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solidshandling Volume 18 Number 1 January/March 1998 Belt
Tension Around a Drive Drum
Modelling Belt TensionAround a Drive Drum
A. Harrison, USA
SummaryAnalysis of the drive drum friction problem by classical
calculusprovides a slip test that allows a designer to determine a
maxi-mum T^/T2 for a conveyor. In the classical slip calculation,
adrive friction coefficient of 0.25 to 0.35 is used. In this paper,
amechanical model is developed to simulate the starting of adrive
drum with a distributed mass-spring system for the belt-ing. The
model allows the belt to be pre-tensioned, then torqueis applied to
develop a 71 and T2 tension. The torque is appliedup to the point
of drum slip. The model produces the tensiondistribution in the
belt around the drum face. The model pro-vides a clearer
understanding of belt tensions around a drivedrum, including
requirements for viscoelastic contraction whendesigning a drive
system.
1. IntroductionThe problem of defining the way in which belt
tension changesfrom its tight-side 71 tension to the outgoing
slack-side 72 ten-sion has been contemplated by a number of
authors. The basicrelationship that defines the amount of 71
tension in relation toT9 tension at the point of drive drum slip
is
n = e (D
The derivation of this equation is based on a simple calculus
forthe problem, where small elements are integrated around thewrap
angle 6 from 7"2 to 7r with an effective coefficient of fric-tion u
defined at the point of limiting friction or the onset of bod-ily
slip. The right hand side of the equation defines the amountof TJTZ
that can be supported by a frictional drive drum againstthe belt.
In the problem, u is the static friction of the belt on thedrive
drum. It should be remembered that the actual 7, and 72tension at
the drive drum is purely the result of conveyor line re-sistance
and material lift, and not a function of the drive drumcoupling to
the belt by friction.Typical values quoted for the effective
friction factor u betweenbelt and pulley are 0.25 < u < 0.35
for steel pulley (lower value)and lagged pulley (higher value).
However, actual measured val-ues of friction coefficient for rubber
on steel and rubber on rub-
Dr. ALEX HARRISON, President of Scientific Solutions Inc.,
Denver, Colorado,USA, 2200 Chambers Road, Unit J, Aurora, CO 80011,
USA. Tel.: +1 303344 90 24; Fax: +1 303 344 91 02.
Details about the author on page 167.
ber range from 0.7 to 1.2, much higher than the effective
frictionu used today for drive slip calculations.Clearly the u used
above is considered an effective frictionvalue over the entire drum
wrap length. When 7", becomes largeby comparison to T2, slip has a
higher probability of occurring,and so for design purpose the
relationship
7,(2)
is applied as a test for the point of slip. For dual drive
drums,other relationships have been developed to accommodate
dif-ferent wrap angles [1], The basic model does not take into
ac-count the following important parameters that are present in
areal system: Contraction of the belting from 7, to 72 (the belt's
elasticity). The real value of u between drum and belting (0.7 to
1.2). Localised sliding as torque is applied to the drum.
Variations in the normal force due to the contraction prob-
lem. Variations in friction coefficient around the drum. Belt
speed exiting the drum will be less than drum speed.A good deal of
recent research has been conducted on theproblem of drum friction,
with two papers by HARRISON [2, 3] andanother by ZEDIES [4], In
references [2] and [3] the problem ofsurface roughness and friction
tests are addressed. In reference[4], actual instrumentation is
used to monitor the way the drivedrum encounters the belting. Based
on surface roughness andrubber hardness tests [2], a new equation
was developed to de-scribe the manner in which 7/72 varies around
the drive wrap,
- = 1 + A BT2
(3)
where A = 2.0 for rubber on steel, and 6 is the wrap angle
[2].The equivalent friction coefficient developed in this research
isnot constant but drops from about 0.86 to 0.56 between 100degrees
and 220 degrees of wrap, with ^ = 0.7 at 150 degreesof wrap.
Therefore, n(6) is a curve that shows equivalent frictiondecreasing
non-linearly as wrap increases, with the model de-scribed by Eq.
(3).A conclusion of the research discussed above was that
themechanism that describes the way in which 71 evolves to 72around
a drive drum is complex and not well understood at thebelt/drum
interface. Friction coefficients are clearly much higherthan used
in design tests for slip. This paper describes anotherapproach to
the problem that involves physical modelling andsimulation.
75
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Belt Tension Around a Drive Drumbulk
Volume 18 Number 1 January/March 1998 SOlMS
2. Physical ModellingDetermining the way that 71 tension evolves
to a lower 7~2 valuearound a drive drum is one of the subjects of
this paper. A phys-ical model using masses and damped springs has
been devel-oped to simulate a belt wrapped around a drive drum.
Themodel may be generally applied to a drive design review
oraudit.
Distributed mass and elasticity elements are developed thatform
the input to a simulation engine [5], The simulation engineallows
the model to run under a set of program control func-tions, written
so that the time when torque is applied can beuser-input.For
example, the rate of pre-tension applied to the model canbe
controlled so that initial conditions are stabilised. After
initialpre-tensioning oscillations are stabilised, the rate of
torque ap-plication is controlled to prevent drum slip while the
model dif-ferentially tensions all the springs on the drum
face.Fig. 1 illustrates the model of a drive drum, constructed so
thatreal-world parameters are applied. For example, the
followingset of parameters are used in the model that will be
described inthis paper:Belt mass = 25.5 kg/m (fabric belt) - 7
masses, 7 kg/massBelt mass = 36.5 kg/m (steel cord belt) - 5
masses, 14 kg/massDrive drum = 1.0 m diameter, wrap angle 210
degreesStatic friction = 0.9, kinetic friction = 0.7 (rubber on
steel)Spring stiffness k = belt stiffness, length L/Q per element
spring
6k, L/6
k , L 6k
k , L
Fig. 1: Physical model of a drive drum for torque application
and simulation ofspring tensions
Fig. 2: Procedural functions for the simulation
Physical Model Inputs \Control Parameters for
Simulation J
Working Model Inputs ]Simulation Engine I
Drum and BeltData
Control RampAlgorithms
Runge-KuttaAuto Time StopCollision Detection
/SimulationNI Outputs J
Fig. 2 shows the interface between the physical model and
theworking model simulator [5]. All mass positions and spring
ten-sions are tracked during the simulation. All inputs to the
physi-cal model need to be correct so that the simulation will be
sta-ble and converge to real solutions.In establishing the model,
the following procedure is required:a) Each spring is attached to a
mass element at a location that
will not cause significant over turning moments.b) Each spring
has the same length L/6, where L is the length of
the unsupported spring.c) Every mass contacts the pulley face
with even bearing pres-
sure so that sliding forces are rapid to compute and so
thatbouncing is not induced when the drum torque is applied.
d) Damping is applied to each spring to reduce dynamic
inter-actions while pre-tension and torque is being applied.
e) At time t = 0, a displacement is applied to the drum along
itshorizontal axis, to pre-tension the springs to a typical
belttension.
f) At a later time after all initial oscillations die away (the
systemis in static equilibrium and tensioned) the torque is
rampedup slowly to create a 7^ and 7"2 tension. This process
contin-ues until the drum rotates beneath the masses (slip).
Eachmass adjusts its position by sliding on the drumface until
allforces are balanced. Spring forces then represent belt
ten-sions.
3. Low Pre-Tension ExampleTension in a fabric belt is generally
lower than in higher stiffnesssteel cord belts, and therefore
spring stiffness has to be set ac-cordingly. A model was
constructed for a 6 element spring sys-tem, as shown in Fig. 1.
Spring lengths were of the order of0.28 m and the drive drum
diameter was 1.0 m. The exactlength of the spring lengths
determines the initial strain, howeverthese initial conditions are
no longer relevant once the torque isapplied to the drum and spring
tensions start to deviate fromeach other. Static friction was set
at 0.9 and sliding or kineticfriction was set at 0.7.
At time t = 0, a slow rising pre-tension of 5 kN was applied
tothe belting for 2 seconds, and then held at this level until 5
sec-onds. The control logic for the simulation sets a true
conditionby the first value and the false condition by the second
value, asfollows :
time steppretension Force Fxtorque
0.025 s
if (f < 2,5000 * f/2,5000)if (f > 5,50 * (f - 5), 0)
Each mass was set at 25 kg and each spring had a stiffness of120
kN/m. Damping for each spring was set to produce smoothsteady-state
conditions. Setting the damping too large results indynamic
stiffness and possible instabilities due to the produc-tion of high
frequencies in the model.Fig. 3 shows the result of the model
simulation. Soon aftertorque is applied, motion of the drum is
observed in this partic-ular model.
Tensions in the spring elements between 5 and 10 seconds
arebasically locked because there is no relative sliding of
themasses at a low value of applied torque. The upper 2 springsnear
7^ (the highest rising curves in the left hand side (Ihs) box
ofFig. 3) experience a rise in tension before others around
thedrum. The T2 side spring tension starts to fall at the same
timethe 71 spring tension rises (see right hand side (rhs) box
ofFig. 3).
76
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bulksolidshandling Volume 18 Number 1 January/March 1998 Belt
Tension Around a Drive Drum
Fig. 3: Results of a low tension simulation
In general, tensions for springs shown in the Ins box are
orderedfrom top to bottom in the same way they are drawn in Fig.
1,namely the top spring of length L contains the tight side
tension7^, and so on. In the rhs box, the lowest spring tension is
alsothe last long spring, i.e. at 7~2 belt tension.This model slips
early in the simulation, allowing the sliding ofthe drum to drag
the masses to a point of static equilibrium. Thispoint is reached
when the masses locate at a point on the drumsurface where spring
forces across each element is balancedby the drag force at its
particular normal load. The simulation inthis particular model
describes constant creep or sliding of thebelt relative to the
drive drum, typical of many operating fabricbelt drives that show
up to 5% slip when running.Analysis of the data shows that all
masses slide to a point of sta-tic equilibrium, with an active
dynamic friction of uk = 0.7 foreach mass. The drum takes about 1
minute to reach a pointwhere the system is at static equilibrium.
At this time, no morespring extensions occur, and the tensions are
measured:
= 791 7 N 72 = 2411 N.The ratio T^/T2 = 3.2837. From the general
equation for slip, theeffective friction coefficient u, that
supports this tension ratio atslip, with 210 degrees of wrap,
is
M, = {In (7~/72)} / 9 = 0.3257.This result is very interesting
in that the static friction coefficientof each mass is set at near
measured values of \a = 0.9, the slid-ing friction is uk = 0.7, and
the equivalent or effective friction isH = 0.3257.The simulation
has produced a result for \a that the simple the-ory described by
Eq. (1) would require, namely a ^ in the range0.25 < u< 0.35.
The value of the effective friction for the systemis about half the
sliding friction value.By taking the data from Fig. 3, it is
possible to plot the way inwhich the spring element tensions vary
around the drive drum.Fig. 4 shows the graph of the evolution of
spring forces at dif-ferent levels of K = T^/T2, up to the point of
static equilibrium.This graph shows that the tension in a belt
around a drivingdrum evolves with an approximate "S" curve. There
may be
some error in the curve near 7T, just before system static
equi-librium (slip) due to dynamic oscillations in the mass
locations(and hence spring tensions) at the point where the
systemchanges to static equilibrium.
4. High Pre-Tension ExampleThe previous example showed a
relative low tension belt model,typical of those used in
underground coal mines. It showed thateven though friction of a
belt against a drive drum is measuredat about jx = 0.9 at limiting
friction, the effective friction as seenat the point of slip (T,/T2
= 3.2837) is \JL = 0.3257. This simula-tion confirmed the typical
value applied in Eq. (1).Another drive drum model with higher
stiffness springs(k = 1000 kN/m) was constructed to simulate a
steel cord belt.A 4-spring element with 5 masses was used. Each
mass had avalue of M = 14 kg and the same drum parameters were
usedas in the previous example. The friction used was also the
sameas in the previous example, namely us = 0.9 and uk = 0.7.
Fig. 4: Tension in the belt between 7, and 72 for various K
ratios
0 30 60 90 120 150 180 210Degrees
-
Belt Tension Around a Drive Drum Volume 1 8 Number 1
January/March 1 998bulk
Various Tensions around Druml 1TUT2
Tensions!(NJOe+005(N)(N)6e+005(N)""
11.8026+005
1.600 2.400
(N)0e+005(N);1.966e+005
1.802e+005
1.6386+005
i^ TSangefe'Olle+005
1.147e+005
9.830e+004
8.192e+004
4.915s+0V
3.27?e+004
1.638e+OQ4!
t^C
x^
1.600
i J""' >i^
W^C
N.2.400
A-xAft
a200 (a)Fig. 5: Simulation to the point of slip for a 100 kN
model pre-tension
For a model pre-tension of 85 kN, 71 and T2 tensions at
drivedrum were 172 kN and 14.1 kN respectively, at the point ofdrum
slip. The slip ratio T^/T2 = 12.226, which gives an
effectivefriction factor of u = 0.654. As pre-tension in the model
in-creases to 165 kN, the effective friction increases u = 0.8
be-fore the drum physically slides beneath the distributed
masses.Fig. 5 shows the evolution of spring tensions around the
drumwith 4 spring elements and 5 masses. The model was
pre-ten-sioned to 100 kN, and the above simulation techniques
wereapplied. Tensions achieved prior to bodily slip of the drum
were7., = 202 kN and T2 = 8 kN. This gives an effective friction
coef-ficient of u = 0.84. The mass-drum interface friction was
0.9.
Fig. 6: Full time history of the simulation in Fig. 5
-16.000 i =; * i
0.400 1.200
; ^Y
2000 2.800 \ It]
This test reveals that a significant torque can be applied to
thedrum prior to slip, well above the values traditionally used
forEq. (1). Spring tensions stabilise to constant values in this
modelat 3.2 seconds after torque is applied. This is where
constantdrive slip is modelled. The pre-tension is so high that the
modelmaintains stability even though the drum slips. This is not
thecase with lower pre-tensioned models. Increasing the numberof
springs to 6 (as before) results in very high frequencies due tothe
spring stiffness and so the model becomes unstable
duringsimulation.Fig. 6 shows a full time history simulation of the
model. The ini-tial pre-tension causes dynamic overshoot in all
spring tensions.
After the system settles down,torque is applied gently and
thetensions split as shown, up to thepoint the drum slips. The
lowergraph shows the rotation of thedrum and total slip at about
2.8seconds.A number of stable simulationswere conducted for the
drive withthe same mechanical conditionsas above. The model's
pre-ten-sion value was increased for eachsimulation so that K
increased.Fig. 7 shows the way the modelsimulates an effective
friction co-efficient as K = T^/T2 increases.The effective friction
u was calcu-lated for each simulation. Clearlyu increases as K is
increased,
showing that the pre-tension levelaffects the effective friction
coeffi-cient at a given wrap angle.
78
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bulkVolume 18 Number 1 January/March 1998 Belt Tension Around a
Drive Drum
MuEff-
0.90.80.70.6
0.30.20.1
010 15
K = T1/T2 Ratio20 25
Fig. 7: Variation in effective friction M with 7/72 (drum
diameter = 1 m)
5. Application to DesignIn conveyor design various tension
calculation methods areused to find the effective tension 7"e
around the conveyor. Thesemethods include CEMA [1], ISO 5048 or DIN
22121. Whilsteach of these methods have drawbacks they are widely
used inindustry to produce an effective tension Te around a
particularconveyor profile. Commercial models may be more
accuratethan ISO or CEMA.Using the simulation results for a typical
1 m diameter drivedrum, Eq. (1) may be re-written to include the
effect of belt ten-sion on effective friction. For the above
conditions of actual fric-tion, namely us = 0.7 and uk = 0.9, the
equation for u(K) is
\i(K) -aK-e~ (4)
where a = 6 x 10"3 and the equation for the slip test
becomes
< e (5)
A design slip test may be summarised with a 5 point procedureas
follows :1. Compute the effective tension 7e for the conveyor.2.
Select a 7~2 tension based on lowest profile tension for sag.3.
Compute the value of 71 based on a best assessment for Te.4.
Compute drive friction coefficient using Eq. (4)5. Determine
whether 7^ is too great, using Eqs. (5 or 6).There are many
instances where the drive slips beneath thebelt, contrary to design
calculations [1]. Each particular con-veyor drive design may be
analysed using simulation, so that theinfluence of drive diameter
and lagging type may be taken intoaccount. Viscoelastic
considerations of the belting can now beaccommodated since the
tension evolution between 7^ and 7~2is known up to the point of
slip [6]. This process is particularlyimportant since there are new
lagging materials in the marketthat claim a larger coefficient of
driving friction to prevent slip.For example, ceramic lagging is
reported to exhibit an effectivefriction value of M =0.45 to 0.5. A
simulation of ceramic againstthe rubber belting will give the
actual values of friction for the sliptest. Mechanical impression
of the lagging nodules in the caseof ceramic lagging may be the
cause of the apparently higher ef-fective friction factor, and this
problem is being researched.
6. ConclusionsSimulation of a drive drum against a rubber belt
with static andsliding friction values near 0.9 and 0.7
respectively shows thatthe effective friction coefficient u varies
between 0.32 and 0.84,depending on the belt tension. At low belt
tensions in relation tobelt stiffness the slip predicting Eq. (2)
is valid, however as theratio T^/T2 increases, a new effective
friction coefficient that istension dependent needs to be applied
to prove the point ofslip. Another outcome of the modelling is the
tension distributionin the belt around the drive drum. This result,
together with vis-coelastic relaxation rates, may lead to improved
belting materi-als.Although a specific simulation example is
described in thispaper, along with assumptions and conclusions,
every drivedrum system will require individual examination to
determine thereal point of slip.
During design iterations, the tension at some point along a
beltneeds to be set, and usually a value is selected for T2 or tail
ten-sion (in head drive conveyors) to ensure that sag at the
loadpoint is kept within design limits. This may change for tail
drivesystems. In general, setting 7~2 or tail tension is relatively
simpleif gravity take-ups are used.Supposing that 72 is set in a
design to control the sag at theload point, the relationship for
the slip test becomes
where the effective tension due to conveyor resistance isTe =
(T,-T2).The research shows that the slip test depends on the
effectivetension due to conveyor resistance. Larger effective drive
drumfriction values are predicted with this research compared to
tra-ditionally used drive friction values of between 0.25 and 0.35.
Ina traditional application, Te is not one of the parameters that
af-fects the effective friction coefficient.
References[1] CEMA: Belt Conveyors for Bulk Materials Handling;
3rd
Edition, 1995.[2] HARRISON, A. and ROBERTS, A.W.: Mechanisms of
force
transfer on conveyor belt drive drums; bulk solids handlingVol.
12 (1992) No. 4, pp. 581-584.
[3] HARRISON, A. and BARFOOT, G.: Load sharing between mul-tiple
drive conveyors; Procs. SME Conference, Reno,Nevada, 1993.
[4] ZEDIES, H.: Investigation of the strain on
drum-laggingsaiming the optimization of the lagging; Doctoral
thesis, Uni-versity of Hannover, Germany, 1987.
[5] Working Model 2D for Windows 95, Users Manual, Knowl-edge
Rev., Summit software 1989-1996
[6] HARRISON, A.: Lagging and belting dynamics in
conveyordrives; bulk solids handling Vol. 16 (1996) No. 2, pp.
353-359.
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