Ž .Powder Technology 115 2001 290–297www.elsevier.comrlocaterpowtec

On the determination of contact parameters for realisticDEM simulations of ball mills

B.K. Mishra), C.V.R. MurtyDepartment of Materials and Metallurgical Engineering, Indian Institute of Technology Kanpur, India

Received 1 June 1999; received in revised form 1 June 2000; accepted 20 July 2000

Abstract

Ž .The discrete element method DEM has been used extensively in simulating multiple interacting bodies undergoing relative motionand breakage. The key to the success of this method lies in correctly establishing the interaction rules and the associated contactparameters. In this paper, this is done by analyzing the impact behavior of a steel ball. Experimental data using an ultra fast load cellŽ .UFLC allowed the material response at the contact to be adequately modeled by a nonlinear differential equation. In turn, the contactparameters to be used in the DEM were directly extracted from the model.

The nonlinear contact behavior led to values of contact parameters that limit the critical time step in the numerical integration. This isunsuitable for any large simulation involving thousands of interacting balls as in the case of a ball mill. This disadvantage is overcome byusing the equivalent linearization technique to transform the nonlinear contact model to an analogous linear model. Not only did thelinearized parameters allow a larger time step but also the use of a linear spring–dashpot model significantly reduced the overallcomputational effort, which is illustrated by simulating a 54.5-cm diameter ball mill.q2001 Elsevier Science B.V. All rights reserved.

Keywords: DEM; Ball mill; Simulation; Spring–dashpot contact; Equivalent linearization

1. Introduction

One of the successful applications of the discrete ele-Ž .ment method DEM as applied to mineral engineering

research includes prediction of charge motion and estima-tion of power draft of tumbling mills that are used in allmining operations for size reduction operations. Since its

w x w xinception 1 and successful adaptation 2 to solve ballmill problems, many have applied DEM to describe the

w xinternal dynamics of tumbling mills 3–7 . Today, withimproving computer speed, DEM offers an extremely vi-able approach for simulating tumbling mills of any size,thereby lending itself as a practical tool for design andoptimization.

The main objective of simulation is to correctly predictthe dynamic profile of the ball charge and the power draft.For an industrial mill of 13-ft diameter, it is done essen-tially by tracking the motion of approximately 100,000balls. Monitoring the position of all these balls at anyinstant is not a trivial task and it is due to the availability

) Corresponding author.Ž .E-mail address: bk@iitk.ernet.in B.K. Mishra .

of the DEM technique that it has only become possible. Inthe DEM technique where the calculation is done in adiscrete fashion by monitoring the forces at each contact,the total number of contacts to be handled, assuming amaximum of 10 contacts per ball is 106. At each contact,the contact forces, and for each ball, the correspondingvelocities, are to be calculated. The list updating requires acalculation of around 30 unknowns per ball. Altogether, itis estimated that the total number of computations per time

Ž y5 .step 10 s in the case of linear springs is of the order109. Furthermore, it is estimated that there is a six-foldincrease in computation when the nonlinear spring–dash-pot contact models are introduced. Therefore, the sheernumber of calculations required until the mill reachessteady state is overwhelming.

The computational limitations restricted the early DEMŽ .applications to two-dimensional 2D assemblies of discs.

DEM was used as a qualitative tool for the computation ofplanar motion of the charge as a function of linear geome-

Ž .try. Following the development of three-dimensional 3Dsimulation codes, it became possible to directly obtainquantitative information such as axial flow of material,frequency spectrum of the intensity of collisions, power

0032-5910r01r$ - see front matterq2001 Elsevier Science B.V. All rights reserved.Ž .PII: S0032-5910 00 00347-8

( )B.K. Mishra, C.V.R. MurtyrPowder Technology 115 2001 290–297 291

draft, etc. However, the success in this approach lies incorrectly determining the parameters involved in the modelfor two reasons: first, these parameters influence the criti-cal time step used in the simulation procedure, and second,the accuracy of the quantitative results of the simulation issolely dictated by the accuracy of these parameters. Even aslight variation in the values of contact parameters is likelyto give a large variation in the simulation results. Theseparameters essentially embody the contact properties of thesystem: stiffness, damping, and friction. Estimation ofcorrect parameters for realistic simulation is not addressedadequately in the literature. In this paper, a formal proce-dure is presented to determine the contact parameters usingthe experimental data in such a way that a compromise ismaintained between numerical accuracy and computationalexpense.

2. Theoretical background

In the DEM, the trajectory of each distinct elementŽ .particle or ball is tracked incrementally by Newton’slaws of motion. Forces are computed at the contactsbetween particles by suitably describing the contact behav-ior by means of a force–displacement law. It is assumedthat the time step may be so small that no new contacts aregenerated in the course of the motion of the particle in thattime step. Hence, the resultant force acting on any particleover one time step is composed of the forces arisingexclusively from contacts that the particle shared in agiven time step. In calculating the contact forces, theparticles are allowed to overlap; every such overlappingcontact is modeled by a pair of spring and dashpot in boththe normal and shear directions. The shear spring–dashpotcombination generates, at most, the maximum static fric-tional force for the materials involved in contact. Aschematic representation of the contact is given in Fig. 1.

Literature reveals that the contact models that describethe force–displacement relationships exhibit purely linear

w xto highly nonlinear contact response. Dobry and Ng 8provide a critical assessment of these models with respectto various simulation conditions. For a linear spring and

Fig. 1. The spring–dashpot model representation of a contact.

dashpot, the equation of motion of an impacting particlecan be expressed as

mxqqxqkxs0 1Ž .¨ ˙where x, x and x are the displacement, velocity and˙ ¨acceleration of the particle;m is the mass of the particle,qis the damping coefficient, andk is the contact stiffness.In order to determine the contact stiffness,k, it is quitecommon to limit the maximum anticipated inter-particlepenetration,x , to a small fraction of the particle diame-max

ter d and define the contact stiffness as

ks f 2mÕ2rd 2 2Ž .0

whereÕ is an estimated maximum velocity of any particle0

in the system andf is the penetration factor defined bydrx . The damping coefficientq is defined bymax

1r2m mi jk

m qmi jqsy2ln e 3Ž . Ž .22p q ln e� 4Ž .

where m and m are the masses of the particlesi and j,i j

respectively ande is the coefficient of restitution.For the discrete particle simulation of a 2D fluidized

w xbed, Tsuji et al. 9 used the linear model by assuming aconstant value for stiffness. Qualitatively, the results weresatisfactory in many respects, such as circulating motionand mixing of particles, but the model could not predictthe correct force vs. time relationship. Zhang and Whitenw x10 have shown that the expression for damping in thelinear model is incorrect in as much as the dampingcoefficient q should not be a constant; it should be afunction of the amount of deformation of the contact. Theyshowed that when the damping is non-zero, the initialforce is large, which is contrary to experimental measure-ments that show that the contact force, during an impact,rises from zero to a maximum value and then returns tozero.

The nonlinear impact phenomenon was extensivelyw xstudied by Hertz 11 . For linearly elastic and homoge-

neous material with perfectly smooth contact surfaces,Hertz showed that the forceF acting at the contact area is

Fskx 3r2 4Ž .Hence, the undamped equation of motion for head-oncolliding bodies is written as

mxqkx 3r2s0 5Ž .¨w xIt was experimentally proved by Velusami 12 that a

contact damping force also exists at the impact area.According to him, the damping forceF is given byd

F sqx 3r2 x 6Ž .˙d

So, the governing equation of motion becomes

mxqqxx 3r2qkx 3r2s0 7Ž .¨ ˙

( )B.K. Mishra, C.V.R. MurtyrPowder Technology 115 2001 290–297292

The nonlinear impact theory of Hertz is limited tometal–metal impact only. However, realizing its implica-tion, researchers in the field of DEM have applied nonlin-ear force theory to obtain a more realistic model for the

w xcontact force. Tsuji et al. 13 used one such model tosimulate a particulate system where the model used torepresent damping differs greatly from that of Velusamiw x12 . The damping coefficient was found heuristically as

1r2 1r4zasa mk x 8Ž . Ž .where a is a constant which depends on the coefficient ofrestitution e. Accordingly, the equation of motion wasreframed as

mxqz xqkx 3r2s0 9Ž .¨ ˙

The contact stiffnessk was determined by contact theory.In the case of two spheres of same material and size,k isexpressed as

'E 2rks 10Ž .23 1ynŽ .where E is the Young’s modulus, andn is the Poisson’sratio of the particles, andr is the radius of the contact

w xarea. Tsuji et al. 13 used the above nonlinear model tosimulate the plug flow of cohesionless particles in a hori-zontal pipe. This model however showed marked quantita-tive disagreement with the experiment, but it was satisfac-tory in the sense that the method needed fewer empiricalfactors than the existing analyses of plug flow.

The theory of compressive strain wave propagation wasw xused by Goldsmith 14 to develop a different type of

nonlinear model. This allowed studying the impact behav-ior of a spherical elastic particle with the following gov-erning equation

3 mk1r2 3r2mxq x xqkx ymgs0 11Ž .¨ ˙

2 rAc

where g is the acceleration due to gravity,r is the densityof the bar,A is the cross-section area of the bar,csYrr,is the velocity of propagation of strain wave down the bar,and Y is the elastic modulus of the bar. King and Bour-

w xgeois 15 used this model to simulate the energy requiredfor particle breakage. Their experimental results showquantitative disagreement with the numerical model withrespect to the evolution of contact force. Finally, one of

w xthe recent publications of Thornton 16 gives a moredetailed theoretical description of several other variationsof nonlinear contact models.

For the simulation of tumbling mills, a detailed study ofthe parameters involved in the model representing thecontact is necessary. The most important parameters fromthe point of view of power requirement are stiffness anddamping. These parameters control the evolution of thecontact forces and, hence, the overall dynamics of theprocess. In the present work, instead of using the contact

parameters on theoretical grounds, a procedure is estab-lished to extract the relevant contact parameters by experi-mental means. By doing this, certain assumptions pertain-ing to material characteristics, e.g. smooth surface and noprior deformation, are done away with. Moreover, thenonlinear contact model is implemented in such a way thatthe unrealistically high initial force is suppressed. Finally,average values of contact parameters derived from thecontact model that matched the experimental data areselected for simulation of power-draw of a laboratory-sizeball mill.

3. Determination of contact parameters

A series of drop ball tests were carried out by using theŽ .ultra fast load cell UFLC at the Comminution Center of

w xthe University of Utah 17 . The data show variation in theforce with displacement and the time history of the contactforce during impact. These data represent the metalliccollisions of ball–ball and ball–wall type in a ball mill.The impact behavior was studied by using three steel ballsof mass 0.096, 0.252 and 0.647 kg. These balls weredropped from a height of 0.3 m on to a steel rod that has aflat surface. The force–time and force–deformation histo-ries on the top surface of the rod was detected by thesolid-state strain gauges and recorded in the oscilloscope.

Fig. 2. Contact force history for a ball of mass 0.252 kg: comparisonbetween experiment and model.

( )B.K. Mishra, C.V.R. MurtyrPowder Technology 115 2001 290–297 293

A detailed description of the experimental set-up and itsw xtheoretical basis is explained elsewhere 15,18 .

The contact parameters were determined from the ex-perimental data in two steps by solving the model equationof the following nonlinear form

mxqqxx sqkx rs0 12Ž .¨ ˙First, the range of nonlinear parametersr and s werefixed. The nonlinearity in the spring,r, and that of thedashpot,s, were kept in the range 1–2 and 0–1, respec-tively. Second, approximate values of contact parameterskand q were taken as initial guess values. Then the modelequation was solved using the Fourth Order Runge–Kuttamethod. The resulting force–displacement and force–timerelationships were fitted with the corresponding experi-mental results. In case of any mismatch, the values of allthe model parameters were modified and the procedurewas repeated all over again. The optimized results showingthe force–time and force–deformation histories for theboth model and experiment are shown in Fig. 2.

Similar analyses were done for all the three ball masses.In each case, the predicted result matched the experimentaldata for rs1.6 and ss0.8. However, the numericalvalues of k and q were different in each case. Thesevalues of the stiffness and damping parameters obtainednumerically were used to calculate the energy loss due tometal–metal impact. The experimental values of the en-ergy loss were obtained by determining the area enclosedby the force–deformation loop. The calculated values ofenergy of impact were found to be in good agreement withthat of the measured value. A summary of the experimen-tal conditions and the corresponding numerical results arepresented in Table 1. Thus, for the metal–metal impact,the equation of motion took the following form

mxqqxx 0.8qkx1.6s0 13Ž .¨ ˙In the foregoing, it was demonstrated that the force–time

history during the collision could be correctly predicted bythe nonlinear model. However, it was also observed thatthe model parameters that determine the contact responsewere numerically large. For example, as seen from Table1, the nonlinear stiffness parameter values are on anaverage of the order 1010 Nrm1.6. This is the most signifi-cant drawback of employing a nonlinear contact model inthe DEM because this method employs an explicitly condi-

Table 1Contact parameters for the nonlinear model

Experiment no. 1 2 3

Ž .Ball mass kg 0.096 0.252 0.64710 10 10Stiffness,k 4.8=10 5.0=10 5.2=106 6 6Damping,q 4.5=10 4.8=10 4.9=10

Ž . Ž .Energy loss J nonlinear model 0.187 0.454 0.985Ž . Ž .Energy loss J experimental 0.20 0.47 1.08

Ž .Absolute error % 6.5 3.4 8.8

tionally stable time-stepping algorithm where careful selec-tion of the time step is required for numerical stability andaccuracy. For an undamped system, the critical time stepD t is inversely proportional to the square root of stiffnessk as

'D tsC mrk 14Ž .

where m is the mass of the smallest ball andC is anumerical constant, which is typically taken as 0.2. A largevalue of k would require an enormous amount of calcula-tions particularly when the number of balls is large. Thecomputational constraint is relieved if the numerical valueof k is of a lower order. In the present work, a linearmodel is sought from the nonlinear one in such a way thatthe error in the calculation of energy loss during impactand the total computational time are minimized. The com-putational time is minimized by having a reduced numeri-cal value for the stiffness parameter. The methodologyemployed is the equivalent linearization technique.

4. Equivalent linearization

The nonlinear equation of motion for a periodicallyresponding system can be written in its most general formas

xq f x , x sF t ,T 15Ž . Ž . Ž .¨ ˙

whereT is the time period. If this equation is linearized as

xqqX xqkX xsF t ,T , 16Ž . Ž .¨ ˙

the errorE in the approximation isX XE x , x s f x , x yq xqk x . 17Ž . Ž . Ž .˙ ˙ ˙

The average of the square of the error over one cycle is

1 T 2X X2² :E x , x s f x , x yq xqk x dt 18Ž . Ž . Ž .˙ ˙ ˙T HT 0

Minimizing this quantity with respect to the coefficientskX

and qX,

E 1 T2² :E sy xEdts0 19Ž .T HXEk T 0

and

E 1 T2² :E sy xEdts0 20Ž .˙T HXEq T 0

Ž .Eq. 19 implies that

1 T X Xx f x , x yq xqk x dts0, 21Ž . Ž .˙ ˙HT 0

or

T T TX 2xf x , x dty k x dty qxxdts0 22Ž . Ž .˙ ˙H H H0 0 0

( )B.K. Mishra, C.V.R. MurtyrPowder Technology 115 2001 290–297294

Table 2Contact parameters for the equivalent linearized model

Experiment no. 1 2 3

Ž .Ball mass kg 0.096 0.252 0.6478 8 8Ž .Equivalent stiffness Nrm 1.17=10 1.13=10 1.72=10

Ž .Equivalent damping, kgrs 984.74 1361.34 1912.86Ž . Ž .Energy loss J linearized model 0.185 0.451 0.967Ž . Ž .Energy oss J experimental 0.20 0.47 1.08

Ž .Absolute error % 7.5 4.04 10.5

SinceHT xxdts0 for a periodic system,˙0

Txf x , x dtŽ .˙H

X 0k s 23Ž .T 2x dtH

0

Ž .Similarly, from Eq. 20 , it can be shown that,

Txf x , x dtŽ .˙ ˙H

X 0q s 24Ž .T 2x dtH

0

Having minimized the error, it can be neglected, therebyreducing the nonlinear governing equation to an equivalentlinear equation

xqqX xqkX xsF t ,T 25Ž . Ž .¨ ˙This method of equivalent linearization is primarily

intended for mildly nonlinear systems with periodic behav-ior. Even though the ball mill system is far from beingperiodic or mildly nonlinear, the technique is being ex-plored.

Using the equivalent linearization technique, the stiff-ness and damping parameters were recalculated. Thesedata along with predicted energy loss for the case ofmetal–metal impact are presented in Table 2. It is ob-served that the value of the equivalent stiffness is of theorder 108, which is significantly lower than that of 1010

obtained by employing the nonlinear model. Although theabsolute error in the energy loss slightly increased in each

Ž .case from its previous value see Table 1 , it remainedwithin acceptable limits. From these observations, it is

believed that the equivalent linearized model is a goodcompromise between purely linear and nonlinear modelsas far as accuracy and computational time requirements areconcerned.

5. Numerical simulation

A 3D DEM code is developed to compare the relativeperformance of various contact models. This computercode has the option of employing either a linear or anonlinear contact model. The particle interaction rules arebased on the contact mechanics theories, which relate thecontact force to the relative approach of particles. For

w xexample, in the nonlinear model, the Hertzian theory 11is used to describe the elastic particle interaction where thenormal force–displacement relationship for spherical parti-cles 1 and 2 with elastic moduliE and E , Poisson’s1 2

ratios n and n , and radii R and R is given as1 2 1 2

4) ) 3r2Ps E R a 26Ž .

3

In the above expression,a is the relative approach that isrelated to the contact radiusa by

)'as aR 27Ž .) wŽ 2. x wŽ 2. x )where 1rE s 1yn rE q 1yn rE , and 1rR1 1 2 2

s1rR q1rR .1 2

The tangential interaction for the nonlinear model isconsidered according to the theory developed by Mindlinw x19 . In the case of a linear contact model, the contactforce is directly proportional to the relative approach andthe shear interaction is controlled by a frictional law. Inboth these models, viscous damping at the contact isallowed. The damping coefficient is related to the coeffi-

Ž .cient of restitution as given in Eq. 3 . The computer codealso provides the energy dissipated due to friction anddamping, which in turn, is utilized to compute the powerdraft of the mill.

In order to assess the accuracy of prediction of individ-ual contact models in terms of power draw, a 54.5=30.4cm ball mill is simulated and the results of the simulation

Table 3Data used in the simulation of the ball mill

Model type Linear model Equivalent linearized model Nonlinear model

Ž .Modulus of elasticity GPa – – 206Poisson’s ratio – – 0.3

5 8Ž .Normal stiffness Nrm 4.0=10 1.53=10 –5 8Ž .Shear stiffness Nrm 2.7=10 1.02=10 –

Coefficient of restitution 0.31 0.31 0.31Coefficient of friction 0.7 0.7 0.7

y4 y5 y6Ž .Time step s 1.01=10 1.03=10 1.56=10Number of balls 324 324 324Mill filling 40% 40% 40%

( )B.K. Mishra, C.V.R. MurtyrPowder Technology 115 2001 290–297 295

Fig. 3. Snapshots of the charge motion at 90% of critical speed: results ofŽ . Ž . Ž .simulation using a linear b equivalent linear, and c nonlinear contact

models.

are compared with the available experimental data ofw xLiddell and Moys 20 . The grinding media used in the

simulation was made up of graded charge of 0.029-, 0.04-and 0.054-m diameter balls having masses 0.096, 0.252and 0.647 kg, respectively. Altogether, 324 balls wererandomly generated to ensure 40% mill filling and a ballload of 140 kg. The stiffness value in the case of a linearmodel was obtained by considering the initial slope of

experimental force–deformation curve for a ball impactinga layer of particles placed on a flat platen. The correspond-ing value for the equivalent nonlinear model was taken asan average of the linearized values of stiffness shown inTable 2. For the nonlinear model, material properties suchas modulus of elasticity and Poisson’s ratio were collectedfrom the literature. Constant values of coefficient of resti-tution and friction were used in all the simulations. Adetailed list of the model parameters used in the respectivesimulations is presented in Table 3.

Numerical simulations were carried out by employingthree contact models: linear, equivalent linear, and nonlin-ear. The simulation results provided the dynamic profile ofthe ball charge. The charge profiles obtained correspond-ing to the three different contact models are shown in Fig.3 for 75% of the critical speed. Visual inspection of theball charge motion in each case, at any given mill speed,did not provide any evidence of discrepancy. However, theindividual trajectories of balls do differ from case to case.

The total energy expenditure of the mill was accountedfor by keeping track of the amount of energy dissipateddue to contact damping and friction. This way, the varia-tion in the power draw of the mill with the mill speed wascomputed. A comparison of the numerical data with that of

w xthe experiments of Liddel and Moys 20 is presented inFig. 4. It is seen from this figure that the linear modelover-estimates the power up to 70% critical speed andagain beyond 90% of critical speed. The equivalent linearand nonlinear models predict the power quite satisfactorilyexcept at 50% of critical speed. Furthermore, all themodels show the same trend in the variation in powerdraw: the power draw reaches its peak value correspondingto a particular mill speed.

It would appear from the data presented in Fig. 4 thatthe power draw of the mill is independent of the contactstiffness. However, careful analysis of the numerical datareveals that it is quite essential to use the correct value forthe stiffness as the material response to impact is predomi-nantly controlled by it. In order to illustrate this idea, the

Fig. 4. Comparisons of power draw of a ball mill of 54.5-cm diameter.

( )B.K. Mishra, C.V.R. MurtyrPowder Technology 115 2001 290–297296

Ž .Fig. 5. Force–displacement relationship at a contact spring force .

force–displacement behavior at a contact is monitored fora ball of mass 0.252 kg, impacting another ball at avelocity of 2.21 mrs. Fig. 5 shows the evolution of the

Ž .force at the contact spring force obtained by using thethree different contact models. The figure on the leftclearly shows a very soft contact response for the linearmodel that is uncharacteristic of the material under test.Moreover, the peak force is significantly lower than thatdue to the other two models. Hence, this data cannot beused to determine the frequency spectrum of the intensityof collisions. The figure on the right-hand side shows acomparison between the force–displacement behavior atthe contact due to the nonlinear and equivalent linearmodel. Here, both the models predict comparable values ofpeak force and deformation.

Examination of the results of simulation leads to thefollowing two important observations with regard to com-putational efficiency. First, the use of the linearized modelfor the description of contact behavior allowed a reducedtime step of integration without any loss of accuracy, andsecond, retaining the linear spring–dashpot type DEM

Ž .Fig. 6. Variation in contact parameters nonlinear with particle layers.

model for the simulation of the mill reduced the overallcomputational expense. This is a point worth noting partic-ularly when DEM is used for the simulation of large mills.

Several other factors affect the contact parameters thatmay have overriding importance with regard to computa-tional efficiency, particularly during simulation of indus-trial mills. These are ball size, impact velocity, amount andthe type of material caught between colliding balls, etc. Atypical variation in the contact parameters with number oflayers captured between the colliding balls is shown in Fig.6. Here, the contact parameters are calculated by fitting the

w xexperimental data of Hoffler 18 to the nonlinear model as¨described earlier. Drop ball test data also show that as theball size increased, the material stiffness and damping didnot vary much for a given drop height. However, theseparameters significantly change with the impacting veloci-

Žties. It is possible to estimate an average velocity drop.height of the ball and considering an average mass of the

balls, the contact parameters can be estimated from dropball tests as discussed earlier.

6. Conclusions

The DEM has become a powerful design and diagnostictool for systematic analysis of the charge behavior in ballmills. However, the accuracy of the method relies heavilyon the model parameters that describe the contact responseto impact. This research work was undertaken to establisha procedure to correctly determine the parameters viz.,contact stiffness, damping, etc., while keeping the numeri-cal and computational expense manageable.

Analysis of the experimental data on drop ball testsusing an UFLC indicated that the deformation behavior ofthe contact is nonlinear. Using a nonlinear contact model,the associated model parameters were extracted aftermatching the model response with the experimental data.The energy dissipated during the collision was very accu-rately predicted by the nonlinear model. While the model

( )B.K. Mishra, C.V.R. MurtyrPowder Technology 115 2001 290–297 297

predicted the contact response quite accurately, the associ-ated contact stiffness parameter turned out to be of theorder 1010 Nrm1.6. It is felt that use of such a high valueof stiffness in a DEM code would reduce the time step ofintegration adding to the computational expense. In orderto circumvent this problem, the nonlinear contact model istransformed to an analogous linear model. This methodprovided a linear stiffness value of 108 Nrm, which is twoorders in magnitude lower than that of the correspondingnonlinear stiffness parameter. Thus, the computational timewas significantly lowered by adapting the equivalent lin-earization technique.

A 0.545-m diameter ball mill is simulated to assess theperformance of the three different contact models: linear,equivalent linear, and nonlinear. The material stiffness wasthe only parameter that was varied in all these simulations.It was observed that all the three models predicted thepeak power within 5%. However, it is misleading toassume that any arbitrary value of the contact stiffness isappropriate for ball mill simulation as it was shown that aAsoftB contact exhibited excessive amount of deformationand did not predict the peak force. Therefore, the numeri-cal data may prove erroneous in determining the frequencyspectrum of the intensity of collisions, abrasion wear ofballs, and breakage of particles.

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