DESIGN PARAMETERS AFFECTING TUMBLING MILL NATURAL FREQUENCIES

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Joseph Baldwin
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ca Abstract Tumblig mills describe a class of mechaical systems defied by a cylidrical chamber filled with balls ad/or rock that rotates aroud its ow logitudial axis.

1 DESIGN PARAMETERS AFFECTING TUMBLING MILL NATURAL FREQUENCIES P. Radziszewski, Y. Y. Qua, J. Poirier Departmet of Mechaical Egieerig, McGill Uiversity, 817 Sherbrooke str west, Motreal, Quebec, CANADA, H3A K6 Abstract Tumblig mills describe a class of mechaical systems defied by a cylidrical chamber filled with balls ad/or rock that rotates aroud its ow logitudial axis. This class of mechaical systems is used to grid to a desired quality differet material i at least three idustries: miig, cemet ad metal powders idustries (ore, cliker ad metal powder). These tumblig mills rage i size from small 1 ft diameter lab mills to a 4 ft diameter semi-autogeous idustrial mill ad are composed of three mai iteractive ad iterdepedet elemets: the mill shell, liers/lifters ad charge. These three elemets work together to impact eergy to the mill charge through the rotatioal motio of the mill shell. As with ay mechaical system, the atural frequecy of that system is essetially dictated by its mass ad stiffess. I the case of a tumblig mill, a rotatig system, the atural frequecy of a give mill, as see by a statioary observer, is also a fuctio of the rotatio speed. The aim of this paper The cosequece for a rotatig mill, ad for that matter ay rotatig system, is that if the atural frequecy of the mill should happe to match the operatig rotatig speed of the mill, resoace of the mill will occur. The objective of this paper is to explore the effect of mill diameter, legth o mill atural frequecies as well as outlie further work. tumblig mills rage i size from small 1 ft diameter lab mills to a 4 ft diameter semi-autogeous idustrial mill ad are composed of three mai iteractive ad iterdepedet elemets: the mill shell (figure 1), liers/lifters (figure ) ad charge (figure 3). These three elemets work together to impact eergy to the mill charge through the rotatioal motio of the mill shell. The amout of eergy imparted to the mill charge is depedat o the lier/lifters used as well as the wor state of these. Figure 1: 4 ft diameter SAG mill Keywords: Tumblig mills, resoace, desig 1. Itroductio Tumblig mills describe a class of mechaical systems defied by a cylidrical chamber filled with balls ad/or rock that rotates aroud its ow logitudial axis. This class of mechaical systems is used to grid to a desired quality differet material i at least three idustries: miig, cemet ad metal powders idustries (ore, cliker ad metal powder). These Figure Typical mill liers With the developmet ad larger scale use of autogeous (AG) ad semi-autogeous (SAG) mills i the miig idustry characterized by icreased mill diameter (1 m) [1] ad varyig ore breakage

characteristics, variable mill speed drives have allowed the adjustmet of mill operatig speed to match the varyig characteristics of the ore as well as couter the effect of lier wear [] (figure 4).

mill radius ad illustrates the speed at which a particle i the mill will cetrifuge: ω g mill cr = (1) R mill I this latter case, the mill could oly operate i the permitted rages rampig through the

2 characteristics, variable mill speed drives have allowed the adjustmet of mill operatig speed to match the varyig characteristics of the ore as well as couter the effect of lier wear [] (figure 4). However, i the case where operatig speed match the atural frequecy of a give mill, the resultig resoace ca shake ot oly the foudatios lose but also create a very usafe work eviromet. mill radius ad illustrates the speed at which a particle i the mill will cetrifuge: ω g mill cr = (1) R mill I this latter case, the mill could oly operate i the permitted rages rampig through the blocked out speeds. Needless to same, each of these solutios to tumblig mill resoace has implicatios either to mill bearig ad foudatio itegrity ad maiteace or to operatig efficiecy. The objective of this paper is to explore the effect of mill diameter, legth o mill atural frequecies as well as outlie further work Figure 3: Simulated mill charge motio [3] Mill Motor Speed [rpm] % crit. Speed Figure 5: Mill motor speeds as a fuctio of blocked out mill critical speeds. Backgroud Figure 4: Lier wear profile [] This is the suspected case for two curretly operatig mills where each of these mills has addressed this problem differetly. I the case of the 1 m mill, extra mass was added to the mill shell effectively lowerig the mills atural frequecy out of the operatig rage. I the case of the 8 m diameter mill [4], the rotatig speeds at which resoace occurred were block out from the possible operatig rage of the mill (figure 5). It should be oted that for a tumblig mill the operatig speed of a mill is defied as a fuctio of critical speed. This speed is defied as a fuctio of I preparig the fiite elemet model of a tumblig mill, two similar systems were studied: thi disk rotatio ad thi shelled rotatig cyliders. Some of the observatios made are outlied below..1 Thi Disk The case of the thi disk as described by the work of Mote ad Szymai [5, 6] illustrates that the disk vibrates trasversely alog odal lies. The mode of vibratio of cosequece is the diameter, circumfereces. It is this mode of vibratio that ofte limits the rotatio speed of the disk whe heated [6]. The atural frequecy of a thi disk rotatig at speed ω d as see by a statioary observer, ω s, ca be determied by the equatio () ad illustrated i figure 6.

3 ω ω s = ± ωd () I this case, the critical speed of the disk is the speed at which it becomes ustable. It is defied as a fuctio of the vibratio mode : ω ω cr = (3) At this speed the backwards travelig wave appears statioary to a statioary observer, resultig i the resoatig state. The ANSYS simulatio produced similar results for mode shapes ad frequecies [8]. The forward ad backward movig waves were illustrated usig the calculatio determied by Szymai, Mote [6]. ω atural frequecy as see by a statioary observer (forward movig wave) atural frequecy as fuctio of rotatio speed atural frequecy as see by a statioary observer (backward movig wave) Figure 6: Rotatig thi disk atural frequecy as see by a statioary observer. Cylidrical Tube Sice the first cylidrical shell problem was ivestigated by Aro i 1974, may efforts have bee devoted to ivestigate how the Coriolis acceleratios ad large deformatios affect the vibratio modes of high speed rotatig hollow cylidrical. Che et al. use a ie odes curviliear super parametric fiite elemet method to solve the problems of vibratios of high speed rotatig shells of revolutio [9]. They also gave the formulas for calculatig the resoace frequecies of cylidrical shells. The two low resoace frequecies are showed as below: ω = Ω ± + 1 ( 1) Ek Ω ρ(1 µ ) ( + 1) (4) I this equatio, the first term shows the affect of the rotatio speed o the frequecies. The secod is due to the ormal bedig stiffess, ad the third caused by the Coriolis force ad large deformatio [9]. The goal ω d of the cylidrical tube simulatio was to reproduce the same mode shapes as those illustrated by [7]. It should be oted that this relatioship is applicable to thi shelled cyliders where thi is defied as follows: t sh / r <= 1/ (5 %) (5) with t sh is shell thickess ad r is tube radius. 3. Mill Natural Frequecy Calculatio I explorig the effect of tumblig mill desig parameters o mill atural frequecy, it was decided to develop a model of the mill startig with the cylidrical shell followed by icremetally addig the differet elemets that affect mill desig ad performace. I the preset case, oly oe iteratio was accomplished startig with the mill as a tube of fiite legth ad the as a capped tube. Mill legth ad diameter (Table 1) was varied i such a way that mill charge volume (3%), rotatio speed (75 % crit.) ad power (76 kw) were costat for all cases. I this case, the mill shell was costraied at both outside edges. Thus, the atural frequecy of mode 3 icreased with icreasig legth to diameter aspect ratio (aspect) over the typical operatig rage of a mill. Table 1. Simulated Mill Legth ad Diameter Mill Diameter [m] Mill Legth [m] Mill Aspect Ratio [1] Oce the models were prepare ad atural frequecies determied, the forward ad backward waves were calculate usig equatio () for the mill tube model results ad usig equatio (4) for the capped tube mill model. 3.1 Cylidrical Tube Model Typical mode shapes for mode ad 3 are foud i figures 7 ad 8. The atural frequecies as a fuctio of mill speed ad aspect ratio ca be foud i figures 9 ad 1. For the 8.5 m diameter (L/D =.5 i figure 9) mill ad over a larger rotatio speed rage, the effect of rotatio speed o the atural frequecy is ot that importat. Whe determiig the forward ad backward movig wave, it is possible to observe the effect o the atural

frequecy of mode 3 as see by a statioary observer (figure 11) usig equatio ().

speed) Figure 11: Forward/Backward frequecy for mode 3 3.

eds were added as show i figure 1 ad the alog with the differet mode shapes (see figure 13).

usig this time equatio (4) (see figure 14).

5%) which is greater that the thi shell defiitio.

4 frequecy of mode 3 as see by a statioary observer (figure 11) usig equatio (). Natural frequecy (Hz) w 6 forward 4 backward % 1% % 3% 4% 5% 6% Mill rotatio speed (% crit speed) Figure 11: Forward/Backward frequecy for mode 3 3. Capped Cylidrical Tube Model Figure 7: Mode shape After completig these iitial set of simulatio, the mill eds were added as show i figure 1 ad the alog with the differet mode shapes (see figure 13). Fially, the atural frequecies were determied for these shapes ad the forward ad backward wave calculated usig this time equatio (4) (see figure 14). It should be oted that i this case, the t sh / r ratio of equatio (5) is equal to.95 (9.5%) which is greater that the thi shell defiitio. Figure 8: Mode 3 shape 7 Natural frequecy chagig with speed for differet models (m=) atural frequecy (HZ) L/D=.64 L/D=.88 L/D=.4167 L/D=.5 L/D=.688 L/D=1.9 L/D=1.7 Figure 1: Mill model rotatio speed (rad/s) Figure 9: Mode frequecy as a fuctio of speed Figure 1: Mode as a fuctio of mill aspect ratio Figure 13: Mode 4 shape

5 6 5 Forward ad backward wave (m=1) of assembled parts. This differece would result i a stiffer system i the curretly simulated case which would result i higher atural frequecies for the simulated mills. frequecy (HZ) With these modificatios, it is expected that the atural frequecy of simulated tumblig mills will approach that observed. 5. Coclusio Figure 14: Mode 1 shape forward ad backward waves 4. Discussio % critical speed The two mai observatios of this iitial exploratory effort are that depedig vibratio mode, the atural frequecy of that mode ca either icrease or decrease with the mill aspect ratio ad operatig speed will create a forward ad backward movig wave that affec ts the mill atural modal frequecies as see by a statioary observer. However, the case simulated is that of the kow mill from Bruswick Mi es where resoace occurs several times betwee 5% ad 75% critical sp eed as opposed to several times greater tha that. If we cosider a simple mass sprig system with stiffess k (N/m) ad mass m (kg), the atural frequecy of this system is: ω = k m (5) I a more complex system, its atural frequecy will be described i a more complex form. This more complex form, but essetially still be defied by some complex stiffess fuctio over some complex mass fuctio. This leads to the first of two explaatios. The mill system is a large, complex system where the resoace frequecy depeds o the whole system. Sice oly a part of this system is modelled ad simulated, the mass is lower thus it should have higher frequecies. After ivestigatig a rotatig thi disk ad the the kow cases of a cylidrical tube, it was possible to explore the effect of mill legth ad diameter o mill atural frequecies. Further, for a kow case, it was possible to ivestigate the evolutio of mill atural frequecy as a fuctio of rotatio speed. These iitial results show that the atural frequecies of a give mill as see by a statioary observer is a fuctio of the mill aspect ratio as well as the operatig speed. These two observatios ted to support the idea that part of the resoace problem of the three mill cases described is the atural frequecy of a give mill is a fuctio of the desig ad operatig parameters of that mill. Although the tedecies i mill atural frequecies as a fuctio of mill desig (aspect ratio) ad rotatio speed are promisig, further work is eeded. Here, oe ca cite completig the mill model by describig it as a assembly of joited parts, itegrate pulp lifters ad the mill charge. Also, experimetatio o a real mill to determie ad validate vibratio modes is eeded. I experimetig o a real tumblig mill system, it would be ecessary to cosider the behaviour of the excitig force from the drive system ad thus describe the whole system. Oce this work is completed ad validated, it becomes possible to address aveues to correctig mill resoace i existig mills or to prevet the desig ad fabricatio of mills that would show uwated resoat behaviour. It should be oted that to date oly, oly the mass ad stiffess of the system has bee cosidered. However, system dampeig plays a importat role ad should be addressed i the future. The secod explaatio relates to the stiffess of the tumblig mill system. The mill model as developed illustrates a soled whole while the real system is a set

6 Ackowledgemets The authors would like to thak the collaboratio of Bruswick Mies for the data o blocked out speeds ad NSERC for the support of this study. Refereces [1] Radziszewski, P., Valery, W., (1999), CADIA SAG Mill Simulated Charge Behaviour, Caadia Mieral Processors Coferece, Ottawa, Jauary. [] Radziszewski, P., Tarasiewicz, S. (1993), Simulatio of Ball Charge ad Lier Wear, It'l J. o Wear, Frictio ad Lub., 169, [3] Herbst, J.A., Nordell, L.K. (1). Optimizatio of the Desig of SAG Mill Iterals Usig High Fidelity Simulatio, SAG 1 Proceedigs, Vacouver, British Columbia, pp. IV-15-IV-164. [4] Orford, I, (5), private commuicatio, Bruswick Mies. [5] Mote Jr., C.D., (1966) Theory of Thermal Natural Frequecy Variatios i Disks Iteratioal Joural of Egieerig Sciece, vol. 8, pp [6] Szymai, R., Mote Jr. C.D. (1977) Priciple Developmets i Thi Saw Vibratio ad Cotrol Research Part 1: Vibratio of Circular Saws Holz als Roh-ud Werstoff, vol 35. pp [7] Liew, K. M.; Ng, T. Y.; Zhao, X. (), Vibratio of Axially Loaded Rotatig Cross-Ply Lamiated Cylidrical Shells via Ritz Method, J. Eg. Mechaics, Vol. 18, No. 9, [8] Poirier, J., Radziszewski, P. (5), Compesatig Stiffess i a Peripherally Heated Rotatig Thi Disk, CANCAM, Motreal. [9] Y. Che, H.B. Zhao ad Z. P. She, Joural of Soud ad Vibratio (1993), Vibratio of High Speed rotatig Shells with Calculatios for Cylidrical Shells

EXPERIMENT OF SIMPLE VIBRATION

EXPERIMENT OF SIMPLE VIBRATION. PURPOSE The purpose of the experimet is to show free vibratio ad damped vibratio o a system havig oe degree of freedom ad to ivestigate the relatioship betwee the basic