dynamic characteristics of a vibrating flip-flow...

Research ArticleDynamic Characteristics of a Vibrating Flip-Flow Screen andAnalysis for Screening 3mm Iron Ore

Chi Yu Xinwen Wang Kunfeng Pang Guofeng Zhao and Wenpeng Sun

School of Chemical and Environmental Engineering China University of Mining and Technology (Beijing) Beijing 100083 China

Correspondence should be addressed to Xinwen Wang xinwenwcumtbeducn

Received 15 December 2019 Revised 29 April 2020 Accepted 4 May 2020 Published 20 May 2020

Academic Editor C M Wang

Copyright copy 2020 Chi Yu et al is is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Deep dry screening is the key unit in mineral processing A vibrating flip-flow screen (VFFS) can provide effective solutionsfor screening fine-grained minerals and it has been extensively used in many industrial fields An accurate dynamic modelof VFFS considering the influence of materials is significant for its dynamic analysis and screening process research but ithas rarely been studied in detail In this paper an improved dynamic model of VFFS is proposed and its dynamic equationsare solved to find the reasonable operating condition and experiments are carried out to verify the reasonability of theproposed model under no-load and loading materials conditions Furthermore the method of multistage sampling andmultilayer screening is also applied to evaluate the screening performance of iron ore at 3 mm cut size on VFFS Results showthat when the mass of materials relative amplitude and operating frequency have values of 107 kg about 6 mm and8079 rads respectively the screening efficiency gradually increases with an increase of screening length reaching 8905however it does not change much when the screening length exceeds 19008 mm Additionally the misplaced materials ofcoarse particles will continue to increase as the screening length increases is provides theoretical and technical supportfor the optimization of the length of the VFFS

1 Introduction

Screening is a vital unit in mineral processing and utili-zation [1 2] In recent years the dry deep screening ofmoist and fine-grained minerals has become increasinglyessential in many industries e dry deep screeningtechnology which can simplify the cleaning process hasbecome the technology of choice for power coal prepa-ration plants [3] In construction of waste industry it alsocan increase the recovery rate of recycled aggregate In theconcentrator plant deep screening of products of a high-pressure roller mill can increase the processing capacity ofthe ball mill due to the finer materials and reduce theconsumption of medium However adjacent moist par-ticles combine to form a covering film on the sieve ap-erture and thus block the aperture which seriouslyreduces the screening performance [4 5] erefore the

vibrating flip-flow screen (VFFS) with elastic sieve matshas been widely used for screening fine-grained mineralsdue to its good performance [6ndash10]

It is difficult to obtain large acceleration for materials onthe ordinary screen therefore the VFFS with double vi-bration principle from a single drive was developed to solvethis problem In order to be ejected the fine particles thatblock the aperture require greater force A single driveprovides two vibration movements and thus the sieve matsare stretched and slackened achieving high accelerationvalues e most striking feature of VFFS is that only 2sim3vibration intensity is needed for the screen frame but themaximum acceleration on the elastic sieve mats can reach50 g thus increasing the service life of the screen machinee dynamic characteristics of VFFS determine the move-ment and spatial distribution of materials during thescreening process which attract considerable attentions

HindawiShock and VibrationVolume 2020 Article ID 1031659 12 pageshttpsdoiorg10115520201031659

Gong et al [11] analyzed the nonlinear characteristics ofVFFS based on the Duffing equation and then discussed theinfluence of the nonlinear stiffness and materials on thesystem response Xiong et al [12] analyzed the dynamics of abanana flip-flow screen with linear springs and proposed ananalytical model for an elastic sieve mat based on catenarytheory Yu et al [13] investigated the influence of severalparameters on the dynamic characteristics of VFFS andproposed a method to adjust its amplitude Zhang et al[14] reported the influence of eccentric mass inclinationand the size composition on the screening efficiency onVFFS by the mean of EDEM simulation Dong et al [15]revealed the complicated influence of aperture shape onthe screening performance of the vibrating screen inwhich the elongation of the rectangular aperture will leadto the increase of the percentage passing especially forlarger particles Jiang et al [16 17] reported that theequal-thickness vibrating screen has better screeningperformance than the normal vibrating screen especiallywhen dealing with a large amount of materials with highmoisture Cleary et al [18 19] investigated the separationperformance of a full industrial-scale double-deck bananascreen for a peak acceleration of 5 g Zhou et al [20]reported that due to the effects of collisions and theresonance the average vibration intensities of the elasticscreen rod and tube were larger than 20 Akbari et al [21]evaluated the dry screening efficiency of the Liwell flip-flow screen at 1mm and 2mm cut sizes Differing from theVFFS the Liwell flip-flow screen is driven by a crank andconnecting rod and its dynamic characteristics are stableduring screening process

Many studies have shown that the screening efficiencyon ordinary screen decreases sharply with the cutting sizebelow 6mm let alone with the 3mm screening [22ndash25] Infact the dynamic response of VFFS interacts with themotion of materials Currently the existing dynamicmodel of VFFS usually did not consider the influence ofmaterials and most of studies were just focused on the-oretical analyses but less on experimental verification Inaddition to improve the screening efficiency the classi-fication performance of VFFS is required to be betterunderstood while it has also rarely been studied in detailat present

In this paper the VFFS was employed for screening3mm iron ore and an improved dynamic model of VFFS isproposed considering the effects of loading materials andverified by analyzing the vibration data obtained from avibration test and analysis unit Furthermore the particledistribution characteristics of various size fractions andscreening performance of different sections on VFFS areinvestigated in the screening process is study providestheoretical and technical support for optimally structuraldesign and industrial applications of VFFS

2 Experimental

21 Materials e raw materials of iron ore used in thisstudy were provided by Heishangou (Shanxi China) withthe total mass of 10700 kg e characteristics of the sample

screening materials are showed in Figure 1 demonstratingthat the dominant size fractions are 25ndash13mm and13ndash6mm with the total contents covering more than 50 ofthe sample e size fractions of 3ndash0mm 6ndash3mm and50ndash25mm take up over 10 of the sample and the corre-sponding yields account for 1176 1889 and 1730respectively It is worth noting that the moisture contents ineach size fraction take up more than 312 and increase asthe particle size decreases and the moisture contents6ndash3mm and 3ndash0mm of samples account for 731 and772 respectively

22 Experimental Test System e experimental test systemconsists of a silo a receiver and the VFFS as shown inFigure 2 e materials of iron ore were fed into the VFFSfrom the silo and the receiver was grouped into fivesections the first four sections to collect the undersizedmaterials and the last one to collect the oversized materialse VFFS consists of the main and floating screen framesrubber shear springs elastic sieve mats support springsand supporting frames with a width and length of 800mmand 2624mm Each elastic sieve mat is 328mm wide soeight pieces mats can be installed in the VFFS e elasticsieve mats have the rectangle array and the shape of thesieve aperture is the straight slot with a length and width of10mm and 3mm respectively e beams of the main andfloating screen frames are arranged alternately and theelastic sieve mats are mounted on two adjacent beamsBesides a vibration test and an analysis unit also are in-cluded in the experimental system as shown in Figure 3which consists of two triaxial acceleration transducers anda multichannel signal acquisition unit with analysis soft-ware and a computer for receiving storing and analyzingthe acceleration signals collected from the measuringpoints

23 Evaluation Since screening is a very complicatedprocess there are always some misplaced materials existingin the oversized and undersized products as shown inFigure 4 e screening efficiency and total misplaced ma-terials were used to evaluate the screening performance inthis paper which is calculated with equations (1) and (2)respectively [26 27]

η Ec + Ef minus 100

Ec co times Oc

Frc

times 100

Ef Fr

f minus co times Of

Frf

times 100

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(1)

Mo Mc + Mf

Mc 100 times cuUc

Mf 100 times coOf

⎧⎪⎪⎨

⎪⎪⎩(2)

2 Shock and Vibration

where η is the screening efficiency () Ec represents theeffective placement efficiency of coarse particles () Ef

represents the effective placement efficiency of fine particles() Mo is the total misplaced materials () Mc is themisplaced materials of coarse particles () Mf is themisplaced materials of fine particles () co is the yield of theoversized product () cu is the yield of the undersizedproduct () Of is the ratio of fine particles in the oversizedproduct () Oc is the ratio of coarse particles in theoversized product () Fr

c is the ratio of coarse particles inthe feeding () and Fr

f is the ratio of fine particles in thefeeding ()

3 Theoretical Analysis of the DynamicCharacteristics of VFFS

Relative movement along the direction of the elastic sievemat will periodically stretch and slacken the mat therebyaffecting the movement of particles on its surface e vi-bration along the vertical elastic sieve mat has little effect[28 29] erefore this paper studies the dynamic responseof the coordinate system with the x-axis along the elasticsieve mat and the dynamic model of VFFS is built as shownin Figure 5 It is necessary to consider the damping effects ofthe rubber shear springs and the support springs but therotation of the VFFS is small and negligible In additionloading materials on VFFS will generate additional mass onthe main and floating screen frames respectively Fur-thermore the materials on the sieve mat will also cause itselastic deformation resulting in an additional stiffness anddamping in the vibration system and these influencescannot be ignored erefore the dynamic equations of the

VFFS are established by analyzing the viscously dampedtwo-degree-of-freedom spring-mass system which could beexpressed as [30]

m1 +Δm2

1113874 1113875 eurox1 + c1x + c2x + Δc( 1113857 _x1 minus c2x + Δc( 1113857 _x2

+ k1x + k2x + Δk( 1113857x1 minus k2x + Δk( 1113857x2

m0ω2r cosωt

(3)

m2 +Δm2

1113874 1113875 eurox2 minus c2x + Δc( 1113857 _x1 + c2x + Δc( 1113857 _x2

minus k2x + Δk( 1113857x1 + k2x + Δk( 1113857x2 0

(4)

where m1 and m2 are the masses of the main and floatingscreen frame respectively (kg) k1x is the stiffness of thesupport springs and k2x is the stiffness of the rubber shearsprings along the x-axis (Nm) and c1x and c2x are theresistance coefficients of the support springs and rubbershear springs along the x-axis respectively (Nsm) m0 is theeccentric mass (kg) and r is the eccentric radius (m) ω is thevibration circular frequency (rads) t is the time (s) x1 _x1and eurox1 are the displacement velocity and acceleration of thecentroid of the main screen frame along the x-axis (m msms2) respectively x2 _x2 and eurox2 are the displacementvelocity and acceleration of the centroid of the floatingscreen frame along the x-axis (m ms ms2) Δm is theadditional mass in the vibration system caused by thematerials which is evenly divided into the mass of the main

40

35

30

25

20

15

10

5

0

Size fraction (mm)

Con

tent

()

3ndash0 6ndash3 13ndash6 25ndash13 50ndash25

YieldMoisture

Figure 1 Materials properties of the samples

Shock and Vibration 3

and the floating screen frame Δk and Δc are the additionalstiffness and damping respectively in the vibration systemFor Δm 0 Δk 0 and Δc 0 this model represents amodel without materials

ey represent a system of two coupled second-orderdifferential equations erefore we can expect that themotion of the mass m1 will influence the motion of the massm2 and vice versa Equations (3) and (4) can be written inmatrix form as

Meurox + C _x + Kx F (5)

where M C and K are called the mass damping andstiffness matrices respectively and are given by

M

m1 +Δm2

0

0 m2 +Δm2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C

c1x + c2x + Δc minus c2x minus Δc

minus c2x minus Δc c2x + Δc

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦

K

k1x + k2x + Δk minus k2x minus Δk

minus k2x minus Δk k2x + Δk

⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦

(6)

Silo

Main frame

Floating frame

Rubber shear springs

Support springs

Drives

Supporting frames

Elastic sieve mats

Receivers

Figure 2 Schematic diagram of VFFS and the screening system

The accelerationtransducers

The frequencyconverter

The vibratingflip-flow screen

The computer andanalysis software

The multichannelsignal acquisition

Figure 3 Schematic diagram of VFFS and the vibration test system


here x and F are called the displacement and force vectorsrespectively and are given by

x Xjeiωt

X1

X2

⎡⎣ ⎤⎦eiωt

j 1 2 (7)

F m0ω2r

01113896 1113897eiωt

(8)

erefore the steady state complex velocity and accel-eration vectors can be written as

_x iωXjeiωt

iωX1

iωX2

⎡⎣ ⎤⎦eiωt

(9)

eurox minus ω2Xje

iωt

minus ω2X1

minus ω2X21113890 1113891eiωt

(10)

Substituting equations (7) (9) and (10) into equation(5) we obtain

X1 m0ω2rc + id

a + id

X2 m0ω2rl + if

a + id

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(11)

where a (k1x + k2x + Δk minus (m1 + Δm2)ω2)(k2x + Δkminus

(m2 + Δm2)ω2) minus (k2x + Δk)2 minus (c1x + c2x + Δc)(c2x + Δc)

ω2 + (c2x + Δc)2ω2 b (k1x + k2x + Δk minus (m1 + Δm2)ω2)

(c2x + Δc)ω + (k2x + Δk minus (m2+ Δm2)ω2)c1xω minus 2(k2x+

Δk)(c2x + Δc)ω and c k2x + Δk minus (m2 + Δm2)ω2d (c2x+ Δc)ω l minus k2x minus Δk f (c2x + Δc)ω

en the actual values of amplitudes X1 and X2 areexpressed respectively as

X1 m0ω2r

c2 + d2

a2 + b2

1113971

X2 m0ω2r

l2 + f2

a2 + b2

1113971

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(12)

e phase angles between two screen frames and theexciting force are written as

ϕ1 arctanbc minus ad

ac + bd

ϕ2 arctanlb minus fa

la + fb

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(13)

en the phase angle between the main and floatingscreen frames is given by

Δϕ ϕ2 minus ϕ1 (14)

e relative amplitude between the main and floatingscreen frames is written as

|X|

X11113868111386811138681113868

11138681113868111386811138682

+ X21113868111386811138681113868

11138681113868111386811138682

minus 2 X11113868111386811138681113868

1113868111386811138681113868 X21113868111386811138681113868

1113868111386811138681113868cos(Δϕ)

1113969

(15)

eparameters of VFFS for this experiment are shown inTable 1

Fcr

FfOfOc

r

Uc

α = 15deg

Flat shales material over6 and less than 13mm

γo

γu

Figure 4 Schematic diagram of screening process on VFFS

y

xO

Support springs

Rubber shearsprings

c1x

m0

m2

m1

c2x

k2x

k1x

22

2 2

∆m2

∆m2

∆c

∆kc2x

k2x

Figure 5 e dynamic model of VFFS


Substituting the parameters in Table 1 into equations (3)and (4) we can obtain the theoretical amplitude of the mainand floating screen frames and the phase angle and therelative amplitude between two frames under no-load andload conditions

It can be observed from Figures 6(a) and 6(b) thatloadingmaterials will change the dynamic response of VFFSIn detail the amplitude of the main screen frame will riseslightly the amplitude of the floating screen frame willdecrease and the phase angle and relative amplitude be-tween the two screen frames also decrease

4 Results and Discussion

41 Vibration Tests and Analyses under No-Load Conditionedesigned centroids of themain and floating screen frameare usually regarded as measuring points respectively be-cause the influence of manufacturing and particles on theposition of the centroids of two frames can be ignored evibration test and an analysis unit were used to collect andanalyze the acceleration signals of two points along the x-axis direction Based on the double integral principle theamplitudes of displacement were obtained from the steadyacceleration signal of the measuring points

Due to the limitation of the structure of the VFFS thehighest vibration frequency point that can be measured isonly 9737 rads at which frequency the amplitude of themain screen frame is 189mm the amplitude of the floatingscreen frame is 1489mm and the relative amplitude is1590mm erefore the experimental dynamic response ofVFFS can be obtained as shown in Figure 7

A comparison of the measured data with theoreticalvalues under no-load condition is illustrated in Figure 8 Ascan be observed the maximum relative errors between themeasured amplitudes of the main screen frame floatingscreen frame and the theoretical values are 5624 and3734 respectively and the corresponding maximumrelative error of the relative amplitude is 6444 Besides themaximum relative error between the measured data of thephase angle and the theoretical value is 12620 e cor-relation between measured amplitudes of the main screenframe floating screen frame relative amplitude and theo-retical values is very strong with the coefficient of deter-mination (R2) being 09988 09982 and 09976 respectivelyIn addition the R2 of measured phase angle and the the-oretical value is 09906 Clearly a few slight differences areobserved between them verifying the reasonability of thedynamic model of the VFFS under no-load condition

42 Analysis of Loading Materials Experiment Because theoperating amplitude of VFFS applied in industries is about 6mm and the frequency is in the range from 7749

to8378 rads [31 32] In this experiment the operatingfrequency of the VFFS was 8079 rads and dynamic re-sponse of the displacement signals in time domain at twomeasuring points is shown in Figure 9 e whole process oftime domain response of the VFFS is divided into five stagesnamely the start stage the steady-state stage the loadingmaterials stage the steady-state stage and the end stageSeveral phenomena need to be noticed in this process In thestart stage and the end stage the amplitude of the main andfloating screen frames and the relative amplitude will rocketwhen the exciting frequency reaches the natural frequency ofthe VFFS which is called ldquoresonancerdquo After the start stagethe VFFS will work in the steady-state stage where theamplitudes maintain basically at a constant value In thestage of loading materials the amplitude of the main screenframe will ascend from 197mm to 224mm and the am-plitude of the floating screen frame will slide from 801mmto 771mm Meanwhile the phase angle decreases from0076 π to 0074 π and the relative amplitude decreasesslightly from 619mm to 562mm when the materials areloaded on the VFFS which indicates that loading materialhas an effect on the stability of amplitude in the system butthe impact is weak and acceptable for practical productionese phenomena are fit with the results obtained from thetheoretical analysis of the dynamics characteristics of VFFS(Figure 6) Furthermore the amplitude will increase to thesteady-state amplitude with the materials decrease

43 Screening Experiments and Analyses e purpose ofstudying the dynamic characteristics of VFFS is to ensure ithas better screening performance in the screening processemethod of multistage sampling andmultilayer screeningwas used to analyze the screening process and classificationperformance of VFFS [33] e undersized materials weredivided into four sections equally along the direction ofmaterials flow Two elastic sieve mats corresponded to onesection so there were four sections Since the width of eachsieve mat was 328mm and the inclination of the VFFS was15 degrees the width of one section in the horizontal di-rection was 6336mm and the screen lengths of sections I IIIII and IV were 0ndash6336mm 6336ndash12672mm12672ndash19008mm and 19008ndash25344mm respectively Inaddition Section V corresponded to the oversized materialse yield and screening percentages of each size fraction indifferent sections and lengths of the VFFS are presented inFigures 10 and 11

Figure 10 illustrates that particles of 3ndash0mm size fractionare the dominant particles in both sections I and II with theyields of 7337 and 6475 respectively and so are par-ticles of 6ndash3mm size fraction in sections III and IV with thecorresponding yields of 6131 and 8000 It is also worthnoticing that the particles of size fraction 13ndash6mm are

Table 1 e parameters of the vibrating flip-flow screen

Symbol m1 m2 m0 r k1x k2x c1x c2x Δm Δk Δc

Unit Kg kg kg mm kNm kNm Nsm Nsm kg kNm NsmValue 916 310 4878 8545 6022 2700 9866 2605 107 270 2605


mostly concentrated in Section V accounting for 9988 ofthis size fraction and only a tiny number of materials be-come undersized products which can be observed in sec-tions I II III and IV e 3D structure of the undersizedparticles of 13ndash6mm is flat shale as shown in Figure 4 emajor size fraction is over 6mm in Section V covering8580 and comparatively 3ndash0mm size fraction in Section Vtakes up a tiny of it with 221 Furthermore covering4061 of 6ndash3mm size fraction particles pass through thescreen apertures and become the undersized products thatis the misplaced materials Figures 10(b) and 11 demon-strate that the size fractions of 50ndash25mm and 25ndash13mm all

enter into Section V and become oversized particlestherefore the screening percentage of these two size frac-tions are all zero in different sections and lengths of VFFS

e 3ndash0mm size fraction particles pass through thescreen apertures and enter in Section I firstly due to the largethickness of the materials layer at the feeding end With thedecrease in the thickness of the materials layer the yield of3ndash0mm size fraction gradually decreases e 6ndash3mm sizefraction particles are relatively difficult to pass through theapertures when the materials layer is thick so the proportionof 6ndash3mm size fraction gradually increases with the thinnessof the materials layer e VFFS is an approximate sieving

40

35

30

25

20

15

10

5

0

Vibration circular frequency (rads)

60

Am

plitu

de (m

m)

0

4

8

12

X

X2

X1

70 80 90

0 40 80 120 160 200 240

No-load conditionLoad condition

(a)

10

08

06

04

02

00

Phas

e ang

le (π

)

70

016

012

008

004

000 75 80 85 90

Vibration circular frequency (rads)0 40 80 120 160 200 240


(b)

Figure 6 eoretical dynamic response of VFFS (a) Amplitude (b) Phase angle

16

14

12

10

8

6

4

2

0


Am

plitu

de (m

m)

0 20 40 60 80 100

Main screen frameFloating screen frameRelative amplitude

(a)

10

08

06

04

02

00

Phas

e ang

le (π

)

Vibration circular frequency (rads)0 20 40 60 80 100

Experimental phase angle

(b)

Figure 7 Experimental dynamic response of VFFS (a) Amplitude (b) Phase angle


form because the sieve apertures have a straight slot shapewith length and width of 10mm and 3mm respectivelyerefore the 6ndash3mm size fraction particles and the13ndash6mm size fraction of flat shale-shaped materials can passthrough the sieve aperture and be observed in the sections III III and IV

Figures 12 and 13 show the screening performance ofdifferent sections and lengths in the VFFS Figure 12 showsthat the sectional finer materials placement efficiency Ef firstincreases and then gradually decreases while there is a littlechange in the coarser materials placement efficiency alongthe direction of materials flow which indicates that the

screening efficiency has a similar change law with Ef emajority of 3ndash0mm size fraction materials firstly passthrough the apertures in sections I and II Furthermore theamount of 3ndash0mm fine particles decreases significantly afterSection II and some of them do not pass through the ap-ertures leading to the decrease of the finer materialsplacement efficiency in the sections III and IV e coarsermaterials placement efficiency Ec decreases and finer ma-terials placement efficiency Ef increases gradually along theflow direction of materials as shown in Figure 13(a)Meanwhile the misplaced materials of fine particles Mf

gradually decrease (Figure 13(b)) In detail Mf successively

16

18

14

12

10

8

6

4

2

0

0 2 4 6 8 10 12 14 16 18Measured amplitude (mm)

Theo

retic

al am

plitu

de (m

m)


(a)

08

06

04

02

0000 02 04 06 08

Theo

retic

al p

hase

angl

e (π)

Measured phase angle (π)


(b)

Figure 8 Comparison of the measured data of (a) amplitudes and the (b) phase angle with the theoretical values

Start Steady-state Steady-state EndLoading materials

20

15

10

5

0

ndash5

ndash10

Time (s)

Disp

lace

men

t (m

m)

0 20 40 60 80 100

Displacement of the main screen frameDisplacement of the floating screen frameRelative displacement

Figure 9 Dynamic response of the displacements of two frames and relative displacement in time domain at the frequency of 8079 rads


subside in the sections I II and III and creep down betweensections III and IV indicating that most of the fine particleshave passed through the apertures before Section III Besidesthe misplaced materials of coarse particles Mc increase withthe increase of screening length and the longer the materialsstay on the screen the easier they pass through the aperturesIt also can be observed that the screening efficiency increasesand the total misplaced materials first decrease and thenincrease with an increase in the screen length is is mainlydue to the reason that the screening process on VFFS is an

approximate screening form with the straight slot of the sieveaperture With the increase of the screening length on VFFSsome of 6ndash3mm size fraction materials lose their ways andbecome the undersized products It is also worth noticing thatthe screening efficiency η shows a little increase from 8808to 8906 however total misplaced materials increase from718 to 796 as the length changes from 19008mm to25344mm erefore the screen surface of the VFFS needsan appropriate length to ensure better screening efficiencyand lower misplaced materials

Yiel

d ac

coun

ted

for s

cree

ning

()

100

90

80

70

60

50

40

30

20

10

0

Section of the screenI II III IV V

50ndash2525ndash1313ndash6

6ndash33ndash0

(a)Yi

eld

acco

unte

d fo

r siz

e fra

ctio

n (

)

100

80

60

40

20

0

Size fraction (mm)50ndash25 25ndash13 13ndash6 6ndash3 3ndash0

IIIIII

IVV

(b)

Figure 10 Distribution of various size fractions in different sections of the VFFS

100

80

010

005

00013ndash6

Size fraction (mm)

60

40

Scre

enin

g pe

rcen

tage

()

Scre

enin

g pe

rcen

tage

()

20

050ndash25 25ndash13 13ndash6

Size fraction (mm)6ndash3 3ndash0

III

IIIIV

(a)

Scre

enin

g pe

rcen

tage

()

Scre

enin

g pe

rcen

tage

()

100

80

020

015

010

005

000

60

40

20


Size fraction (mm)

Size fraction (mm)

6ndash3 3ndash0

13ndash6

6336mm12676mm

19008mm25344mm

(b)

Figure 11 Screening percentages of various size fractions in different areas and length of the VFFS


5 Conclusions

To date the dynamic characteristics of VFFS has beenstudied and analyzed by many scholars however theexisting dynamic model of VFFS usually did not consider theinfluence of materials and there is a little report onscreening experimental research of VFFS

In this paper an improved dynamic model of VFFS isproposed considering the effects of loading materials Forthe validation of this model no-load and loading materialsexperiments on the VFFS were both conducted with results

indicating that the proposed model is capable to describe itsdynamic characteristics in the operating frequency range

Secondly the method of multistage sampling andmultilayer screening was used to analyze the screeningprocess and classification performance of VFFS When themass of materials relative amplitude and operating fre-quency have values of 107 kg about 6mm and 8079 radsrespectively the VFFS has good screening performance inscreening 3mm iron ore with the screening efficiency up to8906 e screening efficiency gradually increases with anincrease of the screening length however it does not change

Effe

ctiv

e pla

cem

ent e

ffici

ency

()

100

80

60

40

20

0I II III

Section of screen (-)IV

EcEfη

(a)M

ispol

aced

mat

eral

()

10

8

6

4

2

0I II III


McMfMo

(b)

Figure 12 Screening performance of different sections of the VFFS

Effe

ctiv

e pla

cem

ent e

ffici

ency

()

100

80

60

40

20

06336 12672 19008

Screen length (mm)25344

EcEfη

(a)

Misp

olac

ed m

ater

al (

)

10

8

6

4

2

0

Screen length (mm)6336 12672 19008 25344

McMfMo

(b)

Figure 13 Screening performance of different lengths in the VFFS (a) Effective placement efficiency (b) Misplaced material


much when the screening length exceeds 19008mm eappropriate screening length is essential to achieve betterscreening performance thereby optimizing the structuraldesign

Yet the screening efficiency can be easily affected byseveral operating factors of VFFS such as frequency am-plitude inclination and feeding rate In future work we willinvestigate the effects of these factors on the screeningprocess and screening efficiency

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

e authors declare that they have no conflicts of interest inthis work

Acknowledgments

e authors would like to thank the TianGong Technologyfor their support in enabling this research e authorsgratefully acknowledge the support from the National In-novation Training Project (C201803813)

References

[1] M F Eskibalci and M F Ozkan ldquoComparison of conven-tional coagulation and electrocoagulation methods for dew-atering of coal preparation plantrdquo Minerals Engineeringvol 122 pp 106ndash112 2018

[2] G Chalavadi R K Singh M Sharma R Singh and A DasldquoDevelopment of a generalized strategy for dry beneficiationof fine coal over a vibrating inclined deckrdquo InternationalJournal of Coal Preparation and Utilization vol 36 no 1pp 10ndash27 2016

[3] P Zhao Y Zhao Z Chen and Z Luo ldquoDry cleaning of finelignite in a vibrated gas-fluidized bed segregation charac-teristicsrdquo Fuel vol 142 pp 274ndash282 2015

[4] X Yu Z Luo H Li et al ldquoEffect of vibration on the separationefficiency of high-sulfur coal in a compound dry separatorrdquoInternational Journal of Mineral Processing vol 157pp 195ndash204 2016

[5] L Peng F Li H Dong C Liu Y Zhao and C DuanldquoCharacteristics analysis of a novel centralized-driving flip-flow screenrdquo International Journal of Mining Science andTechnology vol 24 no 2 pp 195ndash200 2014

[6] H X Zhai ldquoDetermination of the operation range for flip-flow screen in industrial scale based on amplitude-frequencyresponserdquo Journal of China Coal Society (China) vol 32no 7 pp 753ndash756 2007

[7] J W Fernandez P W Cleary M D Sinnott andR D Morrison ldquoUsing SPH one-way coupled to DEM tomodel wet industrial banana screensrdquo Minerals Engineeringvol 24 no 8 pp 741ndash753 2011

[8] C Sheng C Duan Y Zhao P Zhang and L Dong ldquoSep-aration and upgrading of fine lignite in a pulsed fluidized bed2 Experimental study on lignite separation characteristics andimprovement of separation efficiencyrdquo Energy amp Fuelsvol 32 no 1 pp 936ndash953 2018

[9] C Y Zhou X C Fan C L Duan et al ldquoA method to improvefluidization quality in gasndashsolid fluidized bed for fine coalbeneficiationrdquo Particuology vol 43 pp 181ndash192 2019

[10] Y M Zhao C S Liu and C Y Zhang ldquoProgressing of coalscreening theory and equipmentrdquo Coal (China) vol 2pp 15ndash18 2008

[11] S P Gong X W Wang and S Oberst ldquoNon-linear analysisof vibrating flip-flow screensrdquo MATEC Web of conferencesvol 221 p 04007 2018

[12] X Xiong L Niu C Gu and Y Wang ldquoVibration charac-teristics of an inclined flip-flow screen panel in banana flip-flow screensrdquo Journal of Sound and Vibration vol 411pp 108ndash128 2017

[13] C Yu X W Wang H B Wei et al ldquoResearch on debuggingcondition of vibrating flip-flow screen based on amplitude-frequency characteristicsrdquo Journal of Mining Science andTechnology (China) vol 4 no 4 pp 365ndash374 2019

[14] X Zhang B Wu L K Niu et al ldquoDynamic characteristics oftwo-way coupling between flip-flow screen and particlesbased on DEMrdquo Journal of China Coal Society (China)vol 44 no 06 pp 1930ndash1940 2019

[15] K Dong A H Esfandiary and A B Yu ldquoDiscrete particlesimulation of particle flow and separation on a vibratingscreen effect of aperture shaperdquo Powder Technology vol 314pp 195ndash202 2017

[16] H Jiang Y Zhao C Duan et al ldquoDynamic characteristics ofan equal-thickness screen with a variable amplitude andscreening analysisrdquo Powder Technology vol 311 pp 239ndash2462017

[17] H Jiang Y Zhao J Qiao et al ldquoProcess analysis and op-erational parameter optimization of a variable amplitudescreen for coal classificationrdquo Fuel vol 194 pp 329ndash3382017

[18] P W Cleary M D Sinnott and R D Morrison ldquoSeparationperformance of double deck banana screens - Part 1 flow andseparation for different accelerationsrdquo Minerals Engineeringvol 22 no 14 pp 1218ndash1229 2009

[19] P W Cleary M D Sinnott and R D Morrison ldquoSeparationperformance of double deck banana screens - Part 2 quan-titative predictionsrdquo Minerals Engineering vol 22 no 14pp 1230ndash1244 2009

[20] Z Zhou L Huang H Jiang et al ldquoKinematics of elasticscreen surface and elimination mechanism of plugging duringdry deep screening of moist coalrdquo Powder Technologyvol 346 pp 452ndash461 2019

[21] H Akbari L Ackah and M Mohanty ldquoPerformance opti-mization of a new air table and flip-flow screen for fineparticle dry separationrdquo International Journal of Coal Prep-aration and Utilization vol 39 pp 1ndash23 2017

[22] R Weinstein and R J Snoby ldquoAdvances in dry jigging im-proves coal qualityrdquo Mining Engineering vol 59 no 1pp 29ndash34 2007

[23] R Q Honaker M Saracoglu E ompson R BrattonG H Luttrell and V Richardson ldquoUpgrading coal using apneumatic density-based separatorrdquo International Journal ofCoal Preparation and Utilization vol 28 no 1 pp 51ndash672008

[24] B Zhang H Akbari F Yang et al ldquoPerformance optimi-zation of the FGX dry separator for cleaning high-sulfur coalrdquoInternational Journal of Coal Preparation and Utilizationvol 31 no 3-4 pp 161ndash186 2011

[25] H Akbari L A Ackah and M K Mohanty ldquoDevelopment ofa new fine particle dry separatorrdquo Minerals amp MetallurgicalProcessing vol 35 no 2 pp 77ndash86 2018


[26] K Liu ldquoSome factors affecting sieving performance and ef-ficiencyrdquo Powder Technology vol 193 no 2 pp 208ndash2132009

[27] H Jiang Y Zhao C Duan et al ldquoKinematics of variable-amplitude screen and analysis of particle behavior during theprocess of coal screeningrdquo Powder Technology vol 306pp 88ndash95 2017

[28] S P Gong X W Wang C Yu et al ldquoDynamic analysis ofvibrating flip-flow screen based on a nonlinear model of shearspringrdquo Journal of China Coal Society (China) vol 44 no 10pp 3241ndash3249 2019

[29] S P Gong S Oberst and X W Wang ldquoAn experimentallyvalidated rubber shear spring model for vibrating flip-flowscreensrdquo Mechanical Systems and Signal Processing vol 139pp 1ndash15 2020

[30] B C Wen H Zhang S Y Liu et al ldquoVibrating machinerytheoryrdquo Techniques and Applications (in China) 2012

[31] X B Zhang J Y Xie F K Li et al ldquoResearch and applicationof BFS3080RDD flip flow screen to coal preparation plantrdquoCoal Science and Technology vol 44 pp 150ndash154 2016

[32] G F Zheng J B Zhu W D Xia and S L Liu ldquoBanana flip-flow screen benefits coal preparationrdquo Filtration + Separationvol 53 no 4 2016

[33] H Jiang Y Zhao C Duan et al ldquoProperties of technologicalfactors on screening performance of coal in an equal-thick-ness screen with variable amplituderdquo Fuel vol 188pp 511ndash521 2017


Gong et al [11] analyzed the nonlinear characteristics ofVFFS based on the Duffing equation and then discussed theinfluence of the nonlinear stiffness and materials on thesystem response Xiong et al [12] analyzed the dynamics of abanana flip-flow screen with linear springs and proposed ananalytical model for an elastic sieve mat based on catenarytheory Yu et al [13] investigated the influence of severalparameters on the dynamic characteristics of VFFS andproposed a method to adjust its amplitude Zhang et al[14] reported the influence of eccentric mass inclinationand the size composition on the screening efficiency onVFFS by the mean of EDEM simulation Dong et al [15]revealed the complicated influence of aperture shape onthe screening performance of the vibrating screen inwhich the elongation of the rectangular aperture will leadto the increase of the percentage passing especially forlarger particles Jiang et al [16 17] reported that theequal-thickness vibrating screen has better screeningperformance than the normal vibrating screen especiallywhen dealing with a large amount of materials with highmoisture Cleary et al [18 19] investigated the separationperformance of a full industrial-scale double-deck bananascreen for a peak acceleration of 5 g Zhou et al [20]reported that due to the effects of collisions and theresonance the average vibration intensities of the elasticscreen rod and tube were larger than 20 Akbari et al [21]evaluated the dry screening efficiency of the Liwell flip-flow screen at 1mm and 2mm cut sizes Differing from theVFFS the Liwell flip-flow screen is driven by a crank andconnecting rod and its dynamic characteristics are stableduring screening process

Many studies have shown that the screening efficiencyon ordinary screen decreases sharply with the cutting sizebelow 6mm let alone with the 3mm screening [22ndash25] Infact the dynamic response of VFFS interacts with themotion of materials Currently the existing dynamicmodel of VFFS usually did not consider the influence ofmaterials and most of studies were just focused on the-oretical analyses but less on experimental verification Inaddition to improve the screening efficiency the classi-fication performance of VFFS is required to be betterunderstood while it has also rarely been studied in detailat present

In this paper the VFFS was employed for screening3mm iron ore and an improved dynamic model of VFFS isproposed considering the effects of loading materials andverified by analyzing the vibration data obtained from avibration test and analysis unit Furthermore the particledistribution characteristics of various size fractions andscreening performance of different sections on VFFS areinvestigated in the screening process is study providestheoretical and technical support for optimally structuraldesign and industrial applications of VFFS

2 Experimental

21 Materials e raw materials of iron ore used in thisstudy were provided by Heishangou (Shanxi China) withthe total mass of 10700 kg e characteristics of the sample

screening materials are showed in Figure 1 demonstratingthat the dominant size fractions are 25ndash13mm and13ndash6mm with the total contents covering more than 50 ofthe sample e size fractions of 3ndash0mm 6ndash3mm and50ndash25mm take up over 10 of the sample and the corre-sponding yields account for 1176 1889 and 1730respectively It is worth noting that the moisture contents ineach size fraction take up more than 312 and increase asthe particle size decreases and the moisture contents6ndash3mm and 3ndash0mm of samples account for 731 and772 respectively

22 Experimental Test System e experimental test systemconsists of a silo a receiver and the VFFS as shown inFigure 2 e materials of iron ore were fed into the VFFSfrom the silo and the receiver was grouped into fivesections the first four sections to collect the undersizedmaterials and the last one to collect the oversized materialse VFFS consists of the main and floating screen framesrubber shear springs elastic sieve mats support springsand supporting frames with a width and length of 800mmand 2624mm Each elastic sieve mat is 328mm wide soeight pieces mats can be installed in the VFFS e elasticsieve mats have the rectangle array and the shape of thesieve aperture is the straight slot with a length and width of10mm and 3mm respectively e beams of the main andfloating screen frames are arranged alternately and theelastic sieve mats are mounted on two adjacent beamsBesides a vibration test and an analysis unit also are in-cluded in the experimental system as shown in Figure 3which consists of two triaxial acceleration transducers anda multichannel signal acquisition unit with analysis soft-ware and a computer for receiving storing and analyzingthe acceleration signals collected from the measuringpoints

23 Evaluation Since screening is a very complicatedprocess there are always some misplaced materials existingin the oversized and undersized products as shown inFigure 4 e screening efficiency and total misplaced ma-terials were used to evaluate the screening performance inthis paper which is calculated with equations (1) and (2)respectively [26 27]

η Ec + Ef minus 100

Ec co times Oc

Frc

times 100

Ef Fr

f minus co times Of

Frf

times 100

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(1)

Mo Mc + Mf

Mc 100 times cuUc

Mf 100 times coOf

⎧⎪⎪⎨

⎪⎪⎩(2)









m1 +Δm2



m0ω2r cosωt

(3)

m2 +Δm2



(4)


40

35

30

25

20

15

10

5

0

Size fraction (mm)

Con

tent

()


YieldMoisture







M

m1 +Δm2

0

0 m2 +Δm2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C



⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦

K



⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦

(6)

Silo

Main frame

Floating frame


Support springs

Drives

Supporting frames

Elastic sieve mats

Receivers










x Xjeiωt

X1

X2

⎡⎣ ⎤⎦eiωt

j 1 2 (7)

F m0ω2r

01113896 1113897eiωt

(8)


_x iωXjeiωt

iωX1

iωX2

⎡⎣ ⎤⎦eiωt

(9)

eurox minus ω2Xje

iωt

minus ω2X1

minus ω2X21113890 1113891eiωt

(10)


X1 m0ω2rc + id

a + id

X2 m0ω2rl + if

a + id

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(11)







X1 m0ω2r

c2 + d2

a2 + b2

1113971

X2 m0ω2r

l2 + f2

a2 + b2

1113971

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(12)



ac + bd


la + fb

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(13)




|X|

X11113868111386811138681113868

11138681113868111386811138682

+ X21113868111386811138681113868

11138681113868111386811138682

minus 2 X11113868111386811138681113868

1113868111386811138681113868 X21113868111386811138681113868

1113868111386811138681113868cos(Δϕ)

1113969

(15)


Fcr

FfOfOc

r

Uc

α = 15deg


γo

γu


y

xO

Support springs

Rubber shearsprings

c1x

m0

m2

m1

c2x

k2x

k1x

22

2 2

∆m2

∆m2

∆c

∆kc2x

k2x




















40

35

30

25

20

15

10

5

0


60

Am

plitu

de (m

m)

0

4

8

12

X

X2

X1

70 80 90

0 40 80 120 160 200 240


(a)

10

08

06

04

02

00

Phas

e ang

le (π

)

70

016

012

008

004

000 75 80 85 90



(b)


16

14

12

10

8

6

4

2

0


Am

plitu

de (m

m)

0 20 40 60 80 100


(a)

10

08

06

04

02

00

Phas

e ang

le (π

)



(b)







16

18

14

12

10

8

6

4

2

0


Theo

retic

al am

plitu

de (m

m)


(a)

08

06

04

02

0000 02 04 06 08

Theo

retic

al p

hase

angl

e (π)



(b)



20

15

10

5

0

ndash5

ndash10

Time (s)

Disp

lace

men

t (m

m)

0 20 40 60 80 100






Yiel

d ac

coun

ted

for s

cree

ning

()

100

90

80

70

60

50

40

30

20

10

0



6ndash33ndash0

(a)Yi

eld

acco

unte

d fo

r siz

e fra

ctio

n (

)

100

80

60

40

20

0


IIIIII

IVV

(b)


100

80

010

005

00013ndash6

Size fraction (mm)

60

40

Scre

enin

g pe

rcen

tage

()

Scre

enin

g pe

rcen

tage

()

20



III

IIIIV

(a)

Scre

enin

g pe

rcen

tage

()

Scre

enin

g pe

rcen

tage

()

100

80

020

015

010

005

000

60

40

20


Size fraction (mm)

Size fraction (mm)

6ndash3 3ndash0

13ndash6

6336mm12676mm

19008mm25344mm

(b)



5 Conclusions





Effe

ctiv

e pla

cem

ent e

ffici

ency

()

100

80

60

40

20

0I II III


EcEfη

(a)M

ispol

aced

mat

eral

()

10

8

6

4

2

0I II III


McMfMo

(b)


Effe

ctiv

e pla

cem

ent e

ffici

ency

()

100

80

60

40

20

06336 12672 19008


EcEfη

(a)

Misp

olac

ed m

ater

al (

)

10

8

6

4

2

0

Screen length (mm)6336 12672 19008 25344

McMfMo

(b)





Data Availability




Acknowledgments


References











































m1 +Δm2



m0ω2r cosωt

(3)

m2 +Δm2



(4)


40

35

30

25

20

15

10

5

0

Size fraction (mm)

Con

tent

()


YieldMoisture







M

m1 +Δm2

0

0 m2 +Δm2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C



⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦

K



⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦

(6)

Silo

Main frame

Floating frame


Support springs

Drives

Supporting frames

Elastic sieve mats

Receivers










x Xjeiωt

X1

X2

⎡⎣ ⎤⎦eiωt

j 1 2 (7)

F m0ω2r

01113896 1113897eiωt

(8)


_x iωXjeiωt

iωX1

iωX2

⎡⎣ ⎤⎦eiωt

(9)

eurox minus ω2Xje

iωt

minus ω2X1

minus ω2X21113890 1113891eiωt

(10)


X1 m0ω2rc + id

a + id

X2 m0ω2rl + if

a + id

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(11)







X1 m0ω2r

c2 + d2

a2 + b2

1113971

X2 m0ω2r

l2 + f2

a2 + b2

1113971

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(12)



ac + bd


la + fb

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(13)




|X|

X11113868111386811138681113868

11138681113868111386811138682

+ X21113868111386811138681113868

11138681113868111386811138682

minus 2 X11113868111386811138681113868

1113868111386811138681113868 X21113868111386811138681113868

1113868111386811138681113868cos(Δϕ)

1113969

(15)


Fcr

FfOfOc

r

Uc

α = 15deg


γo

γu


y

xO

Support springs

Rubber shearsprings

c1x

m0

m2

m1

c2x

k2x

k1x

22

2 2

∆m2

∆m2

∆c

∆kc2x

k2x




















40

35

30

25

20

15

10

5

0


60

Am

plitu

de (m

m)

0

4

8

12

X

X2

X1

70 80 90

0 40 80 120 160 200 240


(a)

10

08

06

04

02

00

Phas

e ang

le (π

)

70

016

012

008

004

000 75 80 85 90



(b)


16

14

12

10

8

6

4

2

0


Am

plitu

de (m

m)

0 20 40 60 80 100


(a)

10

08

06

04

02

00

Phas

e ang

le (π

)



(b)







16

18

14

12

10

8

6

4

2

0


Theo

retic

al am

plitu

de (m

m)


(a)

08

06

04

02

0000 02 04 06 08

Theo

retic

al p

hase

angl

e (π)



(b)



20

15

10

5

0

ndash5

ndash10

Time (s)

Disp

lace

men

t (m

m)

0 20 40 60 80 100






Yiel

d ac

coun

ted

for s

cree

ning

()

100

90

80

70

60

50

40

30

20

10

0



6ndash33ndash0

(a)Yi

eld

acco

unte

d fo

r siz

e fra

ctio

n (

)

100

80

60

40

20

0


IIIIII

IVV

(b)


100

80

010

005

00013ndash6

Size fraction (mm)

60

40

Scre

enin

g pe

rcen

tage

()

Scre

enin

g pe

rcen

tage

()

20



III

IIIIV

(a)

Scre

enin

g pe

rcen

tage

()

Scre

enin

g pe

rcen

tage

()

100

80

020

015

010

005

000

60

40

20


Size fraction (mm)

Size fraction (mm)

6ndash3 3ndash0

13ndash6

6336mm12676mm

19008mm25344mm

(b)



5 Conclusions





Effe

ctiv

e pla

cem

ent e

ffici

ency

()

100

80

60

40

20

0I II III


EcEfη

(a)M

ispol

aced

mat

eral

()

10

8

6

4

2

0I II III


McMfMo

(b)


Effe

ctiv

e pla

cem

ent e

ffici

ency

()

100

80

60

40

20

06336 12672 19008


EcEfη

(a)

Misp

olac

ed m

ater

al (

)

10

8

6

4

2

0

Screen length (mm)6336 12672 19008 25344

McMfMo

(b)





Data Availability




Acknowledgments


References








































M

m1 +Δm2

0

0 m2 +Δm2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

C



⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦

K



⎡⎢⎢⎢⎢⎢⎣⎤⎥⎥⎥⎥⎥⎦

(6)

Silo

Main frame

Floating frame


Support springs

Drives

Supporting frames

Elastic sieve mats

Receivers










x Xjeiωt

X1

X2

⎡⎣ ⎤⎦eiωt

j 1 2 (7)

F m0ω2r

01113896 1113897eiωt

(8)


_x iωXjeiωt

iωX1

iωX2

⎡⎣ ⎤⎦eiωt

(9)

eurox minus ω2Xje

iωt

minus ω2X1

minus ω2X21113890 1113891eiωt

(10)


X1 m0ω2rc + id

a + id

X2 m0ω2rl + if

a + id

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(11)







X1 m0ω2r

c2 + d2

a2 + b2

1113971

X2 m0ω2r

l2 + f2

a2 + b2

1113971

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(12)



ac + bd


la + fb

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(13)




|X|

X11113868111386811138681113868

11138681113868111386811138682

+ X21113868111386811138681113868

11138681113868111386811138682

minus 2 X11113868111386811138681113868

1113868111386811138681113868 X21113868111386811138681113868

1113868111386811138681113868cos(Δϕ)

1113969

(15)


Fcr

FfOfOc

r

Uc

α = 15deg


γo

γu


y

xO

Support springs

Rubber shearsprings

c1x

m0

m2

m1

c2x

k2x

k1x

22

2 2

∆m2

∆m2

∆c

∆kc2x

k2x




















40

35

30

25

20

15

10

5

0


60

Am

plitu

de (m

m)

0

4

8

12

X

X2

X1

70 80 90

0 40 80 120 160 200 240


(a)

10

08

06

04

02

00

Phas

e ang

le (π

)

70

016

012

008

004

000 75 80 85 90



(b)


16

14

12

10

8

6

4

2

0


Am

plitu

de (m

m)

0 20 40 60 80 100


(a)

10

08

06

04

02

00

Phas

e ang

le (π

)



(b)







16

18

14

12

10

8

6

4

2

0


Theo

retic

al am

plitu

de (m

m)


(a)

08

06

04

02

0000 02 04 06 08

Theo

retic

al p

hase

angl

e (π)



(b)



20

15

10

5

0

ndash5

ndash10

Time (s)

Disp

lace

men

t (m

m)

0 20 40 60 80 100






Yiel

d ac

coun

ted

for s

cree

ning

()

100

90

80

70

60

50

40

30

20

10

0



6ndash33ndash0

(a)Yi

eld

acco

unte

d fo

r siz

e fra

ctio

n (

)

100

80

60

40

20

0


IIIIII

IVV

(b)


100

80

010

005

00013ndash6

Size fraction (mm)

60

40

Scre

enin

g pe

rcen

tage

()

Scre

enin

g pe

rcen

tage

()

20



III

IIIIV

(a)

Scre

enin

g pe

rcen

tage

()

Scre

enin

g pe

rcen

tage

()

100

80

020

015

010

005

000

60

40

20


Size fraction (mm)

Size fraction (mm)

6ndash3 3ndash0

13ndash6

6336mm12676mm

19008mm25344mm

(b)



5 Conclusions





Effe

ctiv

e pla

cem

ent e

ffici

ency

()

100

80

60

40

20

0I II III


EcEfη

(a)M

ispol

aced

mat

eral

()

10

8

6

4

2

0I II III


McMfMo

(b)


Effe

ctiv

e pla

cem

ent e

ffici

ency

()

100

80

60

40

20

06336 12672 19008


EcEfη

(a)

Misp

olac

ed m

ater

al (

)

10

8

6

4

2

0

Screen length (mm)6336 12672 19008 25344

McMfMo

(b)





Data Availability




Acknowledgments


References





































x Xjeiωt

X1

X2

⎡⎣ ⎤⎦eiωt

j 1 2 (7)

F m0ω2r

01113896 1113897eiωt

(8)


_x iωXjeiωt

iωX1

iωX2

⎡⎣ ⎤⎦eiωt

(9)

eurox minus ω2Xje

iωt

minus ω2X1

minus ω2X21113890 1113891eiωt

(10)


X1 m0ω2rc + id

a + id

X2 m0ω2rl + if

a + id

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(11)







X1 m0ω2r

c2 + d2

a2 + b2

1113971

X2 m0ω2r

l2 + f2

a2 + b2

1113971

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎩

(12)



ac + bd


la + fb

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(13)




|X|

X11113868111386811138681113868

11138681113868111386811138682

+ X21113868111386811138681113868

11138681113868111386811138682

minus 2 X11113868111386811138681113868

1113868111386811138681113868 X21113868111386811138681113868

1113868111386811138681113868cos(Δϕ)

1113969

(15)


Fcr

FfOfOc

r

Uc

α = 15deg


γo

γu


y

xO

Support springs

Rubber shearsprings

c1x

m0

m2

m1

c2x

k2x

k1x

22

2 2

∆m2

∆m2

∆c

∆kc2x

k2x




















40

35

30

25

20

15

10

5

0


60

Am

plitu

de (m

m)

0

4

8

12

X

X2

X1

70 80 90

0 40 80 120 160 200 240


(a)

10

08

06

04

02

00

Phas

e ang

le (π

)

70

016

012

008

004

000 75 80 85 90



(b)


16

14

12

10

8

6

4

2

0


Am

plitu

de (m

m)

0 20 40 60 80 100


(a)

10

08

06

04

02

00

Phas

e ang

le (π

)



(b)







16

18

14

12

10

8

6

4

2

0


Theo

retic

al am

plitu

de (m

m)


(a)

08

06

04

02

0000 02 04 06 08

Theo

retic

al p

hase

angl

e (π)



(b)



20

15

10

5

0

ndash5

ndash10

Time (s)

Disp

lace

men

t (m

m)

0 20 40 60 80 100






Yiel

d ac

coun

ted

for s

cree

ning

()

100

90

80

70

60

50

40

30

20

10

0



6ndash33ndash0

(a)Yi

eld

acco

unte

d fo

r siz

e fra

ctio

n (

)

100

80

60

40

20

0


IIIIII

IVV

(b)


100

80

010

005

00013ndash6

Size fraction (mm)

60

40

Scre

enin

g pe

rcen

tage

()

Scre

enin

g pe

rcen

tage

()

20



III

IIIIV

(a)

Scre

enin

g pe

rcen

tage

()

Scre

enin

g pe

rcen

tage

()

100

80

020

015

010

005

000

60

40

20


Size fraction (mm)

Size fraction (mm)

6ndash3 3ndash0

13ndash6

6336mm12676mm

19008mm25344mm

(b)



5 Conclusions





Effe

ctiv

e pla

cem

ent e

ffici

ency

()

100

80

60

40

20

0I II III


EcEfη

(a)M

ispol

aced

mat

eral

()

10

8

6

4

2

0I II III


McMfMo

(b)


Effe

ctiv

e pla

cem

ent e

ffici

ency

()

100

80

60

40

20

06336 12672 19008


EcEfη

(a)

Misp

olac

ed m

ater

al (

)

10

8

6

4

2

0

Screen length (mm)6336 12672 19008 25344

McMfMo

(b)





Data Availability




Acknowledgments


References





















































40

35

30

25

20

15

10

5

0


60

Am

plitu

de (m

m)

0

4

8

12

X

X2

X1

70 80 90

0 40 80 120 160 200 240


(a)

10

08

06

04

02

00

Phas

e ang

le (π

)

70

016

012

008

004

000 75 80 85 90



(b)


16

14

12

10

8

6

4

2

0


Am

plitu

de (m

m)

0 20 40 60 80 100


(a)

10

08

06

04

02

00

Phas

e ang

le (π

)



(b)







16

18

14

12

10

8

6

4

2

0


Theo

retic

al am

plitu

de (m

m)


(a)

08

06

04

02

0000 02 04 06 08

Theo

retic

al p

hase

angl

e (π)



(b)



20

15

10

5

0

ndash5

ndash10

Time (s)

Disp

lace

men

t (m

m)

0 20 40 60 80 100






Yiel

d ac

coun

ted

for s

cree

ning

()

100

90

80

70

60

50

40

30

20

10

0



6ndash33ndash0

(a)Yi

eld

acco

unte

d fo

r siz

e fra

ctio

n (

)

100

80

60

40

20

0


IIIIII

IVV

(b)


100

80

010

005

00013ndash6

Size fraction (mm)

60

40

Scre

enin

g pe

rcen

tage

()

Scre

enin

g pe

rcen

tage

()

20



III

IIIIV

(a)

Scre

enin

g pe

rcen

tage

()

Scre

enin

g pe

rcen

tage

()

100

80

020

015

010

005

000

60

40

20


Size fraction (mm)

Size fraction (mm)

6ndash3 3ndash0

13ndash6

6336mm12676mm

19008mm25344mm

(b)



5 Conclusions





Effe

ctiv

e pla

cem

ent e

ffici

ency

()

100

80

60

40

20

0I II III


EcEfη

(a)M

ispol

aced

mat

eral

()

10

8

6

4

2

0I II III


McMfMo

(b)


Effe

ctiv

e pla

cem

ent e

ffici

ency

()

100

80

60

40

20

06336 12672 19008


EcEfη

(a)

Misp

olac

ed m

ater

al (

)

10

8

6

4

2

0

Screen length (mm)6336 12672 19008 25344

McMfMo

(b)





Data Availability




Acknowledgments


References








































16

18

14

12

10

8

6

4

2

0


Theo

retic

al am

plitu

de (m

m)


(a)

08

06

04

02

0000 02 04 06 08

Theo

retic

al p

hase

angl

e (π)



(b)



20

15

10

5

0

ndash5

ndash10

Time (s)

Disp

lace

men

t (m

m)

0 20 40 60 80 100






Yiel

d ac

coun

ted

for s

cree

ning

()

100

90

80

70

60

50

40

30

20

10

0



6ndash33ndash0

(a)Yi

eld

acco

unte

d fo

r siz

e fra

ctio

n (

)

100

80

60

40

20

0


IIIIII

IVV

(b)


100

80

010

005

00013ndash6

Size fraction (mm)

60

40

Scre

enin

g pe

rcen

tage

()

Scre

enin

g pe

rcen

tage

()

20



III

IIIIV

(a)

Scre

enin

g pe

rcen

tage

()

Scre

enin

g pe

rcen

tage

()

100

80

020

015

010

005

000

60

40

20


Size fraction (mm)

Size fraction (mm)

6ndash3 3ndash0

13ndash6

6336mm12676mm

19008mm25344mm

(b)



5 Conclusions





Effe

ctiv

e pla

cem

ent e

ffici

ency

()

100

80

60

40

20

0I II III


EcEfη

(a)M

ispol

aced

mat

eral

()

10

8

6

4

2

0I II III


McMfMo

(b)


Effe

ctiv

e pla

cem

ent e

ffici

ency

()

100

80

60

40

20

06336 12672 19008


EcEfη

(a)

Misp

olac

ed m

ater

al (

)

10

8

6

4

2

0

Screen length (mm)6336 12672 19008 25344

McMfMo

(b)





Data Availability




Acknowledgments


References






































Yiel

d ac

coun

ted

for s

cree

ning

()

100

90

80

70

60

50

40

30

20

10

0



6ndash33ndash0

(a)Yi

eld

acco

unte

d fo

r siz

e fra

ctio

n (

)

100

80

60

40

20

0


IIIIII

IVV

(b)


100

80

010

005

00013ndash6

Size fraction (mm)

60

40

Scre

enin

g pe

rcen

tage

()

Scre

enin

g pe

rcen

tage

()

20



III

IIIIV

(a)

Scre

enin

g pe

rcen

tage

()

Scre

enin

g pe

rcen

tage

()

100

80

020

015

010

005

000

60

40

20


Size fraction (mm)

Size fraction (mm)

6ndash3 3ndash0

13ndash6

6336mm12676mm

19008mm25344mm

(b)



5 Conclusions





Effe

ctiv

e pla

cem

ent e

ffici

ency

()

100

80

60

40

20

0I II III


EcEfη

(a)M

ispol

aced

mat

eral

()

10

8

6

4

2

0I II III


McMfMo

(b)


Effe

ctiv

e pla

cem

ent e

ffici

ency

()

100

80

60

40

20

06336 12672 19008


EcEfη

(a)

Misp

olac

ed m

ater

al (

)

10

8

6

4

2

0

Screen length (mm)6336 12672 19008 25344

McMfMo

(b)





Data Availability




Acknowledgments


References




































5 Conclusions





Effe

ctiv

e pla

cem

ent e

ffici

ency

()

100

80

60

40

20

0I II III


EcEfη

(a)M

ispol

aced

mat

eral

()

10

8

6

4

2

0I II III


McMfMo

(b)


Effe

ctiv

e pla

cem

ent e

ffici

ency

()

100

80

60

40

20

06336 12672 19008


EcEfη

(a)

Misp

olac

ed m

ater

al (

)

10

8

6

4

2

0

Screen length (mm)6336 12672 19008 25344

McMfMo

(b)





Data Availability




Acknowledgments


References






































Data Availability




Acknowledgments


References




































dynamic characteristics of a vibrating flip-flow...

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