Jtam.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 48, 1, pp. 155-172, Warsaw 2010 INFLUENCE OF COLLISIONS WITH A MATERIAL FEED ON COPHASAL MUTUAL SYNCHRONISATION OF DRIVING VIBRATORS OF VIBRATORY MACHINES Jerzy Michalczyk Piotr Czubak AGH University of Science and Technology, Cracow, Poland e-mail: michalcz@agh.edu.pl The relation between angular oscillations of vibratorymachine bodies – disturbing the vibratory transport – and the loss of cophasal of driving vibrators was indicated in this paper. It was shown that the loss of cophasal running could be caused by periodical collisions of the body with amaterial feed. The mathematical model of this phenomenon was developed. The obtained analytical dependencies, allowing one to assess disphasingof vibratorsand to estimate amplitudes of angularoscillations of themachine,were verifiedby comparisonwith the results obtained by digital simulation of the system behaviour. Key words: vibratorymachines, cophasal run, synchronisation 1. Problem formulation Several essential processing and transportation processes are realised in the industrybymeans of vibratorymachines anddevices, suchasvibrating screens and conveyers, foundry shake-out grids, vibrating tables for production of concrete prefabricates aswell asvibratorydevices for synchronouseliminations of vibrations. Correct performance of this type of machines depends on obtaining syn- chronous, cophasal, angular motion of unbalanced masses constituting the source of the needed dynamic forces (Lavendel, 1981;Michalczyk andCieplok, 1999). As an example, let us discuss the scheme of a vibratorymachine of a linear trajectory of vibrations –Fig.1, inwhich thedrive constitutes two independent inertial vibrators set in motion bymeans of induction motors. 156 J. Michalczyk, P. Czubak Fig. 1. Calculation diagram of the two-vibrator vibratorymachine InFigure 1: Mk,Jk –mass and centralmoment of the body inertia, m, e – mass and eccentricity of the single vibrator, mn – mass of a material feed, kx, ky – coefficients of elasticity of the body suspension in directions x and y. The desirable situation is that both vibrators are counter and cophasal runninggenerating the resulting force in the direction ofworking vibrations ζ. Thedirection of this force shouldpass through themachinemass centre,which ensures lack of excitations (when the systemof elastic supports is symmetrical) for angular oscillations. Conditions for occurrence of tendency for the desirable synchronous and cophasal vibrator running can be determined on the basis of the integral cri- terion formulated by Blekhman (1994), Blekhman and Yaroshevich(2004) D(ϕ1−ϕ2,ϕ1−ϕ3, . . . ,ϕ1−ϕn)= 1 T [ T ∫ 0 (E−V ) dt− T ∫ 0 (Ew−Vw) dt ] =min (1.1) According to this criterion, the set of phase angles is stable around values ∆ϕ12,∆ϕ13, . . . ,∆ϕ1n, if the function D, determined by Equation (1.1) for these values, acquires the local minimum, where: ϕ1,ϕ2, . . . ,ϕn – angles of rotation of individual vibrators versus their initial positions, T – period of forced vibrations, T =2π/ω, Influence of collisions with a material feed... 157 E – kinetic energy of the machine body with rotor masses con- centrated in the pivoting point, V – potential energy of the support systemof themachine body, Ew,Vw – kinetic and potential energy of constrains between vibra- tors, respectively. For machines operating in a far-over-resonant mode, for which influence of elastic forces in the suspension can be neglected and which corresponds to the scheme presented in Fig.1, the above given condition leads to (Lavendel, 1981) D> 0 (1.2) However, the above criterion does not determine occurrence and precision of synchronisation in the case when counter-acting factors exist. The range of allowable disphasing angles of vibrators – for various types of vibratory machines given inLavendel (1981) paper– indicates the importanceof vibrator dissynchronisation for the working process: ∆ϕ¬        3◦−5◦ for vibrating screens 5◦−12◦ for feeders 12◦−16◦ for vibratory conveyers. The thesis that one of the factors disturbing synchronisation of drives is the influence of instantaneous forces originated from colliding of the material feed with the machine will be stated and verified in the paper. The depen- dences between the material feed mass and the work cycle character and the disphasing angle of vibrators causing angular oscillations of the body being responsible for irregular transportation of materials along the machine body will be also determined. 2. Analysis of influence of collisions with the material feed on cophasal running of vibrators Analysis of undisturbed running of vibrators will be performed by means of an averaging method. It allows one to write equations of motion of the ma- chine body, shown in Fig.1, separating the ”quickly” and ”slowly” variable phenomena. Thus, assuming for synchronous running the equality of angular velocities of both vibrators ϕ̇1 = ϕ̇2 and their ”slow” variation (ω≈ const), 158 J. Michalczyk, P. Czubak we can write approximate equations of motion of the body in between the collisions with thematerial feed in the absolute system ξ, η in the form Mξ̈+kξξ=meω 2(sinϕ1+sinϕ2) Mη̈+kηη=meω 2(cosϕ2− cosϕ1) (2.1) Jα̈+kyl 2α=meω2r(sinϕ2− sinϕ1)+meω2R(cosϕ1− cosϕ2) where ξ,η – absolute coordinates determining the position of the body mass centre, α – angle of rotation of the machine body, M – mass of a vibrating part of the machine, M =Mk+2m, J – central moment of inertia of the machine with unbalanced masses brought to the axis of rotation of the vibrators, kξ,kη – coefficient of elasticity in direction ξ and η, respectively, and kξ = kxcos 2β+ky sin 2β kη = ky cos 2β+kx sin 2β the remaining markings are given in Fig.1. Denoting ϕ1 −ϕ2 = ∆ϕ = const and assuming ∆ϕ ≪ 1 and ϕ2 = ωt, ω= const, Equations (2.1) can be presented in the approximated form Mξ̈+kξξ=2meω 2 sin ( ωt+ ∆ϕ 2 ) Mη̈+kηη=meω 2∆ϕsin(ωt) (2.2) Jα̈+kyl 2α=−meω2D∆ϕsin(ωt+γ) where tanγ = r/R (Fig.1). Particular integrals of the above equations, determining the steady state, are of the following form ξ(t)= 2meω2 kξ −Mω2 sin ( ωt+ ∆ϕ 2 ) η(t)= −meω2∆ϕ Mω2−kη sin(ωt) (2.3) α(t)= meω2D∆ϕ Jω2−kyl2 sin(ωt+γ) Let us also determine the second time derivatives in stationary motion – for those coordinates ξ̈(t)= 2meω4 Mω2−kξ sin ( ωt+ ∆ϕ 2 ) η̈(t)= meω4∆ϕ Mω2−kη sin(ωt) (2.4) α̈(t)= −meω4D∆ϕ Jω2−kyl2 sin(ωt+γ) Influence of collisions with a material feed... 159 Dynamic equations analysed up to the presentwere describingbodyvibra- tions on theassumption that theangularmotionof vibrators canbeconsidered as uniform for the steady state. This assumption is equivalent to disregarding the influence of body vibrations on vibrators running. Presently, wewill deve- lop equations ofmotion of vibrators taking into consideration those couplings, it means in a non-inertial coordinate system related to themachine body per- forming vibrations described above. Applyingmoments from the inertial forces resulting fromvibration of their axis to vibrators, we obtain equations of angular motion in the form J0ϕ̈1 =Mz1−meξ̈1cosϕ1−meη̈1 sinϕ1 (2.5) J0ϕ̈2 =Mz2−meξ̈2cosϕ2+meη̈2 sinϕ2 where Mz1,Mz2 – external moments (difference of the driving moment and the moment of friction), J0 – inertial moment of the vibrator versus its axis of rota- tion. Let us mark the vibratory moments Mwi, i=1,2, as expressions Mw1 =−me(ξ̈1cosϕ1+ η̈1 sinϕ1) (2.6) Mw2 =−me(ξ̈2cosϕ2− η̈2 sinϕ2) On the basis of the previously determined solutions of the body motion (not taking into account any influences of vibratorymoments on the vibrators run- ning) wewill determine components of accelerations of axes of both vibrators, disregarding centripetal accelerations as being small as compared with the remaining ones ξ̈1 = ξ̈− α̈Dsinγ η̈1 = η̈− α̈Dcosγ ξ̈2 = ξ̈+ α̈Dsinγ η̈2 = η̈− α̈Dcosγ (2.7) Taking into consideration in the above presented expresions Equations (2.4) and substituting them into (2.6) we will obtain the following equations to the vibratory moments Mw1 =−m2e2ω4 [ 2 Mω2−kξ sin ( ωt+ ∆ϕ 2 ) cos(ωt+∆ϕ)+ + D2∆ϕsinγ Jω2−kyl2 sin(ωt+γ)cos(ωt+∆ϕ)+ + ∆ϕ Mω2−kη sin(ωt)sin(ωt+∆ϕ)+ D2∆ϕcosγ Jω2−kyl2 sin(ωt+γ)sin(ωt+∆ϕ) ] 160 J. Michalczyk, P. Czubak Mw2 =−m2e2ω4 [ 2 Mω2−kξ sin ( ωt+ ∆ϕ 2 ) cos(ωt)+ (2.8) − D2∆ϕsinγ Jω2−kyl2 sin(ωt+γ)cos(ωt)+ − ∆ϕ Mω2−kη sin2(ωt)− D2∆ϕcosγ Jω2−kyl2 sin(ωt+γ)sin(ωt) ] Wewill calculate now the value, averaged for the period T =2π/ω, of the vibratorymoment acting on vibratorNo. 1, applying the assumption ∆ϕ≪ 1 Mw1av = 1 T T ∫ 0 Mw1(t) dt= =−m2e2ω4 ω 2π [ 2 Mω2−kξ 2π/ω ∫ 0 sin ( ωt+ ∆ϕ 2 ) cos(ωt+∆ϕ) dt+ + D2∆ϕsinγ Jω2−kyl2 2π/ω ∫ 0 sin(ωt+γ)cos(ωt+∆ϕ) dt+ + ∆ϕ Mω2−kη 2π/ω ∫ 0 sin(ωt)sin(ωt+∆ϕ) dt+ + D2∆ϕcosγ Jω2−kyl2 2π/ω ∫ 0 sin(ωt+γ)sin(ωt+∆ϕ) dt ] = (2.9) = −m2e2ω4 2 [ 2 Mω2−kξ sin ( − ∆ϕ 2 ) + D2∆ϕsinγ Jω2−kyl2 sin(γ−∆ϕ)+ + ∆ϕ Mω2−kη cos(∆ϕ)+ D2∆ϕcosγ Jω2−kyl2 cos(γ−∆ϕ) ] = = −m2e2ω4 2 [ −∆ϕ Mω2−kξ + D2∆ϕsinγ Jω2−kyl2 (sinγ−∆ϕcosγ)+ + ∆ϕ Mω2−kη + D2∆ϕcosγ Jω2−kyl2 (cosγ+∆ϕsinγ) ] Disregarding terms containing (∆ϕ)2, we will finally obtain Mw1av = −m2e2ω4 2 ( D2 Jω2−kyl2 + 1 Mω2−kη − 1 Mω2−kξ ) ∆ϕ (2.10) Influence of collisions with a material feed... 161 In a similar fashion, calculating the averaged within the period T =2π/ω value of the vibratory moment for vibrator No. 2, we will obtain Mw2av = 1 T T ∫ 0 Mw2(t) dt= =−m2e2ω4 ω 2π [ 2 Mω2−kξ 2π/ω ∫ 0 sin ( ωt+ ∆ϕ 2 ) cos(ωt) dt+ − D2∆ϕsinγ Jω2−kyl2 2π/ω ∫ 0 sin(ωt+γ)cos(ωt) dt+ (2.11) − ∆ϕ Mω2−kη 2π/ω ∫ 0 sin2(ωt) dt− D2∆ϕcosγ Jω2−kyl2 2π/ω ∫ 0 sin(ωt+γ)sin(ωt) dt ] = = −m2e2ω4 2 [ 2 Mω2−kξ sin (∆ϕ 2 ) − D2∆ϕsinγ Jω2−kyl2 sinγ− ∆ϕ Mω2−kη + − D2∆ϕcosγ Jω2−kyl2 cosγ ] = m2e2ω4 2 ( D2 Jω2−kyl2 + 1 Mω2−kη − 1 Mω2−kξ ) ∆ϕ As it can be seen, the values of bothmoments are equal while their direc- tions reverse. Thus, their difference equals ∆Mw =Mw2av −Mw1av = (2.12) =m2e2ω4 ( D2 Jω2−kyl2 + 1 Mω2−kη − 1 Mω2−kξ ) ∆ϕ The above expression constitutes the measure of the ability of the sys- tem to generate the synchronisingmoment, when due to a certain reason the systemwith natural tendency for synchronous cophasal running operates dis- synchronised by an angle ∆ϕ. Let us now consider the influence of periodical collisions with the feed material on vibrators running. After satisfying certain, given below, limitations for the machine motion, the feedmaterial performsperiodicalmotion.Theperiod of thismotion equals the vibration period of themachine. Typical motion of the system is shown in Fig.2. It is being proven, in the theory of motion of a material point on a plate vibrating with a harmonic translatory motion, that the time instant of the 162 J. Michalczyk, P. Czubak Fig. 2. Feed andmachine bodymotion; yn, ym are vertical displacements of the feed and the body, respectively feed material falling on the machine body t3 is a function of a dimensionless parameter kp called the coefficient of throw (Czubak and Michalczyk, 2001; Michalczyk, 1995). This parameter, determining the ratio of the perpendicular component of the machine body vibration accelerations to the acceleration of gravity is expressed as follows kp = Aω2 sinβ g (2.13) where A – vibration amplitude along the ξ axis, g – acceleration of gravity, β – inclination angle of body vibrations versus the horizontal line, Fig.1. Since for m≪M the disturbances ofmachinemotion caused by collisions with the feed are quite small, it is allowed to use the equation for the time of falling t3 t3 = 1 ω arcsin ( 1 kp ) + 2π ω n (2.14) where the first component determines the time t2 of the feedmaterial detach- ment from the body, the second component – the time of a free flight, while n is the root of the equation developed byA. Czubak (Czubak andMichalczyk, 2001) kp = √ [cos(2πn)+2πn2−1 2πn− sin(2πn) ]2 +1 (2.15) The time t3 is counted versus the initialmoment t=0assumed in the instant when themachine body achieves itsmaximumvelocity ξ̇max. This description Influence of collisions with a material feed... 163 is correct for one-strokemotion (whichmeans that the flight is not longer than for 1 vibration period of the machine), which occurs for 1 0, when the feed falls on the body being below the state of static equilibrium, the collision constitutes a higher load for vibrator No. 2. In order to maintain equality of the average angular velocity, the dissynchronisation of vibrators (of the type analysed pre- viously) must occur. Vibrator No. 1 has to run with a lead ∆ϕ as compared to vibrator No. 2, which causes diversification of vibratory moments origina- ted by vibrations of axes in the opposite direction than those originated from collisions with the material feed. Thus, equating modules of Equations (2.12) and (2.23), it is possible to determine the disphasing angle ∆ϕ of vibrators and then – on the basis of (2.3)3 – the time history of body oscillations re- sulting from collisions with a material feed. It should be emphasised, that in consideration of the equality of the average angular velocity of both vibrators, the static characteristic inclination of the driving motors (which shapes the cumulative value of the power input) does not participate in transmission of an increased power into the more loaded vibrator. The form of Equation (2.23) indicates the possibility of equalisation of loads of vibrators originated from collisions with the material feed and thus avoiding dissynchronisations of vibrators and angular oscillations of the body. To this end, it is enough to assure that sinϕ0 = 0 at the moment when the material feed falls on the body. For a single-stroke motion this happens when t3 =π/ω or t3 =2π/ω, i.e. when the collision occurs at themomentwhen the body passes through the balance point. It can be stated, on the basis of Equations (2.14) and (2.15), that such a case occurs for the coefficient of throw: kp =1.14 and kp =2.97. Since thevalueof kp =1.14 ismost oftennot sufficient for aneffective tech- nological process, the assumption of kp = 2.97 is recommended for avoiding body oscillations which cause irregular distribution of vibration amplitudes along the body. 166 J. Michalczyk, P. Czubak 3. Simulation investigations Simplifying assumptions adopted in the given above analysis indicate use- fulness of the verification of the obtained equations by means of computer simulation of the system motion. The model of the system presented in Fig.3 was used for the numerical simulation. Fig. 3. Model of the feeder together with the material feed The model consists of: two inertial vibrators of an independent induction drive (described by static characteristics), themachine bodyperforming plane motion and supported by a system of vertical coil springs and five four-layer models of the loose material feed (Czubak and Michalczyk, 2001; Michalczyk andCieplok, 2006) arranged in differentpoints of themachineworking surface. The effect of the gravity force on angularmotion of the vibrators is taken into account in this model. Themathematicalmodel of such a systemconsists ofmatrix equation (3.1) describing the machine motion, equations (3.6) concerning electromagnetic moments of drivingmotors, equations (3.5) determiningmotions of successive layers of thematerial feed as well as Equations (3.3) and (3.4) describing nor- Influence of collisions with a material feed... 167 mal and tangent interactions in between thematerial feed layers and between thematerial feed and the machine body Mq̈=Q (3.1) where M=        Mk+m1+m2 0 m1h1+m2h2 m14 m15 0 Mk+m1+m2 −m1a1−m2a2 m24 m25 m1h1+m2h2 −m1l1−m2l2 m33 m34 m35 m41 m42 m43 J01 0 m51 m52 m53 0 J02        q̈= [ẍ, ÿ, α̈, ϕ̈1, ϕ̈2] ⊤ (3.2) Q= [Q1,Q2,Q3,Q4,Q5] ⊤ where m14 =m41 =m1e1cos(β+ϕ1) m15 =m51 =m2e2cos(ϕ2−β) m24 =m42 =m1e1 sin(β+ϕ1) m25 =m52 =−m2e2 sin(ϕ2−β) m33 =m2h 2 2+m2l 2 2 +m1h 2 1+m1l 2 1 +Jk m34 =m43 =m1h1e1cos(β+ϕ1)−m1l1e1 sin(β+ϕ1) m35 =m53 =m2h2e2cos(ϕ2−β)+m2l2e2 sin(ϕ2−β) Q1 =−m2e2ϕ̇22 sin(ϕ2−β)−m1e1ϕ̇ 2 1 sin(β+ϕ1)−kx(x+Hα)+ −bx(ẋ+Hα̇)−T101−T102−T103−T104−T105 Q2 =m2e2ϕ̇ 2 2 sin(ϕ2−β)+m1e1ϕ̇ 2 1cos(β+ϕ1)− 1 2 ky(y+ l1α)+ − 1 2 ky(y− l2α)− 1 2 by(ẏ+ l1α̇)− 1 2 by(ẏ− l2α̇)+ −F101−F102−F103−F104−F105 Q3 =−m1h1e1ϕ̇21 sin(β+ϕ1)−m1l1e1ϕ 2 1cos(β+ϕ1)+ −m2h2e2ϕ̇22 sin(ϕ2−β)+m2l2e2ϕ̇ 2 2cos(ϕ2−β)−kxH 2α−kxHx+ −bxHẋ−bxH2α̇− 1 2 ky(y+ lα)l+ 1 2 ky(y− lα)l− 1 2 by(ẏ+ lα̇)l+ + 1 2 by(ẏ− lα̇)l+(T101+T102+T103+T104+T105)Hn+F1012d+ +F102d−F104d−F1052d Q4 =Mel1− bs1ϕ̇21 sgn(ϕ̇1)−m1ge1 sin(β+ϕ1) Mel2− bs2ϕ̇22 sgn(ϕ̇2)−m2ge2cos(ϕ2−β) 168 J. Michalczyk, P. Czubak and Fj,j−1,k – normal component of the j-th layer pressure on the j− 1 layer in the k-th column, Tj,j−j,k – tangent component of the j-th layer pressure on the j− 1 layer in the k-th column, j – material feed index (j=0 concerns the machine body), k – material feed column index J0ic – central moment of inertia of mi, i=1,2 mie 2 i +J0ic = J0i i=1,2 It was further assumed that J01 = J02 = J0 If successive layers of thematerial feed j and j−1 (in the given column) are not in contact, the contact force in the normal direction Fj,j−1,k and in the tangent direction Tj,j−1,k between these layers equals zero Fj,j−1,k =0 Tj,j−1,k =0 for ηj,k ­ ηj−1,k Otherwise, the contact force in the normal direction between the layers j,k and j−1,k of the material feed occurs (or in the case of the first layer: between the layer and the body), themodel of which (Michalczyk, 2008) is of the form Fj,j−1,k =(ηj−1,k−ηj,k)pkH { 1− 1−R2 2 [1−sgn(ηj−1,k−ηj,k)sgn(η̇j−1,k−η̇j,k)] } (3.3) and the force originated from friction in the tangent direction Tj,j−1,k =−µFj,j−1,k sgn(ξ̇j,k− ξ̇j−1,k) (3.4) where R is the restitution coefficient of normal impulses at collision, kH, p – Hertz-Stajerman constants. The form of dependence (3.3) was developed in Michalczyk (2008) on the basis of the Hertz-Stajerman contact forces model modified by taking into account material damping. Parameters of the hysteresis loop were assumed in such a way as to have the ratio of the bodies relative velocity after the collision to their velocity before the collision equal to R. It means that formula (3.3) ensures that this ratio is equal to the assumed restitution coefficient. Influence of collisions with a material feed... 169 Equations of motion of individual layers in the directions ξ and η, with taking into consideration the influence of the conveyer on the lower layers of the material feed are in the following form mnj,kξ̈=Tj,j−1,k−Tj+1,j,k (3.5) mnj,kη̈=−mnj,kg+Fj,j−1,k−Fj+1,j,k Meli – electromagneticmoment generated by the i-thmotor assumed in the form corresponding to the static characteristic of the motor Meli = 2Mut(ωss− ϕ̇i1)(ωss−ωut) (ωss−ωut)2+(ωss− ϕ̇i)2 i=1,2 (3.6) where: Mut – stalling torque of the driving motors, ωss – their synchronous frequency and ωut – stalling frequency. The simulation was performed for the following parameters: l = 0.5m, l1 =1m, l2 =0.5m, H =0.0m, h1 =0.5m, h2 =1m, bx = by =400Ns/m, kx = ky = 150000N/m, m1 = m2 = 5kg, Mk = 120kg, J01 = J02 = J0 = variable, Jk = 25kgm 2, e1 = e2 = variable e(kp)m, D = 1.118m, mut = 50Nm, ωss =50π rad/s, ωut =15.9 ·2π rad/s, bs1 = bs2 =0.00009Nms2. The simulationmodel developed for the verification of analytical solutions takes into consideration not only factors included in the analytical solutions but also other phenomena of essential meaning for the process of vibrators synchronisation, such as e.g. force of gravity. In addition, no limitations for disphasing angles were introduced and the vibratorymoments were treated as variables (not averaged) within the period of machine vibrations. 4. Conclusions In order to verify the analytical solution, the amplitudes of angular oscillations of the body Aα, being the result of vibrators disphasing due to collisions with the material feed were determined on the basis of equations (2.12), (2.23) and (2.3)3. The obtained values were compared with the simulation results. The calculations and simulations were performed for various values of the coefficient of throw kp from the range [1, √ π2+1] and for two masses of the feed: mn =20kg and 60kg.The results obtained byanalytical and simulation methods are presented in Figs.4, 5, 6 and 7. 170 J. Michalczyk, P. Czubak Fig. 4. Angular amplitudes of the body Aα versus coefficient of throw kp for the material feed in a lump form of mass of 20kg, (a) theoretical curve, (b) digital simulation Fig. 5. Angular amplitudes of the body Aα versus coefficient of throw kp for the material feed in a lump form of mass of 60kg, (a) theoretical curve, (b) digital simulation Fig. 6. Angular amplitudes of the body Aα versus coefficient of throw kp for the loose material feed of mass of 20kg, (a) theoretical curve, (b) digital simulation Influence of collisions with a material feed... 171 Fig. 7. Angular amplitudes of the body Aα versus coefficient of throw kp for the loose material feed of mass of 60kg, (a) theoretical curve, (b) digital simulation The comparison of results obtained analytically and by means of digital simulation leads to the following conclusions: • The mathematical model, developed in this paper, properly describes the influence of a lumpedmaterial feed on diversification of phase angles of the vibrators and the resulting angular oscillations of the body for the coefficient of throw within the range kp =1.5 to 3.3, corresponding to variability of this parameter in industrial conditions. It allows one to predict, with high accuracy, the maximum value of body angular oscillations and its position (kp ≈ 1.75). It also indicates the existence of the amplitude minimum for kp ≈ 3.0. However, the value of this minimum is not zero, as indicates the theory, but approximately 20% of the peak value. • In the case of a loose material feed, the analytical dependencies allow prediction, satisfactory for the practice, of the maximum body angular oscillation and its position. However, in this case, the minimum of body oscillations does not occur for kp ≈ 3.0 as predicts the theory, but near kp = 2.7, and its value varies by 20% (for a material feed constituting app. 15% of the body mass) up to 47% (for a material feed constituting app. 46% of the body mass) of the peak value. 172 J. Michalczyk, P. Czubak References 1. Blekhman I.I., 1994,Vibracyonnaya mekhanika, Nauka, Moskwa 2. Blekhman I.I., Yaroshevich N.P., 2004, Extension of the domain of appli- cability of the integral stability criterion in synchronization problems, Journal of Applied Mathematics and Mechanics, 68, 6, 839-846 3. CzubakA.,Michalczyk J., 2001,Teoria transportu wibracyjnego,Wyd. Po- litechniki Świętokrzyskiej, Kielce 4. Lavendel E.E. (edit.), 1981, Vibračıi v Tehhnikie, 4, Mashinostroenie, Moskwa 5. Michalczyk J., 1995,Maszyny wibracyjne, WNT,Warszawa 6. Michalczyk J., 2008, Phenomenon of force impulse restitution in collision modelling, Journal of Theoretical and Applied Mechanics, 46, 4 7. Michalczyk J., Cieplok G., 1999, Wysokoefektywne układy wibroizolacji i redukcji drgań, CollegiumColumbinum, Kraków 8. Michalczyk J., Cieplok G., 2006, Model cyfrowy przesiewacza wibracyj- nego,Modelowanie Inżynierskie, 32, 1 Wpływ zderzeń z nadawą na współfazowość synchronizacji wzajemnej wibratorów napędowych maszyn wibracyjnych Streszczenie Wpracywskazanona istnienie związku poomiędzy zakłócającymiprzebieg trans- portu wibracyjnego wahaniami korpusów maszyn wibracyjnych a utratą współfazo- wości wibratorów napędowych. Wykazano, że utrata współfazowości spowodowana być może przez okresowe zderzenia korpusu z nadawą i zbudowano model matema- tyczny tego zjawiska. Uzyskane zależności analityczne, pozwalające na oszacowanie rozfazowaniawibratorów i ocenę amplitudywahańkątowychmaszyny, zweryfikowano przez porównanie z rezultatami otrzymanymi na drodze symulacji cyfrowej zachowa- nia układu. Manuscript received May 4, 2009; accepted for print June 26, 2009