research article nonlinear model of vibrating screen to...

Research ArticleNonlinear Model of Vibrating Screen to Determine PermissibleSpring Deterioration for Proper Separation

Cristian G. Rodriguez, Manuel A. Moncada, Emilio E. Dufeu, and Mario I. Razeto

Department of Mechanical Engineering, University of Concepcion, Edmundo Larenas 219, 4070409 Concepcion, Chile

Correspondence should be addressed to Cristian G. Rodriguez; [email protected]

Received 4 March 2016; Revised 2 August 2016; Accepted 8 August 2016

Academic Editor: Samuel da Silva

Copyright © 2016 Cristian G. Rodriguez et al. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

Springs of vibrating screens are prone to fatigue induced failure because they operate in a heavy duty environment, with abrasivedust and under heavy cyclic loads. If a spring breaks, the stiffness at supporting positions changes, and therefore the amplitudeof motion and the static and dynamic angular inclination of deck motion also change. This change in the amplitude and in theinclination ofmotion produces a reduction in separation efficiency. Availablemodels are useful to determinemotion under nominaloperating conditions when angular displacement is not significant. However in practice there is significant angular motion duringstartup, during shutdown, or under off-design operating conditions. In this article, a two-dimensional three-degree-of-freedomnonlinear model that considers significant angular motion and damping is developed. The proposed model allows the predictionof vibrating screen behavior when there is a reduction in spring stiffness. Making use of this model for an actual vibrating screen inoperation in industry has permitted determining a limit for spring’s failure before separation efficiency is affected.This informationis of practical value for operation and maintenance staff helping to determine whether or not it is necessary to change springs, andhence optimizing stoppage time.

1. Introduction

Vibrating screens are important machines used to separategranulated ore materials based on particle size. In copperindustry, themost used vibrating screens are of linear motionand with horizontal, sloped, or multisloped (banana) screen.Most of vibrating screens in copper industry are supportedat four positions of the vibrating screen deck, and at eachof these supporting positions there are two or more steelcoil springs (depending on vibrating screen size there maybe more supporting positions and springs). The springs ofvibrating screens are prone to fail due to fatigue becausethey operate continuously in a heavy duty environment, withabrasive dust and under heavy cyclic loads.

If a spring in one supporting position breaks, then thestiffness of the supporting position decreases. If the stiffnessdecreases, then the amplitude of motion and the static anddynamic angular inclination of deck motion are expected tochange in comparison to those expected at design operatingconditions. A change in amplitude motion of 15% could

produce a loss of separation efficiency of 5% [1]. A changein the angular inclination of deck motion produces a changein separation efficiency; for example, there are cases wherethe efficiency is 86% at design operating conditions and fora change of 2∘ in angle the separation efficiency drops to55% [2, 3]. Xiao and Tong [4] show a drop of separationefficiency from 55% to 35% with a change of only 1∘. Theexact loss of efficiency due to amplitude or inclinationdeviation depends on the specific vibrating screen and theore material characteristics. In order to ensure separationefficiency, the vibrating screen vendor provides a range ofadmissible amplitude and angle deviation for deck motion tooperate. The proper amplitude and angle are checked duringcommissioning making measurements with accelerometerslocated at particular positions of vibrating screen deck.

In order to assure operation under design conditions (ata high separation efficiency), the evaluation of amplitudeand angle of inclination due to spring condition wouldallow limiting spring deterioration. To be able to determineamplitude and angle of inclination due to spring condition

Hindawi Publishing CorporationShock and VibrationVolume 2016, Article ID 4028583, 7 pageshttp://dx.doi.org/10.1155/2016/4028583

2 Shock and Vibration

from vibration measurements, it is necessary to have a modelable to predict how the deck will move when all springs areworking at design conditions andhow itwillmovewhen thereis a loss of stiffness due to deteriorated springs.

Models of vibrating screens in literature are focused onseparation [4–10], particles motion [11, 12], and failures indeck structure [13, 14]. Regarding vibrating screen motion,He and Liu [15] developed a three-degree-of-freedom linearmodel of a vibrating screen supported at two positions withequal stiffness for a circular motion vibrating screen. Thismodel assumes stationary motion and a phase relationshipbetween force and vibration of 0∘ in horizontal and verticalmotion. With this assumption, a phase relationship of 90∘between horizontal and vertical displacement is imposed, andthe movement of vibrating screen center of mass results in anellipse or a circle with a vertical or horizontal principal axiswith no inclination. Liu et al. [16] developed a linear modelof a vibrating screen supported in four different positionswith different stiffness under a vertical force. This model didnot consider the possibility of lateral motion of vibratingscreen deck. They simulate supporting positions with looseof stiffness but because of model formulation they obtain achange in amplitude of the vertical displacement of the centerof mass but movement could not change its inclination. Liuet al. [17] considered a linear model similar to He and Liu[15] but with a quadruple exciter mechanism. He and Liu[15], Liu et al. [16], and Liu et al. [17] consider no damping intheir models. L. I. Slepyan and V. I. Slepyan [18] considereda linear model with damping and a tensile force for a linearmotion vibrating screen with equal stiffness in all supportingpositions. These linear models assume that angular motionof vibrating screen deck is low (sin 𝜃 ≈ 𝜃 and cos 𝜃 ≈

1) and are useful for determining motion under nominaloperating conditions because angular displacement is notsignificant but are not able to predict deck motion duringthe startup, during shutdown, or under off-design operatingconditions such as the loss of stiffness in supporting positions.In practice, there is significant angular motion that occurs invibrating screens during startup, during shutdown, andunderoff-design operating conditions.

In this article, a two-dimensional three-degree-of-freedom nonlinear model that considers significant angularmotion and damping is developed in order to predictand evaluate the vibrating screen motion during startup,during shutdown, and under stationary operation conditionswith deteriorated springs. The model simulates differentdeterioration levels of springs in order to determine its effecton amplitude and inclination of deck motion.The simulationis performed considering empty screen and full load. Emptyscreen is simulated because, after commissioning, thevibrating screen is tested empty in order to verify adequateamplitude and angle of inclination of motion. Full load issimulated because it is the normal operating condition andit is also the condition where springs operate under higherdynamic forces.With this simulation, it is possible to predict arange for acceptable spring deterioration in order tomaintainproper motion to achieve an adequate separation efficiency.Finally, the model is compared with field measurementstaken from a vibrating screen in use at a copper mine.

ExciterDeck

SpringsCenter of mass

A

BRear

Front

Figure 1: Vibrating screen.

Rear

Front

y

x

a

z

b

f 𝛼

d

c

A

B

𝛽

M, IkBy, cBy

kBx, cBx

kAy, cAy

kAx, cAx

Figure 2: Model geometry.

2. The Model

In linear motion vibrating screens, motion is obtained by theaction of a linear dynamic force produced by one or moreexciter mechanisms. The linear force in the exciter mecha-nism is obtained by pairs of unbalanced masses mounted inpairs of different shafts that rotate at the same velocity but indifferent directions. Figure 1 shows a schematic of a vibratingscreen with the exciter mechanism at the top of the deck.

The deck is supported by springs. Figure 2 shows themodel of vibrating screen considered in this study, where 𝑀

is the mass of the vibrating screen when in operation; 𝐼 isthe moment of inertia of the center of mass of the vibratingscreen with respect to 𝑧-axis; 𝑘A𝑥, 𝑘A𝑦, 𝑘B𝑥, and 𝑘B𝑦 are thestiffness of springs in horizontal and vertical directions infront and rear positions, respectively; 𝑐A𝑥, 𝑐A𝑦, 𝑐B𝑥, and 𝑐B𝑦 arethe damping of springs in horizontal and vertical directionsin front and rear positions, respectively; 𝑓 is the force dueto the rotation of the unbalanced masses which is equal to𝑛𝑚𝜔2

𝑟 sin𝜔𝑡, where 𝑛 is the number of unbalancedmasses,𝑚the unbalancedmass, 𝜔 the rotating speed, and 𝑟 the distancefrom the axis of rotation to the unbalance mass. The lineAB is the line that connects the middle point of supportingposition at the front (A) with the middle point of supportingposition at the rear (B); 𝑎 and 𝑏 are the distances measuredon the line AB between the front supporting position andthe center of mass and between the center of mass and therear supporting position, respectively; 𝑐 is the distance fromthe center of mass to the line AB; 𝑑 is the distance from thecenter of mass to the point that connects the projection of theline that defines the distance from the center of mass to theline AB, with the projection of the line where the excitationforce 𝑓 is acting; 𝛼 is the angle of inclination of the force inrelation to line AB; and 𝛽 is the angle of inclination of line

Shock and Vibration 3

Rear

Front

y

xz

f 𝛼

ABM, I

fBy

fBx

fAy

fAx𝛽 + 𝜃

yc.m.

xc.m.

Figure 3: Model forces.

AB. To determine the equations of motion, it is assumed thatthe vibrating screen deck center of mass moves a distance𝑥c.m. in the horizontal direction and a distance 𝑦c.m. in thevertical direction and that the screen rotates an angle 𝜃 (seeFigure 3). With these considerations, the displacements 𝑥A,𝑦A, 𝑥B, and 𝑦B at supporting positions are obtained as follows(see Figure 3)

𝑥A = 𝑥 + cos 𝜃 [−𝑎 cos𝛽 + 𝑐 sin𝛽]

+ sin𝛽 [𝑎 sin𝛽 + 𝑐 cos𝛽] + 𝑎 cos𝛽 − 𝑐 sin𝛽

𝑦A = 𝑦 + cos 𝜃 [−𝑐 cos𝛽 − 𝑎 sin𝛽]

+ sin 𝜃 [𝑐 sin𝛽 − 𝑎 cos𝛽] + 𝑐 cos𝛽 + 𝑎 sin𝛽

𝑥B = 𝑥 + cos 𝜃 [𝑐 sin𝛽 + 𝑏 cos𝛽]

+ sin 𝜃 [𝑐 cos𝛽 − 𝑏 sin𝛽] − 𝑐 sin𝛽 − 𝑏 cos𝛽

𝑦B = 𝑦 + cos 𝜃 [−𝑐 cos𝛽 + 𝑏 sin𝛽]

+ sin 𝜃 [𝑐 sin𝛽 + 𝑏 cos𝛽] + 𝑐 cos𝛽 − 𝑏 sin𝛽.

(1)

And consequently, the velocity of these positions is obtainedas follows:

��A = �� − �� sin 𝜃 [−𝑎 cos𝛽 + 𝑐 sin𝛽]

+ �� cos 𝜃 [𝑎 sin𝛽 + 𝑐 cos𝛽]

��A = �� − �� sin 𝜃 [−𝑐 cos𝛽 − 𝑎 sin𝛽]

+ �� cos 𝜃 [𝑐 sin𝛽 − 𝑎 cos𝛽]

��B = �� − �� sin 𝜃 [𝑐 sin𝛽 + 𝑏 cos𝛽]

+ �� cos 𝜃 [𝑐 cos𝛽 − 𝑏 sin𝛽]

��B = �� − �� sin 𝜃 [−𝑐 cos𝛽 + 𝑏 sin𝛽]

+ �� cos 𝜃 [𝑐 sin𝛽 + 𝑏 cos𝛽] .

(2)

In linear models, the terms cos 𝜃 and sin 𝜃 are linearized,which is accurate when angular displacements are notsignificant. Nevertheless, these angular displacements aresignificant when the vibrating screen is operating duringstartup, during shutdown, or with deteriorated springs. To

be able to predict the behavior under significant angulardisplacements, the equations of motion are as follows:

𝑀�� + �� [𝑐A𝑥 + 𝑐B𝑥] − �� sin 𝜃 [𝑐A𝑥 (−𝑎 cos𝛽 + 𝑐 sin𝛽)

+ 𝑐B𝑥 (𝑐 sin𝛽 + 𝑏 cos𝛽)] + ��

⋅ cos 𝜃 [𝑐A𝑥 (𝑎 sin𝛽 + 𝑐 cos𝛽)

+ 𝑐B𝑥 (𝑐 cos𝛽 − 𝑏 sin𝛽)] + 𝑥 [𝑘A𝑥 + 𝑘B𝑥]

+ cos 𝜃 [𝑘A𝑥 (𝑐 sin𝛽 − 𝑎 cos𝛽)

+ 𝑘B𝑥 (𝑐 sin𝛽 − 𝑏 cos𝛽)]

+ sin 𝜃 [𝑘A𝑥 (𝑐 cos𝛽 + 𝑎 sin𝛽)

+ 𝑘B𝑥 (𝑐 cos𝛽 − 𝑏 sin𝛽)] + 𝑘A𝑥 (𝑎 cos𝛽 − 𝑐 sin𝛽)

+ 𝑘B𝑥 (−𝑐 sin𝛽 − 𝑏 sin𝛽) = 𝐹

⋅ sin𝜔𝑡 [cos 𝜃 cos (𝛼 + 𝛽) − sin 𝜃 sin (𝛼 + 𝛽)]

𝑀�� + �� [𝑐A𝑦 + 𝑐B𝑦] − �� sin 𝜃 [𝑐A𝑦 (−𝑐 cos𝛽 − 𝑎 sin𝛽)

+ 𝑐B𝑦 (−𝑐 cos𝛽 + 𝑏 sin𝛽)] + �� cos 𝜃 [𝑐A𝑦 (𝑐 sin𝛽

− 𝑎 cos𝛽) + 𝑐B𝑦 (𝑐 sin𝛽 − 𝑏 cos𝛽)] + 𝑦 [𝑘A𝑦 + 𝑘B𝑦]

+ cos 𝜃 [𝑘A𝑦 (−𝑐 cos𝛽 − 𝑎 sin𝛽) + 𝑘B𝑦 (−𝑐 cos𝛽

+ 𝑏 sin𝛽)] + sin 𝜃 [𝑘A𝑦 (𝑐 sin𝛽 − 𝑎 cos𝛽)

+ 𝑘B𝑦 (𝑐 sin𝛽 + 𝑏 cos𝛽)] + 𝑘A𝑦 (𝑐 cos𝛽 + 𝑎 sin𝛽)

+ 𝑘B𝑦 (𝑐 cos𝛽 − 𝑏 sin𝛽) = 𝐹 sin𝜔𝑡 [cos 𝜃 sin (𝛼 + 𝛽)

+ sin 𝜃 cos (𝛼 + 𝛽)]

𝐼�� + �� sin 𝜃 cos 𝜃

⋅ {𝑐A𝑦 [(𝑐 sin𝛽 − 𝑎 cos𝛽) (−𝑎 sin𝛽 − 𝑐 cos𝛽)

+ (−𝑐 cos𝛽 − 𝑎 sin𝛽) (𝑎 cos𝛽 − 𝑐 sin𝛽)]

− 𝑐B𝑦 [(𝑐 sin𝛽 + 𝑏 cos𝛽) (−𝑐 cos𝛽 − 𝑏 sin𝛽)

− (𝑐 cos𝛽 + 𝑏 sin𝛽) (−𝑐 sin𝛽 + 𝑏 cos𝛽)]

− 𝑐A𝑥 [(𝑎 sin𝛽 + 𝑐 cos𝛽) (𝑐 sin𝛽 − 𝑎 cos𝛽)

− (𝑎 cos𝛽 + 𝑐 sin𝛽) (𝑐 cos𝛽 + 𝑎 sin𝛽)]

+ 𝑐B𝑥 [(𝑐 cos𝛽 − 𝑏 sin𝛽) (𝑐 sin𝛽 + 𝑏 cos𝛽)

+ (𝑐 sin𝛽 − 𝑏 cos𝛽) (−𝑐 cos𝛽 + 𝑏 sin𝛽)]} + ��

⋅ cos2𝜃 {𝑐A𝑦 [𝑐 sin𝛽 − 𝑎 cos𝛽] [𝑎 cos𝛽 − 𝑐 sin𝛽]

− 𝑐B𝑦 [𝑐 sin𝛽 + 𝑏 cos𝛽] [−𝑐 sin𝛽 + 𝑏 cos𝛽]

− 𝑐A𝑥 [𝑐 sin𝛽 + 𝑎 cos𝛽] [𝑐 cos𝛽 + 𝑎 sin𝛽]

+ 𝑐B𝑥 [𝑐 cos𝛽 − 𝑏 sin𝛽] [−𝑐 cos𝛽 + 𝑏 sin𝛽]} + ��


⋅ sin2𝜃 {𝑐A𝑦 [−𝑐 cos𝛽 − 𝑎 sin𝛽] [−𝑎 sin𝛽 − 𝑐 cos𝛽]

− 𝑐B𝑦 [−𝑐 cos𝛽 + 𝑏 sin𝛽] [−𝑐 cos𝛽 − 𝑏 sin𝛽]

− 𝑐A𝑥 [−𝑎 cos𝛽 + 𝑐 sin𝛽] [𝑐 sin𝛽 − 𝑎 cos𝛽]

+ 𝑐B𝑥 [𝑐 sin𝛽 + 𝑏 cos𝛽] [𝑐 sin𝛽 + 𝑏 cos𝛽]} − ��

⋅ cos 𝜃 [−𝑐A𝑥 (𝑐 cos𝛽 + 𝑎 sin𝛽)

+ 𝑐B𝑥 (−𝑐 cos𝛽 + 𝑏 cos𝛽)] − ��

⋅ sin 𝜃 [−𝑐A𝑥 (𝑐 sin𝛽 − 𝑎 cos𝛽)

+ 𝑐B𝑥 (𝑐 sin𝛽 + 𝑏 cos𝛽)] − ��

⋅ cos 𝜃 [𝑐A𝑦 (𝑎 cos𝛽 − 𝑐 sin𝛽)

− 𝑐B𝑦 (−𝑐 sin𝛽 + 𝑏 cos𝛽)] − ��

⋅ sin 𝜃 [𝑐A𝑦 (−𝑎 sin𝛽 − 𝑐 cos𝛽)

− 𝑐B𝑦 (−𝑐 cos𝛽 − 𝑏 sin𝛽)] + 𝑥

⋅ cos 𝜃 [𝑘A𝑥 (𝑎 sin𝛽 + 𝑐 cos𝛽)

+ 𝑘B𝑥 (𝑏 sin𝛽 − 𝑐 cos𝛽)] + 𝑥

⋅ sin 𝜃 [𝑘A𝑥 (𝑎 cos𝛽 − 𝑐 sin𝛽)

+ 𝑘B𝑥 (𝑏 cos𝛽 + 𝑐 sin𝛽)] + 𝑦

⋅ cos 𝜃 [𝑘A𝑦 (𝑎 cos𝛽 − 𝑐 sin𝛽)

+ 𝑘B𝑦 (𝑏 cos𝛽 + 𝑐 sin𝛽)] + 𝑦

⋅ sin 𝜃 [𝑘A𝑦 (−𝑎 sin𝛽 − 𝑐 cos𝛽)

+ 𝑘B𝑦 (−𝑏 sin𝛽 + 𝑐 cos𝛽)] + cos 2𝜃

⋅ {𝑘A𝑥 [𝑎𝑐 (−cos2

𝛽 + sin2𝛽)

+ (𝑐2

− 𝑎2

) sin𝛽 cos𝛽] + 𝑘A𝑦 [𝑎𝑐 (−cos2

𝛽 + sin2𝛽)

+ (𝑐2

− 𝑎2

) sin𝛽 cos𝛽] + 𝑘B𝑥 [𝑏𝑐 (−cos2

𝛽 + sin2𝛽)

+ (−𝑐2

+ 𝑏2

) sin𝛽 cos𝛽]

+ 𝑘B𝑦 [𝑏𝑐 (−cos2

𝛽 + sin2𝛽)

+ (−𝑐2

+ 𝑏2

) sin𝛽 cos𝛽]} + cos 𝜃 sin 𝜃

⋅ {𝑘A𝑥 [4𝑎𝑐 cos𝛽 sin𝛽 + (−𝑐2

+ 𝑎2

) sin2𝛽

+ (𝑐2

− 𝑎2

) cos2𝛽] + 𝑘A𝑦 [4𝑎𝑐 cos𝛽 sin𝛽

+ (−𝑐2

+ 𝑎2

) sin2𝛽 + (𝑐2

− 𝑎2

) cos2𝛽]

+ 𝑘B𝑥 [4𝑏𝑐 cos𝛽 sin𝛽 + (−𝑐2

+ 𝑏2

) sin2𝛽

+ (𝑐2

− 𝑏2

) cos2𝛽] + 𝑘B𝑦 [4𝑏𝑐 cos𝛽 sin𝛽

Motionangle ofinclination

Motionamplitude

Deck staticangle ofinclination

y

x

0

0

Figure 4: Motion characteristics measured in linear vibratingscreens.

+ (−𝑐2

+ 𝑏2

) sin2𝛽 + (𝑐2

− 𝑏2

) cos2𝛽]} + cos 𝜃

⋅ {𝑘A𝑥 [𝑎𝑐 (cos2

𝛽 − sin2𝛽)

+ (−𝑐2

+ 𝑎2

) sin𝛽 cos𝛽] + 𝑘A𝑦 [𝑎𝑐 (cos2

𝛽 − sin2𝛽)

+ (−𝑐2

+ 𝑎2

) sin𝛽 cos𝛽] + 𝑘B𝑥 [𝑏𝑐 (cos2

𝛽 − sin2𝛽)

+ (𝑐2

− 𝑏2

) sin𝛽 cos𝛽] + 𝑘B𝑦 [𝑏𝑐 (cos2

𝛽 − sin2𝛽)

+ (𝑐2

− 𝑏2

) sin𝛽 cos𝛽]} + sin 𝜃 {𝑘A𝑥 [𝑎 cos𝛽

− 𝑐 sin𝛽]2

+ 𝑘A𝑦 [−2𝑎𝑐 sin𝛽 cos𝛽 − 𝑐2cos2𝛽

− 𝑎2sin2𝛽] + 𝑘B𝑥 [−2𝑏𝑐 sin𝛽 cos𝛽 − 𝑐

2sin2𝛽

− 𝑐𝑏2cos2𝛽] + 𝑘B𝑦 [−2𝑏𝑐 sin𝛽 cos𝛽 + 𝑐

2cos2𝛽

− 𝑏2sin2𝛽]} = −𝐹 sin𝜔𝑡 ⋅ 𝑑.

(3)

These three equations correspond to a second-order nonlin-ear system of equations, which can be reduced in a series offirst-order differential equations solvedmaking use of Runge-Kutta method with variable time step.

3. Vibrating Screen Motion

Themotionmeasured on the linear vibrating screen is mainlydefined by two variables: the amplitude and the angle ofinclination of motion as shown in Figure 4. In this section,the behavior of these two variables under different stiffness atsupporting positions is analyzed.

As stated above, the present simulations make use of thedata obtained from a vibrating screen in use at a copper mine


Figure 5: Simulated vibrating screen, front view.

(Figure 5). The characteristics of the vibrating screen whenoperating without load are 𝑎 = 2054mm; 𝑏 = 3896mm; 𝑐 =

298mm; 𝑑 = 1372mm; 𝛽 = 4,8∘; 𝛼 = 135∘; 𝑘A𝑦 = 𝑘B𝑦 =

4992000N/m; 𝑘A𝑥 = 𝑘B𝑥 = 2064000N/m; 𝑀 = 24468 kg;𝐼 = 622443 kgm2; 𝑛 = 6; 𝑚 = 100 kg; 𝜔 = 98,96 rad/s; and𝑟 = 0,225m.

The vibrating screen motion is obtained in the modelafter applying the excitation force 𝑓. To simulate startup ofvibrating screen, the frequency of 𝑓 is considered as variablefrom zero to nominal rotating speed. The rise of frequencyfrom zero to nominal rotating conditions is considered asa normal for an induction motor startup. The amplitudeof force 𝑓 is determined by the unbalance force that isproportional to 134,4𝜔2 N. Approximately 20 seconds afterforce𝑓 reaches nominal rotating speed, the stationarymotionis obtained. This procedure is repeated for the nominalstiffness of springs and for all combinations from 100% to10% of stiffness in decrements of 10% in front and rearsupporting positions. With this, the motion of vibratingscreen for 81 combinations of different stiffness in front andrear supporting positions is obtained. Figure 6 shows themotion obtained in the front supporting position, center ofmass, and rear supporting position for no deterioration ofsprings, thus for 100% of nominal stiffness 𝑘A and 𝑘B.

Figure 6 shows the deviation in angle of inclination fromthe angle at design operating conditions for the center ofmass. The same angle lines in Figure 7 are obtained afterinterpolating selected angles between obtained angles for the81 simulations. Angle deviation at front and rear supportingpositions behaves similarly and for that reason graphs forthese positions are not shown. For all the cases simulatedconsidering empty screen, the amplitude of motion changesbetween a –2,3% and a +0,9% which is below vendor limitsfor amplitude deviation and for that reason it is not shown ina graph.

After empty vibrating screen is simulated, the model issolved considering a full load of mineral. For this case, theposition of the center of mass, total mass, and moment ofinertia change from those considered for the case of emptyvibrating screen simulation.The values used in simulations offull load operation of the vibrating screen are 𝑎 = 2917mm;𝑏 = 3033mm; 𝑐 = 0mm; 𝑑 = 667mm; 𝑀 = 48497 kg; and𝐼 = 1156474 kgm2.

In Figure 8, the motions obtained in the front supportingposition, center of mass, and rear supporting position for nodeterioration of springs and full load are shown. Figure 9shows the deviation of angle from the angle at designoperating conditions. For all the simulated cases consideringfull load screen, the amplitude of motion changes betweena −1,9% and a +0,0% which is below vendor limits foramplitude deviation, and for that reason it is not graphicallyshown herein. In Figure 9, the limit of operation indicated bythemanufacturer (±2∘) is depicted by a wider solid line. It canbe observed that with these two curves a limit of stiffness canbe defined in order to ensure that the loss of stiffness in thesupporting positions has no influence in the proper operationof the vibrating screen.

In order to maintain separation efficiency, this modelis capable of defining a limit for spring stiffness. It hasbeen shown that adequate stiffness at supporting positions isimportant tomaintain a proper operation of vibrating screen.It has also been observed that, for an actual vibrating screen,amplitude variations due to loss of stiffness are not significantwith variations differing from −2,3% to +0,9%; this differencehas negligible effect on separation efficiency; in practicewhenthere are differences in the amplitude of motion, this issolved by modifying unbalance masses configuration. Fromthe values obtained from Figure 9 it can be observed that, inthis vibrating screen, half of stiffness in the front supportingposition (46% 𝑘A) and almost a third of stiffness in therear supporting position (38% 𝑘B) can be allowed.Therefore,maintenance can be performed after one of each two or oneof each three springs is broken at the front and the rearsupporting positions, respectively. Because in the simulatedvibrating screen each supporting position has eight springs inparallel, then the separation efficiencywill not be significantlyaffected before 54% and 62% of springs are broken at the frontand the rear supporting positions, respectively, which leadsto a maximum of four springs broken in each supportingposition.The number of springs to break before maintenancewill depend on the particular vibrating screen characteristicsand therefore should be calculated for each particular case.

4. Conclusions

Available models of vibrating screen motion are useful topredict its behavior under nominal operation conditions.

Themodel proposed in this article can predict behavior ofvibrating screen motion under nominal operating conditionsas well as under significant angular displacement: transientconditions, resonance conditions, and different stiffness atsupporting positions.

These novel capabilities of the proposed model allow theprediction of vibrating screen behavior when there is damagein spring stiffness and, therefore, allow evaluating whetherit is necessary to replace springs or not in order to achievean adequate separation efficiency, or to reduce overhaul time,thus increasing vibrating screen availability.

Making use of this model for an actual vibrating screenhas permitted determining a limit for spring’s stiffness reduc-tion before separation efficiency is affected. For the simulatedvibrating screen, the minimum allowed stiffness is 46% and


115∘

12.0mm

120∘

11.1mm

130∘

9.6mm

−6

−4

−2

0

2

4

6

yB

−6

−4

−2

0

2

4

6

yc.m.

−4 −2 0 2 4 6−6

xc.m.

−4 −2 0 2 4 6−6

xB

−6

−4

−2

0

2

4

6

yA

−4 −2 0 2 4 6−6

xA

Figure 6: Movement in mm front supporting position (A), center of mass (c.m.), and rear supporting position (B) for nominal operatingconditions, empty screen.

+2.0∘

+4.0∘ +1.0∘ +0.1∘

0∘

−0.1∘

−1.0∘

−2.0∘

−4.0∘

0

10

20

30

40

50

60

70

80

90

100

kA

10 20 30 40 50 60 70 80 90 1000

kB

Figure 7: Angle deviation inmovement of center ofmass for different stiffness in front and rear supporting positions as percentage of nominalstiffness, empty screen.

131∘

6.0mm

−6

−4

−2

0

2

4

6

yA

−4 −2 0 2 4 6−6

xA

134∘

5.9mm

−6

−4

−2

0

2

4

6

yc.m.

−4 −2 0 2 4 6−6

xc.m.

136∘

5.3mm

−6

−4

−2

0

2

4

6

yB

−4 −2 0 2 4 6−6

xB

Figure 8: Movement in front supporting position (A), center of mass (c.m.), and rear supporting position (B) for nominal operatingconditions, full load.


0

10

20

30

40

50

60

70

80

90

100

kA

10 20 30 40 50 60 70 80 90 1000

kB

+2.0∘

+4.0∘+1.0∘

+0.1∘0∘

−0.1∘

−1.0∘

−2.0∘

−4.0∘

46% kA

38% kB

Figure 9: Angle deviation in the movement of the center of mass,full load.

38% of nominal stiffness in the front and rear supportingposition, respectively. Because each supporting position haseight springs in parallel, then the failure of four springs canbe allowed in each supporting position.

This information is of practical value for operation andmaintenance staff helping to determine whether or not it isnecessary to change springs, and hence optimizing stoppagetime.

Additional Points

Highlights. (i) A two-dimensional three-degree-of-freedomnonlinearmodel is developed. (ii)Themodel is used to evalu-atemotion under off-design operating conditions. (iii) A limitfor springs failure is determined before separation efficiencyis affected. (iv) The model allows evaluating whether it isnecessary to replace springs or not.

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

References

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[3] B. A. Wills, Wills’ Mineral Processing Technology: An Intro-duction to the Practical Aspects of Ore Treatment and MineralRecovery, Butterworth-Heinemann, Oxford, UK, 7th edition,2006.

[4] J. Xiao and X. Tong, “Particle stratification and penetrationof a linear vibrating screen by the discrete element method,”International Journal of Mining Science and Technology, vol. 22,no. 3, pp. 357–362, 2012.

[5] P. W. Cleary, M. D. Sinnott, and R. D. Morrison, “Separationperformance of double deck banana screens—part 1: flow andseparation for different accelerations,” Minerals Engineering,vol. 22, no. 14, pp. 1218–1229, 2009.

[6] P.W.Cleary,M.D. Sinnott, andR.D.Morrison, “Separation per-formance of double deck banana screens—part 2: quantitativepredictions,”Minerals Engineering, vol. 22, no. 14, pp. 1230–1244,2009.

[7] I. M. Cotabarren, J. Rossit, V. Bucala, and J. Pina, “Modeling ofan industrial vibrating double-deck screen of a urea granulationcircuit,” Industrial and Engineering Chemistry Research, vol. 48,no. 6, pp. 3187–3196, 2009.

[8] G. Wang and X. Tong, “Screening efficiency and screen lengthof a linear vibrating screen using DEM 3D simulation,”MiningScience and Technology, vol. 21, no. 3, pp. 451–455, 2011.

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[11] K. J. Dong, A. B. Yu, and I. Brake, “DEM simulation of particleflow on a multi-deck banana screen,”Minerals Engineering, vol.22, no. 11, pp. 910–920, 2009.

[12] L. Zhao, C. Liu, and J. Yan, “A virtual experiment showing singleparticle motion on a linearly vibrating screen-deck,” MiningScience andTechnology (China), vol. 20, no. 2, pp. 276–280, 2010.

[13] Y.-M. Zhao, C.-S. Liu, X.-M. He, C.-Y. Zhang, Y.-B. Wang, andZ.-T. Ren, “Dynamic design theory and application of largevibrating screen,” Procedia Earth and Planetary Science, vol. 1,no. 1, pp. 776–784, 2009, Proceedings of the 6th InternationalConference on Mining Science & Technology.

[14] Z. Tang, Z. Yin, T. Han, X. Sun, and L. Zhang, “Research ondynamic characteristics of elliptical vibrating screen,” in Pro-ceedings of the International Conference on Mechanic Automa-tion and Control Engineering (MACE ’10), pp. 2366–2369, IEEE,Wuhan, China, June 2010.

[15] X.-M. He and C.-S. Liu, “Dynamics and screening character-istics of a vibrating screen with variable elliptical trace,”MiningScience and Technology (China), vol. 19, no. 4, pp. 508–513, 2009.

[16] C. Liu, L. Peng, and F. Li, “Survey of signal processing methodsand research on vibrating screen fault diagnosis,” in Proceedingsof the 2nd International Conference on Mechanic Automationand Control Engineering (MACE ’11), pp. 1709–1712, IEEE,Hohhot, China, July 2011.

[17] C.-S. Liu, S.-M. Zhang, H.-P. Zhou et al., “Dynamic analysis andsimulation of four-axis forced synchronizing banana vibratingscreen of variable linear trajectory,” Journal of Central SouthUniversity, vol. 19, no. 6, pp. 1530–1536, 2012.

[18] L. I. Slepyan and V. I. Slepyan, “Coupled mode parametricresonance in a vibrating screenmodel,”Mechanical Systems andSignal Processing, vol. 43, no. 1-2, pp. 295–304, 2014.

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