dynamic modeling and parameters optimization of large...

Research ArticleDynamic Modeling and Parameters Optimization of LargeVibrating Screen with Full Degree of Freedom

Xiaodong Yang1 Jida Wu 1 Haishen Jiang2 Wenqiang Qiu1 and Chusheng Liu1

1School of Mechatronic Engineering China University of Mining and Technology Xuzhou 221116 China2School of Chemical Engineering China University of Mining and Technology Xuzhou 221116 China

Correspondence should be addressed to Jida Wu wujidacumt126com

Received 21 October 2018 Revised 19 December 2018 Accepted 1 January 2019 Published 31 January 2019

Academic Editor Davood Younesian

Copyright copy 2019 Xiaodong Yang et al +is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Dynamic characteristic and reliability of the vibrating screen are important indicators of large vibrating screen Considering theinfluence of coupling motion of each degree of freedom the dynamic model with six degrees of freedom (6 DOFs) of the vibratingscreen is established based on the Lagrange method and modal parameters (natural frequencies and modes of vibration) of therigid body are obtained +e finite element modal analysis and harmonic response analysis are carried out to analyze the elasticdeformation of the structure By using the parametric modeling method beam position is defined as a variable and an orthogonalexperiment on design is performed+e BP neural network is used to model the relationship between beam position andmaximalelastic deformation of the lateral plate Further the genetic algorithm is used to optimize the established neural network modeland the optimal design parameters are obtained

1 Introduction

+e vibrating screen is one of the key equipment for coalprocessing which is widely used in grading deslimingsculpting and dewatering of coal [1 2] Due to the devel-opment trend of large scale and intensification of the coaldressing plant a large vibrating screen is urgently needed tosimplify the production system and to reduce plant volumeconstruction investment and operation cost At the sametime the demand on performances and reliability of a largevibrating screen becomes higher It has a very importantsignificance to improve the theoretical research and level ofthe design of the large vibrating screen Baragetti and Villaproposed a study of the dynamics of a heavy loaded vibratingscreen based on a 3-DOF dynamic model and optimized thedesign parameters in order to minimize the pitching angle ofthe screen [3] Wang et al presented an optimization of thelinear vibrating screen based on MATLAB OptimizationToolbox which leads to the optimal productivity per unitpower increase 285 in comparison with the initial value[4] Li et al [5] established the nonparametric modelmapping the vibrating screen efficiency and operating

parameters using the discrete element method and supportvector machines and optimized the parameters with particleswarm optimization He and Liu proposed a theoreticalmodel for the dynamic behavior evaluation of vibratingscreens and presented a new screen with elliptical trace [6]Based on the experiments Zhang established a least squaresupport vector machine model to predict the sieving effi-ciency and optimize the parameters using the adaptive ge-netic algorithm and cross-validation algorithm [7] As thevibrating screen belongs to the group of vibration andutilization machines the structure is subjected to a largedynamic load which can easily cause local elastic de-formation and lead to fatigue failure in the structure Toimprove the reliability of large vibrating screens Zhao et alpresented a new design of a hyperstatic net-beam structureand analyzed dynamic characteristic of it based on the finiteelement method and the new screen proves to have muchhigher structural strength with an enhanced dynamic be-havior [8] Su et al proposed an improved scheme of beamsection for large-scale vibrating screen structure based onstatic analysis and dynamic analysis which amelioratedstress distribution of the vibrating screen is in working

HindawiShock and VibrationVolume 2019 Article ID 1915708 12 pageshttpsdoiorg10115520191915708

process and the fatigue life is increased [9] Baragettipresented a structural solution for high loaded vibratingscreens with new modified side walls and studied the be-havior of the original and modified structure by means oftheoretical and numerical models [10] Peng et al con-ducted a systematic mechanics analysis of the beamstructures and improved the design method consideringbending and random vibration [11] Wang et al presented anovel large vibrating screen with a duplex statically in-determinate mesh beam structure +rough the modelanalysis result comparisons with the traditional vibratingscreen the superiority of this structure was verified [12]Jiang et al established a dynamic model and stabilityequations of the variable linear vibration screen and in-vestigated the motion behavior of screen face as well asconfirmed best range of exciting position [13] Du et alproposed a single-deck equal-thickness vibrating screendriven externally by an unbalanced two-axle excitationwith a large span and presented 3-degree-freedom dynamicequations [14] However the traditional two-degree-of-freedom or three-degree-of-freedom dynamic model canonly reflect the motion in plane but cannot reflect thecomplex motion in space +e traditional optimizationmethod requires iterative solution and is not suitable formultivariable nonlinear model optimization which hashigh computational cost and is not suitable for multivar-iable nonlinear model optimization

At present ANNs have become a preferred alternativeway to solve any of complex highly coupled and nonlinearproblems Rad et al constructed an expert system usedBayesian regulation back-propagation neural network andvibration monitoring data for electric motor status diagnosis[15] Tian developed a LevenbergndashMarquardt artificial neuralnetwork-based method for achieving accurate remaininguseful life prediction of equipment subject to conditionmonitoring [16] Meruane and Mahu trained the neuralnetwork using a noise-injection learning algorithm to reducethe effects of experimental noise [17] Sun and Han proposeda UAV aerial photography monitoring method based ongradient descent with the momentum neural network whichimproved the efficiency of automatic extraction and classi-fication of image features [18] Taking training time iterationtimes and error performance as indicators Zhang et alcompared the training efficiency of BFGS quasi-Newton al-gorithm resilient algorithm LM algorithm and FletcherndashReeves update algorithm for the neural network model in theclassification process of OCT images [19]

+e article is structured as follows In Section 2 thedynamic model with six degrees of freedom (6 DOFs) of thevibrating screen based on the Lagrange method wasestablished and multinatural frequency and natural modesof vibration of the vibrating screen were calculated based onrigid body modal analysis In Section 3 the finite elementmethod was used to analyze dynamic performance of thelarge vibrating screen In Section 4 the BP neural networkwas used to establish the nonlinear mapping between po-sition parameters of stiffening beams and dynamic perfor-mance of vibrating screen and the sensitivity analysis wascarried out bases on the model +e optimal combination of

structural parameters was obtained using the genetic algo-rithm Finally the conclusion is drawn in Section 5

2 Rigid Body Dynamics Analysis

21 e Structure of Vibrating Screen ZS2560 is a linearvibrating screen widely used in coal separation As shown inFigure 1 the ZS2560 linear vibrating screen is mainlycomposed of vibration exciters screen box and spring +escreen box is the main working part of the vibrating screenand accordingly it is vulnerable to the damage +e screenbox includes both left and right sides of the plate the tailgatespring support group and sill and stiffening beams In orderto adapt to heavy load conditions the screen box of ZS2560adopts the double-bottom beam structure instead of thetraditional single-layer beam structure which can achievehigher structural stiffness and lower center of mass +evibration exciters are set up on the left and right sides of theplate which can generate sinusoidal excitation force alongthe normal direction using two eccentric block synchronousreverse rotations Under the action of excitation force theentire screen box is vibrated and a sieving of the material isrealized when in operation

22 Rigid Body Dynamics Modeling At present the sim-plified 2-degree or 3-degree freedom mass-spring vibrationmodel is often used in dynamic analysis of the vibratingscreen [20 21] However the 2-degree vibration model canonly represent the linear motion of the vibrating screen intwo directions and the 3-degree freedom vibration modelcan represent the swing around a certain axis in additionand it is difficult to fully reflect the complex motion of thevibrating screen Considering the translation and rotationaround the center of mass in x-y x-z and y-z planes at thesame time this paper establishes a 6-degree freedom vi-bration model of the vibrating screen (Figure 2) wherein thescreen box is equivalent to the rigid body in space motion

As shown in Figure 2 the origin of the x-O-y coordinatesystem is located at static equilibrium position of the screenbox (the center of mass) m is the mass of the vibratingscreen when it is in operation Jx Jy and Jz are the momentsof inertia of the center of mass of the vibrating screen withrespect to x y and z axes respectively and Kx Ky and Kz arethe stiffness of springs at front and rear positions re-spectively +e distance from each spring to the center ofmass is lij where i represents the spring force direction and jrepresents the spacing direction Besides the displacementson each degree of freedom are labeled as x y and z and theangles are labeled as ψx ψy and ψz According to the me-chanical model the following equations can be obtained

Kinetic energy of the system can be expressed as

T 12

m _x2

+ _y2

+ _z2

1113872 1113873 +12

Jxx _φ2x + Jyy _φ2

y + Jzz _φ2z1113872 1113873

+ Jxy _φx _φy + Jxz _φx _φz + Jyz _φy _φz

(1)

Potential energy of the system can be expressed as

2 Shock and Vibration

U 12

11138901113944 kx x + φylxz minusφzlxy1113872 11138732

+ 1113944 ky yminusφxlyz + φzlyx1113872 11138732

+ 1113944 kz z + φxlzy minusφylzx1113872 111387321113891

(2)+e Lagrange equation with 6 degrees of freedom is

shown as followsd

dt

zT

z _q1113888 1113889minus

zT

zq+

zU

zq F (3)

where q is the generalized coordinates matrix of the vi-bration system q [x y zφxφyφz]T _q and euroq are thegeneralized velocities matrix and generalized acceleration

matrix of the system respectively F is the excitation matrixof the system F [fx fy fz Mx My Mz]T

Substituting Equations (1) and (2) into Equation (3) thecontent of each element in the mass matrix and the stiffnessmatrix is calculated according to the following formula

mkl mlk z2T

z _qkz _ql

kkl klk z2U

zqkzql

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(4)

Accordingly the mass matrix and stiffness matrix of thesystem are constructed as follows

M

m 0 0 0 0 0

0 m 0 0 0 0

0 0 m 0 0 0

0 0 0 Jxx 0 0

0 0 0 0 Jyy 0

0 0 0 0 0 Jzz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

K

1113944 kx 0 0 0 1113944 kxlxz minus1113944 kxlxy

0 1113944 ky 0 1113944 kylyz 0 1113944 kylyx

0 0 1113944 kz 1113944 kylzy minus1113944 kzlzx 0

0 1113944 kylyz 1113944 kzlzy 1113944 kyl2yz + 1113944 kzl

2zy minus1113944 kzlzxlzy minus1113944 kylyzlzx

1113944 kxlxz 0 minus1113944 kzlzx minus1113944 kzlzxlzy 1113944 kxl2xz + 1113944 kzl

2zx minus1113944 kxlxzlxy

minus1113944 kxlxy minus1113944 kylyx 0 minus1113944 kylyzlzx minus1113944 kxlxzlxy 1113944 kyl2yx + 1113944 kxl

2xy

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(5)

Springsupport

Localreinforcement

plate

Strengtheningbeam

Tailgate

Bottombeam

Vibrationexciter

Sideplate

Dischargeport

y Px

PxP0

P0

PyPyx

o

Figure 1 Schematic diagram of the ZS2560 linear vibrating screen

Shock and Vibration 3

+e vibration equation of the linear vibration screen isestablished based on the Lagrange method and it is shownthat

Meuroq + Kq F (6)

where F m0eω2 cos α m0eω2 sin α 0 0 0 m0eω2δ1113858 1113859T

and m0e is the product of the eccentric mass diameter ofthe exciter kgm ω is the rotation speed rads α is thevibrating direction angle rad δ is the normal distancebetween the center of mass and the excitation force di-rection m

23 Rigid BodyModalAnalysis Modal analysis is a commonmethod for studying dynamic characteristics of mechanicalstructures and design optimization Modal analysis of thevibrating screen can be used to obtain modal parameterssuch as natural frequency and natural vibration mode aswell as to provide a reference for the revision and structuredesign optimization of the subsequent simulation calcula-tion model [22 23]

Assuming that the excitation force is equal to zero thefree vibration equation of vibrating screen is defined by thefollowing equation

Meuroq + Kq O (7)

According to the vibration theory we can assume thatthe solution of Equation (7) is as follows

q Aeiλt

(8)

where A is the amplitude vector of the system in the case offree vibration and λ is the modal frequency

Substituting Equation (8) into Equation (7) we can getthe following equation

Kminus λ2M1113872 1113873A O (9)

In order to obtain the generic natural frequencies of thedynamic system the eigenvalues problem has to be solved bythe following equation

Det Kminus λ2M1113872 1113873 0 (10)

+e above equation is the algebraic equation of the 6power real coefficient of λ2 and the natural frequencies ofthe system can be obtained by solving it +e vibration modeof the structure can be obtained by substituting λ intoEquation (9)

In this paper we use a large linear vibrating screen withthe size of 2500mmtimes 6000mm as a research object +ekinetic parameters are shown in Table 1

+e calculated natural frequency and correspondingsystem vibration are shown in Table 2

+e first six vibration modes of the system are presentedgraphically in Figures 3 and 4 +ey are rigid modal and themodal frequency is lower which depends on the vibrationmass and stiffness of spring With the aim to simplify theanalysis both translational vibration modes and rotationalvibration modes are plotted respectively In Figures 3 and 4it can be seen that the fourth-order vibration mode (373Hz)is mainly represented by translation in x direction+e othermodes are obtained by the coupling effect of translation androtation which is mainly because of unsymmetrical in-stallation position of spring from the center of mass on the x-y plane and there is an offset between the spring mountingplane and the center of mass in the z plane which leads to theexistence of a nondiagonal element in the stiffness matrixtherefore the translational and rotational motions are notcompletely decoupled

3 Analysis of Finite Element Model Structure

To verify the results obtained by theoretical analysis and tooptimize the structure dynamics the finite element model ofvibrating screen was set up using the finite element softwareANSYS (Figure 5)+e vibration exciters were replaced by theconcentrated mass points with the same quality because theelasticity of vibrating vibrator was very small +e materialproperties defined the elastic modulus as E 2030times1011 Paand Poisson ratio was u 03 +e modal analysis was per-formed for the vibrating screen structure and the first tenorder modes were obtained However only the first six orderrigid body modal shapes were compared (Figure 6)

+e comparison of results obtained by theoreticalanalysis and finite element analysis for the first six ordermodes is shown in Table 3 wherein it can be seen that thevalue of natural frequency obtained by finite elementanalysis is in good agreement with the value obtained bytheoretical calculation and the maximal relative error is569

In particular the error of the torsional mode (3rd 5thand 6th orders) is greater than that of the translational mode(1st 2nd and 4th orders) which is due to the linearizationsimplification of torsion in the theoretical model +emotion trend of the sieve as shown in Figure 6 is the same asthat shown in Figures 3 and 4+erefore it is shown that theboundary conditions of the finite element model are con-sistent with the theoretical model +e finite element modelcan reflect the dynamic characteristics of the system whichcan be used as a basis for further dynamic analysis andstructural optimization

4 Structural Optimization Analysis

41 Establishment and Fitting of BP Neural Network ModelFor analysis of the elastic deformation of the vibration screenin the movement process the 7th 8th 9th and 10th order

Vibrationscreen

y Spring

xO

z

kz

ψz

ψy

ψx

ky

lxz = lyz

lzx = lyx

ndashlxy = ndashlzy

kx

Figure 2 Sketch map of 6 degrees of freedom vibration model


modes were extracted which are deformable modes andmodal frequency lies on structural stiffness of the vibratingscreen In Figure 7 and Table 4 there are two elastic modalsaround the working frequency (16Hz) that are the ninthmodal (14697Hz) and the tenth modal (21968Hz) whichtake a central part to the total deformation of the vibratingscreen and are easy to lead to resonance We can see that themain elastic deformation takes place at the side plate whichhas low torsional stiffness and bending stiffness In additionthe bending stiffness of the lower part is related to the screenframe tailgate and torsional stiffness of the lower part isrelated to the sieve frame at the front outlet +e main

deformation occurs on the lateral plate so it is necessary tooptimize the strength by adjusting the beam position inorder to improve lateral stiffness

On the basis of modal analysis the harmonic responseanalysis was carried out based on the modal superpositionmethod and the exciting force F 150000N was applied tothe installation position of the exciter while the workingfrequency was 16Hz

+e relationship between structure design variables ofvibration system and its dynamic characteristics is highlynonlinear For complex systems the relation between theseparameters and performance cannot be expressed explicitlyby a linear function and direct optimization of these pa-rameters is costly to calculate However the artificial neuralnetworks (ANNs) have a very strong nonlinear mappingability [24] so ANNs are very suitable for establishing themodel of the vibration system Currently the BP neuralnetwork is the most widely used neural network modelMoreover it has been proven theoretically that a three-layerBP network can approximate any rational function [25 26]

Table 1 +e kinetic parameters of the ZS2560 vibrating screen

Parameter Numerical valuem 7290 kgJxx 11875 kgmiddotm2

Jyy 19632 kgmiddotm2

Jzz 27233 kgmiddotm2

kx ky kz 12times106Nm4times106Nm 12times106Nmlxy minus0543m

lzx lyx252m rear front233m rear back

lxz lyz 154m

Table 2 Modal calculation results of the vibrating screen (1sim6order)

Order Natural frequency (Hz) Natural modes of vibration1 19947 [0 0 1 00834 00157 0]T

2 20346 [minus1 minus0002 0 0 0 00151]T3 35803 [0 0 00383 00262 minus1 0]T4 37300 [0 1 0 0 00 0]T

5 46042 [0 0 minus01491 1 00152 0]T6 66498 [00566 00429 0 0 0 1]T

1 2 3

ndash1

0

1

1st2nd3rd

4th5th6th

Figure 3 Translational vibration mode of 1ndash6 orders 1 x directiondegree of freedom 2 y direction degree of freedom 3 z directiondegree of freedom

4 5 6

ndash1

0

1

1st2nd3rd

4th5th6th

Figure 4 Torsional vibration mode of 1ndash6 orders 4 ψx directiondegree of freedom 5 ψy direction degree of freedom 6 ψz directiondegree of freedom

Figure 5 Grid partition of vibrating screen and boundary con-dition setting


+erefore the BP neural network shown in Figure 8 is usedto model the relationship between elastic deformation ofvibrating screen and its design variables

+e sigmoidal function (sigmoid) was selected as anactivation function of the hidden layer in BP ANN and thelinear function (pureline) was used as a transfer function ofthe output layer +erefore the entire network transferfunction can be expressed as

G(X) b2k + 1113944

NI

i1υ b1i + 1113944

NJ

j1ηijxj

⎛⎝ ⎞⎠ (11)

where ηij is the network connection weight value b1i and b2kare threshold values for network connections υ is the im-plicit excitation function NI is the number of input layer

nodes NI 12 and NJ is the number of hidden layer nodeswhich is calculated according to the formula

NJ leNJ + q

1113969+ s (12)

where q is the number of output elements q 1 and s is theconstant between 0 and 10 s 2

+e input parameters of the network are the horizontaland vertical installation position of beams which is shown inFigure 9 while the output parameter is the maxelastic de-formation of the screen box of ZS2560

Design of the neural network model requires ANNstraining with a certain number of samples with a properdistribution to make the neural network learn and expressthe relation accurately In order to meet the requirements of

040204 max039379038553037728036092

035252036077

034426033601032775 min

(a)

038091 max037745037399037053036707036361036015035669035322034976 min

(b)

0634705564504782103999603217202434701652300869840008739 min

071294 max

(c)

03813 max037822037514037207036899036591036283035975035667035359 min

(d)

062081 max055214048346041479034611027744020876014009007141500027399 min

(e)

080053 max0711680622830533980445130356270267420178570089717000086521 min

(f )

Figure 6 Modal picture of vibrating screen (1sim6 orders) (a) 1st order (b) 2nd order (c) 3rd order (d) 4th order (e) 5th order (f ) 6th order

Table 3 Comparison of FEM results with theoretical values

Order 1 2 3 4 5 6FEM results (Hz) 19774 20166 36141 37023 4 8486 62717+eoretical values (Hz) 19947 20346 35803 37300 46042 66498Error () minus087 minus088 094 minus074 531 minus569


both test workload and design accuracy the orthogonal testdesign method is used to select a few representative schemesfrom a large number of test schemes

In design optimization first the design variables weredefined and then positions of 6 strengthening beams in xand y dimensions were labeled as andashf and set as designparameters as shown in Figure 9

+rough the orthogonal design with 12 factors and 3levels 27 sets of parameter combinations were obtained+efinite element model was established and the correspondingelastic deformation was calculated

In order to more intuitively reveal the trend of the testresults changing with the level of each factor the rangeanalysis method was used to analyze the results of theorthogonal test directly +e trend of the 12 factors in thisexperiment is shown in Figure 10 For each factor theabscissa is the horizontal number and the ordinate is thecorresponding mean According to Figure 10 for factorsa_x and b_y as the value increases the elastic deformationof the structure increases For the factors a_y b_x c_x andc_y as the value increases the elastic deformation of thestructure increases first and then decreases Besides forother factors as the value increases the elastic deformationof the structure decreases In particular for the factor d_xas the value increase the elastic deformation of the

structure keeps 5644mm which means the effect of thevariable on the elastic deformation in the first interval isweak

On the basis of the orthogonal test the qualitative lawof structural deformation was obtained through the rangeanalysis method In order to more accurately design thestructure neural network modeling and parameter opti-mization are required According to the structure shown inFigure 8 a neural network model with 12 inputs and 1output was constructed Twenty-seven groups of orthog-onal test data were divided into training group and testgroup in which the training group contains 20 groups ofdata and the test group contains 7 groups of data +eLevenbergndashMarquardt algorithm was used as the trainingfunction of the neural network to improve the convergencespeed of the model After the iterative training the sim-ulation results and error distribution were as shown inFigure 11 where the error between simulation results andtraining results is acceptable After training we tested thenetwork with the samples which were not used for thetraining and the agreement of ANN output and real outputis presented in Figure 12 wherein it can be seen that themaximal error is less than 16 thus a high-degreeagreement is achieved

42 Genetic Algorithm Optimization +e GA simulates theDarwinian evolution genetic selection ie the evolutionprocess of survival of the fittest rules with the same group ofchromosome by a random search algorithm combining theinformation transformation mechanism initialization pa-rameter coding and initial population and then by using thecrossover operation and mutation natural selection oper-ator parallel iteration and optimization solutions [27]

10217 max0909190796720684250571780459303468302343601218900094136 min

(a)

12955 max115161007708637307198057587043194028801014407000014222 min

(b)

092434 max082165071895061626051357041087030818020549010279000010089 min

(c)

084769 max075643066516057390482640391380300110208850117590026322 min

(d)

Figure 7 Modal picture of vibrating screen (7sim10 order) (a) 7 order (b) 8 order (c) 9 order (d) 10 order

Table 4 Modal calculation results of vibrating screen (7sim10 order)

Order Natural frequency f (Hz)7 110288 133729 1469710 21968


Since GA is based on random operation there are no specialrequirements for search space and derivation is not neededDue to the advantages of simple operation and fast con-vergence speed GA has quickly developed in recent yearsand it has been widely used in combinatorial optimizationadaptive control and many other engineering fields Based

on the research results in Section 41 the optimization flowchart of this paper is shown in Figure 13

In order to reduce the elastic deformation by searchingthe optimal beams position the genetic algorithm is selectedto optimize the BP network model established and it isdefined by

Structural elastic deformation BP neural network

Beam positions

Elastic deformation

(a)

Inputlayer

Hiddenlayer

Outputlayer

Output data

Input data

Input data

Input data

(b)

Figure 8 BP neural network prediction model for structural elastic deformation (a) Input and output parameters of BP neural networkmodel (b) BP neural network model structure

f_x

e_xo

a_y

y

a_x

b_xc_x

d_x

b_y

c_y

e_y

d_y

f_y

x

Figure 9 Schematic diagram of design variable for the vibrating screen

1a_x a_y b_x b_y c_x c_y

2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3

560

561

562

563

564

565

566

Aver

age v

alue

of e

lasti

c def

orm

atio

n (m

m)

Orthogonal level

(a)

1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3556

558

560

562

564

566

568

Aver

age v

alue

of e

lasti

c def

orm

atio

n (m

m)

d_x d_y e_x e_y f_x f_yOrthogonal level(b)

Figure 10 Trend diagram of relationship between beam positions and elastic deformation


Find X x1 x2 x121113864 1113865T

min G(X)

st xLi lexi lexU

i

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(13)

where X x1 x2 x121113864 1113865T is defined design variables

x1simx2 correspond to the position of a sim f beam G(X) is theoptimized target function xL

i and xUi are the upper and

lower limits of the design variables Considering the sideplate shape and the sieving position xL

i and xUi are defined as

shown in Table 5Sensitivity analysis can determine the gradient re-

lationship between the design variable and the objective

function and reflect the contribution of the design variable tothe change of the objective function

+e differential form sensitivity of the design variable isdefined by the following equation

zG

zxi

G XminusΔxie( 1113857minusG(X)

Δxi

(14)

where e is the array with the same number of X vectors 1 atxi and 0 at the rest and Δxi is the change in design variables

Based on the BP neural network model established inSection 41 the differential form sensitivity of the designvariable has been obtained by perturbing the design variableto calculate the change degree of the objective function +efinal result is shown in Figure 14

As shown in Figure 14 the elastic deformation is moresensitive to the x-coordinate of the beams than the y-coordinate+erefore this indicates that more attention should be paid tothe horizontal coordinates of beams in the further optimizationprocesses Besides except the beam a the sensitivity of elasticdeformation to the position of other beams is negative whichmeans that the increase of beam coordinates will lead to thedecrease of elastic deformation +e sensitivity analysis resultscan be used as a criterion to evaluate the optimization results

+e genetic algorithm parameter settings are shown inTable 6 and the estimated and target values were calculatedfor the minimum of objective function in the feasible regionshown in Table 5

+e iterative histories for objective function are shown inFigure 15 In the initial state the results obtained by thegenetic algorithm fluctuate greatly After 20 generations withthe increase of the number of iterations the results convergegradually Finally the optimization results tend to be stable

+e optimal design variables obtained by the geneticalgorithm are compared with the original design variables inTable 7 +e obtained design parameters are added to thefinite element model and the optimized results are obtained

0 2 4 6 8 10 12 14 16 18 20ndash05

00

05

10

5

6

Elas

tic d

efor

mat

ion

(mm

)

e number of training data

Training resultSimulation resultError between the two

Figure 11 Comparison between the training results and thesimulation results

Define designparameters and

optimizationobjective

Orthogonalexperimental

design

FEM analysis

BP neuralnetwork model

Final designparameters

Geneticalgorithm

optimization

Sensitivityanalysis and

range analysis

Figure 13 Optimization flow chart of the large vibrating screen

1 2 3 4 5 6 7ndash05

00

05

10

5

6

Elas

tic d

efor

mat

ion

(mm

)

e number of testing data

Testing resultSimulation resultError between the two

Figure 12 Comparison between the testing results and the sim-ulation results


After optimization the maximal elastic deformation was5283mm and compared to the initial value it was reducedfor 699

From Table 7 it can be seen that the factors a_x and b_yare lower than the initial values after optimization and otherfactors are higher than the initial values +e variation law ofeach factor is consistent with the qualitative law obtained bythe range analysis method +erefore the optimization re-sults obtained by neural networks and genetic algorithms arecredible

5 Conclusion

A new structural solution for high loaded vibrating screenswas proposed in this paper and the following conclusionsare obtained

(1) +e dynamic model of the vibrating screen with 6degrees of freedom is established based on theLagrange equation By numerical calculation thefirst six orders natural frequency and the corre-sponding vibration mode of rigid body as well as thedistribution law of rigid body modes are obtained bynumerical calculation Among them the 1st mode(19947Hz) 2nd mode (20346Hz) and 4th mode(35803Hz) are mainly translational motion and the3rd mode (35803Hz) 5th mode (46042Hz) and6th mode (66498Hz) are mainly translational mo-tion By means of the vibration mode the complexspace motion of the vibrating screen is dynamicallydecoupled and the dynamic decoupling condition ofthe vibrating screen is given

(2) +e excitation frequency range that influences thesteady state operation of the vibrating screen isanalyzed and the maximum error between the finiteelement model and the theoretical value is 569 thefeasibility and accuracy of the finite element model indescribing the vibration characteristics of the systemare verified

Table 5 +e upper and lower bounds of design variables (mm)

i 1 2 3 4 5 6 7 8 9 10 11 12xL

i (mm) 800 1200 17224 1596 4020 1705 5107 1219 900 500 1100 120xU

i (mm) 1200 1600 21224 1996 4620 2005 5607 1619 1400 800 1500 240

ndash014

ndash012

ndash010

ndash008

ndash006

ndash004

ndash002

000

002

004

006

Sens

itivi

ty

Design variablesa_x a_y b_x b_y c_x c_y d_x d_y e_x e_y g_x g_y

Figure 14 +e sensitivity of design variables

Table 6 +e genetic algorithm parameter settings

GA parameters ExplainCoding method Binary codingOrdering method Ranking compositorSelection method RouletteCross method Random multipoint intersectionVariation method Disperse multipoint variationPopulation size 200Crossover probability 08Mutation probability 01Evolutionary algebra 200

0 20 40 60 80 100 120 140 160 180 20052

53

54

55

56

57

58

59

60

Max

imal

elas

tic d

efor

mat

ion

(mm

)

Evolutionary algebra

Figure 15 +e iterative histories for objective function

Table 7 Optimized design variables

Parameters Original value(mm)

Optimal value(mm)

Round(mm)

a_x 1126 8144550945 8145a_y 1400 1274839705 1275b_x 19224 1751020621 1751b_y 1696 1688835332 1689c_x 4120 4587176421 4587c_y 1805 1924636191 1925d_x 5307 5571593556 5572d_y 1419 1432600318 1433e_x 9396 1319516474 1320e_y 570 7868099837 787f_x 1300 1355805261 1356f_y 170 2192089448 219Maximal elasticdeformation 568 mdash 5283


(3) Based on the orthogonal test and finite elementanalysis the BP neural network is used to model therelationship between design variables and elasticdeformation of the vibrating screen And the sen-sitivity of each variable to structural elastic de-formation is analyzed the elastic deformation ismore sensitive to the x-coordinate of the beams thanthe y-coordinate

(4) Genetic algorithm is used for optimizationAccording to the obtained results the maximumelastic deformation was reduced by 699 comparedto the original design

+is optimization design is simple and flexible itshortens the design period and applies to other similarproducts

Nomenclature

A +e amplitude vector of the system in the case offree vibration

b1i b2k +reshold values for network connectionsF +e excitation matrix of the systemJxx +e moments of inertia of the vibrating screen

with respect to x-axis (kgmiddotm2)Jyy +e moments of inertia of the vibrating screen

with respect to y-axis (kgmiddotm2)Jzz +e moments of inertia of the vibrating screen

with respect to z-axis (kgmiddotm2)Jxy +e product of inertia in the x-y plane (kgmiddotm2)Jyz +e product of inertia in the y-z plane (kgmiddotm2)Jxz +e product of inertia in the x-z plane (kgmiddotm2)kx +e stiffness of springs in the x direction (Nm)ky +e stiffness of springs in the y direction (Nm)kz +e stiffness of springs in the z direction (Nm)lij +e spatial position of the spring (m)m +e mass of the vibrating screen (kg)m0e +e product of eccentric mass diameter of the

exciter (kgmiddotm)NI +e number of input layer nodesNJ +e number of hidden layer nodesq +e number of output elementsx y z +e translational degrees of freedom in the x y

and z directions (m)ψx ψyψz

+e rotation degrees of freedom in the x y and zdirections (rad)

ω +e rotation speed (rads)α +e vibrating direction angle (rad)δ +e normal distance between the center of mass

and the excitation force direction (m)λ +e modal frequency (Hz)ηij +e network connection weight valueυ +e implicit excitation function

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 there are no conflicts of interestregarding the publication of this paper

Acknowledgments

+is work has been supported in part by the NationalNatural Science Foundation of China (Project nos 51775544and U1508210)

References

[1] O A Makinde B I Ramatsetse and K Mpofu ldquoReview ofvibrating screen development trends linking the past and thefuture in mining machinery industriesrdquo International Journalof Mineral Processing vol 145 pp 17ndash22 2015

[2] S Huband D Tuppurainen L While L Barone P Hingstonand R Bearman ldquoMaximising overall value in plant designrdquoMinerals Engineering vol 19 no 15 pp 1470ndash1478 2006

[3] S Baragetti and F Villa ldquoA dynamic optimization theoreticalmethod for heavy loaded vibrating screensrdquo Nonlinear Dy-namics vol 78 no 1 pp 609ndash627 2014

[4] Y Y Wang J B Shi F F Zhang et al ldquoOptimization ofvibrating parameters for large linear vibrating screenrdquo Ad-vanced Materials Research vol 490 pp 2804ndash2808 2012

[5] Z F Li X Tong B Zhou and X Wang ldquoModeling andparameter optimization for the design of vibrating screensrdquoMinerals Engineering vol 83 pp 149ndash155 2015

[6] X M He and C S Liu ldquoDynamics and screening charac-teristics of a vibrating screen with variable elliptical tracerdquoInternational Journal of Mining Science and Technologyvol 19 no 4 pp 508ndash513 2009

[7] B Zhang J K Gong W H Yuan J Fu and Y HuangldquoIntelligent prediction of sieving efficiency in vibratingscreensrdquo Shock and Vibration vol 2016 Article ID 91754177 pages 2016

[8] Y M Zhao C S Liu X M He Z Cheng-yong W Yi-binand R Zi-ting ldquoDynamic design theory and application oflarge vibrating screenrdquo Procedia Earth amp Planetary Sciencevol 1 no 1 pp 776ndash784 2009

[9] R H Su L Q Zhu and C Y Peng ldquoCAE applied to dynamicoptimal design for large-scale vibrating screenrdquo in Pro-ceedings of Acis International Symposium on Cryptographypp 316ndash320 Heibei China 2011

[10] S Baragetti ldquoInnovative structural solution for heavy loadedvibrating screensrdquo Minerals Engineering vol 84 pp 15ndash262015

[11] L-p Peng C-s Liu B-c Song J-d Wu and S WangldquoImprovement for design of beam structures in large vibratingscreen considering bending and random vibrationrdquo Journal ofCentral South University vol 22 no 9 pp 3380ndash3388 2015

[12] Z Q Wang C S Liu J Wu H Jiang B Song and Y ZhaoldquoA novel high-strength large vibrating screen with duplexstatically indeterminate mesh beam structurerdquo Journal ofVibroengineering vol 19 no 8 pp 5719ndash5734 2017

[13] 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

[14] C L Du K D Gao J Li and H Jiang ldquoDynamics behaviorresearch on variable linear vibration screen with flexiblescreen facerdquo Advances in Mechanical Engineering vol 6pp 1ndash12 2014


[15] M K Rad M Torabizadeh and A Noshadi ldquoArtificial NeuralNetwork-based fault diagnostics of an electric motor usingvibration monitoringrdquo in Proceedings of 2011 InternationalConference on Transportation Mechanical and ElectricalEngineering (TMEE) pp 1512ndash1516 IEEE ChangchunChina 2011

[16] Z G Tian ldquoAn artificial neural network method forremaining useful life prediction of equipment subject tocondition monitoringrdquo Journal of Intelligent Manufacturingvol 23 no 2 pp 227ndash237 2012

[17] V Meruane and J Mahu ldquoReal-time structural damage as-sessment using artificial neural networks and antiresonantfrequenciesrdquo Shock and Vibration vol 2014 Article ID653279 14 pages 2014

[18] Y Sun and J Y Han ldquoMonitoring method for UAV image ofgreenhouse and plastic-mulched Landcover based on deeplearningrdquo Journal of Agricultural Machinery and Machinery(in Chinese) vol 49 pp 133ndash140 2018

[19] S Zhang Y M Liang and J Y Wang ldquoApplication of BP-ANN in optical coherent tomography images classificationrdquoJournal of Optoelectronics Laser vol 23 pp 391ndash395 2012

[20] C-s Liu S-m Zhang H-p Zhou et al ldquoDynamic analysisand simulation of four-axis forced synchronizing bananavibrating screen of variable linear trajectoryrdquo Journal ofCentral South university vol 19 no 6 pp 1530ndash1536 2012

[21] Y Y Li C X He X D Song et al ldquoStudy on dynamic analysisand parameter influence of vibrating screen system of asphaltmixing equipmentrdquo Chinese Journal of Computational Me-chanics (in Chinese) vol 32 pp 565ndash570 2015

[22] W Y Li C Y Mi W G Wu et al ldquoApplication of exper-imental modal analysis technique to structural design of linearvibrating screenrdquo Journal of Coal Science amp Engineering (inChinese) vol 1 pp 88ndash90 2004

[23] Y Hou L Y Wang and J Ma ldquoFree vibration analysis ofshale shaker laminated flat-screenrdquo Journal of MechanicalEngineering vol 51 no 23 pp 60ndash65 2015

[24] S Umehara T Yamazaki and Y Sugai ldquoA precipitationestimation system based on support vector machine andneural networkrdquo Electronics and Communications in Japan(Part III Fundamental Electronic Science) vol 89 no 3pp 38ndash47 2006

[25] M Y Rafiq G Bugmann and D J Easterbrook ldquoNeuralnetwork design for engineering applicationsrdquo Computers ampStructures vol 79 no 17 pp 1541ndash1552 2001

[26] M T Hagan M Beale and M Beale Neural Network DesignChina Machine Press Beijing China 2002

[27] A Brabazon O M Neill and S McGarraghy Genetic Al-gorithm Springer Berlin Germany 2015


International Journal of

AerospaceEngineeringHindawiwwwhindawicom Volume 2018

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Hindawiwwwhindawicom Volume 2018


Active and Passive Electronic Components

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics

Hindawiwwwhindawicom

Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Control Scienceand Engineering

Journal of



Journal ofEngineeringVolume 2018

SensorsJournal of



RotatingMachinery


Modelling ampSimulationin EngineeringHindawiwwwhindawicom Volume 2018


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation


Hindawi

wwwhindawicom Volume 2018

Advances in

Multimedia

Submit your manuscripts atwwwhindawicom

process and the fatigue life is increased [9] Baragettipresented a structural solution for high loaded vibratingscreens with new modified side walls and studied the be-havior of the original and modified structure by means oftheoretical and numerical models [10] Peng et al con-ducted a systematic mechanics analysis of the beamstructures and improved the design method consideringbending and random vibration [11] Wang et al presented anovel large vibrating screen with a duplex statically in-determinate mesh beam structure +rough the modelanalysis result comparisons with the traditional vibratingscreen the superiority of this structure was verified [12]Jiang et al established a dynamic model and stabilityequations of the variable linear vibration screen and in-vestigated the motion behavior of screen face as well asconfirmed best range of exciting position [13] Du et alproposed a single-deck equal-thickness vibrating screendriven externally by an unbalanced two-axle excitationwith a large span and presented 3-degree-freedom dynamicequations [14] However the traditional two-degree-of-freedom or three-degree-of-freedom dynamic model canonly reflect the motion in plane but cannot reflect thecomplex motion in space +e traditional optimizationmethod requires iterative solution and is not suitable formultivariable nonlinear model optimization which hashigh computational cost and is not suitable for multivar-iable nonlinear model optimization

At present ANNs have become a preferred alternativeway to solve any of complex highly coupled and nonlinearproblems Rad et al constructed an expert system usedBayesian regulation back-propagation neural network andvibration monitoring data for electric motor status diagnosis[15] Tian developed a LevenbergndashMarquardt artificial neuralnetwork-based method for achieving accurate remaininguseful life prediction of equipment subject to conditionmonitoring [16] Meruane and Mahu trained the neuralnetwork using a noise-injection learning algorithm to reducethe effects of experimental noise [17] Sun and Han proposeda UAV aerial photography monitoring method based ongradient descent with the momentum neural network whichimproved the efficiency of automatic extraction and classi-fication of image features [18] Taking training time iterationtimes and error performance as indicators Zhang et alcompared the training efficiency of BFGS quasi-Newton al-gorithm resilient algorithm LM algorithm and FletcherndashReeves update algorithm for the neural network model in theclassification process of OCT images [19]

+e article is structured as follows In Section 2 thedynamic model with six degrees of freedom (6 DOFs) of thevibrating screen based on the Lagrange method wasestablished and multinatural frequency and natural modesof vibration of the vibrating screen were calculated based onrigid body modal analysis In Section 3 the finite elementmethod was used to analyze dynamic performance of thelarge vibrating screen In Section 4 the BP neural networkwas used to establish the nonlinear mapping between po-sition parameters of stiffening beams and dynamic perfor-mance of vibrating screen and the sensitivity analysis wascarried out bases on the model +e optimal combination of

structural parameters was obtained using the genetic algo-rithm Finally the conclusion is drawn in Section 5

2 Rigid Body Dynamics Analysis

21 e Structure of Vibrating Screen ZS2560 is a linearvibrating screen widely used in coal separation As shown inFigure 1 the ZS2560 linear vibrating screen is mainlycomposed of vibration exciters screen box and spring +escreen box is the main working part of the vibrating screenand accordingly it is vulnerable to the damage +e screenbox includes both left and right sides of the plate the tailgatespring support group and sill and stiffening beams In orderto adapt to heavy load conditions the screen box of ZS2560adopts the double-bottom beam structure instead of thetraditional single-layer beam structure which can achievehigher structural stiffness and lower center of mass +evibration exciters are set up on the left and right sides of theplate which can generate sinusoidal excitation force alongthe normal direction using two eccentric block synchronousreverse rotations Under the action of excitation force theentire screen box is vibrated and a sieving of the material isrealized when in operation

22 Rigid Body Dynamics Modeling At present the sim-plified 2-degree or 3-degree freedom mass-spring vibrationmodel is often used in dynamic analysis of the vibratingscreen [20 21] However the 2-degree vibration model canonly represent the linear motion of the vibrating screen intwo directions and the 3-degree freedom vibration modelcan represent the swing around a certain axis in additionand it is difficult to fully reflect the complex motion of thevibrating screen Considering the translation and rotationaround the center of mass in x-y x-z and y-z planes at thesame time this paper establishes a 6-degree freedom vi-bration model of the vibrating screen (Figure 2) wherein thescreen box is equivalent to the rigid body in space motion

As shown in Figure 2 the origin of the x-O-y coordinatesystem is located at static equilibrium position of the screenbox (the center of mass) m is the mass of the vibratingscreen when it is in operation Jx Jy and Jz are the momentsof inertia of the center of mass of the vibrating screen withrespect to x y and z axes respectively and Kx Ky and Kz arethe stiffness of springs at front and rear positions re-spectively +e distance from each spring to the center ofmass is lij where i represents the spring force direction and jrepresents the spacing direction Besides the displacementson each degree of freedom are labeled as x y and z and theangles are labeled as ψx ψy and ψz According to the me-chanical model the following equations can be obtained

Kinetic energy of the system can be expressed as

T 12

m _x2

+ _y2

+ _z2

1113872 1113873 +12

Jxx _φ2x + Jyy _φ2

y + Jzz _φ2z1113872 1113873

+ Jxy _φx _φy + Jxz _φx _φz + Jyz _φy _φz

(1)

Potential energy of the system can be expressed as


U 12





shown as followsd

dt

zT

z _q1113888 1113889minus

zT

zq+

zU

zq F (3)




mkl mlk z2T

z _qkz _ql

kkl klk z2U

zqkzql

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(4)


M

m 0 0 0 0 0

0 m 0 0 0 0

0 0 m 0 0 0

0 0 0 Jxx 0 0

0 0 0 0 Jyy 0

0 0 0 0 0 Jzz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

K


0 1113944 ky 0 1113944 kylyz 0 1113944 kylyx







2xy

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(5)

Springsupport

Localreinforcement

plate

Strengtheningbeam

Tailgate

Bottombeam

Vibrationexciter

Sideplate

Dischargeport

y Px

PxP0

P0

PyPyx

o




Meuroq + Kq F (6)





Meuroq + Kq O (7)


q Aeiλt

(8)



Kminus λ2M1113872 1113873A O (9)


Det Kminus λ2M1113872 1113873 0 (10)











Vibrationscreen

y Spring

xO

z

kz

ψz

ψy

ψx

ky

lxz = lyz

lzx = lyx

ndashlxy = ndashlzy

kx













lxz lyz 154m




5 46042 [0 0 minus01491 1 00152 0]T6 66498 [00566 00429 0 0 0 1]T

1 2 3

ndash1

0

1

1st2nd3rd

4th5th6th


4 5 6

ndash1

0

1

1st2nd3rd

4th5th6th






G(X) b2k + 1113944

NI

i1υ b1i + 1113944

NJ

j1ηijxj

⎛⎝ ⎞⎠ (11)



NJ leNJ + q

1113969+ s (12)




040204 max039379038553037728036092

035252036077

034426033601032775 min

(a)

038091 max037745037399037053036707036361036015035669035322034976 min

(b)

0634705564504782103999603217202434701652300869840008739 min

071294 max

(c)

03813 max037822037514037207036899036591036283035975035667035359 min

(d)

062081 max055214048346041479034611027744020876014009007141500027399 min

(e)

080053 max0711680622830533980445130356270267420178570089717000086521 min

(f )












10217 max0909190796720684250571780459303468302343601218900094136 min

(a)

12955 max115161007708637307198057587043194028801014407000014222 min

(b)

092434 max082165071895061626051357041087030818020549010279000010089 min

(c)

084769 max075643066516057390482640391380300110208850117590026322 min

(d)









Beam positions

Elastic deformation

(a)

Inputlayer

Hiddenlayer

Outputlayer

Output data

Input data

Input data

Input data

(b)


f_x

e_xo

a_y

y

a_x

b_xc_x

d_x

b_y

c_y

e_y

d_y

f_y

x



2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3

560

561

562

563

564

565

566

Aver

age v

alue

of e

lasti

c def

orm

atio

n (m

m)

Orthogonal level

(a)

1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3556

558

560

562

564

566

568

Aver

age v

alue

of e

lasti

c def

orm

atio

n (m

m)




Find X x1 x2 x121113864 1113865T

min G(X)

st xLi lexi lexU

i

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(13)










zG

zxi


Δxi

(14)







0 2 4 6 8 10 12 14 16 18 20ndash05

00

05

10

5

6

Elas

tic d

efor

mat

ion

(mm

)







design

FEM analysis



Geneticalgorithm

optimization


range analysis


1 2 3 4 5 6 7ndash05

00

05

10

5

6

Elas

tic d

efor

mat

ion

(mm

)







5 Conclusion





i 1 2 3 4 5 6 7 8 9 10 11 12xL

i (mm) 800 1200 17224 1596 4020 1705 5107 1219 900 500 1100 120xU

i (mm) 1200 1600 21224 1996 4620 2005 5607 1619 1400 800 1500 240

ndash014

ndash012

ndash010

ndash008

ndash006

ndash004

ndash002

000

002

004

006

Sens

itivi

ty





0 20 40 60 80 100 120 140 160 180 20052

53

54

55

56

57

58

59

60

Max

imal

elas

tic d

efor

mat

ion

(mm

)





Optimal value(mm)

Round(mm)






Nomenclature











Data Availability




Acknowledgments


References
































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


U 12





shown as followsd

dt

zT

z _q1113888 1113889minus

zT

zq+

zU

zq F (3)




mkl mlk z2T

z _qkz _ql

kkl klk z2U

zqkzql

⎧⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎩

(4)


M

m 0 0 0 0 0

0 m 0 0 0 0

0 0 m 0 0 0

0 0 0 Jxx 0 0

0 0 0 0 Jyy 0

0 0 0 0 0 Jzz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

K


0 1113944 ky 0 1113944 kylyz 0 1113944 kylyx







2xy

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(5)

Springsupport

Localreinforcement

plate

Strengtheningbeam

Tailgate

Bottombeam

Vibrationexciter

Sideplate

Dischargeport

y Px

PxP0

P0

PyPyx

o




Meuroq + Kq F (6)





Meuroq + Kq O (7)


q Aeiλt

(8)



Kminus λ2M1113872 1113873A O (9)


Det Kminus λ2M1113872 1113873 0 (10)











Vibrationscreen

y Spring

xO

z

kz

ψz

ψy

ψx

ky

lxz = lyz

lzx = lyx

ndashlxy = ndashlzy

kx













lxz lyz 154m




5 46042 [0 0 minus01491 1 00152 0]T6 66498 [00566 00429 0 0 0 1]T

1 2 3

ndash1

0

1

1st2nd3rd

4th5th6th


4 5 6

ndash1

0

1

1st2nd3rd

4th5th6th






G(X) b2k + 1113944

NI

i1υ b1i + 1113944

NJ

j1ηijxj

⎛⎝ ⎞⎠ (11)



NJ leNJ + q

1113969+ s (12)




040204 max039379038553037728036092

035252036077

034426033601032775 min

(a)

038091 max037745037399037053036707036361036015035669035322034976 min

(b)

0634705564504782103999603217202434701652300869840008739 min

071294 max

(c)

03813 max037822037514037207036899036591036283035975035667035359 min

(d)

062081 max055214048346041479034611027744020876014009007141500027399 min

(e)

080053 max0711680622830533980445130356270267420178570089717000086521 min

(f )












10217 max0909190796720684250571780459303468302343601218900094136 min

(a)

12955 max115161007708637307198057587043194028801014407000014222 min

(b)

092434 max082165071895061626051357041087030818020549010279000010089 min

(c)

084769 max075643066516057390482640391380300110208850117590026322 min

(d)









Beam positions

Elastic deformation

(a)

Inputlayer

Hiddenlayer

Outputlayer

Output data

Input data

Input data

Input data

(b)


f_x

e_xo

a_y

y

a_x

b_xc_x

d_x

b_y

c_y

e_y

d_y

f_y

x



2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3

560

561

562

563

564

565

566

Aver

age v

alue

of e

lasti

c def

orm

atio

n (m

m)

Orthogonal level

(a)

1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3556

558

560

562

564

566

568

Aver

age v

alue

of e

lasti

c def

orm

atio

n (m

m)




Find X x1 x2 x121113864 1113865T

min G(X)

st xLi lexi lexU

i

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(13)










zG

zxi


Δxi

(14)







0 2 4 6 8 10 12 14 16 18 20ndash05

00

05

10

5

6

Elas

tic d

efor

mat

ion

(mm

)







design

FEM analysis



Geneticalgorithm

optimization


range analysis


1 2 3 4 5 6 7ndash05

00

05

10

5

6

Elas

tic d

efor

mat

ion

(mm

)







5 Conclusion





i 1 2 3 4 5 6 7 8 9 10 11 12xL

i (mm) 800 1200 17224 1596 4020 1705 5107 1219 900 500 1100 120xU

i (mm) 1200 1600 21224 1996 4620 2005 5607 1619 1400 800 1500 240

ndash014

ndash012

ndash010

ndash008

ndash006

ndash004

ndash002

000

002

004

006

Sens

itivi

ty





0 20 40 60 80 100 120 140 160 180 20052

53

54

55

56

57

58

59

60

Max

imal

elas

tic d

efor

mat

ion

(mm

)





Optimal value(mm)

Round(mm)






Nomenclature











Data Availability




Acknowledgments


References
































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



Meuroq + Kq F (6)





Meuroq + Kq O (7)


q Aeiλt

(8)



Kminus λ2M1113872 1113873A O (9)


Det Kminus λ2M1113872 1113873 0 (10)











Vibrationscreen

y Spring

xO

z

kz

ψz

ψy

ψx

ky

lxz = lyz

lzx = lyx

ndashlxy = ndashlzy

kx













lxz lyz 154m




5 46042 [0 0 minus01491 1 00152 0]T6 66498 [00566 00429 0 0 0 1]T

1 2 3

ndash1

0

1

1st2nd3rd

4th5th6th


4 5 6

ndash1

0

1

1st2nd3rd

4th5th6th






G(X) b2k + 1113944

NI

i1υ b1i + 1113944

NJ

j1ηijxj

⎛⎝ ⎞⎠ (11)



NJ leNJ + q

1113969+ s (12)




040204 max039379038553037728036092

035252036077

034426033601032775 min

(a)

038091 max037745037399037053036707036361036015035669035322034976 min

(b)

0634705564504782103999603217202434701652300869840008739 min

071294 max

(c)

03813 max037822037514037207036899036591036283035975035667035359 min

(d)

062081 max055214048346041479034611027744020876014009007141500027399 min

(e)

080053 max0711680622830533980445130356270267420178570089717000086521 min

(f )












10217 max0909190796720684250571780459303468302343601218900094136 min

(a)

12955 max115161007708637307198057587043194028801014407000014222 min

(b)

092434 max082165071895061626051357041087030818020549010279000010089 min

(c)

084769 max075643066516057390482640391380300110208850117590026322 min

(d)









Beam positions

Elastic deformation

(a)

Inputlayer

Hiddenlayer

Outputlayer

Output data

Input data

Input data

Input data

(b)


f_x

e_xo

a_y

y

a_x

b_xc_x

d_x

b_y

c_y

e_y

d_y

f_y

x



2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3

560

561

562

563

564

565

566

Aver

age v

alue

of e

lasti

c def

orm

atio

n (m

m)

Orthogonal level

(a)

1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3556

558

560

562

564

566

568

Aver

age v

alue

of e

lasti

c def

orm

atio

n (m

m)




Find X x1 x2 x121113864 1113865T

min G(X)

st xLi lexi lexU

i

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(13)










zG

zxi


Δxi

(14)







0 2 4 6 8 10 12 14 16 18 20ndash05

00

05

10

5

6

Elas

tic d

efor

mat

ion

(mm

)







design

FEM analysis



Geneticalgorithm

optimization


range analysis


1 2 3 4 5 6 7ndash05

00

05

10

5

6

Elas

tic d

efor

mat

ion

(mm

)







5 Conclusion





i 1 2 3 4 5 6 7 8 9 10 11 12xL

i (mm) 800 1200 17224 1596 4020 1705 5107 1219 900 500 1100 120xU

i (mm) 1200 1600 21224 1996 4620 2005 5607 1619 1400 800 1500 240

ndash014

ndash012

ndash010

ndash008

ndash006

ndash004

ndash002

000

002

004

006

Sens

itivi

ty





0 20 40 60 80 100 120 140 160 180 20052

53

54

55

56

57

58

59

60

Max

imal

elas

tic d

efor

mat

ion

(mm

)





Optimal value(mm)

Round(mm)






Nomenclature











Data Availability




Acknowledgments


References
































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia












lxz lyz 154m




5 46042 [0 0 minus01491 1 00152 0]T6 66498 [00566 00429 0 0 0 1]T

1 2 3

ndash1

0

1

1st2nd3rd

4th5th6th


4 5 6

ndash1

0

1

1st2nd3rd

4th5th6th






G(X) b2k + 1113944

NI

i1υ b1i + 1113944

NJ

j1ηijxj

⎛⎝ ⎞⎠ (11)



NJ leNJ + q

1113969+ s (12)




040204 max039379038553037728036092

035252036077

034426033601032775 min

(a)

038091 max037745037399037053036707036361036015035669035322034976 min

(b)

0634705564504782103999603217202434701652300869840008739 min

071294 max

(c)

03813 max037822037514037207036899036591036283035975035667035359 min

(d)

062081 max055214048346041479034611027744020876014009007141500027399 min

(e)

080053 max0711680622830533980445130356270267420178570089717000086521 min

(f )












10217 max0909190796720684250571780459303468302343601218900094136 min

(a)

12955 max115161007708637307198057587043194028801014407000014222 min

(b)

092434 max082165071895061626051357041087030818020549010279000010089 min

(c)

084769 max075643066516057390482640391380300110208850117590026322 min

(d)









Beam positions

Elastic deformation

(a)

Inputlayer

Hiddenlayer

Outputlayer

Output data

Input data

Input data

Input data

(b)


f_x

e_xo

a_y

y

a_x

b_xc_x

d_x

b_y

c_y

e_y

d_y

f_y

x



2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3

560

561

562

563

564

565

566

Aver

age v

alue

of e

lasti

c def

orm

atio

n (m

m)

Orthogonal level

(a)

1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3556

558

560

562

564

566

568

Aver

age v

alue

of e

lasti

c def

orm

atio

n (m

m)




Find X x1 x2 x121113864 1113865T

min G(X)

st xLi lexi lexU

i

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(13)










zG

zxi


Δxi

(14)







0 2 4 6 8 10 12 14 16 18 20ndash05

00

05

10

5

6

Elas

tic d

efor

mat

ion

(mm

)







design

FEM analysis



Geneticalgorithm

optimization


range analysis


1 2 3 4 5 6 7ndash05

00

05

10

5

6

Elas

tic d

efor

mat

ion

(mm

)







5 Conclusion





i 1 2 3 4 5 6 7 8 9 10 11 12xL

i (mm) 800 1200 17224 1596 4020 1705 5107 1219 900 500 1100 120xU

i (mm) 1200 1600 21224 1996 4620 2005 5607 1619 1400 800 1500 240

ndash014

ndash012

ndash010

ndash008

ndash006

ndash004

ndash002

000

002

004

006

Sens

itivi

ty





0 20 40 60 80 100 120 140 160 180 20052

53

54

55

56

57

58

59

60

Max

imal

elas

tic d

efor

mat

ion

(mm

)





Optimal value(mm)

Round(mm)






Nomenclature











Data Availability




Acknowledgments


References
































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




G(X) b2k + 1113944

NI

i1υ b1i + 1113944

NJ

j1ηijxj

⎛⎝ ⎞⎠ (11)



NJ leNJ + q

1113969+ s (12)




040204 max039379038553037728036092

035252036077

034426033601032775 min

(a)

038091 max037745037399037053036707036361036015035669035322034976 min

(b)

0634705564504782103999603217202434701652300869840008739 min

071294 max

(c)

03813 max037822037514037207036899036591036283035975035667035359 min

(d)

062081 max055214048346041479034611027744020876014009007141500027399 min

(e)

080053 max0711680622830533980445130356270267420178570089717000086521 min

(f )












10217 max0909190796720684250571780459303468302343601218900094136 min

(a)

12955 max115161007708637307198057587043194028801014407000014222 min

(b)

092434 max082165071895061626051357041087030818020549010279000010089 min

(c)

084769 max075643066516057390482640391380300110208850117590026322 min

(d)









Beam positions

Elastic deformation

(a)

Inputlayer

Hiddenlayer

Outputlayer

Output data

Input data

Input data

Input data

(b)


f_x

e_xo

a_y

y

a_x

b_xc_x

d_x

b_y

c_y

e_y

d_y

f_y

x



2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3

560

561

562

563

564

565

566

Aver

age v

alue

of e

lasti

c def

orm

atio

n (m

m)

Orthogonal level

(a)

1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3556

558

560

562

564

566

568

Aver

age v

alue

of e

lasti

c def

orm

atio

n (m

m)




Find X x1 x2 x121113864 1113865T

min G(X)

st xLi lexi lexU

i

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(13)










zG

zxi


Δxi

(14)







0 2 4 6 8 10 12 14 16 18 20ndash05

00

05

10

5

6

Elas

tic d

efor

mat

ion

(mm

)







design

FEM analysis



Geneticalgorithm

optimization


range analysis


1 2 3 4 5 6 7ndash05

00

05

10

5

6

Elas

tic d

efor

mat

ion

(mm

)







5 Conclusion





i 1 2 3 4 5 6 7 8 9 10 11 12xL

i (mm) 800 1200 17224 1596 4020 1705 5107 1219 900 500 1100 120xU

i (mm) 1200 1600 21224 1996 4620 2005 5607 1619 1400 800 1500 240

ndash014

ndash012

ndash010

ndash008

ndash006

ndash004

ndash002

000

002

004

006

Sens

itivi

ty





0 20 40 60 80 100 120 140 160 180 20052

53

54

55

56

57

58

59

60

Max

imal

elas

tic d

efor

mat

ion

(mm

)





Optimal value(mm)

Round(mm)






Nomenclature











Data Availability




Acknowledgments


References
































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia









10217 max0909190796720684250571780459303468302343601218900094136 min

(a)

12955 max115161007708637307198057587043194028801014407000014222 min

(b)

092434 max082165071895061626051357041087030818020549010279000010089 min

(c)

084769 max075643066516057390482640391380300110208850117590026322 min

(d)









Beam positions

Elastic deformation

(a)

Inputlayer

Hiddenlayer

Outputlayer

Output data

Input data

Input data

Input data

(b)


f_x

e_xo

a_y

y

a_x

b_xc_x

d_x

b_y

c_y

e_y

d_y

f_y

x



2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3

560

561

562

563

564

565

566

Aver

age v

alue

of e

lasti

c def

orm

atio

n (m

m)

Orthogonal level

(a)

1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3556

558

560

562

564

566

568

Aver

age v

alue

of e

lasti

c def

orm

atio

n (m

m)




Find X x1 x2 x121113864 1113865T

min G(X)

st xLi lexi lexU

i

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(13)










zG

zxi


Δxi

(14)







0 2 4 6 8 10 12 14 16 18 20ndash05

00

05

10

5

6

Elas

tic d

efor

mat

ion

(mm

)







design

FEM analysis



Geneticalgorithm

optimization


range analysis


1 2 3 4 5 6 7ndash05

00

05

10

5

6

Elas

tic d

efor

mat

ion

(mm

)







5 Conclusion





i 1 2 3 4 5 6 7 8 9 10 11 12xL

i (mm) 800 1200 17224 1596 4020 1705 5107 1219 900 500 1100 120xU

i (mm) 1200 1600 21224 1996 4620 2005 5607 1619 1400 800 1500 240

ndash014

ndash012

ndash010

ndash008

ndash006

ndash004

ndash002

000

002

004

006

Sens

itivi

ty





0 20 40 60 80 100 120 140 160 180 20052

53

54

55

56

57

58

59

60

Max

imal

elas

tic d

efor

mat

ion

(mm

)





Optimal value(mm)

Round(mm)






Nomenclature











Data Availability




Acknowledgments


References
































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia






Beam positions

Elastic deformation

(a)

Inputlayer

Hiddenlayer

Outputlayer

Output data

Input data

Input data

Input data

(b)


f_x

e_xo

a_y

y

a_x

b_xc_x

d_x

b_y

c_y

e_y

d_y

f_y

x



2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3

560

561

562

563

564

565

566

Aver

age v

alue

of e

lasti

c def

orm

atio

n (m

m)

Orthogonal level

(a)

1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3556

558

560

562

564

566

568

Aver

age v

alue

of e

lasti

c def

orm

atio

n (m

m)




Find X x1 x2 x121113864 1113865T

min G(X)

st xLi lexi lexU

i

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(13)










zG

zxi


Δxi

(14)







0 2 4 6 8 10 12 14 16 18 20ndash05

00

05

10

5

6

Elas

tic d

efor

mat

ion

(mm

)







design

FEM analysis



Geneticalgorithm

optimization


range analysis


1 2 3 4 5 6 7ndash05

00

05

10

5

6

Elas

tic d

efor

mat

ion

(mm

)







5 Conclusion





i 1 2 3 4 5 6 7 8 9 10 11 12xL

i (mm) 800 1200 17224 1596 4020 1705 5107 1219 900 500 1100 120xU

i (mm) 1200 1600 21224 1996 4620 2005 5607 1619 1400 800 1500 240

ndash014

ndash012

ndash010

ndash008

ndash006

ndash004

ndash002

000

002

004

006

Sens

itivi

ty





0 20 40 60 80 100 120 140 160 180 20052

53

54

55

56

57

58

59

60

Max

imal

elas

tic d

efor

mat

ion

(mm

)





Optimal value(mm)

Round(mm)






Nomenclature











Data Availability




Acknowledgments


References
































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


Find X x1 x2 x121113864 1113865T

min G(X)

st xLi lexi lexU

i

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

(13)










zG

zxi


Δxi

(14)







0 2 4 6 8 10 12 14 16 18 20ndash05

00

05

10

5

6

Elas

tic d

efor

mat

ion

(mm

)







design

FEM analysis



Geneticalgorithm

optimization


range analysis


1 2 3 4 5 6 7ndash05

00

05

10

5

6

Elas

tic d

efor

mat

ion

(mm

)







5 Conclusion





i 1 2 3 4 5 6 7 8 9 10 11 12xL

i (mm) 800 1200 17224 1596 4020 1705 5107 1219 900 500 1100 120xU

i (mm) 1200 1600 21224 1996 4620 2005 5607 1619 1400 800 1500 240

ndash014

ndash012

ndash010

ndash008

ndash006

ndash004

ndash002

000

002

004

006

Sens

itivi

ty





0 20 40 60 80 100 120 140 160 180 20052

53

54

55

56

57

58

59

60

Max

imal

elas

tic d

efor

mat

ion

(mm

)





Optimal value(mm)

Round(mm)






Nomenclature











Data Availability




Acknowledgments


References
































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




5 Conclusion





i 1 2 3 4 5 6 7 8 9 10 11 12xL

i (mm) 800 1200 17224 1596 4020 1705 5107 1219 900 500 1100 120xU

i (mm) 1200 1600 21224 1996 4620 2005 5607 1619 1400 800 1500 240

ndash014

ndash012

ndash010

ndash008

ndash006

ndash004

ndash002

000

002

004

006

Sens

itivi

ty





0 20 40 60 80 100 120 140 160 180 20052

53

54

55

56

57

58

59

60

Max

imal

elas

tic d

efor

mat

ion

(mm

)





Optimal value(mm)

Round(mm)






Nomenclature











Data Availability




Acknowledgments


References
































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





Nomenclature











Data Availability




Acknowledgments


References
































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


















RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


dynamic modeling and parameters optimization of large...

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