THEORETICAL RESEARCH OF STRESS IN RUBBER-FABRIC CONVEYOR BELTS

 

ABSTRACT

Under the combined action of stretching, bending and circumferential force, compressive stresses arise in the rubber layers, caused by the action of the overlying stretched gaskets on the underlying ones. The article is devoted to the fact that, in terms of the available methods, it is assumed that there is no influence of normal compressive stresses arising between the layers of the tape moving along the drum. To determine the bending and tensile stresses in the tape, it can be concluded that it is necessary to develop more deliberate methods, these circumstances forced us to carry out an independent analysis using the provisions of the theory of elasticity

АННОТАЦИЯ

При совместном действии растяжения, изгиба и окружного усилия в резиновых прослойках возникают сжимающие напряжения, вызванные действием вышележащих растянутых прокладок на нижележащие.

Статья посвящена, что в части имеющихся методов предполагается отсутствие влияния нормальных сжимающих напряжений, возникающих между слоями ленты, движущейся по барабану. Для определения изгибных и растягивающих напряжений в ленте можно сделать вывод о необходимости выработать более совещенные методы, эти обстоятельства заставили выполнять самостоятельный анализ с использованием положений теории упругости.  

 

Keywords: belt conveyor, belt, rubber-fabric belt, drum, gasket, interlayer, stress, slippage, movement, bending, stretching, tension

Ключевые слова: ленточный конвейер, лента, резино-тканевая лента, барабан, прокладка, прослойка, напряжения, проскальзывания, перемещение, изгиб, растяжения, натяжения.

 

Introduction

As a result of the analysis of methods for determining bending and tensile stresses in the tape, it can be concluded that it is necessary to develop more advanced methods, which will take into account more factors than until now [1-9].

In this regard, it can be recalled that in some of the available methods, it is assumed that there is no influence of normal compressive stresses arising between the layers of the tape moving along the drum [10]. In other works [11, 12], the conclusions of the theory of composite rods are used, but the rods are not considered as complex, statically indeterminate elements.

Despite the severity of the approach to the problem and the depth of research [11, 12], these theories provide solutions that are sufficiently simple for practical application only for beams, rods and packages composed of 2-3 elements and, in addition, they solve only direct problems, where given external loads are looking for deformations and stresses.

The above circumstances forced us to carry out an independent analysis using the provisions of the theory of elasticity. To solve the problem, the following simplifications were adopted:

- the problem is considered under conditions of a plane stress state, i.e. consider the stresses along the areas parallel to the plane to be zero;

- the material of fabric pads and rubber interlayers is considered to obey Hooke's law [13];

- each layer is represented by a model of an extensible thread that does not resist bending; we neglect the change in its thickness during deformation;

- the thickness of the layer, the thickness of the adhesive layer and the thickness of the tape are small compared to the length of the contact zone;

- we consider the problem in a quasi-static setting, at low speeds running around.

Stress in a single layer tape.

Consider a single layer rubber-fabric tape in the area of its contact with the drum [14, 15].

We place the origin of coordinates in the middle of the contact zone (Fig. 1). the X coordinate (curvilinear coordinate) is directed along the tape. It is curvilinear within the range –ƪ0≤x≤ƪ0, outside the segment –ƪ0, ƪ0 - rectilinear, 2ƪ0 - the zone of the tape wrapping around the drum, 2ƪ - the length of the tape, ƪ >> ƪ0.

Consider the picture of the interaction between the tape and the drive drum from a qualitative point of view. The drum stretches the belt in front of it and at the –ƪ0 point the drum hits the stretched belt. For a while, the points of the drum and the belt move together (adhesion with ideal roughness), after which slippage occurs and Coulomb friction arises in contact. But with real ratios of geometric and mechanical characteristics, the friction force transmitted in the adhesion area is insignificant and can be ignored.

 

Figure. 1 Coordinate system

 

To prove the last statement, consider two options. First, assume that there is no slippage.

Consider the tape in the main and auxiliary states (Fig. 2)

 

Figure 2 Estimated belt states:

a - main, b - auxiliary.

 

The ground state is characterized by the fact that on the segment –ƪ0, ƪ0 tangential displacement is given in the form of a linear function and unknowns: normal P (x) and tangent τ (x) stresses. The auxiliary state is characterized by the fact that at some point t a unit force is applied (Fig. 3), which causes tangential displacements and normal stresses due to the curvature of the tape.

 

Figure 3.  Application of a unit force at point t

 

The displacement of point X from a single concentrated force applied at point t of the interlayer is the sum of the displacements of the cord and the displacement of the interlayer with a thickness  on a non-deformable base - a cord. Let us determine the elongation Δ in the cord, terminated by two ends, from the unit force. Discarding the fittings and replacing their action with unknown reactive forces A₁ and A₂, we compose the equilibrium equation:

                                               (1)

In this equation, there are two unknown forces A₁ and A₂, which cannot be found from one equation. To solve it, it is necessary to compose an additional deformation equation. The total length of the tape cannot change, therefore

Δ = 0                                                (2)

The total elongation Δ can be expressed as the sum of the elongations from the force A₁ and on the strength of A₂:

                                     (3)

from here

 

Then we get

                                        (4)

and the elongations will be equal:

                               (5)

To determine the displacement at point X of the cord, terminated by two ends, from a unit force, we use the equation of a straight line passing through two points:

                                       (6)

from here

                                              (7)

Substituting the value (5) into the formula (7), we obtain

  

Let us choose the Dirac delta function δ (x) as a characteristic of the interlayer displacement from a single concentrated force.

Displacement of point X from application to point t of elastic layer with thickness  a unit concentrated force is taken in the form

                                                  (8)

where δ (x) is the generalized Dirac function, which is zero for X ≠ 0 and is equal to infinity X = 0. Formula (8) can be obtained from the passage to the limit when considering the contact problem for a layer when tending 0, where α is the half-length of the contact line. A similar conclusion is given in [16]. Let us check the accepted function (8), considering the uniform shear deformation of a layer of infinite length. Consider a layer with a thickness rigidly coupled with the boundaries G₁ and G₂ bounding this layer (Fig. 4)

 

Fig. 4. Rigidly linked layer with bounding boundaries G₁ and G₂.

 

If  G₂ has no displacement, and G₁ has horizontal displacement F (x) = const, then the tangential stresses τ (x) = const are related to “F” by Hooke's law at shear

On the other hand, this displacement will be obtained by integrating the kernel (8) over the entire line G₁... using the theorems of integration of generalized functions [15], we obtain G₁:

which proves the possibility of using function (8). Therefore, the total displacement at point X from a unit concentrated force applied at point t of the interlayer is equal to:

                           (9).

In formula (9), the first term is the displacement of an elastic layer with a thickness , rigidly linked to the base - the cord, the second - the movement of the cord - fabric.

 where δ (x) is the Dirac delta function;

h is the layer thickness;

G - rubber shear modulus;

F - cross section of the cord layer;

ƪ - half-length of the tape.

Based on Betty's theorem, we obtain

                                   (10)

                                       (11)

A and ε are constants  

Differentiating (10) twice, taking into account (11), we obtain

                                            (12)

where

To determine arbitrary constants of integration, let us consider the process of tensioning the belt by a drum as the process of rolling a drum on a belt. When rolling, the shear stresses in the belt at the point of collision are zero. Therefore, the conditions for finding arbitrary constants will be and equilibrium conditions:

                                   (13)

We get

                       (14)

This function differs little from zero almost throughout the entire length of the contact line and increases strongly to the point ƪ0, which indicates that normal stresses cannot provide perfect contact and there must necessarily be slippage in the contact zone.

In the second version, we will also consider a single-layer tape under the condition of adhesion on the site  and in the slip section in the presence of Coulomb friction ... Therefore, denoting the friction coefficient by ƒ, we obtain, based on Betty’s theorem, the system:

          (15)

System (15) supplemented by equilibrium conditions and conditions of limiting equilibrium at the point α.

From the 4th equation of the system, we obtain by differentiating

                                            (16)

where do we get

                                 (17)

From the first equation by differentiation, we obtain

                                          (18)

Its interval

,                             (19)

To determine C1 and C2, we have the last two equations of system (15)

,

                                    (20)

To find α, one should use the condition , which will lead to a transcendental equation. From the limiting ratios, the following can be indicated:

1.                                               (21)

those. there is no slippage in the contact.

2.                                                          (22)

those. slipping of the tape along the drum occurs within the entire wrap angle.

3.                                                                (23)

those. on the plotthere is a zone of no friction forces and a slipping zone.

At real values G, E, h, h1, ƪ0, ƪ we find that in the adhesion section, the tangential stresses are very small and sharply increase to their limiting value directly near the point α. The latter confirms the rule accepted in technology to calculate the friction arc from condition (17), neglecting the clutch friction (18).

Stress in multilayer tape.

Let us transfer the conclusions about the forces of contact between the drum and a single-layer tape to a multilayer tape, assuming that the shear stresses in the zone of contact between the tape and the drum are completely determined by Coulomb friction. When running around a multilayer tape around the drum, the stress state of the latter is caused by the initial tension of the tape, bending of the tape around the drum ("pure" bending of the tape) and transmission of the circumferential force. The first type of stress state is with uniform stretching by the force To. When the number of layers is n + 1, the tensile force of each layer is equal to ... No tangential stresses. Consider a "clean" bend in the tape. To find tangents stresses between the layers, we divide the tape into separate layers and put them on the drum (Fig. 5).

 

Figure. 5 Dismembered multilayer tape

 

The ends of the tape form steps with a step h α (25), where 2 α is the wrap angle. We apply some normal stresses to the line (they are insignificant) and to the cut lines Ɩ tangents, which should combine the points that diverged as a result of the curvature of individual layers. The normal stresses appearing in the same way will not be included in the equations due to point 4 of the initial prerequisites.

Consider the K-th layer in the main and auxiliary states (Fig. 6). In the ground state, it is acted upon by the stresses τk-1 (x) and τk (x). The displacement will be Vk-1 (x) and Vk (x) - unknown functions. Moving the edges along the layer is denotedIn the auxiliary state, we apply a unit force on top of the K-th layer at the point t. We write the displacement function (2) in the form:

where k is the compliance coefficient

F (t, x) is determined by formula (11).

Using the reciprocity theorem for jobs, we obtain the following relations: for the layer К

 (26) for K + 1 layer (Fig. 7)

 (27) Add (26) and (27)

 (28)

 

Figure 6 Estimated belt states:

a - main, b - auxiliary.

Figure 7. Estimated belt states:

a - main, b - auxiliary.

 

From this it is clear that  - geometric mismatch of points in the dismembered K and K + 1 layers,  - a step between the layers (24). Therefore, the right side of the equation is

Thus, the equation for the junction of K and K + 1 layers is:

 

K = 1, 2, 3 ... ... n.                             (29)

For K = 1, it should be assumed that and the equation contains only two integral terms; for K = n, it should be assumed thatand the equation will also have two integral terms. Thus, the system is complete (i.e., the number of unknown functions is equal to the number of equations):

           (30)

When transferring the circumferential force, the system of resolving equations retains its form (8). Only  and is an unknown function. To solve problem (8), it is necessary to supplement the equilibrium equations for each of the layers, after which the system has a unique solution at  where Рn is the normal pressure on the belt, f is the coefficient of friction.          The system of integral equations for the transfer of the circumferential force has the form f (t) = 0:

         (31)

The resulting system contains n + 2 unknown functions for n + 1 equations, i.e. it is incomplete. This system should be supplemented with equilibrium conditions. For each of the layers, the following equalities are valid:

                                      (32)

where K = 1, 2, 3, ……, n + 1 and the boundary conditions for T:

                                    (33)

where  - circumferential set effort.

System (9) together with the equilibrium equation (32), (33) should be supplemented with the conditions coulomb friction

                                       (34)

The system of equations for determining becomes rather cumbersome, so we will adopt an iterative solution method. At the first stage, we assume that for a given coefficient of friction and tension of the branches T_I and T_II shear stress distribution law within is given by Euler's formula. In this case, we can discard the first equations of system (31) and in the second equation the integral containing, move to the right. After solving the system and findingby formulas (32) we find ... By and  find taking into account pure bending, and, consequently, in a new approach. With a big difference between and newly found  should be considered as the new right side of the system, and so on.

It is interesting to note that with this In the iterative approach, the resolving system of equations has the same form as for the case of “pure” bending. This makes it possible, if necessary, to immediately take into account both the bending stresses and the stresses from the transfer of the circumferential force. For this, functions (28) should be added to the right-hand side of the equation. When solving the problem programmatically The iterative process simplifies the task by using the inverse matrix of the main system. To solve the system of integral equations, we derive some auxiliary formulas. Due to the fact that the kernel of the integral equation is linear in the sections x˂t or x ˃t with respect to "X" has the form

                                  (35)

we derive formulas for definite integrals.

We denote           

Then we get for t outside the segment α, β:

where                           

and

Let's change the coordinates using the formula

x = U - α cos θ, dx = α sin θ d θ                                      (36)

where

                                                               

Then at х = α, θ = 0 and at х = β, θ = π.

Substitute the sought for on the segment α, β function in the form of a trigonometric interpolation Lagrange polynomial with interpolation nodes at Chebyshev points:

                          (37)

where τj is the coefficient of the polynomial; n is the number of interpolation points on the segment α, β θj = - nodes of interpolation.

Representation (37) makes it possible to calculate the previously introduced functions τ * (x) and τ ** (x):

where

             

                      (38)

Wherein

                           (39)

where

 or                                            (40)

The same limiting relations are valid for τ ** (x). The formulas introduced above make it possible to apply the collocation method to the integration line divided into a number of sections, on each of which the sought function has a smooth structure. In this problem, the boundaries of the sections will be the points of change of external influences acting on it:

 - 1st section;

 - 2nd section;

 - 3rd section;

 - 4th section.

It should be noted that the interpolation points are condensed towards the boundaries of the parcels. So, we split the line of integration 2 Ɩ into m sections, on each of which we take the expansion (14) with nq, q = 1, 2, 3, …… m points of interpolation. Then the coefficients of the interpolation formulas will be indexed, where q = 1, 2, 3,…., m is the number of the section on the integration line; m is the number of sections of the integration line break; k is the number of the layer in the multilayer tape; k - 1, 2, 3,…., n, n + 1 - the number of layers in the tape; j is the number of interpolation points on the q^th section. To apply the above approach to solving a system of integral equations, we rewrite it in the form:

       (41)

where f (t) - is determined from formula (6), and τ0 in the first approximation by the formula obtained from (12):

                                       (42)

Substituting formulas (18) into the system and giving θt values for the same interpolation points, we obtain a system of equations of the order... To describe it, we will use a continuous numbering with the unknown xj. The system of linear equations has the form

 , i = 1, 2, 3,…., N                                     (43)

Here

           (44)

Taking into account the structure of system (43) and the adopted indexation, to find δij we use the following computational procedure:

I. We represent i and j as

                                                 (45)

Then at
at
at

where  is the Kronecker symbol.

To calculate F1 representation (45) gives:

k¹1 +1 - number of the integration section;

j¹ is the number of the interpolation coefficient of the polynomial;

k¹ + 1 is the number of the unit force application area;

i'- number of the point of application of a unit force on the site.

Formula (*) taking into account (22) can be written in the form

 ti = Uk1 + 1-αk1 + 1 cosθi                                            (46)

We also introduce the conditional distance to the point xj:

Xj = Uk`1 + 1-αk`1 + 1 cosθi                                                                    (47)

Then, taking into account (nine) we obtain for k`1 ≠ k`:

,       (48)

where z = Sign (t-xj).

for k'1 = k '

  (49)

The above formulas make it possible to find the coefficient of interpolation polynomials, and hence the magnitudes of shear stresses. It can also be seen that the task is easy to program. Knowing the shear stresses τк (х) and tensile forces in each layer, we find by the formula Тk:

                                     (50)

which, with the accepted interpolation formulas for the integration section q`, takes the form

       (51)

After that, for the site - Ɩ₀, + Ɩ0, one can construct a recursive formula for sequentially finding the compression force of a fiber:

                                                       (52)

From the given iterative process, it becomes clear that, in accordance with premise (3), one can take into account the compression deformation of the layers across the fibers.

Conclusion

An analytical study has shown that a complex stress state arises in a conveyor belt operating under bending, stretching and when transmitting a circumferential force from a drum, which can be described by the proposed system of integral equations. Tangential, normal and compressive stresses in the tape depend on its thickness and bending radius, thickness of spacers and rubber layers, elasticity and shear moduli, as well as the angle of its wrap around a drum or roller on the amount of tension, on the circumferential force. The nature of the change in the tangential and normal stresses along the arc of the girth and along the thickness of the tape from bending and from the circumferential force is determined by propositions of integral equations. They also determined the nature of the change in the compressive stresses in each gasket from the circumferential force and under the combined action of tension and from the circumferential force.

 

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