Determining the shape of a flexible thread in the field of horizontal and vertical forces

Abstract

In theoretical mechanics, the equilibrium of a flexible, inextensible thread is considered, to which the tension force of its ends and the distributed force of weight along the length of its arc are applied. An unsolved problem is finding the shape of the thread under the action of other distributed forces. This study has considered the equilibrium of a completely flexible thread, to which, in addition to this force, a transverse distributed force is applied. A sail serves as an example. Wind of equal intensity in the plane of the orthogonal section of the sail can be considered a distributed force. The sail can be cut into narrow strips with the same shape of the curves of the cross-section, which are equal to the cross-section of the sail as a whole. The theory of flexible thread is applied in the current study. The task is reduced to finding the curve of the cross-section of the sail. The object of research is the formation of a cylindrical surface from a flexible thread under the action of distributed forces applied to it. An important characteristic of the shape of a flexible thread is its curvature. Its dependence on the length of the arc was found and it was established that the found curve is a chain line (catenary). This is the feature of the current research and its distinguishing characteristics. The significance of the results stems from the derived analytical dependences, according to which the change in the ratio between the distributed forces acting on the flexible thread deforms it, but it retains the shape of the catenary. At the same time, the angle of deviation of its axis of symmetry from the vertical changes. In the absence of a horizontal distributed force and the presence of only a distributed force of weight, the axis of symmetry of the chain line is directed vertically – at an angle of 90° to the horizontal. If they are equal, this angle is 45°. Scope of application includes structures with stretched supporting wires, conveyor belts, flexible suspended ceilings, the shape of which can be calculated by using our results