Analytical study of the cord path character in rubber-cord flexible elements

The mathematical model of the cord path character in rubber-cord flexible elements is investigated. The applicability boundaries of known classical expressions for the determining of the cord path character are analysed. It is shown that the nominal initial cord angle on the assembly drum exists and is uniquely determined for the given assembly drum radius, the element carcass radius as well as the ply inclination to the meridian at the element surface equator. Sinus multiplication formulas which relate the initial cord angle on the drum, fiber tilt to the manufactured element meridian and the fiber tilt at an arbitrary point of manufactured flexible element carcass are obtained. The dependence of the technological constant on the initial ply angle is analysed. The coefficient of dependence between the ply angle and the effective radius in manufactured flexible element are obtained.


Introduction
Rubber-cord flexible elements (RCE) are widely used due to their elastic properties, which allow considerable mechanisms movement under the influence of external loads [1]. The RCE carcass consists of several layers or plies of rubberized cord forming a fiber network where the plies are superimposed at different angles. Usually, the carcass is a surface of revolution, working under the distributed loads and loads, caused by the axial, radial and other forces in its embedding. This model case is well described in terms of pneumatic tires theory with its basic equations [2][3][4][5] which can be used to: -determine the fiber tilt angle β at any point of the manufactured element if the angle β k at the element surface equator (see fig. 1) is known: -determine the slope angle of the normal at an arbitrary point of the element surface In the expressions (1)   -technological constant (TC), determined for the fixed inclination α to the meridian of the assembly drum with R sb radius and the cord stretch δ.
Taking into account (4), the law of the fibers arrangement in the fiber network (on the surface of the element) is obtained: . sin R χ β = (5) The above equations have their own applicability boundaries, which are primarily related to the element and the assembly drum dimensions and the fiber inclination on the assembly drum.

The application range of the tire geometry basic equations
The first step is to determine the pneumatic tires theory basic assumptions which are limiting its usage [6,7]. Let us consider the RCE condition on the assembly drum. The conditional cord stretch δ is taken equal to zero. We will also assume that the element manufacturing process is not causing the fibers elongation. The change in technological constant χ is due to a change in its argument α, determined on a finite interval [0, π/2]: If R sb is measured in meters, the base unit of χ will be [1/m]. The assembly drum radius R sb is constant, therefore, the minimum and maximum values of χ (α) are attainable at the end points of the α variation interval. The minimum value is reached when α o =0, in this case χ(0) = 0. The maximum value is reached for α = π / 2: Wherein χ (α) is the assembly drum radius reverse value: Thus, the graph of χ (α) monotonically increasing from zero to the assembly drum radius R sb reverse value: Reverse values of χ (α) will have the base unit [m] and can be represented as where R p (α) is the calculated reverse value of the χ (α) from expression (10) . It should be noted that Maximum of R p (α) grows without bound while α approaches zero; minimum value is achievable when Thus, the R p (α) reaches a minimum value equal to the assembly drum radius R sb : Under given inclination to the meridian at the element surface equator β k there is the following dependence between the calculated value R p and the RCE equator radius R k :

Nominal angle of the cut on the assembly drum
Under given angle β k , there is an initial fiber inclination on the assembly drum α H for which the following relation comprising the sin β k , R k and R sb radius exists: Angle value α H satisfying the equation (15) for given R sb , R k and sin β k will be called the nominal initial fiber inclination on the assembly drum or simply the initial angle.
If put the equator radius R k (α H ) (corresponding to theinitial angle α H ) in correspondence to the following value of calculated radius R p (α d ): then, with an account to (10), the following equation will take a place: The obtained equation (18.2) allows the calculation of the fiber inclination α d under given values of the assembly drum radius R sb (or diameter D sb ) and radius R K (or diameter D K ) of the RCE carcass.

The interconnection between the angles α Н , α D and β K
With the obtained expression (18.2), the RCE carcass geometry can be fully described by its fiber angles. Converting (15), using expression (18), we obtain (21) In expression (21), the fiber inclination to the meridian on the assembly drum is less than 1. Consequently, the angle exists and is uniquely determined for a given radius of the assembly drum R sb , the radius of the equator R k , and the value β k of the fiber inclination to the meridian of the manufactured RCE. The expression (20) will be called the first sinus multiplication formula which determines the sine of the fiber nominal angle on the assembly drum. All obtained expressions were considered for a constant value of fiber inclination to the manufactured RCE meridian and different values of fiber inclination on the assembly drum. It also allows to formulate a number of new equations-consequences, which are given below. The position of the considered point α g with respect to the point α m on the interval [0, π / 2] is defined by the following equations: Taking into account (30), theequation (31) can be represented as following: (32)

Change of technological constant
Consideration of the technological constant as the angle α function leads to a visual In the table 1 the geometric values of the RCE equator radius and assembly drum radius in R k (α) function correspond to the specified initial conditions, while the fiber inclination to the RCE surface equator and initial fiber inclination on the assembly drum are interchanged relatively to named radiuses.
To match the angles β on the RCE surface with their radiuses, the following transformation can be applied:

The RCE effective radius
For further consideration of the RCE geometric parameters, three specific radiuses determining the RCE shape as shown in fig. 2 can be pointed out.
The obtained formulas are equivalent and allow estimation of the effective radius R o changes caused by changing of angle α from zero to α d . Consideration of the α changing interval starting point obtains: α = 0, β = 0 while С 1 = С 2 = 0,5. For both formulas (33) -(34) the following expression is actual: Consideration of the α changing interval end point obtains: α = α d , β = π/2 while R r = R H , C 1 = 1, C 2 = 0 and R o = R sb . Thus, when the angle α changes from zero to α d , the effective radius decreases with a coefficient . sin

Conclusion
The obtained equations can be used during the RCE design process for the selection of the optimal fiber initial angle under given angles in manufactured RCE. Analysis of the given expressions applicability boundaries shows the character of RCE carcass working conditions changing depending on the initial angles laid during the design. The angles exit beyond the permissible values leads to loss of the RCE structural stability under the influence of external loads [8].