Aerican Journal of Applied Sciences 6 (): 48-56, 9 ISSN 546-939 9 Science Publications Coputer Model For Sieves Vibrations Analysis, Using an Algorith Based on the False-Position Method Dinu I. Stoicovici, Miorita Ungureanu, Nicu Ungureanu and Mihai Banica Departent of Engineering and Technological Manageent, North University of Baia Mare, Str. Dr. Victor Babes, Nr. 6A, 4383 Baia Mare, Roania Abstract: The analysis of the sieves vibrations in the case of screening civil engineering construction bul aterials is usual ade by using soe differential equations depending on paraeters related to different aterial and sieves characteristics. One of those paraeters is the throwing coefficient -cthat is the ratio between the force capable to throw up the particle fro the sieve surface, and the gravity of this particle. The throwing coefficient is one of the ost iportant characteristics of a sieve dynaic behavior and its values are often used to establish a particular case to the sieve oscillations. In order to find the position of a particle that jup on the screen surface, a syste of 6 differential equations with 6 unnown integrating constants can be established. All the involved equations are in transcendent for and it is necessary to solve the syste by coputer algoriths. First of all, the 6 unnown integrating constants are replaced with related linear relations depending on the throwing coefficient. Secondly, an original coputer algorith based on the so-called false-position ethod is proposed. In order to validate it, the new syste of 6 differential equations depending on the throwing coefficient is solved for soe particular cases of the particle jups. Finally, the solutions are copared for the sae conditions of the initial used syste. The conclusion is that in the case of the construction bul aterials, the two systes give alost siilar solutions. In this case, the new syste depending on the throwing coefficient is uch easier to wor with that the initial syste. Another advantage is that in the very first steps one can choose the throwing coefficient and establish the best vibrating regie. Key words: Bul construction aterial, sieves, throwing coefficient, false-position ethod INTRODUCTION Establishing the ost favorable vibration operation conditions for swinging screens is the essential proble when devising such equipent. The aplitude and the frequency of vibrations are the decisive factors that influence the vibrating conditions. The sieves operate best in over-critical angular resonant regie, at high frequencies coupled with sall aplitudes for aterials with a ainly fine grading, and at sall frequencies coupled with high aplitudes for sorting aterials with ainly coarse grading. In the literature [,], there are guiding principles that recoend how to adopt the frequencies, the aplitudes, and how to adopt the dynaical conditions. The ost used vibrating syste in screening construction bul aterial is the inertia one. In Fig. the ain coponents of such equipent are presented. Sieve Screen t Vibration generator Fig. : Inertia vibrating one-dec screen coponents For the case of sorting construction bul aterial on vibrating screens with inertia driving syste, the overcritical regie is generally adopted, because it has lower sensitivity to perturbations. For exaple, when Corresponding Author: D.I. Stoicovici, Departent of Engineering and Technological Manageent, North University of Baia Mare, Str. Dr. Victor Babe, Nr. 6A, 4383 Baia Mare, Roania Tel: +4 6 89 Fax: +4 6 7653 48
A. J. Applied Sci., 6 (): 48-56, 9 A n g ul ar res o n an t freq u e n cy Aplitude U n d ercri ti cal freq u en cy O v ercri t i cal freq u en c y F req u en cy Fig. 4: The particle jup in the case when the jup is as long as the coplete sieve oscillation Fig. : Over-critical vibratory regie ( t ) ( t) ξ = a sin ω + ε η = b sin ω () Fig. 3: The particle jup in the case when the jup is less than a coplete sieve oscillation an over-dose of bul aterial occurs accidentally on the screen, the equipent presents a tendency to go to lower values of frequencies, and also, the aplitude to go to higher values Fig.. There is the advantage of obtaining an intense regie, and that will free faster the surface of the screen fro the over-dose. During a noral vibratory functioning regie, there are two periods of tie fro the point of view of the oveents of the sieve: the first one (Interval index I in Fig. 3-4) in which the oveent of the screen aes possible the acceleration, the deceleration and/or the stop of the particle oveents on the screen surface, but the particle will stay all the tie on this surface. And a second one (Interval index II) in which the oveent of the screen aes possible the jup of the particle fro the surface of the screen. Generally is accepted [-3] that the sieve oves describe an ellipse. In this case, the equestion are Fig. 6, 49 In forula (), ξ-represents the elongation of the sieve otion on Oξ axis, η-represents the elongation of the sieve otion on Oη axis, a-represents the half of the aplitude of the oveent on ξ-line, b-represents the half of the aplitude of the oveent on η-line; ω- represents the angular frequency of the oveent; ε- the difference of phase. The acceleration in this case on η-line is: η = b ω sin( ωt) () The axiu value of expression () is: η = b ω (3) The expression of gravitation acceleration coponent on η-line is: g = η g cos α (4) In order to be able to define the oveent conditions for each of those intervals, the throwing coefficient (sybol c) can be used. The throwing coefficient is defined as the iniu value of the ratio between the force capable to throw up the particle, and the gravity of this particle: b ω sin ωt b ω c = = = g cosα g cosα K (5) In this inertia syste, the particle juping state often used is one single jup corresponding to one, two
A. J. Applied Sci., 6 (): 48-56, 9 or several coplete oscillations of the screen. This state of the jup is established [,3] by the values of the throwing coefficient: for instance, the particle will begin to ove only when the throwing coefficient c is greater that. That is because in sorting bul aterial only a dynaical regie where the particle jups over the surface is possible to adopt. That will give the first condition: η Z O F η sin ωt S F ξ sin (ωt+ε) ξ c > (6) Also, this jup occurs [3] and taes place in the sae tie when the screen aes a coplete oscillation. That will give the second condition: O g η ξ c = + π = 3.9 (7) There are also other conditions that ust be realized, that iposed for the throwing coefficient other values, in order to obtain the best dynaical behavior for the screen. Other conditions to ipose are: The particle in its jup ust ove higher than the thicness of the wire fro which the sieve is ade The length of the particle jup ust be big enough the particle to pass at least in the next eye of the sieve, The dynaical regie ust be sufficient to avoid a wea jup of the particle So, fro all these considerations the throwing coefficient ust have, for the construction bul aterials, values in-between:.5 c 3.5 (8) We have two possible situations fro the point of view of the particle s oveents on the sieve: The particle jup is less than a coplete sieve oscillation, so the particle will ove together with the sieve before another jup will occurs (Fig. 3) The particle jup is as long as the one coplete sieve oscillation is. This is the ideal case, and eans that the particle stays on the sieve only in the oent of its fall on the sieve, and iediately after the particle is thrown again by the sieve oscillation (Fig. 4) The first regie is an ideal one. It ight be present soeties, but even if this regie is the one that is wanted and calculated, due to nuerous variable factors 5 Fig. 5: Mobile axis (Z, S) and static axis (η, ξ) for the sieve-particle syste (such as the thicness of the aterial layer on the sieve, and the collisions in-between the particles inside the aterial layer, the oist of the aterial, etc.) after an ideal, long jup, several others short ones follow. The second regie is present alost peranently on the particle evolution on the sieve. So, it is preferable to adopt fro the very beginning the first regie (Fig. 3), which is closer to the real phenoenon. In order to build a atheatical odel we consider such a obile particle-screen coplex as in Fig. 5, oving in the dynaic regie as in Fig. 3. In Fig. 5: the ηoξ axes are the otionless ones, and the zo s axes are the ones that ove together with the sieve, represents-the total ass of the screen, -the ass of the particle, η, ξ - the springs rates on Oη, respectively, on Oξ axes. Also, generally the inertia screens have a gradient to assure a better fluidity of the aterial on the sieve. In the Interval I (the particle and the sieve stay in contact), in the case of the oveent of one single particle on the sieve of an inertia screen, as in Fig. 5, the screen oveents are described by the next differential equation [,3] : η Fη η + η = sin ωt g cosα (9) + + In Eq. (9), new diensionless variables are introduced, variables defined in (). In (): z- represents diensionless elongation, ω-represents the angular frequency of the oveent of the screen, ω - represents the angular resonant frequency of the oveent of the screen, -frequency coefficient, - ass coefficient.
A. J. Applied Sci., 6 (): 48-56, 9 ω η z = η ; τ = ωt ; ω = ; F η ω g cos α = ; = ; n = F ω η () The diensionless for of the Eq. (9) is defined in (). z + z = sin τ n + + The Eq. () has the following solution: z A sin I = I τ + + BI cos + sin τ n + + τ () () In the above Eq. (), A I and B I represent the integrating constants. The Eq. () represents the oveent of the screen with the particle on the sieve in Interval I. In Interval II, we have the equation of the oveent of the sieve, without the presence of the particle on it: η Fη η + η = sin ω t g cos α (3) With the sae new variables fro (), the Eq. (3) becoes: z + z = sin τ n (4) This equation has the following solution (5): z = A sin τ + B cos τ II II II sin τ n (5) The Eq. (5) represents the oveent of the screen without the particle on the sieve in Interval II. A II and B II represent the integrating constants. Siilar considerations [3] lead us to the equation of particle oveent during the jup over the sieve (6). z = A τ τ cos τ S II B τ τ sin τ II ( τ τ ) cos τ AII sin τ sin τ BII cos τ cos τ + + ( sin τ sin τ ) n τ τ + z τ τ S (6) The Eq. (6) represents the jup of the particle on the screen in Interval II. In this equation τ represents the oent when the jup-starts and z SO represents the speed of the particle in this oent. In order to find the values of the constants A I, B I, A II, B II, fro the Eq., 5 and 6 the initial states are [3] : The continuity of the space: When the particle start the jup fro the sieve surface (τ ), that is the screen coordinates before (), and iediately after the jup (5) are the sae: z ( τ ) = z ( τ ) (7) I II The continuity of the space: When the particle falls on the sieve surface (τ c ), after the jup, that is the screen position before (5), and iediately after the falls (), are the sae: z ( τ ) = z ( τ ) (8) II C I C The continuity of the screen speed at the oent (τ ): ( τ ) = z ( τ ) (9) z I II The expression of the sieve speed after the particle falls (a perfect plastic shoc between the sieve surface and the particle is considered): z z z ( τ π ) = ( τ ) + ( τ ) I C II C S C + () The obile syste acceleration in the oent of the jup of the particle is equal and opposite to the gravity coponent on Oη axes: 5
A. J. Applied Sci., 6 (): 48-56, 9 Fig. 6: The A I values (fro [3] pg 75, Fig.7.4) Fig. 8: The A II values (fro [3] pp: 75, Fig. 7.7) Fig. 7: The B I values (fro [3] pp: 75, Fig. 7.5) Fig. 9: The B II values (fro [3] pp: 75, Fig. 7.8) z τ = n () I The displaceent of the particle in the oent of the fall on the sieve is null: S ( τ ) = () z C The initial states 7- will lead us to a syste of 6 differential equations with 6 unnown paraeters A I, B I, A II, B II, τ, τ c. All the involved equations are in transcendent for and it is necessary to solve the syste by coputer algoriths. Solving each ties this syste of equations is intricate, of course. There are soe graphs [3] in order to find the necessary values for A I, B I, A II, B II, (Fig. 6-9). There is an iportant difficulty to assure the integration of all those factors in a one coprehensive atheatical odel. The ain object of the present case of study is to adapt the existing syste of Eq. 7- in order to be 5 able to build a nuerical odel to establish a best inertia vibratory regie in the case of sorting construction bul aterials. Starting fro the aboveentioned syste, a new syste is ade using linear expressions depending on throwing coefficient to replace the syste integrating constants. The two systes are solved using the so-called false-position ethod for soe usual sorting cases, and using the sae initial conditions. Finally the results in those two cases are copared. MATERIALS AND METHODS It is obviously that the wor with the diagras to find values of interest of A I, B I, A II, B II, is not very precisely, and also, woring with the six equations syste is not realistic in a day-to-day design wor. So, in order to ae it easier to wor with all these equations, the actual values of the integrating constants (corresponding to the construction bul aterial) A I, B I, A II, B II, obtained fro the graphs in Fig. 6-9 was replaced with a linear interpolation of the. The
A. J. Applied Sci., 6 (): 48-56, 9 Table : The values of integrating constants considered for linear interpolations c A I B I A II B II.5.4 -.4 -.5 -.68.55.5 -.45 -.4 -.73.6.6 -.44 -.35 -.75.65.8 -.45 -.3 -.85.7.9 -.465 -.5 -.9.75. -.475. -.95.8. -.49.3 -..85.4 -.5.4 -.5.9.43 -.55.4 -..95.47 -.55.48 -.5 3..5 -.545.5 -. 3.5.54 -.555.53 -.5 3..56 -.558.55 -.5 3.5.59 -.575.58 -. 3..6 -.585.59 -. 3.5.6 -.6.6 -. 3.3.65 -.6.68 -.5 paraeters in Fig. 6-9 considered for linear interpolation are [4] : the throwing coefficient.5 c 3.3, the frequency coefficient =., the ass coefficient =. In Table are the A I, B I, A II, B II, values obtained. The linear interpolations were built with the corresponding built-in functions of MatLab progra. The interpolations are illustrated in Fig. -3 (in the highlight windows are the interpolation equations). The new fors of Eq. are 3, 4 and 5. Values of A..5..5. -. -. y =.7*x -.36 A = f(c).5.6.7.8.9 3 3. 3. 3.3 Throwing coefficient "c" Residuals. Linear: nor of residuals =.538.5.6.7.8.9 3 3. 3. Fig. : The linear interpolation for A I Values of B -.3 -.4 -.5 -.6 -.5 y = -.6*x +.4 B = f(c) A linear B linear -.7.5.6.7.8.9 3 3. 3. 3.3 Throwing coefficient "c". Residuals Linear: nor of residuals =.789.5 -..5.6.7.8.9 3 3. 3. τ z = (.7 c.36) sin + (.6 c +.4) + (3) τ + cos + + c z =.6 c.46 sin τ +.6 c. cos ( τ) sin τ c τ τ z.6 c.46 c cos τ.6 c. s = + ( ) sin τ τ τ sin ( ) sin ( ) (.6 c.) cos( ) cos( ) cos τ.6 c.46 τ τ τ τ + + τ τ ( sin sin ) The two systes of equations (first, 5, and 6, -. and second 3, 4, and 5) are both to characterize the oveents of the particle-screen syste at any oent. Fig. 3: The linear interpolation for B II 53 (4) (5) Fig. : The linear interpolation for B I Values of A.5..5 A linear -.5.5.6.7.8.9 3 3. 3. 3.3 Throwing coefficient "c" Residuals. Linear: nor of residuals =.5597. -. -. y =.6*x -.46 A =f(c).5.6.7.8.9 3 3. 3. Fig. : The linear interpolation for A II Values of B -.5 -. -.5.5.6.7.8.9 3 3. 3. 3.3 Throwing coeffcient "c" Residuals..5 -.5 B = f(c) y = -.6*x -. Linear: nor of residuals =.3574 B linear.5.6.7.8.9 3 3. 3.
A. J. Applied Sci., 6 (): 48-56, 9 The two systes are in transcendent for and it is necessary to solve the syste by coputer algoriths. One of the available atheatical nuerical possibilities is the so-called false-position ethod [4], valid because in both systes the functions are differentiable. A coputer algorith was ade to find the values for the oents when the particles start the jup over the sieve (τ ), based on Eq. 7. To apply the false-position ethod in this case, fro (6) we build a new function, ade by the difference between the two functions existing in (6). AI sin τ + BI cos τ + + sin τ n ( + ) = + = A sin τ + B cos τ II II sin τ n (6) The new function -g(τ )-as show bellow, ust be null for the solution (in this case the throwing oent τ ): g( τ ) = AI sin τ + BI cos τ + + sin τ n ( + ) AII sin ( τ ) + B cos τ + sin τ + n = II ( ) (7) Now, a straight-line approxiation to g (τ ) in the range of two solutions [g (τ ), g (τ )] is possible, and we can estiate the root by linear interpolation (Fig. 4). g(τ ) After each iteration, a new doain is defined which incorporates the new found solutions so that [g (τ ), g (τ )] always spans the root. That is, we replace g (τ ) or g (τ ) by g (τ S ), depending on its sign. The τ _S forula is in Eq. (8) below: g( τ _) τ _S = τ _ + ( τ_ τ _) g( τ_ ) g( τ _) RESULTS AND DISCUSSION (8) The values found for τ, depending on the throwing coefficient c, are shown in Table. This calculus is ade for the sae screening conditions already entioned:.5<c<3.3; =., =.4. The analysis that follows of the screen-particle syste behaviour in the case of the two systes was done with values of the throwing coefficient that are iportant for construction bul aterials (that is.5 c 3.3). Using MatLab, the values of the functions z I,, z II, and z S fro Eq., 5 and 6 where overlapped for each of the 7 values of the throwing coefficient in Table (an exaple is illustrated in Fig. 5 for c =.5), and the sae process was done for z, z, and z s (Fig. 6). Another analysis was done also using MatLab, for the axiu values of the Eq., 5, 6, 3, 4 and 5. The results in the case of Eq. 6 and 5 are illustrated bellow in Fig. 7 for.5<c<.75, and in Fig. 8 for.8<c<3.3. Also, starting at the sae jup oents, the particle falling locus was calculated in the two cases, using Eq. 6 and 5 (Fig. 9 are the results for the falling coordinates of the particles on the sieve for.5<c<.75, and in Fig. are the results for the falling coordinates for.8<c<3.3). Fro the analysis of the overlapped graphs results that best trajectory are at values of the throwing coefficient in-between.5 and.75. For the values greater than.75 the trajectory of the particle is not high enough. One of the ost iportant conditions for g(τ _) g(τ s ) g(τ ) Fig. 4: The false-position ethod 54 Table : The values of c.5.55.6 τ 6.546 5.553 4.74 c.65.7.75 τ 3.958 3.3 3.3 c.8.85.9 τ 3.63.84.57 c.95 3. 3.5 τ.496.93.79 c 3. 3.5 3. τ.66.3.47 c 3.5 3.3 τ.38.768
A. J. Applied Sci., 6 (): 48-56, 9 Z I / Z II / Zs.5 -.5 - -.5 - -.5 Particle jup - c =.5 fall oent Zs(tauc)= Z I jup oent z(tauo)=z(tauo) Z II Zs -3 3 4 5 6 7 8 tau axiu values of the jup equations in A,B,A,B equations in "c.5.45.4.35.3.5..5..5 Fig. 5: Overlapping of z I, z II and z S, for c =.5 (Eq., 5 and 6).8.9 3 3. 3. throwing coefficient "c" 3.3 Z I / Z II / Zs.5 -.5 - -.5 - -.5-3 Particle jup - c =.5 Z I Z II Zs -3.5 3 4 5 6 7 8 tau Fig. 6: Overlapping of z, z and zs, for c =.5 (Eq. 3-5) axiu values of the jup.3.5..5..5 -.5 equation in A,B,A,B.5.55.6.65.7.75 throwing coefficient equations in "c" Fig. 7: Maxiu values of z I fro () and z fro (3) an optial functioning regie is that the fall of the particle to be alost noral to the sieve surface. Without a high enough trajectory this request cannot be fulfilled. 55 Fig. 8: Maxiu values of z II fro (5) and z fro (4) ZI(tauc) - falling coordinates -.76 -.77 -.78 -.79 -.8 -.8 -.8 -.83 -.84 equation in c c=.6 c=.65 c=.7 c=.75 equat ions in A,B,A,B 6 6 6 63 c=.65 c=.6 c=.55 c=.5 c=.75 c=.7 c=.55 tauc (falling oent s) c=.5 Fig. 9: The falling coordinates of the particles on the sieve for the initial syste (rhobus) and the new syste (triangle) for.5 c.75 The axiu values of the particle jup are alost siilar for both equations systes. Still, for values of the throwing coefficient in-between.8 and 3.3 there are soe differences in the liits of around %. Regarding the falling locus of the particles, for the values of the throwing coefficient in-between.5 and.75 the spreading is alost siilar in the field of the graph. Instead, in the case of the values of the throwing coefficient in-between.8 and 3.3 it is obviously that the spreading is different, covering two different zones. The conclusion is that in the case of the construction bul aterials, the two systes give
A. J. Applied Sci., 6 (): 48-56, 9 ZI(tauc)-falling coordinates equations in A,B,A,B -.74 c=.85 -.76 c=.8 c=.9 c=.95 c=3.5 -.78 c=3.5 c=3 c=3.5 c=3. c=3. -.8 c=3.3 -.8 -.84 -.86 -.88 58.8 59.8 6.8 6.8 c=3 c=.95 c=3.5 c=3. c=3. c=3.5 c=3.5 c=3.3 tauc (falling oents) equations in c c=.85 c=.8 c=.9 Fig. : The falling coordinates of the particles on the sieve for the initial syste (rhobus) and the new syste (square).8 c 3.3 siilar solutions, especially for the throwing coefficient values in-between.5 and.75. In the case.8c3.3 other values should be considered for and in order to eep the sorting efficiency at high level. Generally, for construction bul aterial the functioning frequency ust be ties greater that the angular resonant frequency of the screen - this in highly necessary in order to avoid a slow crossing by resonant area, that can destroy the screen, so should be around. [4]. Also, the ass of the aterial on the sieve should not be greater because the whole syste becae instable, that is a sall over-dose of aterial can stop the vibrations or disturb the plane-parallel oveents of the screen, so should be around.4 [4]. Finally it results that the throwing coefficient used for equations in the case of construction bul aterial ust also be in the range of.5-.75. The new syste is ore convenient and easy to wor with for devising sorting construction bul aterials processes. Another advantage is that in the very first steps one can choose the throwing coefficient and establish the best oscillations paraeters. REFERENCES. Mihailescu, St., 983. Masini de Constructii si pentru Prelucrarea Agregatelor. Editura Didactica si Pedagogica, pag. 369-384. Machines in Civil Engineering and Aggregate Processing. Didactical and Pedagogical Publishing, pp: 369-384.. Munteanu, M., 986. Introduction in the Dynaics of Vibrating Machines. Editura Acadeiei RSR, (RSR Acadey Publishing, pp: 96-99). 3. Peicu, R.A., 975. Studiul vibraiilor la ciururi în vederea stabilirii unor etode de calcul i proiectare, în scopul îbuntirii coeficientului de calitate a cernerii. Tez de doctorat. Istitutul de Construcii Bucureti 975. (Screens vibrations study with a view to establishing soe calculus and design ethods in order to optiized the sorting quality coefficient, PhD thesis, Civil Engineering University, Bucharest 975). 4. Stoicovici, D., 4. Contribution to the optiization of vibration sieves dynaical paraeters in the case of screening critical huidity aterial. PhD Thesis, Technical University of Cluj Napoca, June 4. 56