PROBLEMS OF INTERACTION OF VIBRATING SURFACES WITH PROCESSABLE MATERIALS

We consider the mathematical model of interaction of a vibrating surface with the load placed on it. With the purpose of accounting for influence on behavior of not only system interaction of a blade with a load but also internal interaction of particles of a material, the load is submitted as a finite number of strips with zero thickness. The carrying blade is represented as a vibrating membrane. It is supposed that the weight of the material is comparable to or considerably surpasses the weight of the blade. Therefore, the model takes into account the inertia of the material. In the model with joint movement of the blade and the load, the separation opportunity of the load from the blade is provided. Therefore, there is a phase of separate movement of the blade and the load, with their subsequent connection accompanied with impact. The process of system movement is represented as alternating sequences of joint and separate movements of the load and the blade. The modeling of the process of the interaction of the load and the blade is represented as an initial-boundary value problem. The method of solution is developed and the exact solution of the set problem is obtained in a class of generalized functions.


Introduction
The experience of industrial application of various mechanisms using processing of materials on a vibrating surface has shown that the efficiency of such machines influences not only interaction of a material with a carrying surface, but also internal interaction of particles of a material. Therefore, for reception of additional parameters for optimization of work of such type of mechanisms, it is necessary to create a mathematical model allowing to dismember processable material on components, and then to provide each of such parts with the required characteristics. the material is comparable to or considerably surpasses the weight of the blade. Therefore, the model takes into account the inertia of the material. In case of a membrane, the movement of system is described by an equation of the kind where ρ i is i's layer of a material density, δ-Dirac function, h i -growing sequence of numbers. The oscillatory movements are imparted to the membrane at its edges, T are the stretching efforts.
The carrying blade has a rectangular form with the sides of the rectangle parallel coordinate axes, and 0 ≤ x ≤ b, 0 ≤ y ≤ L. To the sides x = 0 and x = b of this blade is applied harmonic moving; the parties y = 0 and y = L are free. It is assumed that on a blade along a straight line x = h i , the thin layers of a load of density ρ i are located, the blade is stretched on its edges in a plane xOy by efforts T.

Solution of the first stage of the problem
With the help of principle of superposition, the solution of the given problem is found as the sum of three functions Here the function u 1 (x, y,t) describes stationary oscillations of a blade with load and is represented as satisfying also the homogeneous equation appropriate (2.1), and the boundary conditions (2.2) and (2.3). The function u 2 (x, y,t) describes free oscillations of a blade with a load arising by start from a status of equilibrium and rest, and is the solution of the homogeneous equation appropriate (2.1), satisfying the boundary conditions At last, the function u 3 (x, y,t) describes oscillations of a blade arising owing to the sudden application of a load at t = 0, and is the solution of (2.1) with the initial conditions (2.4) and the boundary conditions (2.3) and (3.3).
To find of all functions u i , i = 1,2,3, there comes up a question on obtaining the general solution of the equation Therefore, first of all, we will obtain the general solution of (3.5), using a method developed in [1].
The real general solution of the homogeneous equation for p > 0 is Z(x) = C 1 sin px + C 2 cos px, (3.8) and for p = 0 is Z(x) = C 1 x + C 2 . Therefore, if z i (x) are the solutions of (3.6) which satisfy the initial conditions then for p < 0, for p > 0, and for p = 0, z i (x) = x − h i . Accordingly to [1], the general solution of (3.5) should be found as where H(x) is the Heavyside function. As a result of substitution of this form of the solution in (3.5), on account of the initial conditions (3.9), properties of δ-function, and following from (3.12), equalities we obtain (3.14) Hence, the general solution of (3.5) is the function Instead of recurrent formulas for factors B i , it is more convenient to use obvious values of these factors, With the purpose of reception of the decision in an obvious and most evident kind, we will state in this paper the solution of the problem for a case of two symmetrically located strips of a load. We assume n = 2, In this special case the function X(x) is represented as follows. For p < 0, For p > 0, For p = 0, We consider the problem of obtaining a function u 1 (x, y,t). Substituting the form of the solution (3.2) in the homogeneous equation (2.1), we obtain the equation for the function U 1 (x, y): To solve (3.21) we will apply a method of separation variables in the form As a result of separation variables, we obtain system of the two equations with the boundary conditions For λ = 0, general solution of the first equation (3.23) turns out from the formula (3.19) at and consequently it looks as From (3.24) we obtain X(0) = A, X(b) = A, whence and from (3.28) we determine constants of integration. Thus, having designated we will have Thus, the solution of a problem for function u 1 looks as where X(x) is determined by formula (3.30), and D by formula (3.29).
We consider now problem on finding function u 2 (x, y,t). Separating variables, we will find function u 2 (x, y,t) as (3.32) The substitution of the form of the solution (3.32) in the homogeneous equation (2.1) results in the two equations Solution of the first equation in (3.33) also is found by method of separation variables as (3.34) In result there appears a boundary problem of Sturm and Liouville about integration of system of the equations with boundary conditions Therefore the question on existence of eigenvalues of a considered boundary problem for p < 0 is reduced to a question on existence of positive roots of function G(z). In [3], it is shown that at 1/α < γµ n b < 1/(α(1 − α)) such positive roots z n exist. In this case boundary problem for X(x) has the eigenvalues λ n = a 2 µ n − z n b 2 (3.42) and the eigenfunctions (3.43) (b) p = 0, that is, λ n = (πn(a/L)) 2 , n = 0,1,2,.... Denoting q = (1/T)(πn(a/L)) 2 ρ H , we will have in this case that the general solution of the second equation in (3.35) looks as (3.20) at S 1 = S 2 = 0, q 1 (h) = q 2 (h) = q. The substitution of this solution in boundary conditions (3.37) gives Roots of function in square brackets are q 1 = b/h(b − 2h) and q 2 = 1/h, whence Generally, numbers n 1 and n 2 will not be natural, and it will mean that for p = 0 the boundary problem (3.35), (3.37) has no eigenvalues. But, if one of numbers n 1 or n 2 , or both these numbers are integers, the boundary problem (3.35), (3.37) will have one or two eigenvalues λ n1 = (πn 1 (a/L)) 2 and λ n2 = (πn 2 (a/L)) 2 and one or two eigenfunctions (3.46) (c) p > 0, that is, λ > (πn(a/L)) 2 , n = 0,1,2,.... In this case general solution of the second equation in (3.35) looks as (3.19) Therefore, substituting (3.19) in boundary conditions (3.37), we will have C 2 = 0, (3.47) In [3], it is shown that the function G(z) has at each n a countable set of positive roots in points z nm , n = 0,1,2,..., m = 1,2,.... Here roots z nm are supplied with an index m in order of their increase. Then (p nm ) 1/2 = (z nm /b), λ nm = a 2 ((z nm /b) 2 + (π(n/L)) 2 ), q nm = a 2 (ρ H /T)((z nm /b) 2 + (π(n/L)) 2 ), n,m = 0,1,2,....
The eigenfunctions of a boundary problem for the second equation (3.35) with boundary conditions (3.37), corresponding to eigenvalues λ nm , look as (3.48) The functions T(t) now are easily determined.

V. A. Ostapenko 403
Hence, solution u 2 can be written as with arbitrary constants A nm , B nm , A n , B n , A ni , B ni . In the general case for obtaining these constants, it is necessary to expand the right parts of the initial conditions It can be shown that the eigenfunctions Xnm(x), responding to various eigenvalues λ nm , are linearly independent, but not orthogonal on an interval [0,b]. Therefore coefficients of expansion b nm and c nm have to be determined from infinite system of the linear equations Determinants of these systems are determinants of Gramme, therefore these systems have unique solution for b nm and c nm , respectively. In a considered special case the kind of the solution u 1 permits the formula (3.49) to be limited only to double sum, and therefore where t OTP is the least positive root of the first equation (4.1), determining the moment of possible separation of a load. If the separation will take place, the structure of system at t > t OTP will change and beginning from t = t OTP a load and blade will sometime move separately. The movement of a blade will be determined by function v(x,t), being on an interval 0 < x < b and at t > t OTP by the solution of the wave equation The movement of a load will be described by the equation The substitution (4.7) in (4.2) results in the equation whose general solution is the function Substituting (4.7) in boundary conditions (4.4) and taking into account (4.9), we will receive system of the equations from which are obtained Therefore, (4.12) Then the function v 2 (x,t) should be the solution of (4.2), satisfying the homogeneous boundary conditions v 2 (0,t) = 0, v 2 (b,t) = 0, (4.13) and the initial conditions (4.14) The problem of searching for the function v 2 (x,t) can be solved by the method of continuations or Fourier method. In case of application of the Fourier method, the function v 2 (x,t) will look like [ Thus, if separation of load and blade will take place, the separate movement of a blade will be described by function In a case of n strips, the problem becomes complicated as separation can take place only in parts of strips, and in this case movement of a blade and part of strips, which has stayed on it, will be described by an equation of the kind (2.1). In this case, problem about movement of a blade is necessary to solve in principle the same way as in case of movement of system without separation. However in each concrete case, the various layers of load can separate from blade and in various sequences, therefore problem becomes multivariant.

Connection of the load and the blade
Further there will be a connection of a load and a blade at the moment of time t cµ , which is determined as the least positive root of the equation z(t) = v(h i ,t), greater than t OTP . At t > t cµ , the system again will change structure, and at this stage again it is necessary to consider a problem about joint movement of a blade and a load. This problem is similar to the one considered at the beginning, only initial conditions in case of two stripes will look as u x,t cµ = v x,t cµ , u t x,t cµ = ψ 2 (x), (5.1) where the function ψ 2 (x) is under construction, proceeding from the following reasons. The blade at t = t cµ has speed v t (x,t cµ ). Besides, on straight lines x = h and x = b − h on a blade, at this moment the load having speed z (t cµ ) falls. Proceeding from preservation of quantity of movement on straight lines x = h and x = b − h, we accept that at the moment of time t = t cµ the speed on these straight lines is v h = ρv t x,t cµ + ρ H z t cµ ρ + ρ H .
(5.2) Therefore ψ 2 (x) is equal to v t (x,t cµ ), if x = h or x = b − h, and is equal v h , if x = h or 408 Interaction of vibrating surfaces with materials

Connected movement of load and blade
The solution of the problem on the connected movement of load and blade after their joining, that is at t > t cµ , is found as the sum of three functions: u(x, y,t) = u 1 (x, y,t) + u 31 (x, y,t) + u 4 (x, y,t). (6.1) Here functions u 1 (x, y,t) and u 31 (x, y,t) are the same ones that have been obtained above with formulas (3.2) and (3.55) accordingly. We only recall that finally function u 1 (x, y,t) is obtained as