Transverse vibrations rope moving in longitudinal direction

The article researches the oscillations of a rope moving in the longitudinal direction. The model takes into account the tension of the rope, flexural stiffness and resistance of the external environment. The object of the study refers to a wide range of oscillating one-dimensional objects with moving boundaries. At a constant speed of longitudinal motion, the rope oscillations are characterized by a set of eigen frequencies. In the absence of medium resistance, a discrete integral Fourier transform is used to solve the problem. As a result, an equation is obtained in the form of series, which makes it possible to find the exact values of the eigen frequencies. The problem in the presence of medium resistance was solved by the Kantorovich-Galerkin method. The equation obtained allows us to find approximate values of the first two eigen frequencies. A comparison of the exact and approximate frequencies estimates the accuracy of the solution obtained by the Kantorovich-Galerkin method. The article analyzes how the speed of longitudinal rope motion affects the shape of natural oscillations. The solution is made in dimensionless variables, which allows us to use the obtained results to calculate the oscillations of a wide range of technical objects.