Lingyuan Kong

Robert G.Parker

1

Professor

e-mail:parker.242@osu.edu

Department of Mechanical Engineering,

Ohio State University,

650 Ackerman Rd.,

Columbus,OH 43202

Mechanics and Sliding Friction in

Belt Drives With Pulley Grooves

The steady mechanics of a two-pulley belt drive system are examined where the pulley

grooves,belt extension and wedging in the grooves,and the associated friction are

considered.The belt is modeled as an axially moving string with the tangential and

normal accelerations incorporated.The pulley grooves generate two-dimensional radial

and tangential friction forces whose undetermined direction depends on the relative

speed between belt and pulley along the contact arc.Different from single-pulley analy-

ses,the entry and exit points between the belt spans and pulleys must be determined in

the analysis due to the belt radial penetration into the pulley grooves and the coupling of

the driver and driven pulley solutions.A new computational technique is developed to

ﬁnd the steady mechanics of a V-belt drive.This allows system analysis,such as speed/

torque loss and maximum tension ratio.The governing boundary value problem (BVP)

with undetermined boundaries is converted to a ﬁxed boundary form solvable by a

general-purpose BVP solver.Compared to ﬂat belt drives or models that neglect radial

friction,signiﬁcant differences in the steady belt-pulley mechanics arise in terms of belt

radial penetration,free span contact points,tension,friction,and speed

variations.DOI:10.1115/1.2168469

1 Introduction

The mechanics between belt and pulley in their contact zones

has attracted attention since Euler 1 published on it in 1762.

Belt-pulley mechanics impact the important industrial consider-

ations of belt tension and life,power transmission efﬁciency,

maximum transmissible moment,and noise.For example,for ser-

pentine belt drives used in the automotive industry,belt tensions

are desired to be as small as possible to reduce belt fatigue and

prolong bearing life,yet power loss from belt slip is unacceptable.

This requires understanding of belt-pulley interactions.Current

practically observed behaviors still differ considerably from theo-

retical prediction for certain belt drives,as communicated by belt

drive manufacturers.Belt-pulley friction modeling and interac-

tions with the grooves appear to be major sources of the error and

perhaps the least understood aspects of the mechanics.

Different theories have been established for the belt-pulley in-

teraction.Comprehensive reviews of belt mechanics can be found

in the works of Fawcett 2 and Johnson 3.Although some mod-

els were developed on the basis of belt shear deformation theory

4–6,belt creep theory is still the most widely adopted.In this

theory,the belt is assumed to be elastically extensible,friction

develops due to the relative slip between the belt and pulley,and

a Coulomb law describes the belt-pulley friction.For a two-pulley

belt drive where the driver and driven pulleys have the same ra-

dius,Gerbert 7 used this theory and established that the contact

zones for a ﬂat belt are divided into slip and adhesion zones.

Bechtel et al.8 and Rubin 9 incorporated belt inertia effects

into this creep theory and presented improved solutions for two-

pulley belt drives.Kong and Parker further extended this model

by incorporating belt bending stiffness and applied it to two-

pulley belt drives 10 and multiple-pulley serpentine belt drives

with tensioner assemblies 11.

All of the above models are for ﬂat belt drives without consid-

eration of the pulley grooves.Fewer researchers have studied

grooved pulley drives such as V-belt systems.Hornung 12 con-

sidered the interaction between a V-belt and the pulley grooves.

Due to computational constraints at the time,only qualitative dis-

cussion and rough approximate solutions are obtained.Gerbert

and Sorge 13 established an effective model to examine sliding

of the V-belt in the grooves.They analyzed individual driver or

driven pulleys isolated from the rest of the system.The governing

equations of the belt on a single pulley are solved by a shooting

technique where the boundary value problem BVP is cast as an

initial value problem IVP and the boundary conditions are speci-

ﬁed at only one point.The equations are then integrated until

another point is found that satisﬁes certain conditions and can

serve as the other boundary.The disadvantage of this method is

that it is difﬁcult to systematically obtain solutions for the physi-

cal inputs that are typically speciﬁed.The limitation to single-

pulley analysis,where one cannot naturally link the driver and

driven pulley solutions,prevents straightforward application-to-

system analysis where multiple pulleys always exist and their so-

lutions are coupled.Accordingly,this method cannot be applied

directly to study system behavior nor calculate system outputs,

such as power efﬁciency and maximum transmitted moment.

In this paper,Gerbert and Sorge’s model is adopted and applied

to a two-pulley system where the belts sliding in the driver and

driven pulley grooves are coupled by the two free spans.The BVP

for the entire drive is solved for speciﬁed span tensions.Belt

radial penetration into the grooved pulleys i.e.,“wedging” leads

to initially unknown contact points between the belt spans and

pulleys.Consequently,the steady motion is governed by a BVP

with unknown boundaries.This is different from many studies in

the literature where the boundaries of the belt-pulley contact arcs

are assumed to be ﬁxed at the points of common tangency of the

driver and driven pulleys 14–18.The tangential friction that

transmits the power and radial friction from seating and unseating

of the belt are modeled.Belt inertia in the tangential and normal

directions is fully considered.A computational approach is devel-

oped to solve the BVP for the entire drive for speciﬁed span

tensions.Based on this model,the steady mechanics of a two-

pulley drive are analyzed and some important design criteria,

including power efﬁciency and maximum tension ratio,are

examined.

2 Governing Equations of a Belt Sliding in Pulley

Grooves

Figure 1a shows cross sections of the belt and the pulley

groove.The groove wedge angle is .Friction between the belt

and pulley develops in the sliding plane,where the belt edge

1

Corresponding author.

Contributed by the Power Transmission and Gearing Committee of ASME for

publication in the J

OURNAL OF

M

ECHANICAL

D

ESIGN

.Manuscript received December

15,2004;ﬁnal manuscript received June 23,2005.Review conducted by Teik C.

Lim.

494/Vol.128,MARCH 2006 Copyright © 2006 by ASME Transactions of the ASME

contacts the pulley groove,because the contacting material par-

ticles and associated relative sliding velocity vector V

˜

s

exist in

this sliding plane Fig.1b.The friction force is projected into

the normal plane,which bisects the belt and is perpendicular to

the pulley axis,to establish the equations of motion.The angle

s

is the angle between V

˜

s

and the normal plane.is the sliding

angle deﬁning the direction of the belt relative sliding velocity

vector projected in the normal plane.Figure 1b shows that the Z

component of the belt relative sliding velocity vector V

˜

s

is

V

s

sin

s

.On the other hand,this velocity component can also be

written as V

s

cos

s

cos tan.Equivalence of these two expres-

sions leads to the relationship between these angles 13

tan

s

= tancos 1

where −/2

s

/2.

Figure 2 shows the free body diagram of a segment of an ex-

tensible belt in the pulley grooves.The belt is modeled as an

axially moving string.An Eulerian formulation is adopted for the

control volume.The model is based on that in 13 except that belt

inertia through longitudinal and centripetal accelerations is con-

sidered here while neglected in 13.Note that Figs.1 and 2 are

similar to those in 13 but with additional vectors GV due to the

consideration of belt inertia.For steady motions,conservation of

mass requires that

G= msVs = const 2

where G is the mass ﬂow rate,ms is the belt mass density per

unit length,Vs is the belt speed,and s is the arclength coordinate

along the belt.Balance of linear momentum projected along the

belt tangential and normal directions in the normal plane leads to

dF − GV

ds

= 2p− sinsin+cos

s

sin+ 3

F − GV

= 2psincos −cos

s

cos+ 4

where F is the belt tension,p is the normal compressive pressure

between the belt and the pulley groove surfaces,is the inclina-

tion angle between the belt velocity and the velocity of the over-

lapping point on the pulley Fig.2,is the Coulomb friction

coefﬁcient,=ds/dis the belt radius of curvature,and is the

natural angular coordinate Fig.2.

The belt radial penetration is governed by 13

x = R − rs =

2p

z

k

=

2p

k

cos +sin

s

5

where R is the constant belt pitch radius,rs is the belt radius

coordinate,k is the radial spring stiffness,and p

z

is the pressure

load component exerted on the belt along the pulley axial direc-

tion Fig.1a.k is determined mainly by the belt cross-sectional

geometry and material properties.Gerbert 7 gives an approxi-

mate estimation k=12H/BE

z

tan,where H is the belt height,B

is the belt width top side of V-belt,and E

z

is the belt modulus of

elasticity in the transverse direction.

r and are polar coordinates with origin at the pulley center

Fig.2.Substitution of the geometric relations ds=rd/cos ,r

=R−x,=ds/d,and =− d=d−d into 3 and 4

leads to the polar coordinate equations

T

= F − GV

= 2p− sinsin+cos

s

sin+

R − x

cos

6

= 1 −

2p

T

sincos −cos

s

cos+

R − x

cos

7

where T=F−GV is the belt tractive tension and

is the deriva-

tive with respect to the angular coordinate .Because tan

=dr/rd=r

/r Fig.2,substitution of x=R−r yields

x

= − R − xtan 8

To complete the problem,a constitutive law relating belt ten-

sion F and velocity V is needed.Following 8–10,19,the consti-

tutive law is

F = EAm

0

V/G− 1 ÞT = EAm

0

− G

2

V/G− EA 9

where EA is the belt longitudinal stiffness and m

0

s is the belt

mass density per unit length in the stress-free state,which can be

measured.The mass ﬂow rate G is not known initially and is

determined in the analysis.Comparison of 9 and Eq.4 in 13

shows that the constitutive laws are consistent with each other.

Velocity analysis from Fig.2b reveals that

V cos = r+ V

s

sin V sin= V

s

cos 10

Elimination of the sliding velocity V

s

and use of r=R−x and 9

lead to

tan=

cos − R − xEAm

0

− G

2

/GT + EA

sin

11

In summary,the motion of the belt sliding in the grooves is

governed by the three differential equations 6–8 and the four

algebraic equations 1,5,9,and 11.These equations apply

to the entire belt-pulley contact zone on a pulley.

The governing equations seem complicated at ﬁrst sight be-

cause they involve coupled differential and algebraic equations

with many variables.The primary variables are T,,and x,whose

behavior is governed by 6–8.All other variables such as

V,p,

s

,,etc. are intermediate variables that can be explicitly

expressed in terms of the three basic variables T,,and x based on

the four algebraic equations 1,5,9,and 11.In other words,

the steady motion of the belt in the belt-pulley contact zone could

Fig.1 Belt sliding in pulley grooves:„a… cross section and

acting forces,and „b… velocities

Fig.2 „a… Free body diagramof a moving curved string includ-

ing belt inertia effect and „b… pulley velocity r,belt segment

velocity V„s…,and relative speed V

s

„s…

Journal of Mechanical Design MARCH 2006,Vol.128/495

be cast as a boundary value problem for T,,and x,governed

solely by three differential equations.Realization of this point aids

understanding of the subsequent solution procedure for the full

two-pulley system.Nevertheless,the formulation 6–8,1,5,

9,and 11 is retained for clarity of equations and convenience

of numerical solution.

Within a contact zone there is no adhesion zone where the belt

penetration,speed,and tension remain constant,as exists in a

ﬂat-belt model 8–10.Gerbert and Sorge 13 gave a mathemati-

cal proof of the nonexistence of an adhesion zone.An alternative

explanation based on physical insight is given here that clearly

shows that an adhesion zone cannot exist in the grooved pulley

model.Taking the driven pulley as an example,suppose there is

an adhesion zone BC in the belt-pulley contact zone Fig.3.The

only possibility is that it exists in the middle part of the contact

zone because belt penetration varies in the entry and exit zones.

For this assumed adhesion zone BC,the belt penetration and ten-

sion must be constant and the belt speed including that at B must

be the same as the linear velocity of the overlapping point B on

the pulley,i.e.,V

B

=r

B

1

,where r

B

is the belt radius at B and

1

is the rotation speed of the driven pulley in this paper,the sub-

scripts 1 and 2 represent the driven and driver pulley,respec-

tively.At an arbitrary point A in the entry zone outside BC,the

belt tension is less than that at point B because the driven pulley

entry zone connects with the slack span.According to the consti-

tutive law 9,the belt velocity at A is also smaller than that at B,

i.e.,V

A

V

B

.Because the belt velocity component along the cor-

responding pulley tangential direction is always less than or equal

to its absolute speed,we have V

A

cos

A

V

A

V

B

.Furthermore,

the speed of the overlapping point B on the pulley is less than that

at A,r

B

1

r

A

1

,due to the lesser belt penetration in the entry

zone.Thus we have V

A

cos

A

r

A

1

,and the belt tangential

speed is less than that of the pulley of the same point.This con-

tradicts the requirement that the tangential friction must be oppo-

site the direction of belt travel on the driven pulley.Consequently,

the existence of an adhesion zone on the driven pulley is not

possible.There is,however,a single point where the belt moves

purely in the pulley tangential direction =0 at the transition from

seating to unseating.Similar reasoning applies to the driver pulley

to rule out the existence of an adhesion zone there.

3 Solution for a Symmetric Two-Pulley Belt Drive

The steady motion analysis is presented for a two-pulley belt

drive.The driver and driven pulleys are assumed to have the same

radius,wedge angle,and friction coefﬁcient.The method pre-

sented,however,extends naturally to a general belt drive with

different pulleys.The speciﬁed parameters are:driver and driven

pulley pitch radius R,center distance between the two ﬁxed pul-

leys L,belt longitudinal stiffness EA,constant rotation speed

2

of the driver pulley,friction coefﬁcient ,pulley wedge angle ,

belt mass density per unit length m

0

s in the stress-free state,

radial spring stiffness k,and belt tractive tensions in the slack and

tight spans T

s

and T

t

,respectively.

Figure 3 shows the belt drive.The belt-pulley contact points

C

1

C

4

are not known a priori and must be determined.To permit

a solution,the governing equations for the steady motion of the

whole system,including the two belt-pulley contact zones with

undetermined boundaries,are transformed into a standard bound-

ary value problem form on a ﬁxed domain,namely,

u

t = Ft,ut a t b

gua,ub = 0 12

where F,u,and g are n-dimensional vectors and F and g may be

nonlinear.

The undeﬁned boundary requires special treatment.The wrap

angles of the belt-pulley contact zones Fig.3 for the driver and

driven pulley are

1

and

2

,respectively.They are not known at

this point.Nevertheless,they are used to deﬁne the following

nondimensional variables

ˆ

1

=

1

1

ˆ

2

=

2

2

0 ˆ

1

,ˆ

2

1 13

Correspondingly,the governing differential equations for the belt

on the driven pulley 0ˆ

1

1 are

dT

1

d

ˆ

1

= 2p

1

− sintan

1

+cos

s1

tan

1

cos

1

+ sin

1

R − x

1

1

14

d

1

dˆ

1

=

1 −

2p

1

T

1

sin−cos

s1

cos

1

− sin

1

tan

1

R − x

1

1

15

dx

1

d

ˆ

1

= − R − x

1

tan

1

1

16

To incorporate the unknown constant

1

in the standard BVP

form 12,it is deﬁned as the unknown function

1

ˆ

1

governed

by

d

1

ˆ

1

dˆ

1

= 0,0 ˆ

1

1 17

Similarly,the governing equations for the driver pulley on 0

ˆ

2

1 are

dT

2

dˆ

2

= 2p

2

− sintan

2

+cos

s2

tan

2

cos

2

+ sin

2

R − x

2

2

18

d

2

d

ˆ

2

=

1 −

2p

2

F

2

sin−cos

s2

cos

2

− sin

2

tan

2

R − x

2

2

19

dx

2

dˆ

2

= − R − x

2

tan

2

2

20

d

2

dˆ

2

= 0 21

The intermediate variables,such as p

1

,p

2

,

s1

,

s2

,

1

,

2

,etc.,are

still governed by the four algebraic equations 1,5,9,and 11

with the subscript 1 or 2 attached for the driven and driver pul-

leys,respectively.

The following boundary conditions are evident for the driven

and driver pulleys

x

1

0 = 0,x

1

1 = 0,T

1

0 = T

s

,T

1

1 = T

t

22

Fig.3 Two-pulley belt drive with belt penetration into pulley

grooves

496/Vol.128,MARCH 2006 Transactions of the ASME

x

2

0 = 0,x

2

1 = 0,T

2

0 = T

t

,T

2

1 = T

s

23

Additional conditions come from the belt in the pulley grooves

being tangent to the free spans at the four belt-pulley contact

points C

1

C

4

Fig.3.Suppose the global coordinate origin is

located at the midpoint of the slack span Fig.3,and the as yet

unknown slack span length is .Both spans are straight for a

string model of the belt no bending stiffness.The coordinates of

the two pulley centers are then

x

o

2

= −

2

− R cos

2

−

2

1

,y

o

2

= R sin

2

−

2

1

24

x

o

1

=

2

+ R cos

2

+

1

0

,y

o

1

= R sin

2

+

1

0

25

The pulley centers have ﬁxed distance L

x

o

1

− x

o

2

2

+ y

o

1

− y

o

2

2

= L

2

26

The coordinates of the two belt-pulley contact points for the tight

span are determined geometrically as

x

C

2

= −

2

− R cos

2

−

2

1

+ R cos

3

2

−

2

+

2

1

27

y

C

2

= R sin

2

−

2

1

+ R sin

3

2

−

2

+

2

1

28

x

C

1

=

2

+ R cos

2

+

1

0

− R cos

3

2

−

1

−

1

0

29

y

C

1

= R sin

2

+

1

0

+ R sin

3

2

−

1

−

1

0

30

The tight span goes through point C

2

,and it is tangent to the

belt in the driver pulley groove.Its slope can be calculated from

the three angles

2

0,

2

1,and

2

on the driver pulley as z

2

=tan−−

2

1−

2

−

2

0.The line of the tight span can then

be written as y−y

c

2

=z

2

x−x

c

2

.Similarly,working from the

driven pulley,the tight span goes through point C

1

and its slope is

z

1

=tan

1

+

1

0−

1

1.The tight span line is also y−y

c

1

=z

1

x−x

c

1

.These two lines must be the same,which requires

z

1

− z

2

= 0 y

1

− x

c

1

z

1

− y

2

− x

c

2

z

2

= 0 31

In the above analysis,the slack span length ,the mass ﬂow

rate G,and the driven pulley rotation speed

1

are unknown.

Analogous to 17,these unknown constants are incorporated into

the standard BVP form 12 by adding three trivial ODEs

d

dˆ

1

= 0

dG

dˆ

1

= 0

d

1

dˆ

1

= 0,0 ˆ

1

1 32

The standard BVP form 12 involves only coupled differential

equations.The algebraic equations 26 and 31 are naturally in-

corporated into the form 12 by treating them as boundary con-

ditions where the unknown constants in 26 and 31 can be

written as the values at either boundary for example,

2

can be

written as either

2

0 or

2

1.The total order of the 11 differ-

ential equations 14–21 and 32 that deﬁne F in 12 equals the

number of boundary conditions 22,23,26,and 31 that

deﬁne g in 12.The algebraic equations 22,23,26,and 31

are incorporated in the deﬁnition of F from the above differential

equations and require no special processing.Although the original

problem has unknown boundaries,it is now deﬁned entirely on

the interval 0,1.This standard BVP form 12 can be solved by

general-purpose two-point BVP solvers.This procedure is

straightforward to implement,and the accuracy of the results are

ensured with use of state-of-the-art solver codes.

The pulley torques are

M

i

=

0

i

2p

i

cos

si

sin

i

R − x

i

2

/cos

i

d

i

i = 1,2

33

They are useful for subsequent calculation of the system power

efﬁciency.The torques can be obtained through direct integration

of 33 once the distributions of belt tension,speed,and radial

penetration have been obtained.Alternatively,by integrating these

terms into the standard BVP form,they are a natural product of

the BVP solution without additional effort.For example,for the

torque on the driven pulley,one deﬁnes I

1

1

=

0

1

2p

1

cos

s

1

sin

1

R−x

1

2

/cos

1

dand adds the following

ODE and boundary condition to the above BVP formulation

dI

1

1

d

1

=

2p

1

cos

s

1

sin

1

R − x

1

2

cos

1

,

34

0

1

1

with I

1

0 = 0

I

1

1

is the desired torque M

1

on the driven pulley and is a direct

output of the solution.Although the added ODE and boundary

condition 34 are written in dimensional form over the range

0,

1

,use of 13 transforms them into the necessary form on

0,1.The torque on the driver pulley can be similarly obtained.

4 Results and Discussion

The belt tension F and tractive tension T=F−GV differ by GV,

which is nearly constant along the belt 10.In the following

analysis,references to belt tension always mean the tractive ten-

sion,T.

Convergence of the numerical BVP solution is not assured be-

cause of its complexity.For such a highly nonlinear problem,an

initial solution guess by intuition or insight is not reliable.Instead,

the initial guess is found using a trial and error method.First,the

driver pulley is arbitrarily speciﬁed a wrap angle

2

and two

boundary tensions T

t

DR

and T

s

DR

;it is not hard to ﬁnd its nu-

merical solution from 6–8 plus G

=0.The equation G

=0 is

added because G is an unknown constant;deﬁning it as the ﬁeld

variable G and enforcing zero derivative enables natural inclu-

sion in the standard form 12.The four boundary conditions are

similar to those in 23 and

2

is speciﬁed.Next,for the driven

pulley,the wrap angle

1

is arbitrarily speciﬁed,and the two

boundary tensions are the same as those for the driver pulley

problem.The governing equations and boundary conditions are

similar to those of the driver pulley except that G

=0 is replaced

by

1

=0.G is speciﬁed as that computed from the driver pulley

and,unlike the driver pulley,the rotation speed of the driven

pulley

1

is not known.Again,the numerical solution can be

found for the driven pulley.After computing the solutions for the

driver and driven pulleys,the geometry of the two pulleys and the

belt in their grooves is plotted with the free spans extending from

the two pulleys such that the two slack spans align.In general,the

two tight spans are not geometrically compatible,i.e.,they do not

overlap with each other Fig.4.The parameters,such as the two

wrap angles,are adjusted until the geometric compatibility condi-

tion i.e.,alignment of the two spans is close to being satisﬁed.At

this stage,the numerical solutions of the two individual pulleys,

together with the wrap angles,can be used as the initial guess for

the solution of the full two-pulley BVP with the same speciﬁed

parameters as those in the ﬁnal step of trial and error.This initial

guess is typically sufﬁcient for the numerical solution to converge.

Acontinuation procedure avoids repetition of the above process

as parameters change.After a numerical solution is obtained from

Journal of Mechanical Design MARCH 2006,Vol.128/497

the above process,the parameters can be changed in small incre-

ments where each numerically exact solution obtained in the pre-

vious step serves as the initial guess for the current step.Even

with such a strategy,not all parameter combinations can be

solved.For instance,in the example problem,when the two span

tensions are out of the range presented in the following ﬁgures,

the above procedure fails due to the sharp changes of the inclina-

tion and sliding angles in the belt-pulley contact zones.Note,

however,the large range of span tensions that can be handled.

Even for the simpler single pulley case using an alternate numeri-

cal method,ﬁnding meaningful solutions involves numerical

troubles and requires careful selection of the parameters 13.In-

clusion of belt bending stiffness might smooth the sharp changes

that can occur in the driver pulley exit zone and improve numeri-

cal performance.

This paper analyzes two-pulley systems.If the belt mechanics

on only a single driver or driven pulley are desired as in 13,the

presented BVP-solver method remains a convenient technique.

This is because the two free span tensions and the wrap angle,

which are the three boundary conditions speciﬁed for single-

pulley analysis 13,can be directly speciﬁed and readily varied as

desired.This is cumbersome for the shooting method in 13 that

requires trial and error.

This section presents steady solution results for a belt drive

with two identical pulleys.The data are speciﬁed in Table 1.Note

that the friction coefﬁcient is adopted from 13.Figure 5 shows

the steady solutions with increasing tight span tension while the

slack span tension remains constant.The belt penetration features

are evident for large tight/slack tension ratio.In particular,note

the distinctly different belt shape and penetration properties be-

tween the two pulleys’ entry and exit zones.The two belt free

spans couple the driver and driven pulley solutions and need to be

tangent to the “wedging” belt in the entry and exit zones.Corre-

spondingly,the two belt free spans are no longer on the line of

common tangency of the two pulleys,as in the corresponding

string models of ﬂat belt drives 8,9.Instead,the two free spans

are nonparallel and this shows why torque loss exits.Figure

6shows the variations of pulley wrap angles and torques with

increasing tight span tension.As the tight/slack span tension ratio

increases,the wrap angles for both pulleys increase considerably.

The wrap angle on the driver pulley increases more quickly than

on the driven pulley.When the tight/slack span tension ratio is

large,the wrap angle on the driver pulley is much larger than that

on the driven pulley,and the two free spans are markedly unpar-

allel to each other.When the tight span tension is close to that of

the slack span,the two wrap angles are close.Even for such a

case,extrapolation of Fig.6a shows that the wrap angles would

be around 190 deg,larger than the 180 deg for ﬂat belt drives or

when belt wedging is ignored.Only when both span tensions drop

to zero do the wrap angles become 180 deg.The torques on the

driver and driven pulley differ from each other Fig.6b,as

compared to the always equivalent driver and driven pulley

Table 1 Physical properties of the example belt drive with two

identical pulleys

R

1

=R

2

=0.25 m L=1.3933 m EA=120 kN

k=900 kN/m

3

1

=1000 pm m

0

=0.108 kg/m

1

=

2

=0.4 =18 deg T

s

=100 N

Fig.4 Search of the initial solution guess by trial and error

Fig.5 Steady solutions for the systemspeciﬁed in Table 1:„a…

T

t

=700 N,„b… T

t

=1200 N,„c… T

t

=3000 N,and „d… T

t

=5000 N

Fig.6 Variation of pulley „a… wrap angles and „b… torques with

tight span tractive tension for the system speciﬁed in Table 1

498/Vol.128,MARCH 2006 Transactions of the ASME

torques for ﬂat belt drives 8–10.The torque difference increases

with the free span tension difference.The two torques are nearly

equal when the tight/slack tension ratio is comparatively small.

Figure 7 shows the tension distributions on the belt-pulley con-

tact zones for the driver and driven pulley,respectively.Although

the variation shapes are quite different from each other,they share

some common characteristics.In the entry or exit zones,both belt

tensions vary slowly.This is because in these zones,the belt radial

penetrations are small;correspondingly,the friction force is small

and does not offer signiﬁcant tangential force to change the belt

tensions.

Figure 8 shows the belt radial penetrations in the belt-pulley

contact zones.The penetration patterns on the driver and driven

pulleys are quite different.For both cases,rapid changes of the

penetrations occur in the entry/exit zones.But in the middle zone,

the belt penetration on the driver pulley varies little,which differs

from the continuously increasing penetration on the driven pulley

also see Fig.5.

The belt inclination angles

2

and

1

in the belt-pulley contact

zones are given in Fig.9.Negative positive belt inclination

angle means that belt penetration increases decreases at the cor-

responding point while the belt penetration reaches the maximum

point when the belt inclination angle is zero.In the entry/exit

zones,the amplitudes of the belt inclination angles are larger than

those in the middle zones because of relatively small pressure

between the belt and pulley,which leads to rapid seating/

unseating of the belt into the pulley grooves.The seating and

unseating rate x

is approximately measured by the belt inclina-

tion angle see 8.

Seating of the belt in the entry zone is determined mainly by the

belt entry tensions given the pulley/groove geometry,friction co-

efﬁcient,and the belt properties.Because the belt entry tension of

the driver pulley is higher than that of the driven pulley,the belt

on the driver pulley is more quickly seated than on the driven

pulley,resulting in larger amplitude belt inclination angles.This

point is most apparent for the driver pulley.For the two extreme

cases T

t

=700 N and T

t

=5000 N,the belt inclination angles at the

entry point differ by more than 10 deg Fig.9a.While for the

driven pulley,although the tight span tensions are very different,

the belt entry tension is the same,i.e.,100 N.Accordingly,the

belt inclination angles do not change much in the entry zones Fig.

9b.

Unseating of the belt in the exit zone is different from the

seating action in the entry zone.The belt unseating rate in the exit

zone depends not only on the belt exit tensions but also on how

deeply the belt is wedged in the pulley grooves in the middle

zone.For pulleys with the same belt penetration in the middle

zones,the smaller the belt exit tension,the larger the belt inclina-

tion angle required to overcome the belt “wedging” and unseat the

belt.To visualize this,imagine that the belt in the exit zone pulley

grooves is pulled out by tugging on the belt in the free spans with

the speciﬁed tensions.On the other hand,if the belt exit tensions

are the same,the deeper the belt penetration in the middle zones,

the larger the belt inclination angles in the exit zones see Fig.

9a where the exit tension is the same for all curves.Although

the belt penetrations of driver and driven pulleys in the middle

zones are comparable Fig.8,the exit tension on the driver pulley

is lower than that of the driven pulley.Consequently,the belt

Fig.7 Variation of belt tractive tensions in belt-pulley contact

zones with tight span tractive tension for the system speciﬁed

in Table 1

Fig.8 Belt radial penetrations along driver and driven contact

arcs for the system speciﬁed in Table 1:„a… driver pulley and

„b… driven pulley

Journal of Mechanical Design MARCH 2006,Vol.128/499

inclination angle in the driver pulley exit zone is higher than its

counterpart on the driven pulley Figs.9a and 9b.This results

in more rapid unseating in the exit zone on the driver pulley than

on the driven pulley Figs.5 and 8.

The above differences in the entry/exit zones on the driver and

driven pulleys cause the two spans to be nonparallel.This effect

becomes more apparent with signiﬁcant tight/slack tension ratio.

The belt sliding angles

2

and

1

in the belt-pulley contact

zones are given in Fig.10.Belt sliding angles indicate the direc-

tion of the friction force relative to the pulley radial direction Fig.

2.They are determined by the belt sliding speed in the pulley

radial direction and the relative speed between belt and pulley

surfaces along the pulley tangential direction.For the driven pul-

ley,where the belt drives the pulley,the belt speed along the

pulley tangential direction is faster than that of the overlapping

point on the groove surface,so the belt sliding angle is in the

range 0–180 deg.For the situation where the pulley drives the

belt on the driver pulley,the belt sliding angle is in the range

180–360 deg.For the driven pulley,when the belt reaches an

extremal of belt penetration and the belt inclination angle is zero,

the belt speed along the pulley radial direction is zero;corre-

spondingly the belt sliding angle is 90 deg Fig.10.At this point,

the friction force fully contributes to overcoming the driven pulley

torque,like the case of a ﬂat belt.Asimilar situation exists on the

driver pulley;at the maximum penetration point,the belt sliding

angle is 270 deg and the friction force fully contributes to resist-

ing the pulley driving torque.When the belt sliding angles are

away from 90 deg or 270 deg,the belt moves in both the radial

and tangential directions relative to the groove surfaces.In the

extreme case of =0 deg or 360 deg,the belt moves only radi-

ally relative to the groove surfaces with decreasing penetration.A

sliding angle of 180 deg corresponds to purely radial belt motion

with increasing penetration.For both extreme cases,there is no

friction contribution to the pulley torque.When the pulley torque

increases decreases,the sliding angles adjust to make greater

shares of the contact zones close to away from 90 deg for the

driven pulley or 270 deg for the driver pulley,as well as increas-

ing the wrap angles.The abrupt changes of the belt sliding angles

in the exit zones are caused by the sharp decreases in belt

penetration.

This study does not consider belt bending stiffness,which is an

important factor in belt-pulley drives 10,11,20.Inclusion of

bending stiffness might make the belt penetrations and inclina-

tions vary more smoothly in the entry/exit zones,resulting in

more parallel spans even with large tension differences.Reducing

these sharp changes may also improve numerical convergence for

less accurate initial guesses.

The power efﬁciency is deﬁned as the ratio between the powers

of the driven and driver pulleys,=M

1

1

/M

2

2

.Figure 11

shows that increasing the tight/slack tension ratio signiﬁcantly de-

creases the rotation speed of the driven pulley and the power

efﬁciency .The rotation speed of the driven pulley is always less

than that of the driver pulley,which is ﬁxed at 1000 rpm.Efﬁ-

ciency decreases because

1

decreases with tension ratio for ﬁxed

2

while the ratio M

1

/M

2

decreases slightly with tension ratio

Fig.6b.For drives with appreciable free span tension differ-

ence,the rotation speed of the driven pulley and the power efﬁ-

ciency are much less than those for ﬂat belt drives,where the

Fig.9 Belt inclination angles

2

and

1

along driver and driven

contact arcs for the system speciﬁed in Table 1:„a… driver pul-

ley and „b… driven pulley

Fig.10 Belt sliding angles

2

and

1

along driver and driven

contact arcs for the system speciﬁed in Table 1:„a… driver pul-

ley and „b… driven pulley

500/Vol.128,MARCH 2006 Transactions of the ASME

driven pulley rotation speed is close to that of the driver pulley

and the power efﬁciency is always close to unity even for maxi-

mum transmitted moment cases 9,10.

In ﬂat belt drives,the maximum transmitted moment,or the

maximum span tension ratio,is reached when all of a belt-pulley

adhesion zone converts to a sliding zone.For drives with pulley

grooves,there are no adhesion zones as discussed earlier,and this

criterion for the maximum transmitted moment does not apply.

Comparison of Figs.6b and 11 shows that the rotation speed of

the driven pulley decreases with the driven pulley torque.Theo-

retically,the maximum transmitted moment is reached when the

rotation speed of the driven pulley is zero,although Fig.11 sug-

gests vanishing driven pulley speed may be reached asymptoti-

cally.For such a case,V

s

=V and

1

=90−

1

deg on the entire

contact arc Fig.2.In this state,friction on the driven pulley

contributes to the torque as much as possible given the seating/

unseating action.Complete contribution of the friction to the

torque is impossible except at the single point where

1

=0 be-

cause only the friction component in the pulley tangential direc-

tion contributes to the torque while some friction in the pulley

radial direction is unavoidable due to belt seating and unseating.

For the driven pulley,nonzero rotation speed always keeps the

belt sliding angle

1

,which gives the direction of the friction,

away from 90 deg;that is,

1

90−

1

for unseating

1

0 or

1

90−

1

for seating

1

0.Neglecting the pulley grooves

can signiﬁcantly underestimate the maximum transmitted

moment.

The above results show that although the present model and

comparable ones for ﬂat belts 7–10 are based on similar creep

theory assumptions where Coulomb friction prevails and its ex-

istence depends on belt extensibility and relative slip between belt

and pulley surface,the consideration of pulley grooves greatly

complicates the model,resulting in a two-dimensional radial and

tangential contact problem between belt and pulley surfaces.This

two-dimensional model is hardly studied in the literature and

poses challenging mathematical obstacles to solve it.On the other

hand,mechanical textbooks and handbooks emphasize only ﬂat

belt conclusions,which are better known because the models are

established and far easier to solve.V-belt mechanics are normally

approximated from ﬂat belt theory.A typical example is the

widely used textbook by Juvinall and Marshek 21.V-belts are

treated only brieﬂy,and the main design equation 19.3a is di-

rectly modiﬁed from equation 19.3 for ﬂat belts with the remark:

“The ﬂat-belt equations can be modiﬁed by merely replacing the

coefﬁcient of friction f with the quantity f/sin.Eq.19.13 then

becomes 19.13a.” The present simulation results show that dis-

tinctive belt behaviors exist that cannot be inferred from ﬂat belt

models such as the belt’s qualitatively different interactions with

the driver and driven pulleys and no differentiation of adhesion/

sliding zones.

To date,no experiments exist in the literature to validate this

two-dimensional 2D model,whose validity must be evaluated

on the underlying mechanics principles and engineering assump-

tions.The model itself is relatively new,originating in 2002 13,

and there is scope for incorporation of belt bending stiffness and

other reﬁnements.A primary purpose of this paper is to advance

numerical solution techniques to generate results for a full two-

pulley drive that can be compared to experiments 13 analyzes

only a single pulley.Subsequent experiments demand careful at-

tention to measuring the belt penetrations,maintaining pulley

alignment,and the like.Nevertheless,this 2D model deepens

knowledge of belt mechanics and explains phenomena that cannot

be explained by classical ﬂat belt models for example,the ﬂat

belt model predicts no torque loss,as indicated here in Fig.5b.

5 Conclusions

Acomputational method based on general-purpose BVP solvers

is proposed to compute the steady mechanics of a two-pulley

V-belt drive.Belt sliding in the pulley grooves leads to two-

dimensional tangential and radial friction.This contrasts sharply

with common textbook/handbook simpliﬁcations that extrapolate

V-belt behavior from ﬂat belt behavior through,for example,use

of a modiﬁed friction coefﬁcient.The belt is modeled as an axially

moving string with belt inertia fully considered.The “wedging” of

the belt in the pulley grooves makes the belt-pulley contact points

unknown a priori.The original BVP on unknown domain is trans-

formed to a standard BVP form on ﬁxed domain.The steady

solutions include belt-pulley contact points,radial penetration in

pulley grooves,the magnitude and direction of the friction forces,

tension,and belt speed.The main ﬁndings include:

1.Wrap angles increase with tight/slack span tension ratio

and are signiﬁcantly larger than those for comparable ﬂat

belt drives.

2.There are no adhesion zones on the driver or driven pul-

ley;the belt slides in the pulley grooves along the entire

contact arc.

3.Large tight/slack tension ratio causes the belt to exit the

pulley grooves abruptly resulting in signiﬁcant nonparal-

lelism of the two free spans that leads to torque loss.

4.The driven pulley rotation speed is lower than for ﬂat

belt drives,especially for heavy loads with signiﬁcant

span tension differences.

5.The theoretical maximum transmitted moment occurs

when the driven pulley rotation speed drops to zero.At

this point,the system has the maximum tight/slack ten-

sion ratio.

6.Neglecting the pulley grooves underestimates the maxi-

mum transmitted moment and overestimates the system

power efﬁciency.

Acknowledgment

The authors thank Mark IVAutomotive/Dayco Corporation and

the National Science Foundation for support of this research.

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502/Vol.128,MARCH 2006 Transactions of the ASME

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