Lingyuan Kong
Robert G.Parker
1
Professor
email: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 twopulley 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 twodimensional radial
and tangential friction forces whose undetermined direction depends on the relative
speed between belt and pulley along the contact arc.Different from singlepulley 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 Vbelt 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
generalpurpose BVP solver.Compared to ﬂat belt drives or models that neglect radial
friction,signiﬁcant differences in the steady beltpulley 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.
Beltpulley 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 beltpulley interactions.Current
practically observed behaviors still differ considerably from theo
retical prediction for certain belt drives,as communicated by belt
drive manufacturers.Beltpulley 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 beltpulley 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 beltpulley friction.For a twopulley
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 multiplepulley 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 Vbelt systems.Hornung 12 con
sidered the interaction between a Vbelt 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 Vbelt 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 applicationto
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 twopulley 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 beltpulley 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 crosssectional
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 Vbelt,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 stressfree 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 beltpulley 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 beltpulley 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
twopulley 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
ﬂatbelt 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 beltpulley 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 TwoPulley Belt Drive
The steady motion analysis is presented for a twopulley 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 stressfree 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 beltpulley 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 beltpulley 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 ndimensional vectors and F and g may be
nonlinear.
The undeﬁned boundary requires special treatment.The wrap
angles of the beltpulley 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 Twopulley 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 beltpulley 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 beltpulley 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
generalpurpose twopoint BVP solvers.This procedure is
straightforward to implement,and the accuracy of the results are
ensured with use of stateoftheart 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 twopulley 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 beltpulley 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 twopulley systems.If the belt mechanics
on only a single driver or driven pulley are desired as in 13,the
presented BVPsolver 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 beltpulley 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 beltpulley
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 beltpulley 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 beltpulley 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 beltpulley 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 beltpulley 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 beltpulley
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 twodimensional radial and
tangential contact problem between belt and pulley surfaces.This
twodimensional 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.Vbelt mechanics are normally
approximated from ﬂat belt theory.A typical example is the
widely used textbook by Juvinall and Marshek 21.Vbelts 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 ﬂatbelt 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
twodimensional 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 generalpurpose BVP solvers
is proposed to compute the steady mechanics of a twopulley
Vbelt 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
Vbelt 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 beltpulley 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 beltpulley 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.
References
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502/Vol.128,MARCH 2006 Transactions of the ASME
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