MATHEMATICAL MODEL FOR ESTABLISHING THE VIBRATING CONDITIONS SPECIFIC TO ELECTROMAGNETIC SIEVES

Transcription

1 THE INTERNATIONAL CONFERENCE OF THE CARPATHIAN EURO-REGION SPECIALISTS IN INDUSTRIAL SYSTEMS 6 th edition MATHEMATICAL MODEL FOR ESTABLISHING THE VIBRATING CONDITIONS SPECIFIC TO ELECTROMAGNETIC SIEVES Mircea Horgos Mihai Banica, Dinu Stoicovici 3 Lecturer Eng.,,3 Lecturer Eng. Ph.D. North University of Baia Mare Abstract: The theoretic bases of soe atheatical odels for the sorting processes on vibrating sieves were established once the odern calculation equipent appeared. The present paper presents a siplified calculation odel for vibrating conditions specific to the sieving of pulverized aterials on electroagnetic sieves. Key words: sorting process, electroagnetic sieves, false position ethod. ESTABLISHING THE JUMP CONDITIONS Let us consider an electroagnetic vibrator, with sieves tilted to angle α in reference to the horizontal line of the place, with the obile coordinate syste axes xo y, solitary with the sieve (axis Ox oriented along the sieve) and axes shown in figure. ξo η of the fixed coordinate syste, as Figure. The fixed and obile reference axes syste ξ 4

2 We shall approach the case of these conditions with a single jup of the grain of aterial at an oscillation period of the perturbing force. Let us further consider types of intervals sybolized with I for the oents when the aterial is on the sieve and ove solitary with it, and sybolized with II those oents when the aterial is in id-jup on the sieve. Thus, for the case in the above figure we shall have the differential equation of the oveent in interval I as follows: The following notations are ade: η Fη η + η= (+ cosωt) g cosα () + + ω dz z = η ; z = ; τ=ωt; ω = F dτ η ω g cosα ; ; n F = = = ω With these notations the above equations becoes: η η ; () The solution of this equation is: z + z = (+ cos τ) n (3) + + z = AI sin τ + BI cos τ cos τ+ n( + ) (4) The oveent of the sieve after the aterial lifts at oent τ 0 is given by equation: η F η η+ η= ( + cos ω t) g cos α (5) We again adopt the notations in (.), and we obtain the following equation: The solution of the equation is: z + z = (+ cos τ) n II = II ( τ ) + II ( τ) τ+ 4 z A sin B cos cos n (6) (7). ESTABLISHING THE MATHEMATICAL MODEL In order to deterine the oscillating oveent equations in the specific field of pulverized aterials sorting, we further deterine the constants A I,B,A I II,B II which appear 4

3 in relations (4) and (7). Deterining the value of these constants was ade using the noogras in []. The values of the constants extracted fro these noogras are given in accordance with the throw coefficient c in the table bellow (Table ). Table : The values of coefficients C A B A B A I, B I, A II, B II Using relation A I sin τ 0 + BI cos τ0 cosτ0 n ( + ) = II ( 0 ) II ( 0 ) 0 4 = A sin τ + B cos τ cos τ + n (8) we will deterine the oent when the throw of the grain off the sieve taes place. The left side eber of the equation represents the oveent of the obile frae with the aterialto-be-sieved on it, and the right side eber represents the oveent of the obile frae without the aterial-to-be-sieved on it. Relation (8) represents the space continuity condition at the oent the grain is thrown, that is to say that the position of the obile frae 43

4 iediately before the grain lifts ust coincide with its position iediately after the grain lifts. Because equation (8) is a transcental one, it can be solved through the so-called ethod of the false position with the help of a proper calculation algorith. The algorith is designed with the help of the prograer MatLab ([], [3], [4]). The false position ethod consists in using an equality between two derivative functions. In this case we use, as entioned, relation (8). In this case the difference between the two functions results fro equation (9), where the unnown isτ 0 - the oent of the grain s jup off the sieve: g( τ ) = A sin τ + B cos τ sinτ n + 0 I 0 I A II sin( τ 0 ) + BII cos( τ0 ) sinτ 0 n = 0 ( ) The solution can be deterined by linear approxiation between two given values of equation (9), as in figure : (9) Figure. Schee for exeplifying the false position ethod The approxiation is done in the interval generated by two values given to the functions within the equations [g(τ 0_ ), g(τ 0_ )]. Generating the values is done precisely through the prograer, and the root can be deterined through linear approxiation. After each iteration, the doain shrins so that the new doain deterined by the new values of [g(τ 0_ ), g(τ 0_ )] contains the root as well. The latter is obtained by replacing g(τ 0_ ) or g(τ 0_ ), deping on their positive or negative sign with a value calculated with the forula: 44

5 the following: g( τ0_) τ 0_S = τ 0_+ ( τ0_ τ0_) g( τ0_) g( τ0_) The calculation prograer for solutions τ 0 of the transcental equation (9) is function vibrating_oveent clc %constants a= 0.04; b=-0.4; a=-0.05; b=-0.68; =0.; =0.4; % solving the equation ind=0; for c=.5:0.05:3.3; ind=ind+; [tauo_fin] = rezolvare_ecuatie_(a,,,b,c,a,b); tauo_fin_(ind)=tauo_fin; %afisare rezultate finale tauo_fin_ disp('finish') save ('atlabm') function [tauo_fin] = rezolvare_ecuatie_(a,,,b,c,a,b) %valoare presupusa tauo_0=pi; tauo_=tauo_0; tauo_=tauo_0; % stabilirea lui Z si Z contor=0; while contor=contor+; tauo_=tauo_+0.*tauo_0; Z=a*sin(*tauo_/sqrt(+))+b*cos(*tauo_/sqrt(+))-^/(4+4*^)*cos(*(tauo_))-(+)/c-(a*sin(*tauo_)+b*cos(*tauo_)-^/(4-^)*cos(*(tauo_))-/c); Z=a*sin(*tauo_/sqrt(+))+b*cos(*tauo_/sqrt(+))- ^/(4+4*+^)*cos(*(tauo_))-(+)/c-(a*sin(*tauo_)+b*cos(*tauo_)-^/(4- ^)*cos(*(tauo_))-/c); if Z*Z<0 brea %etoda falsei poziii contor=0; while contor=contor+; Z=a*sin(*tauo_/sqrt(+))+b*cos(*tauo_/sqrt(+))-^/(4+4*^)*cos(*(tauo_))-(+)/c-(a*sin(*tauo_)+b*cos(*tauo_)-^/(4-^)*cos(*(tauo_))-/c); Z=a*sin(*tauo_/sqrt(+))+b*cos(*tauo_/sqrt(+))- ^/(4+4*+^)*cos(*(tauo_))-(+)/c-(a*sin(*tauo_)+b*cos(*tauo_)-^/(4- ^)*cos(*(tauo_))-/c); if abs(z)<0^- tauo_s=tauo_; brea (0) 45

6 if abs(z)<0^- tauo_s=tauo_; brea tauo_s=tauo_+(tauo_-tauo_)*(-z/(z-z)); Zs=a*sin(*tauo_s/sqrt(+))+b*cos(*tauo_s/sqrt(+))-^/(4+4*^)*cos(*(tauo_s))-(+)/c-(a*sin(*tauo_s)+b*cos(*tauo_s)-^/(4-^)*cos(*(tauo_s))-/c); if abs(zs)<0^- tauo_s=tauo_s; brea if Zs*Z>0 tauo_=tauo_s; else tauo_=tauo_s; %valoarea finala tauo_fin=tauo_s; 3. CONCLUSION With the help of this prograer we calculated the values of the lift oent of the aterial grain lift for values of the throw coefficient fro c=,5 to c=3,3. For each value given to the throw coefficient c, the corresponding values of coefficients A I,B,A I II,B II were given. The values thus obtained for the throw oent are provided in the table bellow (table ): Table. The values of the throw oents of the grain on the sieve. c τ aruncare Table (continuation) c τ aruncare REFERENCES [] Ghinea, M., Fireteanu, V., MATLAB, calcul nueric, grafica, aplicatii. Editura Teora, Bucuresti, 995 [] Peicu, R. A., Masini din industria aterialelor de constructii. Editura Didactica si Pedagogica, Bucuresti, 966 [3] Schilling, R. J., Harris, L. S., Applied nuerical ethods for engineers using Matlab and C. Broos and Cole Publishing Copany, USA, 999 [4] Ursu-Fischer, N., Ursu, M., Metode nuerice in tehnica si prograe in C/C ++. Casa Cartii de Stiinta, Cluj-Napoca,