A new solution of the discrete catenary problem

Abstract

We address the very old and famous catenary problem. However, instead of competing with a flexible and continuous chain, we consider a discrete chain made of N identical links. By using the elementary tools of Newtonian mechanics, we show that there is a simple relation between the angles of the links’ slope that enable us to find the N discrete chain solution. We also investigate the limit of the continuous chain and show that the discrete solution converges into the well-known classical solution of the catenary problem. The advantage of this new method is that it better enables the attainment of a physical insight into the problem in a quite simple way, without using advanced mathematics.

Calculation of Optimal Length for Skyline

“…The results of the numerical analysis of the obtained dependences ( 14) and (15), as well as the proposed formula (16), show that the value L opt is uniquely determined by the geometric characteristics of the cable systems: span length l and slope of the chord to the horizon α [1], and force parameter q (distributed load) affects only the absolute value of the maximum tension T max .…”

“…After evaluating the results of approximation and taking into account the obtained coefficients of the approximating function, an engineering formula is proposed, which with a slight error in the parameter of the length of the skyline (δ max < 0.1% in range of α ∈[0°; 45°] allows to determine the value L opt depending on the characteristics of the cable logging systems. For symmetric systems in which the ends of the skyline are located at the same level (α = 0°), from the algebraic Equation ( 16) we can obtain the known results [1,15,16], where for such cases it is proposed to determine the value L opt by the formula L opt = 1.258 l . This confirms the correctness of the results of the study, and at the same time significantly expands the scope of their application, as symmetrical schemes of cable logging systems in engineering practice, in particular in forest engineering, are used very rarely.…”

“…In case of the uniform distribution of the vertical load (skyline self-weight) along the length of the skyline and neglection of deformations at the first stage (its longitudinal and transverse stiffness, temperature deformation, pliability of supports, etc.) [3,4], [7], [12], the shape of the sagging curve is described by a catenary [1], [9], [15] and Equations ( 1) and ( 2), for the system of flat rectangular coordinates x 0y 0 , which starts at point С (Figure 1):…”

“…Technical means of skyline system are widely used in many branches of engineering [1], [15,16], in particular in the logging and timber industries. At present, in the process of forestry, especially in difficult terrain, soil, and hydrological conditions, cable logging systems and snubbing have been widely used [2], [7], [11], [13,14].…”