Approaches Regarding the Calculus of the Elastic System of the Initial Vibrating Screens

Transcription

1 Approaches Regarding the Calculus of the Elastic System of the Initial Vibrating Screens GHEORGHE ENE, NICOLETA SPOREA* Politehnica University of Bucharest, Faculty of Mechanical Engineering and Mechatronics, Process Equipment Department, 313 Splaiul Independentei, , Bucharest, Romania The papers present the design of the elastic bearing system from the inertial vibrating screens, made of the helical compressed coil springs. The innovations are: the dimensioning of the screen vibrations generator, the establishment of elastic constant from bearing screen s casing and the dimensioning of helical coil springs, so the screen can achieves vibrations on a specified trajectory. Keywords: vibrating screen, sieves, compressed spring The inertial vibrating screens are widely used for screening in many grained material processing. Previous experimental and numerical investigations were performed in several contributions in the literature, from the classical screening devices [1, 2] to new types of vibrating screens. Recent studies have been conducted, by adopting Discrete Element Methods, in order to describe the particulate motions over vibrating sieves, thus identifying some parameters for the optimization of the vibrating screen efficiency, such as the screen-deck inclination angle and the maximum vibration amplitude [3, 4]. A numerical algorithm and a FEM model were developed, in order to analyze the way in which the selected design parameters influence the vibrating screen dynamic output variables of interest, such as the maximum vertical and horizontal displacement amplitudes, and the maximum pitching angle reached in the dynamic excitation of the screen [5]. A 3DOF (3 Degree of Freedom) dynamic model for the vibrating screens was approached [6] with the purpose of determining some impartial criteria for the appreciation of dynamic characteristics of the equipment. The quality of the screening operation is much better if the vibrations are transmitted uniformly over the entire screen surface. On the other hand, the efficiency of sizing, the energy consumption, noise and vibrational pollution are highly affected by the vibrational parameters of screens. By using the discrete element model, the domain of optimal vibrational parameters for efficient screening procedure was more precisely determined [7]. The damping springs, if treated as part of the screen structure, may be detected by the static damage identification methods based on the accurate displacement. A static-based method named as the twicesuspended-mass method (TSMM) was proposed theoretically to diagnose single spring fault of a vibrating screen [8]. With the relationship between changes of spring deformation and changes of stiffness derived from the corresponding static analysis, static deformation data tested before and after a spring fault could be used to locate the failure spring. An experiment was finally carried out to check the feasibility of TSMM. The operating parameters of a frame screen equipped with a vibrating sieve in the process of industrial screening of loose materials with model particle shapes were studied [9]. Two parameters were investigated, the mass of screened material and the height of material on the sieve calculated on this basis. The knowledge of these values is a basis for calculations related to designing of sieve surface size necessary for a given screening process. Typical construction of the inertial vibrating screen, with circular vibrating (fig. 1 - Inertial vibrating screen) includes mainframe 1 where are placed the sieves (1...4 sieves), elastic elements (helical coil springs, air springs or elastic rubber elements i.e.) 2 and the fixed frame 4. Fig. 1. Inertial vibrating sceen [10] 1-mainframe; 2-elastic bearing systems; 3-inertial vibrations generator; 4-the fixed frame The vibrations of the screen s casing are produced by the centrifugal inertial force of unbalanced eccentric masses (counterweights) (fig. 2 - Scheme of inertial vibrations generator), driven in rotation by the inertial vibrations generator 3. Fig. 2. Scheme of inertial vibrations generator The sieves are mounted inclined at an angle of to the horizontal plane, in order to facilitate the advance of material on them. The screen s casing oscillates on circular trajectories due to the vibration generator. Depending on the axial and transverse stiffness ratio of elastic bearing elements, the points on the screen s casing, adjacent with the axis of vibration generator, describe trajectories much closer to the circle, and the most distant can have elliptical or linear trajectories. * nsporea@yahoo.com; Tel.: REV.CHIM.(Bucharest) 67 No

2 The radius of trajectory depends on the ratio between of the sieve s casing weight and counterweights. For the postresonance regime, far away from resonance, ω = (6...10). p, where p is the eigen pulsation of the vibrating system, we can write: (1) where A is the oscillation amplitude of the screen s casing (the radius of the circular trajectory of the vibrations); m - total mass of the screen s casing (including the mass of vibrations generator and the mass of material that will be screening, found on sieve); m 0 counterweights mass; r 0 the mass eccentricity. From the analysis of this relationship, it is observed that the amplitude oscillation is not constant, it depends, for a given screen construction (r 0 and m 0 are imposed), on the amount of material loaded on the sieve. At the overloaded screen, the vibrations amplitude decreases, the damping effect appears, leading to a reduction of screening efficiency. At the under loaded screen with material, efficiency of screening also decreases because of the increase the amplitude oscillation and the velocity material on sieve, the granules of the material jumping over the sieve meshes. Therefore, these sieves require proper and uniform feeding with material. In order to achieve a quality screening, the vibration amplitudes on the horizontal (A x ) and vertical (A y ) direction will be established according to the characteristics of the screened material and the technological conditions. The screen will vibrate with the amplitudes Ax and Ay, if the static moment (m o. r o ) of the eccentric masses, as well as constants k x and k y on the Ox and Oy directions of the elastic bearing system elastic are properly determined. The elastic constants k x and k y, needed to realise the vibration screen with amplitudes A x and A y, shall ensure through proper design of the elastic bearing s components. When the elastic bearing is made of the helical compressed coil springs the elastic constants k x and k y principally depends on the spring coefficient C = D / d (D - winding average diameter), the wire diameter (d) and the number of active windings (i). The diagram presented in figure 4, that establishes a graphical correlation between, k x /k y, h s, / D st, δ st / h ratios ( δ st is the static spring elongation, h s - the compressed coil spring height), is used to calculate the helical compressed coil springs, axial and transverse loaded. This work presents a design for the coil springs of elastic bearing system from the inertial vibrating screens so that they achieve the amplitudes A x and A y on the horizontal and vertical directions. -the number of springs: u; -the spring coefficient: C = D / d (it is, generally, C= [11], for cold wrapped springs, and C= , for hot wrapped springs). The experimental data obtained from field as well as the ones resulted from the fatigue strength studies, lead to values of C = [12]); -the transverse elasticity modulus of the spring, G(G = N/m 2, for steel). The determination of the bearing elastic constants, on the transverse (horizontal) O x and axial (vertical) O y direction is carried on the following sequence: -the mass eccentricity (the distance between the rotation center and the mass centre of the eccentrically paced mass) is adopted: r o (fig. 2); the value of the eccentrically placed mass of vibrator is determined [1, 13]: (1a) - the eccentrically placed mass of vibrator will be corrected: (2) where A oy is the amplification factor [1, 13]: The vibration trajectory is an ellipse with semi-axes A x and A y, the major axis of the ellipse is inclined with respect to the x-axis of the coordinate system by the angle γ, as defined by the relation [1, 13]: When the damping on the horizontal direction is very low (2. n x / p x 0), the corresponding phase angle is set to ϕ x =0; in this case, the expression (4) becomes: The angle of phase ϕ y is determined using the relationship [1]: (3) (4) (5) Establishment of the elastic spring constants The elastic bearing system (made of helical compressed springs) of the casing sieve must achieve certain values of the amplitude of vibration. In order to design it is necessary to know: - the vibration amplitude on the horizontal direction: A x ; - the vibration amplitude on the vertical direction: A y ; - the perturbation force pulsation (the angular velocity of the vibrations generator shaft) ω; - the mass of the vibrating system (including the mass of vibration generator and the material mass that will be screened): m; - appropriate ratio from the post-resonance operating mode, with k ω = ω / p y (where p y is the eigen pulsation of the elastic system, on the vertical direction); - the damping factor on the vertical direction: 2. n y / p y. For the system of springs, the following are adopted, in order to design: Also it can be determined using the relationship [1, 13]: The following relations will be determined: -the eigen pulsation of the elastic system, on the vertical direction: p y = ω / k ω (8) -the elastic constant of the spring system, on the vertical (axial) direction [1, 13, 14]: REV.CHIM.(Bucharest) 67 No (6) (7) (9)

3 - the eigen pulsation of the elastic system, on the horizontal direction (inside the post-resonance domain, the damping effect in the elastic system is insignificant) [12]: (10) -the elastic constant of the spring system, on the horizontal direction: (11) - the ratio of the elastic constants, on the horizontal and vertical directions: (12) Relationships for the dimensioning of the spring The helical compression coil springs are characterized by specific dimensions [1]: -the compressed coil spring height: (13) where: h o is the height of the free spring; δ st is the static spring elongation, on the vertical direction: (14) where: F 1 = m. g / u is the static load of spring; k 1y = k y = / u = - the elastic constant of a spring, on the vertical direction; - the height of the free spring (fig. 3 - Geometrical elements of the helical compressed spring): (15) where: h b is the height of blocked spring, i - the number of active windings, p s - the winding pitch, d - the wire diameter; The resonance amplitude is determined by the relationship: A or = A y. A oy (19) where the amplification factor A o,y is defined by the relationship (3) with ω / p y = 1. Taking into account the expressions (14)... (18), the relationship (13) will becomes: (20) Using the relationship (20), h s, /D and δ st, /h s, are determined. The values calculated on this way will be equals to the corresponding values resulted from the diagram (fig. 4 - The dependence between the ratio of lateral k x and axial k y stiffness and h s, / D, δ, st, /h s, for compression helical coil springs) (k x, determined by the relationship (12)). The following relationships are obtained: where (21) (22) should be determined by the relationship (14) and (19). The compressed coil spring height results from the relationships (21) and (22): (23) (24) By equalizing these relationships, taking into account D = C. d, one obtains: Fig. 3. Geometrical elements of the helical compressed spring (25) - the height of the blocked spring: (16) where i t is the total number of windings; -the total number of windings (for the spring with processed ends) [15]: i t = i (17) -the winding pitch of the spring [16]: The relationship (25) can be used to determining the wire diameter: Other geometrical elements of the spring are: (18) - the winding average diameter: (28) where A r is the resonance amplitude. - the winding angle of the free spring: Considering that the resonance amplitude is about 5 10 times larger than the steady regime amplitude [16], (29) when the non-blocking condition is imposed to the - the winding outer diameter: resonance regime too, in relation (18) the resonance (30) amplitude occurs (especially at the machine stops, when - the winding inner diameter: the passing through resonance takes longer than in the (31) case of machine start) REV.CHIM.(Bucharest) 67 No (26) The number of active windings results from the relationship (23) considering D/d = C: (27)

4 -the eccentrically placed mass of vibrator will be corrected: The constructive solution with two eccentric masses placed at the ends of the same shaft, each of them with m o / 2 = 7kg, is chosen. -The angle of phase is determined using the relationship: Fig. 4. The dependence between the artio of lateral k yx and axial k y stiffness and h ys / D, δ yst / h ys for compresion helical coil springs [15] - the wire length needed to achieve the winding: (32) Example It is considered a vibrating screen excited by an inertial vibrations generator, with eccentrically disposed rotating masses. The elastic system used for the bearing of the sieve frame consists of the compression helical springs (fig. 1). These are known: -the vibration amplitude on the horizontal direction: A x = 3mm; -the vibration amplitude on the vertical direction: A y = 2.5mm; -the perturbation force pulsation: ω = s -1 ( n = 960 rot/min); -the mass of the vibrating system: m = 650 kg; -appropriate ratio from the post-resonance operating mode, with k ω = ω / p y = 5; -the damping factor on the vertical direction: 2. n y / p y = 0.2. For the system of springs, the following are adopted: -the number of springs: u = 8 (four packages, each one contains two springs); -the spring coefficient: C = 7; -the transverse elasticity modulus of the spring G= N/m 2, for steel. The calculations are carried out in the following sequence: The vibration generator design Taking into account the given values for the mass m and the amplitude on vertical direction A y, the calculus will be carried out by adopting a suitable value for the eccentricity r 0, from constructive point of view. So, a suitable value for the eccentric mass m o results: -the mass eccentricity is adopted, constructively: r o =120 mm; -the value of the eccentrically placed mass of vibrator is determined: The vibration trajectory is an ellipse with semi-axes A x = 3mm and A y = 2.5mm, the major axis of the ellipse is inclined with respect to the x-axis of the coordinate system by the angle γ: The elastic constants determination -the eigen pulsation of the elastic system, on the vertical direction: -the elastic constant of the spring system, on the vertical (axial) direction: -the eigen pulsation of the elastic system, on the horizontal direction: -the elastic constant of the spring system, on the horizontal direction: -the ratio of the elastic constants on the horizontal and vertical directions: The wire diameter and number of active windings determination From the diagram shown in figure 4, results (for -The wire diameter of the spring: -the amplification factor is determined: -The amplification factor in resonance regime (ω / p y = 1; 2. n y / p y = 0.2): REV.CHIM.(Bucharest) 67 No

5 The buckling of the spring The spring does not buckle during the operation if h o / D < 3 [15]. Because the ratio of the height of the free spring and the winding average diameter is: -The resonance amplitude, on the vertical direction: -The number of active windings: the spring will not lose the stability during process. Further, the calculus is performed for the values k x =0.75 and k x = 1, in order to emphasize the influence of k x ratio on the vibrations screen and the constructive characteristics of its springs. For k x =0.75 the spring system is characterized by: - the elastic transverse constant: - the vibration pulsation on the transverse direction: The geometrical elements of the spring determination - the winding average diameter: The elliptical trajectory characteristics of the vibration screen are: - the vibration amplitude on the horizontal direction (inside the post-resonance domain, when the damping effect in the elastic system is insignificant): - the compressed coil spring height: ; - the total number of windings: - the height of the blocked spring: - the vibration amplitude on the vertical direction: A y =2.5 mm; - the major axis of the ellipse is inclined with respect to the x-axis of the coordinate system by the angle γ: - the height of the free spring: - the static spring elongation: -the winding pitch: -the winding angle of the free spring: From the diagram shown in figure 4 results (for the value With these values will be determined: - the wire diameter of the spring:. -the winding outer diameter: - the winding inner diameter: - the number of active windings: - the wire length needed to achieve the winding: To obtain vibration circular trajectories, the elastic constants of spring system on the horizontal and vertical REV.CHIM.(Bucharest) 67 No

6 Table 1 THE RESULTS OF CALCULUS directions, must be equal: k x = N/m, that means k x =1. From the figure 4 results (for the value With these values will be determined: - the wire diameter of the spring: - the number of active windings: The others constructive features of the spring will be changed adequately. The results are summarized in table 1. From the analysis of these values results that, with increasing of the transverse stiffness of the springs, tending at the axial stiffness, the wire diameter increases, the number of active windings decreases and vibration trajectory changes its shape from the elliptical to circular one. Conclusions Vibrating regime characteristics of the screen (vibration trajectory shape, pulsation and amplitudes on the horizontal and vertical directions) are determined by the technological parameters, the nature and particularities of the material that will be sieving, affecting the performance of screen: flow, efficiency and screening accuracy. Relations established in this study allow the design of the elastic bearing system of the screen, consists of compressed coil springs. So, the characteristics of the vibrating regime screen, imposed by the nature and particularities of the material, are assured. The presented example leads to the stages of calculus and shows the influence of the transverse and axial stiffness ratio of the elastic bearing has both on the vibrating regime screen characteristics and the structural characteristics of spring components (wire diameter of the spring, number of active windings). The originality of this work consists of: -the dimensioning of the screen vibrations generator; -the establishment of elastic constant from bearing screen s casing; -the dimensioning of helical coil springs so the screen can achieves vibrations on a specified trajectory. References 1. ENE, GH., Echipamente pentru clasarea ºi sortarea materialelor solide polidisperse, Editura Matrix Rom, Bucureºti, JINESCU, V. V., Utilaj tehnologic pentru industrii de proces vol. IV, Editura Tehnicã, Bucure ti, WANG, G., TONG, X., Min.Sci. Technol, 21, nr. 3, 2011, p ZHAO, L., ZHAO, Y., Min. Sci. Technol, 21, nr. 5, 2011, p BARAGETTI, S., VILLA, F., Nonlinear Dyn, 78, nr. 1, 2014, p DRÃGAN, N., The Annals of DUNAREA DE JOS University of Galai, Fascicle XIV, Mechanichal Engineering, 2012, p CSIZMADIA, B., HEGEDUS, A., KEPPLER, I., IUTAM Bookseries, 30, 2011, p PENG, L. P., LIU, C. S., LI, J., J. Cent. South Univ., 21, nr. 4, 2014, p SZYMANSKI, T., WODZINSKI, P., Physicochemical Problems of Mineral Processing, 39, 2005, p *** *** Cap3.pdf 12. MUNTEANU, M., Introducere în dinamica maºinilor vibratoare, Editura Academiei, Bucuresti, ENE, GH., PAVEL, C., Introducere în tehnica izolãrii vibraþiilor ºi a zgomotului, Editura Matrix Rom, Bucuresti, ENE, GH., Rev. Chim. (Bucharest), 60, no. 11, 2009, p HARRIS, C., CREDE, C., ªocuri ºi vibraþii, vol. II, Editura Tehnicã, Bucuresti, 1968, p VAISBERG, L. A., Proektirovanie i rascet vibrationnih grohotov, Izd. Nedra, Moskva Manuscript received: REV.CHIM.(Bucharest) 67 No