Multi-Object

8/13/2019 Multi-Object

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Received 29 September 2007; accepted 09 December 2007

Project 50574091 supported by the National Natural Science Foundation of China

Corresponding author. Tel: +86-516-82841479; E-mail address: [email protected]

Multi-object optimization design for differential and gradingtoothed roll crusher using a genetic algorithm

ZHAO La-la, WANG Zhong-bin, ZANG FengSchool of Mechanical and Electrical Engineering, China University of Mining and Technology,Xuzhou,Jiangsu 221008,China

Abstract: Our differential and grading toothed roll crusher blends the advantages of a toothed roll crusher and a jaw crusher andpossesses characteristics of great crushing, high breaking efficiency, multi-sieving and has, for the moment, made up for the short-comings of the toothed roll crusher. The moving jaw of the crusher is a crank-rocker mechanism. For optimizing the dynamic per-formance and improving the cracking capability of the crusher, a mathematical model was established to optimize the transmissionangle and to minimize the travel characteristic value m of the moving jaw. Genetic algorithm is used to optimize the crushercrank-rocker mechanism for multi-object design and an optimum result is obtained. According to the implementation, it is shownthat the performance of the crusher and the cracking capability of the moving jaw have been improved.Key words:differential and grading toothed roll crusher; crank-rocker mechanism; genetic algorithm; multi-object optimization

1 Introduction

Material crushing is an indispensable process forproduction in many industries (e.g. mining, metal-

lurgy, chemical industry). Traditional crushers (e.g.,jaw crusher, impact hammer crusher, rotary crusherand hammer mill) mainly depend on the workingparts that put the impact pressure on the materiel tobe crushed in order to implement crushing. Crushersare of unwieldiness, low efficiency and high energyconsumption. These disadvantages have greatly re-stricted efforts to improve their crushing capability.The traditional crushers we already have cannot meetthe current needs of production. In recent years, anumber of British MMD toothed roll crushers havebeen used. But from casual investigation, we find

some shortcomings in these toothed roll crushers: 1)Material crushing is realized by meshing teeth. Allthe raw minerals to be crushed are sent into a crush-ing chamber and discharged by force through themeshing teeth including those up to the standard ofparticulates. The crushers, failing to accomplish realgrading crushing, using up a lot of energy, are ineffi-cient and the crushing teeth quickly show damagefrom metal fatigue. 2) The phenomenon of jam oc-curs under two conditions. One is a high flow inwhich large chunks of coal are mixed with smallpieces. The other one is high humidity in which teethbecome clogged with wet coal. Because the toothed

roll crusher has no effective grading mechanism, theresult is unsatisfactory and we cannot depend solelyon the overworked meshing teeth. According to in-vestigations in a number of coal mines using MMD

toothed roll crushers, we found that none of themcould even reach the nominal amount of crushed ma-terial. 3) The crushing capability of MMD crushers isimproved by increasing the power and strength oftransmission parts, resulting in high power consump-tion and costs. Based on these considerations, com-bined with the demand of the China Shenhua EnergyCo., Ltd. and the Shendong Coal Branch, our researchteam designed a new, highly efficient differential andgrading toothed roll crusher to make up the lacksfrom traditional crushers[1].

As Fig. 1 shows, the crusher breaking part is com-

posed of differential teeth and a crank-rocker mecha-nism. A sketch of the moving jaw crank-rockermechanism is shown in Fig. 2. In a search for opti-mization designs of a crank-rocker mechanism, itseems that much of the literature consulted aims onlyat optimizing the transmission angle . For example,Li, et al just opted for minimizing the travel charac-teristic value of the moving jaw m of the jawcrusher[23]. But in a practical and typical crushingprocess, the capability of the moving jaw is closelyrelated to both the transmission angle and the travelcharacteristic value of the moving jaw. We have useda GA (genetic algorithm) in order to optimize the

J China Univ Mining & Technol 18 (2008) 03160320

JOURNAL OF CHINA UNIVERSITY OF

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TECHNOLOGY

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ZHAO La-la et al Multi-object optimization design for differential and grading toothed roll 317

crank-rocker mechanism of the differential and grad-ing toothed roll crusher for the purpose of optimizingthe transmission angle and minimizing the travelcharacteristic value of the moving jaw m.

1.Motor assembly of sieving mechanism; 2.Roller screen axis; 3.Drivinggears; 4.Breaking teeth; 5. Moving jaw; 6.Adjustment mechanism; 7.Frame;

8. Motor assembly of crankshaft; 9.Motor assembly of toothed roll axis

Fig. 1 General structure drawing of differentialand grading toothed roll crusher

Fig. 2 Breaking mechanism sketch of differentialand grading toothed roll crusher

2 Kinetic analysis of the crank-rocker

mechanism

The performance of a hinged four-bar mechanismdepends on the relative length of its bars. If we set thelength of the rocker c equal to 1, then the relativelengths of the crank, the connecting rod and the bodyframe are a, band d. The mathematical model of thedesign of the crank-rocker mechanism is independentof the actual length, which makes it more universal[4].

2.1 Optimal transmission angle

The transmission force of the crank-rocker mecha-nism depends mainly on the transmission angle . Thebigger the transmission angle, the better the transmis-sion force property it has. The transmission anglechanges during the process of transmission. The

choice of a suitable size for each component can op-timize the minimum transmission angle. Therefore, itis necessary to increase the transmission angle in or-der to improve the transmission force and crushing

efficiency.Two possible positions where the minimal trans-

mission angle minmay appear are shown in Fig. 3:

min 1 2min( , 180 ) = (1)

where

)}2/(])(arccos{[ 2221 bcadcb += (2)

)}2/(])(arccos{[ 2222 bcadcb ++= (3)

Fig. 3 Collinear crank and body frame

2.2 Travel characteristic value mof moving jaw

The travel of the moving jaw is divided into ahorizontal part sand a vertical part h. The function ofthe horizontal travel is to crush material, but that ofthe vertical travel can not help crushing but also canintensify the abrasion of the jaw. Diminishing thevalue of m can both reduce the energy consumptionand the abrasion and simultaneously, it can improveproductivity and increase the crushing ratio[2]. Somesteps should be taken to decrease the value of m.

In order to simplify the calculation, point C istaken out for analysis. The geometric relationshipsshown in Fig. 4 are:

Fig. 4 Collinear crank and connecting rod

2 2( ) 2( ) cosd b a c b a c = + (4)


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1

sinarcsin( )

cC AD

d

= (5)

2 2 2

2

( )arccos

2( )

a b d cC AD

a b d

+ + =

+ (6)

2 1( )sin ( )sin( )s b a a b C AD C AD = + +

(7)

2 1( ) cos( ) ( )cosh a b C AD C AD b a = + +

(8)So, the travel characteristic value distance of point

Cis:m=h/s; the angle of the toggle plate is: =+90; the swing angle of the toggle plate is:

22arcsin( 1 2 )s m c = +

where is the dip angle of the connecting rod whosemagnitude depends on the mechanism and thestrength of the moving jaw[3]. Its value ranges usuallybetween 15 to 20. s is the horizontal travel and hthe vertical travel.

3 Mathematical model of differential and

grading toothed roll crusher moving jaw

Depending on the kinetic analysis above, both thetransmission angle and the travel characteristicvalue of point C can be expressed by the relativelength of the crank, the connecting rod and the bodyframe of the crank-rocker mechanism. The designvariables are:

T T1 2 3[ , , ] [ , , ]x x x a b d= =X

The objective function for maximizing the minimaltransmission angle of the moving jaw is:

1 1 2( ) min( , 180 ) maxf x =

The objective function for minimizing the travel

characteristic valuemof the moving jaw is:

max1

)(2 ==h

s

mxf

The following constraints apply:1) Boundary constraint conditions of design vari-ables:

max max ( 1, 2, 3)i i ia x b i =

where aimaxand bimaxare the upper and lower limitsof design variables.

2) Conditions for the crank-rocker mechanism topossess a crank:

ba , 1a , da , dba ++ 1 ,dba ++ 1 , 1++ bda

3) Constraints of the transmission angle: Increasingthe transmission angle can improve the transmission

efficiency and augment the horizontal travel. But toolarge a transmission angle has the reverse effect onthe stress of the moving jaw and the principal axis[3].

A common condition is that:oo 5545

4) Constraints of the travel characteristic value:according to experience, the range of m usually isbetween 1.5 and 2.5.

5) Travel of moving jaw: the horizontal travel shasan obvious effect on productivity. If s is too small itwould reduce the productivity, but in contrast, it willintensify the crushing force and lead to damage byoverloading the equipment[5]. The constraint of thehorizontal travel then becomes:

min(0.3 0.4)s d

where dmin is the minimum dimension of the dis-charge port.

6) Angle constraints of the toggle plate. Usually,the range of is 18 to 23.

In summary, the optimization problem of thecrank-rocker mechanism can be boiled down to thefollowing double programming objective:

T1 2max ( ) ( ( ), ( ))f x f x f x

=X R

(9)

where T1{ | ( ) ( ( ), ..., ( )) 0}n

mx E g x g x g x= = R .

The traditional solutions of multiple objective op-timization problems achieve low efficiency and caneasily lead to an apparent local optimum. But thesearching method of GA is iteration in which the pos-sibility of running into a local, optimal solution isreduced and the process of solution is accelerated bythe method of disposing of many individuals syn-chronously in the population. By using probabletransmission rules to guide searching direction, GAhas no special requirement about searching space (e.g.connectedness, convexity, etc.) and does not need anyadditional information. We have applied GA to opti-mize the crank-rocker mechanism of the differentialand grading toothed roll crusher as a multi-objectdesign.

4 Genetic algorithm

Genetic algorithm is a randomized searchingmethod, which imitates natural, evolutionary laws.GA was, for the first time, presented by Holland in1975 with the following main features: it operates onthe structured object directly, without any restrictionin functional derivation and continuity; it possessesan implicit parallelism and improved capability ofglobal optimization; by using a stochastic optimizingmethod, it can automatically obtain and guide thesearching space for optimization and it can also adjustthe search direction adaptively without specificrules[67]. GA has, of late, been an important technol-ogy in intelligent computing. The main approaches of

GA are: 1) Encoding: before searching in the data ofthe solution space, GA expresses the data as a geno-typic string structure and various combinations can


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ZHAO La-la et al Multi-object optimization design for differential and grading toothed roll 319

obtain different points. 2) Creating an initialized ge-nus group: GA randomly generates N initializedstring data, in which each datum is called an individ-ual and these individuals form a genus group. GAstarts the iteration by using the string data as an initial

point. 3) Estimating individual adaptability: the valueof the adaptability function indicates the quality of anindividual or the solution. For different problems, thedefinition of an adaptability function is different. 4)Selection: the purpose of selection is to choose ex-cellent individuals among present genus group andpermit them the chance to propagate offspring asparents. Selection is the embodiment of this thoughtfor GA. The principle of the selection is that indi-viduals with high adaptability will have a largerprobability to contribute one or more offspring. 5)Crossover: this is the most important operation forGA. Crossover operation can generate new genera-tions that possess the characteristics of the previousgeneration. It embodies the thought of informationexchange. 6) Mutation: at first, GA selects randomlyan individual from the genus group. It then changesthe value of one datum from the string data with agiven probability for the selected individual. Similarto the biological universe, the probability of mutationfor GA is also very low; usually its value ranges from0.001 to 0.01, i.e., mutation provides an opportunityto generate new individuals.

5 Multi-object optimization based on ge-

netic algorithm

We took the crank-rocker mechanism of the dif-ferential and grading toothed roll crusher with acrushing capacity of 4000 t/h as our optimizing object.In order to decide the length of the crank, the con-necting rod and the body frame of the crank-rockermechanism when the objective function (9) obtainsits maximum value, under the condition that the valueof the travel speed ratio coefficient is 1.25, the con-necting rod is 300 mm and the dip angle of the con-necting rod 18.

We used GA to solve this problem by setting thegenus group scale equal to 50, the crossover probabil-ity at 0.8, the mutation probability at 0.005 and thenumber of generations in the evolution at 1000.

The traditional binary coding method is compara-tively convenient when used in a theoretical analysis.But for multidimensional and high precision numeri-cal problems, it tends to low efficiency and inaccu-racy[8]. We used a natural number coding method toset the three variables a, b and d as genes, whichcombine orderly into a chromosome[9]. For example,

p and q are two in-

dividuals.During initialization, GA will generate a genus

group in which 50 individuals are produced on thebasis of a variable range. We chose Eq.(9) as total

objective function whose value denotes individualadaptability. We took individualspand qas examplesto be substituted in Eq.(9), i.e., f(p)=0.8145,

f(q)=0.7887, f(p)>f(q). This indicates that the indi-vidual adaptability ofp is better than that of q.

A roulette method is used in the selection operation,in which we imitate a game of roulette and calculatethe total adaptability and at the same time the relativeand accumulative adaptability for each individual.Then the roulette is turned for 50 times and a randomnumber is created between 0 and 1 for each time. Anindividual can be selected by comparing the numberwith the accumulative adaptability for each individual.For example, if we let rbe a random number, thenfc(i)is the accumulative adaptability of individual i and

fc(i+1) is the accumulative adaptability of individuali+1. If fc(i) r fc(i+1), then individual i+1 will bechosen. By analogy, for all the selected individualswe can compose a new genus group and carry outcrossover and mutation operations.

A single point crossing method is used in crossoveroperations, generating a random number between 0and 1 when searching in the genus group. If thenumber is less than the crossover probability and ithas selected an even number of individuals, then thecrossover operation can be implemented after pairingrandomly. Taking the individuals pand q as a pairedexample, we set the crossing point as 2, meaning thatwe choose the second gene to cross over. After thecrossover operation, the individuals become

p andq .A uniform mutation method is adopted in this mu-

tation operation, generating a random number be-tween 0 and 1 when searching in the genus group. Ifthe number is less than the probability of mutation,the current individual will mutate. The mutation op-eration is similar to initialization in which we rebuildgenes for an individual by the boundary of designvariables.

After selection, crossover and mutation operations,an evaluation function is called to insure that the bestindividual can be preserved. The optimization results

(by conversion) are shown in Table 1.Table 1 Result of genetic optimization

Optimizationvariables

a (mm) b (mm) d (mm) min () l(m)

Optimal values 10.07 690.54 788.01 52.11 0.48

Roundingvalues

10 691 788 52 0.5

The optimization results, which have been appliedin production practice, can satisfy the constraint con-ditions. The Zhengzhou Great Wall MetallurgyEquipment Factory, which cooperated with our re-

search team, has produced corresponding crusherswhich have been sold to a number of mines in China.According to the actual working situation, the differ-ential and grading toothed roll crusher possesses the


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characteristics of great crushing strength, high break-ing efficiency, stable working state, high capability ofclearing blockage and anti-blocking and stablegranularity of crushing products, all of which havemet their expected design target.

6 Conclusions

We have used genetic algorithm to carry out amulti-object optimized design for the breakingmechanism of a differential and grading toothed rollcrusher by selection, crossover and mutation opera-tions under certain constrained conditions. On thebasis of this presentation, we may state the following:

1) Our differential and grading toothed roll crusherblends the advantages of a toothed roll crusher and a

jaw crusher and possesses characteristics of great

crushing, high breaking efficiency, multi-sieving and,for the moment, has made up for the shortcomings ofthe toothed roll crusher.

2) Since it is different from traditional optimizingmethods, GA can carry out heuristic global optimiza-tion. It is a parallel, concurrent and gradually evolu-tion searching process in which a local optimum isavoided.

3) The optimization results have been applied inpractice and the actual crusher working state is stable.The industrial application has been proven to goodeffect. It has also been shown that, in order to opti-mize the crusher crank-rocker mechanism for

multi-object design with optimizing transmission an-gle and minimizing the travel characteristic ofmoving jaw mas objective functions, can obtain op-timum results.

Acknowledgements

The authors wish to express their most sincere ap-preciation to Prof. Huang Jiaxing for giving valuableadvice. The authors are also indebted to the Zheng-

zhou Great Wall Metallurgy Equipment Factory forproviding useful material.

References

[1] Zhao L L, Zang F, Wang Z B, et al. Design and motionsimulation for differential and grading toothed rollcrusher. Coal Mine Machinery, 2007, 28(6): 1921. (InChinese)

[2] Li X, Wang G B. Optimum design of compound pendu-lum jaw crusher with genetic algorithms. ConstructionMachinery, 2006(6): 5562. (In Chinese)

[3] Shen Y J. Optimum design for structure parameters of

PE250

750 compound pendulum jaw crusher.Machin-ery, 1994, 21(4): 2325. (In Chinese)[4] Zheng S H. Application of genetic algorithm in optimum

design of planar crank and rocker mechanisms. Journalof China Institute of Metrology, 1999(2): 7174. (InChinese)

[5] Wang P. Optimum design for crank-rocker mechanism ofcrusher.Journal of South China Construction University,1998, 6(2): 6065. (In Chinese)

[6] Holland J H. Adaptation in Natural and Artificial Sys-tems. Michigan: The University of Michigan Press,1975.

[7] Pan F P, Gong D W, Sun X Y, et al. Research on noveladaptive genetic algorithm. Journal of China Universityof Mining & Technology, 2003, 32(1): 6870. (In Chi-

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