling (fig. 2), and

r - dimensionless coefficient the sign of that is de-

pendent on the direction of motion

Figure 2.

Connection of the coordinate systems { } xOy and

{ }O are given by rotate transform on the angle

[0, 2] that is defined from the values of projec-

tions , x

of the instantaneous velocity of rolling

on the axis x and (fig. 2):

2

cos sin , sin cos ,

cos , sin ,r x r r x y

x y x y

(3)

Substitution of expressions (3) to the formula (2) givesdependence of distribution of the normal contact

stresses on the value and direction of rolling velocity:

( , ) ( , ) 1 ( )r y xr

k x y x y x y

R

(4)

Figure 3.The typical behavior of function (4) for different val-

ues of the rolling coefficient r k at the supposition that

at the absence of rolling distribution of the normal con-

tact stresses (solid line) is described by Hertz low2 2

2 2

3( , ) 1

2

x y x y

R R

(5)

is presented at fig.3 by the dash lines.

Integration of the expressions (1) on the spot contacttaking in account the formula (4) gives the exact inte-

gral model of combined friction of sliding and rolling,

that in dimensionless variables: x xR , yR ,

ˆ ˆ ˆ ˆ ˆ( , ) ( , ) y x y N R in supposition that distribution of

contact stress at the absence of rolling has central sym-

metry , ) ( ) x y r , has in polar coordinate system

with origin at the center of contact circle

cos , sin , [0,1], [0, 2 ] x r y r r following form

2 1

2 2 20 0

2 1 3 2

2 2 20 0

22 1

2 2 20 0

( sin ) ( ) 1 sin

2 sin

( )cos

2 sin

( sin ) ( ) 1 sin

2 sin

r x r

r y

r

r x r

C

v ur r r k r F fN drd

u r v uvr

fNk ur r F drd

u r v uvr

ur vr r r k r drd M fRN

u r v uvr

(6)

where F

and F are the components of the friction

force directed correspondently on the tangent and nor-

mal to the trajectory of motion, andC

is the torque of

whirling respectively the center of circle area directed

perpendicularly to plane of whirling.Transition at the model (6) from the c onsideration of

the connection of the friction of rolling and sliding interm of projection and of the velocity of rolling

to the its absolute value and to the angle between

direction of rolling and sliding gives the equivalentform of this model

2 1

2 2 20 0

2 1 3 2

2 2 20 0

22 1

2 2 2

0 0

( sin ) ( ) 1 sin sin

2 sin

( ) cos cos

2 sin

( sin ) ( ) 1 sin sin

2 sin

r

r

r

C

v ur r r k r F fN dr d

u r v uvr

ur r F fN k dr d

u r v uvr

ur vr r r k r fRN drd

u r v uvr

(7)

One of the distinguish feature of model (6)-(7) is appearance of none zero component of friction force nor-

mally directed to the trajectory of motion. At the pres-


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ence of combined motion of rolling and sliding the net

vector of friction forces is not opposite directed to thevector of sliding velocity.

At supposition that the distribution of the contact

stresses ( , ) x y is play role of density the violation at its

central symmetry defined by the formula (4) leads toshift of the gravity center of contact circle respectively

the geometric centre in the direction of whirling (along

axe (fig.2)) on value , the projections of which toaxes and are defined by the formulas:

13

0

cos , sin ,

( )

y x x r r y r r

r r

s sk sk s sk sk

s R r r dr

(8)

The shift of the center of gravity of contact spot, de-fined by formulas (8) leads to appearance of torque of

rollingr

parallelly directed to the plane of sliding the

projections of that on the directions of the tangent

and normal to the trajectory of motion are defined

by expression:

cos , sin , y x

r r r r

r r

r r

M M M M M M

M sk N

(9)

Thus the net torque of friction forces at rectangular coordinate system one axis of that is directed on thetangent of trajectory of motion is

, , C M M M M (10)Expressions (6)-(7) for torque

C and force compo-

nents , F as function of , v have several significant

properties detailed investigated in [Kireenkov, A.A.,

2008]. These properties allow simplifying the frictionmodeling with the aid of replacing of the exact integralmodels (6)-(7) by the approximate models based on the

Pade approximations of corresponded order. This approach permits to escape the integration over the spot of

contact. In corresponded with results of the

work[Kireenkov, A.A., (2008)] the combined modelfriction rolling and sliding of the first order based on the

partial-linear Pade approximation has form:

0 0

0 0

0 0 0 00 0

00 0 0

sin,

sin,

cos,

, , ,

1 1 1, ,

r ur x r r ur C

r r x r r r

y r r r r

r

r C ur C v uu v

C

u ur r v

M u k M v M u k M v M

u mv u mv

F v k F u F v k F u F

v au v au

u k F u F k F

u bu u bv

F F F F M M M M

F v M u v F

m M u a F v b k F u

(11)

The model of the first order (11) is sufficient for the

dynamics investigation, but for more precise qualitativeanalysis the model of the second order is required. This

model not only good approximates the exact integralmodels (6)-(7) but conserves all their properties such as

behavior of these functions and their first derivatives at

zero and infinity.

2 20

2 2 2 2

2 2

02 2 2 2

2 20

2 2 2 2

2 2

02 2 2 2

2 2

2 2 2

( )

( ) sin

( )

( ) sin

( ) ( ) cos

r ur xC

r

r ur

r r x

r

r r

yr r r r

r

M u muv k M v M

v muv u u v

M u muv k M v

u muv v u v

F v auv k F u F

v auv u u v

F v auv k F u

v auv u u vk F u buv k F u buv

F v buv u v

2

0 00 00

, ,C

u ur r v

buv u

F u v M v F a m b

F v M u k F u

(12)

The comparison of the integral model (solid line) and

models of the first (11) (dash-dot line) and the sec-

ond (12) (dash line) for the Hertz distribution of contactstresses (5) as function of parameter v u is pre-

sented on the fig.4:

Figure 4.


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Models (11)-(12) of combined friction of rolling and

sliding based on the Pade approximations can be con-sidered as reological models, because there are no re-

quired in solving of real problems to calculate the dou-

ble integrals, defined the coefficients of Pade approxi-mations. These coefficients can be defined from theexperiments.

3. Hertz caseIf the distribution of normal contact stresses obeys the

Hertz law (5), then with the aid of the transfer of theorigin of the coordinate system to the instantaneous

center of the velocities 1O (fig.5) to, possibly, obtain

the precise equations of model in the elementary func-tions.

Figure 5.

Normal and tangential components of the net vector of

friction forces in the polar coordinate system { , , }1

O r

with the origin in the instantaneous center of velocities(Fig. 5) being distant behind the geometric center of the

contact area to the value v kR in the direction of

normal to the velocity of sliding speed v are defined by

formulas:2

2

( , ) c o s

( , ) s in

G

G

F fR q q dqd

F fR q q dqd

(13)

Integration limits in formula (13) depend on the ar-

rangement of the instantaneous center of velocities. If the instantaneous center of velocities is located inside

the contact area 1k , then polar angle [0,2 ] ,

while if out of the area of contact that* * *[ , ], sin 1 R h k . Interval of the variation

of the dimensionless radius-vector r R is found

from the conditions of the intersection of polar ray with

the circle of the contact area 2[ , ]q q :

2 2

1

2 2

2

cos 1 sin

cos 1 sin

q k k

q k k

(14)

The distribution of contact stresses, which is obeyedthe Hertz law (13), in the introduced variables takes the

form

2 2

2

3, ) 2 cos( ) 1 1

2

sin ( cos )r y xr

N q q kq k

R

k q k q

(15)

The substitution of expression (15) into formula (13),

taking into account of formulas (14) and location of thecenter of instantaneous velocities, defines tangential and

normal force components of friction force as the piece-wise-continuous functions of the parameter , which

are smooth at joint point:*

2

*1

2

1

*2

*1

2

1

0

2 2

2 2

0

2

1 ( cos ) cos , 1

1 ( cos ) cos , 1

( , ) (sin ) , 1

( , ) (sin ) , 1

32

2

q

r x

r q

q

r x

r q

q

q y

r q

r

q

k q k q dqd k

F k

q k q dqd k

q q dqd k

F k

q q dqd k

N q k

2cos 1q k

(16)

Integrals (16) are calculated in the quadratures

[Kireenkov, A.A., 2008]. The result of integrationrepresents the tangential and normal component of the

friction force as function of two parameters [0, )k

and [ 1,1]k r

.

The whirling torque it is calculated on the basis equal-

ity [Zhuravlev, V.Ph., 1998]: C h M hF , where

is the main torque of friction forces relative to the

instantaneous center of the velocities:*

2

*1

2

1

2

22

0

1 ( cos ) , 1

1 ( cos ) , 1

q

r x

r q

h q

r x

r q

k q k q dqd k

M Rk

q k q dqd k

(17)

Integrals (2.22) are also calculated in the elementaryfunctions. Thus, the transfer of the origin of the coordi-

nate system to the instantaneous center of velocitiesmakes it possible to construct in the Hertz case the pre-

cise coupled model of the rolling and sliding friction,represented in the elementary functions. However, the

obtained result is too lengthy and is inconvenient. Inorder to use it in the dynamics problems it is necessary

to build, at the beginning, the appropriate Pade approximations. Consequently, even if it is possible to

accurately integrate the equations of model, the most

effective approach is the using of developed abovemodels based on direct construction of the Pade’s ap-

proximations.


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References

Kireenkov, A.A., (2008). Coupled Models of Slid-

ing and Rolling Friction in the solids dynamics onthe rubbed plane. Izvest. RAS. MTT., 2008, No. 3,

pp. 116–131.

Svendenius, J. (2003). Tire models for use in braking

application. LICENTIATE THESIS (ISRN

LUTFD2/TFRT--3232--SE). Department of Automatic

Control, Lund Institute of Technology, Box 118, SE-

221 00 Lund, Sweden. 95pZhuravlev, V.Ph., (1998). The model of dry friction in

the problem of the rolling of rigid bodies. Journal of

Applied Mathematics and Mechanics. Vol.62. No.5. pp.705-710Zhuravlev, V.Ph., Kireenkov, A.A. (2005). Pade expan-

sions in the two-dimensional model of coulomb friction.Mechanics of Solids. Vol.40, No.2. pp.1–10.


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