Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

55, 4, pp. 1423-1435, Warsaw 2017
DOI: 10.15632/jtam-pl.55.4.1423

A MODEL FOR THE DYNAMIC ANALYSIS OF A BELT TRANSMISSION

USING THE DAHL FRICTION MODEL

Krzysztof Kubas

University of Bielsko-Biala, Faculty of Mechanical Engineering and Computer Science, Bielsko-Biała, Poland

e-mail: kkubas@ath.bielsko.pl

The paper presents a two-dimensional model of a belt transmission for dynamic analysis.
It is assumed that the belt is modelled by links joined by spring-damping elements with its
rotational and translational stiffness. Normal forces in the contact between the belt and the
pulleys are implemented by assuming its stiffness and damping, whereas friction ismodelled
by the Dahl friction model. The calculation results are also presented in two specific cases
of load of the belt transmission.

Keywords: belt transmission, dynamic analysis, Dahl frictionmodel

1. Introduction

In the author’s earlier papers, two-dimensional models of belt transmissions using the Threlfall
friction model (Kubas, 2015) and a model including microslip were developed (Kubas, 2014).
In the paper (Kubas, 2014), assumptions and requirements made during the process of model
development were presented. In the present paper, the Dahl friction model (Dahl, 1968) is
assumed as one in the group of dynamic friction models that allows one to include stiffness of
contact areas.

Papers should bementioned here in which friction was modelled between rubber and other
materials. The most well known are works related to automotive engineering, especially those
dealingwith themodelling of friction between the tyre and the road, see e.g. often cited work by
Canudas deWit et al. (2003). Among the proposed frictionmodels, also presented in the paper
above was theDahl frictionmodel. The paper (Canudas deWit et al., 2003) introduced another
dynamic friction model – the LuGremodel, which allows one to include the Stribeck effect.
Leamy and Wasfy (2002a,b) presented belt transmission models with a piecewise linear

friction model with the possibility of predicting belt creep. It is called the Coulomb-like tri-
linear creep-rate-dependent frictionmodel. Another samplemodel was presented in (Kim et al.,
2011) and was called the elastic/perfectly-plastic friction law (EPP).
A group of papers should also be mentioned in which the Dahl friction model is applied as

a way of modelling friction in the revolute joint of a belt tensioner, see e.g. Bastien et al. (2007)
and Chatlet et al. (2008).
The changing belt and chain transmissions research objectives over the centuries and more

important works were presented by Fawcett (1981).

2. Mathematical model

The model presented in an earlier paper (Kubas, 2014) of a belt transmission was modified by
changing it into a friction model. As presented in the above-mentioned work, it was assumed
that the belt would be divided into nb bodies. Each neighbouring pair of bodies was joined by
the SDEwith proper translational and bending stiffness and damping parameters (Fig. 1).



1424 K. Kubas

Fig. 1. Assumed belt model with translational and torsion SDEs

For each body i (i=1, . . . ,nb), it was assumed that therewere three generalised coordinates
(Fig. 2): translations xi and yi and rotation by an angle ϕi relative to the mass centre. The
generalised coordinates in the body i are presented in Fig. 2.

Fig. 2. Generalised coordinates of the belt body i

It was also assumed that there were np pulleys in the transmission “lying” in the xy plane
and rotating around the axis parallel to z with a rotation angle θj (j = 1, . . . ,np). Therefore,
the vector of the generalised coordinates takes the form

q
T =
[

q
b
1
T
, . . . ,qbi

T
, . . . ,qbnb

T
,θ1, . . . ,θj, . . . ,θnp

]

(2.1)

where qbi is the vector of generalised coordinates of the body i (presented in Fig. 2).

2.1. Spring-damping elements

Thevalues of forces andtorques in the translational andtorsionSDEconnecting thebody i−1
with the body i are described by the Kelvin-Voigt (Voigt, 1892) relations

FLtrai =F
Rtra
i−1 = ctra∆l

L
i + btra∆l̇

L
i

MLtori =M
Rtor
i−1 = ctor(ϕi−ϕi−1)

(2.2)

whereFLtrai ,F
Rtra
i are values of translational forces in the left SDE (connecting the body iwith

the body i− 1) and the right SDE (connecting the body i with the body i+1), respectively;
MLtori ,M

Rtor
i – bending torques in the left SDE and right SDE, respectively; ctra, btra – trans-

lational stiffness and damping coefficients in SDE; ctor – torsional stiffness coefficient in SDE;
∆lLi ,∆l̇

L
i – translational deformation and deformation speed of SDE.

2.2. Model of the belt-pulley contact

A vector notation of the forces is used to take into account the contact between the belt
bodies and pulleys in the transmission. A diagram of the assumed distribution of forces acting
on the belt body i at the period of contact with the pulley j is presented in Fig. 3.



A model for the dynamic analysis of a belt transmission... 1425

Fig. 3. Assumed configuration of the velocity components of the belt body i and forces acting on this
body from the pulley j

It is assumed that the force components, i.e. normal forceNij and friction forceTij, would
be applied to the mass centre of each body which has contact with one of the pulleys.
As is shown in Fig. 3, the vector rij is orientated from the centre of the pulley j to themass

centre of the body i. If there is contact between the body and the pulley but the value of the
normal force Nij is still zero, the length of this vector equals an arbitrary value rpj. Thus, at
the time of a non-zero normal force, there is an inequality: rij <rpj. The position of the pulley
centre in the global coordinate system is specified via the vector Ppj.
The vector rij can be determined from the following formula

rij =q
b
i −q

p
j (2.3)

A versor (unit vector) according to the direction and sense of the vector rij equals

rij =
rij

|rij|
(2.4)

The penetration depth of the belt body i with the pulley j can be determined from the
formula

pij = rpj −|rij| (2.5)

Assuming that the linear velocity of the centre of the pulley is zero, then the value of
penetration velocity ṗij is equal to the value of the normal velocity componentv

n
i (Fig. 3). This

value is determined on the basis of the following scalar product

ṗij = |v
n
i |=−v

T
i ·rij (2.6)

where vi is the velocity of the mass centre of the body i.
Since the contact forceNij formed during contact between the body i and the pulley j has

a consistent direction with the direction of the versor rij, then

Nij =Nijrij (2.7)

The value of this force is determined in a similar form as given in the paper (Čepon et al.,
2010), in which the authors proved a nonlinear relation between the penetration depth and the
normal force

Nij(pij, ṗij)= c1p
2
ij + c2pij + bṗij (2.8)

where c1, c2 are belt-pulley contact stiffness coefficients, b – belt-pulley contact damping coeffi-
cient.



1426 K. Kubas

2.3. Dahl friction model

The value of the relative velocity vTij between the body i and the pulley j is needed to
calculate the friction force (shown in Fig. 3). Theway of calculating this velocity was presented
in an earlier work (Kubas, 2014).

As was mentioned before, the Dahl model is assumed as presented in (Dahl, 1968)

µij =











(µ0ij −µk)exp
(

− σ
µk
(xij −x0ij)

)

+µk for vTij ­ 0

(µ0ij +µk)exp
(

− σ
µk
(xij −x0ij)

)

−µk for vTij < 0
(2.9)

wherevTij is the relative velocity between twomovingparts,x0ij – startingvalue of displacement
at the moment of change of sense of the relative velocity, µ0ij – starting value of the friction
coefficient (according tox0ij displacement),xij –displacement,µk – dynamic friction coefficient,
σ – stiffness coefficient of the friction joint.

The shape of an exemplary displacement-dependent friction characteristic is presented in
Fig. 4.

Fig. 4. Exemplary shape of a Dahl friction characteristic as a function of displacement

First, the starting value of displacement is x0ij and the starting value of friction coeffi-
cient µ0ij is equal to 0. When the two corresponding layers start to move and the relative
velocity ẋ is positive (in relation to the assumed local coordinate system), the friction coefficient
increases (1st phase shown inFig. 4) according to equation (2.9)1. Sometimes, as shown inFig. 4,
the relative velocity can reach the value of µk (especially when its direction does not change
in a relatively long period of time). When the direction of the velocity changes to the opposite
(negative values), the starting value of the displacement and the starting value of the friction
coefficient should change to the values: x0ij = x

′, µ0ij = µk. The friction coefficient starts to
decrease (2nd phase) according to equation (2.9)2 and can reach an arbitry value of −µk (this
situation is shown in Fig. 4). When the velocity changes again into positive values, the friction
coefficient increases again (3rd phase), and x0ij =x

′′, µ0ij =−µk.

The value of the friction force can be, therefore, described as

Tij =µijNij (2.10)

The direction of the friction force vector is opposite to the relative velocity vTij.



A model for the dynamic analysis of a belt transmission... 1427

2.4. Equations of motion

Equations of motion of the body i take the form

miẍi =X
T
·
[

F
Ltra
i +F

Rtra
i +

np−1
∑

j=0

(Nij +Tij)+mig
]

miÿi =Y
T
·
[

F
Ltra
i +F

Rtra
i +

np−1
∑

j=0

(Nij +Tij)+mig
]

Iziϕ̈i =Z
T
·
(

M
Ltra
i +M

Rtra
i +M

Ltor
i +M

Rtor
i

)

(2.11)

where X
T
= [1,0,0], Y

T
= [0,1,0], Z

T
= [0,0,1] – versors consistent with the axes x, y, z,

respectively, of the global coordinate system, MLtrai = −l
′

i × F
Ltra
i , M

Rtra
i = l

′

i × F
Rtra
i ,

g – vector of gravitational acceleration.
It is assumed that the pulleys motion will take place from the set torques. The equation of

motion of the pulley j has the form

Izj θ̈j =Mdj −
nb−1
∑

i=0

Z
T
·MTij (2.12)

where Izj is the mass moment of inertia of the pulley j,Mdj – the set torque,MTij = rij ×Tij
– friction torque acting from the belt body i.
The resistance torque acting on the pulley can be included in the assumedmodel. Anegative

value ofMdj (e.g. constant or dependent on velocity θ̇j) should be taken in this case.

Case 1: Analysis with a driving torque of 100Nm

A two-pulley belt transmission with the same radius rp =0.1m is taken into analysis. The
scheme of the assumed transmission is presented in Fig. 5. The distance between the pulleys
is lp = 0.290m. The analysed belt is 1.2m in length. The friction coefficient, translational and
rotational stiffness anddamping coefficients are assumedbasedonown relatedworks and (Čepon
andBoltežar, 2009; Čepon et al., 2009, 2010). Thenumberof belt elements is assumedasnb =60
and the coefficient as σ = 10000m−1. One belt element and the neighbouring spring-damping
element (on the left side of the belt element) are chosen to show some example calculation
results. The selected belt element is also presented in Fig. 5 as case 1.

Fig. 5. Assumed belt transmission with a selected belt element

The torque applied in the driving pulley is also assumed as a partially-linear time-depended
function.From0to0.3s, thevalueof the torquedecreased from0to−50Nm.After thatmoment,
it remained constant.



1428 K. Kubas

The resistance torque applied in the driven pulley is described as a function of its angular
velocity

Md2 =
100

2π
θ̇2 (2.13)

All of the presented calculation results omit the stabilisation phase of the analysed model.

The calculated driving and resistance torques are presented in Fig. 6.

Fig. 6. Driving and driven torques applied to the pulleys in case 1

The achieved angular velocities of the driving and driven pulleys are presented in Fig. 7. As
can be observed, after 0.3s the angular velocities remain constant with a value almost equal to
2π rad/s. Some disturbances can also be observed as a result of the assumed discrete model of
the belt. It is worth mentioning that this effect can be reduced even more by assuming more
belt elements; this will of course cause an increase in the number of equations of motion.

Fig. 7. Calculated values of angular velocities of the pulleys

In the next two figures (Figs. 8a and 8b), the longitudinal deformation and resulting force
in a chosen spring-damping element is presented. As has been mentioned, it is located next to
the chosen rigid element shown in Fig. 5. As can be observed, this element is located in the
passive part of the belt at the moment that the transmission starts. The values of deformation
and force decrease. Some disturbances and an increase in the values can be observed just over



A model for the dynamic analysis of a belt transmission... 1429

0.3s. From this moment, the rigid element is located on the driven pulley. This situation lasts
to about 0.82s. After this time period, it moves in the active part of the belt.

Fig. 8. Longitudinal deformation and reaction force in the selected SDE: (a) longitudinal deformation,
(b) reaction force

As can be observed from the shapes in the figures, the reaction force in the spring-damping
element mainly depends on the deformation. The dependence on the deformation velocity is
negligible.
Figure 9 shows the reaction forces on each element at a chosen time of 0.4s. Elements with

numbers 1-9 and 53-60 are located in the passive top part of the belt. The values of the force in
this part are about 120N. Elements 23-39 are in the active bottom part of the belt. The values
of the force in this part are about 1120N. Elements located in the driven pulley correspond
to numbers 10-22 (growing force), whereas those located on the driving pulley correspond to
numbers 40-52 (decreasing force).

Fig. 9. Reaction forces in all SDEs at a selected time

In Figs. 10a and 10b, the calculated values of normal and friction forces acting on a selected
rigid element from the driven pulley are presented. As can be observed, up to about 0.6s,
the friction forces are smaller than the normal forces and do not correspond to the developed
friction or slip (which is consistent with the results of the angular velocities shown in Fig. 7).
From about 0.6s, the values start to be equal or almost equal.



1430 K. Kubas

Fig. 10. Normal and friction forces acting on a selected belt element from the driven pulley:
(a) normal force, (b) friction force

In Fig. 11a, the calculated values of the friction coefficient of the selected belt element are
shown. The values are negative because of the opposite direction of the friction force to the
assumed positive direction of the pulley rotation (Fig. 3). As can be observed, when contact
between the element and the driven pulley starts, the values of the friction coefficient increase
rapidly to a value of about 0.59 and return to zero (fromabout 0.31s to 0.38s). At thismoment,
the element loses contact with the pulley (this can also be observed inFig. 10a). At about 0.39s,
the belt element returns to the area of the pulley. Up to about 0.68s, the values increase to the
assumed value µk. The values remain constant from this time to about 0.82s. At about 0.82s,
the element loses contact with the driven pulley and startsmoving in the active part of the belt.

Fig. 11. Change of the friction coefficient: (a) in a selected belt element, (b) presented as a function of
displacement

In Fig. 11b, the calculated values of the friction coefficient are shown as a function of di-
splacement. As can be observed, the values asymptotically grow to the assumed value µk. Two
moments of changing the values to zero, as identified earlier, can be observed.

The disturbances of the changing values of the friction coefficient result from the changing
direction of the relative velocity, which is presented in Fig. 12.



A model for the dynamic analysis of a belt transmission... 1431

Fig. 12. Calculated relative velocity between a chosen rigid element and the driven pulley

The presented case of load causes a relatively large acceleration of the transmission. After
velocity stabilisation, only friction in one direction can be analysed. Calculations with another
drive and resistance torques are repeated.

Case 2: Oscillating torque

Next, the torque applied to the driving pulley as shown in Fig. 13 is assumed. As can be
observed, first the torque is linear-changing from 0 (at the start) to −50Nm (at time 0.25s).
Next, the values increase to 50Nm (at 0.75s). At the end of analysis, the torque decreases to 0.
Simultaneously, the resistance torque of the passive pulley has opposite values.
This kind of torque loads the transmission in two opposite directions with a relatively small

movement. Thebelt is slightly rotated clockwise. The chosen belt element (whose initial position
is shown in Fig. 5 as case 2) moves toward the right side of the driven pulley.
The rest of the conditions (initial preload of the belt, dimensions of the transmission, etc.)

are the same.

Fig. 13. Driving and driven torques applied to the pulleys in case 2

Asshown inFig. 14, theangularvelocities of thedrivinganddrivenpulleysdiffer significantly.
It can also be observed that the transmission oscillates in moments of the changing direction of



1432 K. Kubas

these velocities (at time 0.25s and 0.75s). Just before the oscillations, the velocities differ about
0.1rad/s.

Fig. 14. Calculated angular velocities of the pulleys in case 2

Figure 15 shows the resulting reaction force in the selected spring-damping element. At the
beginning, the force is about 420N and decreases to about 100N. After changing the direction
of the load torques, the force increases to about 1110N. At the end of the simulation the force
achieves a value of about 580N.

Fig. 15. Calculated values of the reaction force in a selected spring-damping element in case 2

Figure 16 shows the calculated normal and friction forces between the chosen belt element
and the pulley. Because the selected belt element moves slowly to the right side of the driven
pulley, where the normal force is larger, the absolute friction force also increases to about 190N.

After 0.75s, the selected belt element moves back and the absolute values of normal and
friction forces start to decrease.

Figure 17a shows the calculated values of the friction coefficient. In this case, the friction
between the selected belt element and the pulley achieves values of −µk and µk, which means
that in particular moments it is fully developed. It corresponds to intervals 0.16s-0.25s and
0.66s-0.76s, respectively.



A model for the dynamic analysis of a belt transmission... 1433

Fig. 16. Normal and friction forces of a selected belt element in case 2

Fig. 17. Change of the friction coefficient: (a) in a selected belt element in case 2, (b) presented as a
function of displacement in case 2

Figure 17b shows values of the friction coefficient as a function of displacement acting on
a chosen element from the pulley. As can be observed, at the beginning the friction coefficient
decreases to an arbitrary value −µk and the displacement is achieved at about 0.83mm. Then
the direction of the relative velocity changes. After this moment, the values increase to a po-
sitive value of µk. The relative displacement achieves in this phase about 0.45mm (absolute
displacement is about 0.38mm).

It has beendecided to check howmuch the change in theσ coefficient affected the calculation
results. Figure 18 shows the friction coefficient courses as a function of the displacement in two
cases: σ = 10000 and σ = 5000 as assumed earlier. As can be observed, the slope of the curve
has changed.

Figure 19 shows the calculated angular velocities of the pulleys. A comparison of these
courses with the courses fromFig. 14 shows slightly larger amplitudes of the vibrations. Because
of the assumed smaller value of σ, the system became ‘more flexible’. The contact and friction
parameters will be investigated on a research stand in the future.



1434 K. Kubas

Fig. 18. Change of the friction coefficient presented as a function of displacement in case 2, for two
different values of the coefficient σ

Fig. 19. Calculated angular velocities of pulleys in case 2 with the assumed coefficient σ=5000

3. Conclusions

Ashasbeenmentionedbefore, theproposedDahlmodeldoesnot include theStribeckeffect.This
assumption is possible because the static anddynamic friction coefficients are approximately the
same. It would be necessary to assume another frictionmodel if they were different. The LuGre
frictionmodel is especially interesting and very popular (Canudas deWit et al., 1995). In future
works, it will be the main subject of investigation in belt transmission applications. It is also
important to compare the prepared models with early mentioned models, especially with the
EPP friction model, which also includes elasticity of the joint.

Of course, themost important is to compare themwith realmeasurementsmadeona research
stand.Asmentionedbefore, a series of experiments on a specially built research stand is planned.
The main subject of these experiments will be to measure the friction parameters between the
belt and the pulley. The measurements will be done for a clean belt and with some impurities
(oil, water, etc.). It has been noticed from first experiments made on a clean belt that friction
between the belt and the pulley cannot be described with a relatively simple Euler formula. Its
dependence on the belt preload or wrap angle can be more complicated. The friction can also
depend on the rest time between movements of the transmission or belt slip, on the relative
velocity between the belt and the pulley surfaces or even on the acceleration.



A model for the dynamic analysis of a belt transmission... 1435

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Manuscript received September 28, 2016; accepted for print July 18, 2017