# Mechanics and Sliding Friction in Belt Drives With Pulley Grooves

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18 Ιουλ 2012 (πριν από 6 χρόνια και 7 μέρες)

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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
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 1￿a￿ 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
15,2004;ﬁnal manuscript received June 23,2005.Review conducted by Teik C.
Lim.
contacts the pulley groove,because the contacting material par-
ticles and associated relative sliding velocity vector V
˜
s
exist in
this sliding plane ￿Fig.1￿b￿￿.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 1￿b￿ 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
= tan￿cos ￿ ￿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= m￿s￿V￿s￿ = const ￿2￿
where G is the mass ﬂow rate,m￿s￿ is the belt mass density per
unit length,V￿s￿ 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
d￿F − GV￿
ds
= 2p￿− sin￿sin￿+￿cos ￿
s
sin￿￿+￿￿￿ ￿3￿
F − GV
￿
= 2p￿sin￿cos ￿−￿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/d￿is the belt radius of curvature,and ￿is the
natural angular coordinate ￿Fig.2￿.
The belt radial penetration is governed by ￿13￿
x = R − r￿s￿ =
2p
z
k
=
2p
k
￿cos ￿+￿sin￿
s
￿ ￿5￿
where R is the constant belt pitch radius,r￿s￿ 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.1￿a￿￿.k is determined mainly by the belt cross-sectional
geometry and material properties.Gerbert ￿7￿ gives an approxi-
mate estimation k=12￿H/B￿E
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￿− sin￿sin￿+￿cos ￿
s
sin￿￿+￿￿￿
R − x
cos ￿
￿6￿
￿
￿
= 1 −
2p
T
￿sin￿cos ￿−￿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 − x￿tan￿ ￿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 = EA￿m
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.2￿b￿ 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￿
tan￿=
cos ￿− ￿R − x￿￿￿EAm
0
− G
2
￿/￿G￿T + 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￿ = F￿t,u￿t￿￿ a ￿t ￿b
g￿u￿a￿,u￿b￿￿ = 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 0￿￿ˆ
1
￿1 are
dT
1
d￿
ˆ
1
= 2p
1
￿− sin￿tan￿
1
+￿cos ￿
s1
￿tan￿
1
cos ￿
1
+ sin￿
1
￿￿
￿￿R − x
1
￿￿
1
￿14￿
d￿
1
d￿ˆ
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
￿
d￿ˆ
1
= 0,0 ￿￿ˆ
1
￿1 ￿17￿
Similarly,the governing equations for the driver pulley on 0
￿￿ˆ
2
￿1 are
dT
2
d￿ˆ
2
= 2p
2
￿− sin￿tan￿
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
d￿ˆ
2
= ￿− ￿R − x
2
￿tan￿
2
￿￿
2
￿20￿
d￿
2
d￿ˆ
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￿
d￿ˆ
1
= 0
dG
d￿ˆ
1
= 0
d￿
1
d￿ˆ
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
2￿p
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
2￿p
1
cos ￿
s
1
sin￿
1
￿R−x
1
￿
2
/cos ￿
1
ODE and boundary condition to the above BVP formulation
dI
1
￿￿
1
￿
d￿
1
=
2￿p
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.6￿a￿ 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.6￿b￿￿,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.9￿a￿￿.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.
9￿b￿￿.
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.
9￿a￿ 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.9￿a￿ and 9￿b￿￿.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.6￿b￿￿.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.6￿b￿ 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.5￿b￿￿.
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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