Jr. of Industrial Pollution Control 33(1)(017) pp 909-913 www.icontrolpollution.com Reiew PROCESS DUTY EFFECT O THE VIBRATIO GYRATORY-COE CRUSHER DYAMICS EVGEIY VITALIEVICH SHISHKI * AD PAVEL VITALIEVICH SHISHKI St. Petersburg Mining Uniersity 199106, St. Petersburg, Line 1 Vasilyesky Island, bld., Russia (Receied 06 May, 017; accepted 08 May, 017) Key words: Vibration gyratory-cone crusher, Power balance equation, Maximum power, Viscous resistance, process duty ABSTRACT This paper studies the effect of milled material on the dynamics and stability of the ibration gyratory-cone crusher where the body and milling cone oscillations are excited by two unbalance ibration exciters installed on the machine body. The laws of induced oscillations of the actie crushing members with account for the motion iscous resistance as well as the equation of the power balance in the machine operating mode and the equation for determining the specific coefficient of equialent iscous resistance, n, were deduced. The condition of synchronous existence of in-phase ibration exciter rotation mode and steady operation of the crusher was formulated, which imposes certain constraints on the maximum power of the motors in the machine operating mode used for material crushing. ITRODUCTIO The ibration gyratory-cone crusher deeloped in the Research and Engineering Corporation MEKHAOBR-TEKHIKA (Vaisberg, et al., 004) is deeloped for milling arious types of natural and man-made mineral raw material. This machine has such adantages as a high degree of crushing and insignificant content of fine grains in the crushed material, which in its turn has high practical alue for the products manufactured of this material. The crusher consists of a soft reinforced body and a crushing cone attached to the body by means of special helical spring packs. At that, the cone has only one translational degree of freedom. The crusher is set into motion by a pair of self-synchronized inertial ibration exciters installed on the body. With such installation of the ibration exciters, the selfsynchronization margins turn out to be sufficiently high and weakly depend on the machine operation mode. At that, the crusher dynamic scheme is symmetrical and balanced. Apart from this, the fact that no stiff kinematic links between two selfsynchronized ibration exciters are present makes the machine considerably easier to maintain and reliable in operation. The paper considers the effect of the process duty (the material milled in the crushing chamber) on the stea diness of the synchronous-counterphase motion of the body and crushing cone in the operation mode by means of introducing linear-iscous damper between the crushing agents. It seems that such a method of accounting for the effect of the process duty, as an equialent iscous resistance against the actie member oscillations, on the dynamics of a shock-ibrating machine (ibrating grizzly) has been proposed and successfully used in practice (Barzuko Vaisberg, et al., 1978), and then deeloped in a monograph (Vaisberg, 1986) Crusher oscillation equations The dynamic flowchart of the machine considered in this paper is shown in Fig. 1. The crusher body 1, as well as the crushing cone 3, attached to the body by means of the special helical spring pack, can be considered as absolutely solid bodies which moe translationally (straightforwardly) in the operating mode. At that, the crusher body has rubber-in-shear mounting 4 with anishingly small rigidity. The inertial ibration exciters 5, located inside the body (load-bearing body) are unbalanced rotors rotated by *Corresponding authors email: shishkin_e@spmi.ru
910 SHISHKI ET AL. Fig. 1 Dynamic flowchart of the ibration gyratory-cone crusher. two independent asynchronous motors. The axes of the ibration exciters are perpendicular to the plane on which the crusher moes. The dynamics of the considered machine is studied in the paper (Shishkin and Kazako, 015) where the equations of induced crusher oscillations are obtained in the form of the second-kind Lagrange equation. At that, the resistance against the crusher body and cone oscillations proportional to the first degrees of the elocities of their graity centers were accounted for (this is how the presence of material in the crushing chamber was taken into account). It is extremely important to take into account the influence of the crushed material on the stability of the operating mode or determining the conditions of existence and steadiness of the body and crushing cone synchronous-counterphase motion in this mode. Therefore, the model of a machine with lineariscous damper between two crushing agents was used for the study of this issue. If we suppose that the ibration exciter rotors rotate smoothly with a preliminarily unknown synchronous angular rate and phase shifts 1 and, that is, φ 1 = t + 1, φ = t + (1) then the equations of the crusher small oscillations under the effect of harmonic perturbing forces will be written as follows: m1 y1+ β y1 y + c( y1 y) = me sin ( t+ α1) + sin ( t+ α), m y+ β y y1 + c( y y1) = 0, Mx= me cos ( t+ α1) cos ( t+ α), Iϕ = me { b sin ( t+ α1) sin ( t+ α) ( a1+ cos ( t+ α1) cos( t+ α) } () Here, as preiously [4], the following designations are adopted: C xy fixed Cartesian reference system; the point C is the common graity center of the system if the masses of all bodies m 1 and m, as well as the masses of the rotors (ibration exciters) m 1 and m (m 1 = m = m ) are taken into account; y 1 and y are the coordinates of the graity centers of the body, C 1, and the crushing cone, C ; x and φ are the coordinate of C and the rotation angle of the body 1 and system as a single solid body, measured counterclockwise; φ 1 and φ are the rotation angles of the eccentricity-ectors of the exciters relatie to a fixed axis C x, measured in opposite directions: φ 1 counterclockwise, and φ clockwise; M = m 1 + m + m is the total mass of the crusher; 1 and are ertical distances between the system graity center and the body and crushing cone graity centers, respectiely, in the position of the machine static equilibrium; b and d are, respectiely, horizontal and ertical distances between the axes of the exciters and the body graity center; e is the unbalances eccentricity (distance from the rotation axis of the exciters and the unbalance graity center) of ibration exciters; c is the helical spring pack rigidity; β is the coefficient of equialent iscous resistance; I is the equialent central torque of the system inertia; determined under the formula: I= I 1 +I +ma where I 1 and I are the central body and crushing cone inertia torques; m mm 1 = is the reduced mass m m 1+ of the considered two-mass system, a is the ertical distance between the body and the cone mass centers in the position of the machine static equilibrium. Direct soling the system of equations () at arbitrary initial alues of 1 and allows obtaining the laws
PROCESS DUTY EFFECT O THE VIBRATIO GYRATORY-COE CRUSHER DYAMICS 911 of induced oscillations of the body and crushing cone in the following form: α me cos m k m n y1 = 1 sinτ cosτ m1+ m m1 m1 (3) α me cos 3 ( k ) n y = + 1 sinτ cosτ m1+ m Where α + α τ = t + 1 α α α = 1 = ( k ) + 4n Here k= cm is the proper frequency of summary oscillations of the considered two-mass system, β = is the specific iscous resistance coefficient. n m Power balance equation ow let us consider the motion of the ibration exciters (unbalances) (Blekhman, 013; Blekhman, 1971; Blekhman, 000; Fidlin and Drozdetskaya, 015; Sperling, et al., 1997; Sperling, et al., 000). We assume that the torques L 1 and L of the electric motors, as well as the torques in bearings R 1 and R, act along the coordinates φ 1 and φ. Let us consider that the torques are the gien functions of unbalance angular elocities, φ 1 and φ. Then, the equations of the rotation of ibration exciters in the form of the second-kind Lagrange equations will hae the following form: ( I + me ) ϕ1 = L1 ϕ1 R1 ϕ1 me y1+ bϕ cosϕ1 x ( a1+ ϕ sinϕ1 (5) ( I + me ) ϕ = L ϕ R ϕ me y1+ bϕ ϕ x ( a1+ ϕ ϕ cos sin (4) where I is the central torque of the unbalance inertial. Then we assume that the rotors of the exciters rotate almost smoothly, and the links between are weak. To determine the synchronous angular elocity and the phase shift differences, 1, the right parts of the motion equations (5) of the exciters shall Π be aeraged for the period 0 < t < according to the method of small parameter. At that, the expressions (1) and (3) shall be put under the integrals. In the result of aeraging, we get: P i L() R () W i =0, i=1,, (6) where the quantities α m α α W1 = me cos { ( k )sin + ncos b + ( a1 + α m1( m1+ m) + sin } I α m α α W = m e cos [(k )sin ncos + m1( m1+ m ) α m α α b + ( a1+ α W = m e cos [(k )sin ncos + sin (7) m1( m1+ m) I are referred to as ibration torques that characterize the aeraged impact of the torque type which is transmitted from one exciter to the other ia small oscillations of the body (bearing body). The system of two transcendental equations (6) allows the solution =0, which corresponds to the synchronous-sinphase mode, for the determination of unknown and. In fact, after the alue =0 is substituted in the equations (6), we get only one equation for the determination of : L( ) = R( ) + W( ) (8) where the quantity 5 memn W = (9) m 1( m 1+ m ) equals the period-aerage ibration torque that constrains the rotation of the exciter rotors in the synchronous-sinphase mode. By multiplying the equation (8) by we get: L =R+W (10) This relation has the form of the power balance equation in the system. Let us introduce the designation: = W = (L R) (11) ote that the aeraged power =W characterizes the useful power expenditure for crushing the material in the operating chamber. It is important that the alue of the equialent coefficient of iscous resistance β (or n) shall be adopted from the equation (10) (or (8)). For this purpose the aerage motor power L, as well as the power R that characterizes the power loss in bearings, shall be experimentally determined. After the expression (9) is substituted in the equality (11), the power balance equation in the crusher operating mode gains the final form: 6 4memn = (1) m ( m + m ) 1 1 This relation can be considered as the equation for determining the specific coefficient of the equialent iscous resistance, n. At that, the alue of the power,, as well as the synchronous angular elocity, shall be determined experimentally for the gien crusher operation mode. The corresponding equation, following from the expression (4), is a square root and has the following form: 4 mem ( k ) n n+ = 0 (13) m ( m + m ) 4 1 1
91 SHISHKI ET AL. The solution of the quadratic equation (13) has the form as follows: mem mem ( k ) 4 4 4 8 1, = ± m1( m1+ m) 4 m ( m 1 1+ m) 4 n (14) In the case of the symmetrical resonance (=k), the specific coefficient of the equialent iscous resistance cannot equal zero. Therefore, only the sign + in front of the radical has the physical sense. Apart from this, the quadratic discriminant (14) must be positie. The corresponding inequality is written in the following form soled for : 5 mem < max = (15) m ( m + m ) k 1 1 The inequality (15) shall be satisfied to secure the synchronous-sinphase mode of the ibration exciter rotation in the crusher. In other words, the inequality (15) allows determining the summarized maximum motor power in the machine operating mode spent for the material crushing. From (8) it follows that the initial phase ϕ, the alue of which for sinusoidal component A sin(τ + ϕ) is unknown, influences the formation of the amplitudefrequency spectrum. This influence is by way of component 1 [sin( t + ϕ ) sin ϕ ], and is reduced π to exposing formed spectrum Φ=Φ (, t ϕ, ) to pulses with frequency, and offsetting it by alue sin φ, which, due to multiplier 1 decrease their alues π with time. The presence of component 1 [sin(t + ϕ) sin ϕ] π, where phase alue ϕ is unknown, affects subsequent analytical calculations based on formula (8), for example, hampers calculation of amplitudes A with required precision. 1 The influence of component [sin( t + ϕ ) sin ϕ ] π on the alue of spectrum Ф= Ф(t,,φ), which is formed based on formula (8), may be reduced by increasing time interal [0, t], i.e. the time of forming the amplitude-frequency spectrum. With time t that corresponds to periods of the sinusoidal component, the influence of this component on the alue of the formed amplitude-frequency spectrum, according to (11), will not exceed alue Φ 1 Φ π. This allows, through choosing time of spectral analysis t, to reduce the influence of initial phases on the result of calculation by formula (8) to the desired alues. So, for instance, if for forming an amplitude-frequency spectrum at frequency, time t is allocated, which corresponds to = 16 periods, the uncertainty introduced by the influence of the initial phase of the harmonic on the alue of the formed Φ spectrum at this frequency will not exceed 0.01 Φ. Accordingly, if amplitudes A are to be calculated with the use of formula (8) with the relatie error not exceeding δ 0.01, then, in accordance with formula (1) for forming the spectrum, it is necessary to allocate time that corresponds to 16 periods. COCLUSIO The performed theoretical study allows estimating the effect of crushed material on the dynamics and steadiness of the ibration gyratory-cone crusher operating mode. In particular, the deried formula (14), when experimental data is used, enables determination of the numerical alue of the specific equialent iscous resistance coefficient, n, and under the condition that the inequity (15) is satisfied, in the machine designed under the considered scheme, the synchronous sinphase rotation of ibration exciters is realized required for the crusher efficient operation. REFERECES Barzuko, O.P., Vaisberg, L.A., Balabatko, L.K. and Uchitel, A.D. 1978. Vliyanie tekhnologicheskoi nagruzki na samosinkhronizatsiyu ibroozbuditelei [Effect of Process Duty on Self-Synchronizing Vibration Exciters]. Obogashchenie rud. : 31-33. Blekhman, I.I. 1971. Sinkhronizatsiya dinamicheskikh system [Synchronization of Dynamic Systems]. Moscow: auka. p. 896. Blekhman, I.I. 000. Vibrational Mechanics. onlinear Dynamic Effects, General Approach, Applications. Singapore et al.: World Scientific Publishing Co., p. 509. Blekhman, I.I. 013. Teoriya ibratsionnykh protsesso i ustroist. Vibratsionnaya mekhanika i ibratsionnaya tekhnika [Theory of Vibration Processes and Deices. Vibration Mechanics and Vibration Machines]. St. Petersburg: PH Ruda i Metally, p. 640. Fidlin, A., and Drozdetskaya, O. 015. On the Aeraging in Strongly Damped Systems: The General Approach and Its Application to Asymptotic Analysis of the Sommerfeld s Effect]. In Book of Abstracts of IUTAM Symposium on Analytical Methods in onlinear Dynamics (Frankfurt, Germany, July 6-9, 015). Germany: Technische Uniersität Darmstadt, pp. 19-1 Shishkin, E.V. and Kazako, S.V. 015. Vynuzhdennye kolebaniya ibratsion-noi drobilki rezonansnoi oblasti chastot [Induced Oscillations of Vibration Crusher in the Frequency Resonance Area]. Obogashchenie rud. 5 : 4-46.
PROCESS DUTY EFFECT O THE VIBRATIO GYRATORY-COE CRUSHER DYAMICS 913 Sperling, L., Merten, F. and Duckstein, H. 1997. Rotation and ibration in examples of the method of direct moement diision. Technische Mechanik. 17(3) : 31-43. Sperling, L., Merten, F. and Duckstein, H. 000. Self- Synchronization and Automatic Balancing in Rotor Dynamics. International Journal of Rotating Machinery. 6(4) : 75-85. Vaisberg, L.A. 1986. Proektiroanie i raschet ibratsionnykh grokhoto [Designing and Calculation of Vibrating Grizzlies]. Moscow: edra, p. 144. Vaisberg, L.A., Zarogatsky, L.P. and Turkin, V.Ya. 004. Vibratsionnye drobilki. Osnoy rascheta, proektiroaniya i tekhnologicheskogo primeneniya [Vibration Crushers. Basics of Calculation, Designing and In-Process Application]. St. Petersburg: Publishing House of VSEGEI, p. 306.