There is proposed mathematical model of process of vibration movement of flexible particle along harmoniously oscillated surface, with counting possibility of flexible particle and its own oscillations.
The relationship obtained as a result of model analysis establishes instant connection between speed of movement of absolutely solid particle as well as flexible particle depending on physical and mechanical properties of the latter and oscillation frequency of surface.
The obtained results and conclusions are necessary of providing of non-destructive testing of serviceability of industrial equipment for process in principle of which lies movement of flexible material along vibrated surface (oscillating conveyer, oscillating trough, vibrating feeder, vibrating separator, etc.), as well as for adjustment of apparatus for maximum capacity.
At present at the different enterprises there is more widely applied the processing of different waste to re-usable raw material. The creation of lines and technologies on processing of secondary resources proposes usage of different devices for transportation or supply of raw to the equipment envisaged by technological process. The vibrating transportation of the enrichment products has become the most advanced kind in the resource- saving technologies . In process of exploitation of vibration devices there is a necessity of the solution of the important problems of adjustment and providing the non-destructive testing of serviceability of the equipment. For the successful solution of these problems it is necessary to grasp the model of vibrating transportation of the granulated material possessing different resiliency. The mentioned below mathematical model of process of vibration transportation of an elastic material allows to obtain conclusions for a quantitative assessment of speed of vibration transportation of particles of different resiliency and size, and this, without doubt, allows to solve the above-mentioned subjects.
As it is known, all bodies without exception have elastic properties. In process of interaction of an excitation source, ( in this case it is harmonically oscillating plane), with a body, in the latter natural oscillations occur, which depending on properties of a material have different period of damping and amplitude. In a general view the problem of interaction of an excitation source with a system of oscillation was considered earlier by other authors, but from the point of view of influencing natural oscillations of an elastic body on vibration transportation, as far as we know, the given problem was not under decision. As base the model there was used the model of an absolutely solid body.
The model, proposed by us, takes into account a possibility of elastic deformation of a particle on a vibrating plane. Thus, for simplification a certain inadequacy of the proposed model to real process is accepted, however it seems to us, that the given model contains the main features of a phenomenon. It is more important to the fact, that the experimental study of movement of elastic materials on a vibrating plane is extremely hindered because of impossibility of exact separation of elastic properties of a particle from its other physical and mechanical properties (size, shape, density, weight etc.), but at the same time the ratio, obtained in the proposed paper, have received full experimental confirmation.
Let us consider behaviour of an elastic fragment on is harmonically oscillating plane with accounting of fact that the following forces (fig. 1) act on it: gravity 
, force of reaction of a support 
, where
where l- characteristic size of a fragment,
r - density of a fragment.
Fig 1: Forces acting on an elastic particle during oscillations
of a horizontal plane.
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As for it depends on a degree of a roughness of a working surface, i.e. from a coefficient m which is taking into account adhesion of a material with a working surface. If to change angle of a slope of a working surface to a horizontal there are two more forces will occur:
which in its turn depends on force 
and thrust force 
(see fig. 2).
Fig 2: Forces acting on an elastic particle during oscillations
of an inclined plane.
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The origin of force in this case is of no principled value , and, in particular, can be the component of gravity force . From here is apparent, that
At value of vibrating acceleration 
there is a full-contact motion. In this case it is easy to determine an average speed of motion of a particle for one period:
For simplification it is possible to admit, that F = mP,, then
| (1) |
Optimising expression (1) for period we receive average speed of motion of an absolutely solid body over vibrating plane 
:
| (2) |
As it is seen from equation(2) the average speed of motion of a particle over inclined plane depends only on oscillation frequency of a working surface and its roughness (m).
For consideration of behaviour of an elastic particle it is necessary to introduce one more force - elastic force 
, which takes into account natural oscillations of a particle depending on its deformation. The change of elastic force depending on deformation of a particle is advisable to consider according to a Hooke's law. Then, in view of oscillations Z'(t):
We'll obtain:
| (3) |
Where k - elastic coefficient of a fragment, R - characteristic size of a particle.
As it is known, the resiliency of a single particle is connected to a young's modulus by a following ratio:
| (4) |
Where E - Young's modulus, S - contact area of a particle with a surface, l - typical size of a particle (for particles of the spherical shape average diameter is accepted).
Thus, the law of vertical oscillations of a centre of mass on an inclined plane in respect to considered fragment can be described by a following equation:
After transformation we shall receive:
Where we shall introduce identifications:
Having accepted initial conditions Z(0)=R and Z(0)=0 and solving equation analytically, we receive following expression:
| (5) |
Here a is that characteristic parameter, which takes into account elastic properties of a particle characterised by natural oscillations f the latter. The forces of reaction of a support of such particle in view of expression (5) will be equalled to elastic force in each instant of time:
After introducing identifications we obtain:
j - the initial phase with which movement of a particle is considered.
Let's compare force of pressure of an elastic fragment to force of pressure of an absolutely solid par:
| (6)
|
| (7)
|
From equations (6) and (7) follows, that taking into account resiliency of material of a particle results in change of oscillations kind of such particle, and consequently also thrust forces F, which in its turn depends on friction force Ffr and provides movement of a particle along the plane.
Thus, the elastic particle moving downwards (if necessary it can move upwards) on an inclined plane (under condition of F)Ffr ) except of forced oscillations from a vibrating source has also its natural oscillations c(a)sin(at+j) in view of the certain factor h(a ) , which depends on properties of a material such as the size, the resiliency (young's modulus' etc. Besides from ratio (6) and (7) follows, that the elastic particle on a vibrating plane represents a mechanical system possessing natural oscillations on which one the forced oscillations from an excitation source are superimposed. The nature of their interaction also determines specificity of behaviour of an elastic material on an vibrating surface. The average speed of movement of an elastic particle on an inclined is determined as follows:
| (8) |
Let's simplify expression (8) having accepted Ffr=m p. It is permissible, as there were no limitations for value of angle of inclination of a vibrating plane to a horizontal . Then we shall receive expression:
| (9) |
Value of a phase j, having accepted the applicable time zero of movement of a particle, we equate to zero j = 0. After integrating expression (9) we receive:
Having averaged speed on period we shall receive:
Having substituted value 
is received:
| (10) |
It is possible to neglect by relation P/ma, in connection with a small value of the latte, then the expression (10) will become as follows 
:.
In view of expression (1) is received:
| (11) |
Thus, from expression (11) follows, that average speed of motion of an elastic particle on a vibrating surface 
is connected to similar speed of movement of an absolutely solid particle 
by the applicable ratio.
For real materials, including rubber 
, therefore factor 
. In such case, after simplification the expression (11) becomes:
| (12) |
The obtained ratio of speeds of an elastic particle and absolutely solid particle demonstrates, that the vibrating movement of an elastic material is hindered compared to similar movement of particle of an absolutely solid material. Though the considered model not absolutely corresponds to a reality, since it proposes the full-contact nature of its motion on a vibrating plane, i.e., as a matter of fact, there was considered a case of oscillations a material point connected with is harmonically oscillating surface by a certain rigid connection. However the results obtained here were exactly confirmed experimentally during research of vibrating movement of metal and rubber particles along horizontal and inclined planes. In further there is a problem of a quantitative assessment of speeds in view of a proposed relation (12). For this purpose it is advisable to link in one ratio characteristic parameters of a conveyed material (density r, size l, young's modulus E ) with value a characterizing natural oscillations of a fragment. In view of expression (4) we shall receive :
where: r - density of a particle,
v - Volume of a particle.
Since the particle, for example, particle of rubber in rubber-processing production, as a rule has always incorrect shape, then in his case it is approximately possible to accept S ~ l2 and v ~ l3 then
In view of it, the expression (12) is possible to convert into a shape suitable for practical calculations :
| (13) |
Where J - mean horizontal speed of an elastic material,
Jo- Mean horizontal speed of an absolutely solid material,
w - Oscillation frequency of a plane.
Thus, from the obtained ratio (13) follows, that with reduction of rigidity (young's modulus) of particles the speed of horizontal movement of a material on a vibrating trough is slowed down respectively. At the same time, in case of greater grinding of a material (decreasing of parameter l), effects bound with resiliency decrease. From here follows, that at enough small size the material will be moved so as if it is consisted of absolutely solid particles.
Let's apply the obtained expression (13) for a quantitative assessment of speed of vibrating transportation of rubber chip. In view of an industrial current ( v = 50 Hz ) w = 2 p v » 300 Hz,. Density of gum makes r = 1,2 . 103 kg / M3, the young's modulus lays within the limits 1,5-5,0 MPa. The typical size of rubber conveyed after crushing on a vibrating trough in real process makes up l » 10MM = 10-2M.
Then the speed of movement of such rubber chip will be determined as follows:
| (14) |
From (14) follows, that according to adopted allowances the speed of vibrating movement of the given rubber chip of size about 10 mm will be in 0,94 times less, than for an absolutely solid material of the same size. Besides, the expression (13), after relevant transformations, allows to estimate critical size of material, i.e. the maximum size of rubber particles for which vibrating movement along oscillating plane is completely ceased:
| (15) |
From here we receive :
.
Substituting adopted before value of parameters of gum in (15) we shall receive :
| (16) |
From outcome of expression (16) it follows, that in particles of rubber of size 150 - I 60 mm the oscillations from an excitation source pass over due to resiliency of a material not into energy of mechanical movement, but into energy of natural oscillations.
Therefore, for effective vibrating movement of an elastic material, the feed stock should be crushed with the accounting of the fact that during transportation of less rigid material more small-sized grinding of raw is necessary (E (r) 0 and lcrit (r) 0). The quantitative assessments can be conducted according to the formulas (13) an (15).
The calculations carried out here have preliminary nature and do not claim for a split-hair accuracy, as the mathematical model inadequate to real process was initially adopted, since the model proposes full-contact nature of motion of particle along vibrating plane- the fact which does not take place actually. However, namely sufficient simplification of a mathematical model allows to finish calculation up to practically significant quantitative outcomes. The further elaboration of (theoretical) results obtained here should be performed exclusively on the basis of special experimental researches. At the same time, the results and conclusions obtained here are rather important and are indispensable in process of design of vibrating conveyor, selection and layout of a technological and production equipment of a line in rubber-processing and other productions, adjustment of vibrating equipment for maximum productivity, and mainly for providing the non-destructive testing of serviceability of different kinds of the vibrating equipment.
