Paradox of a particle’s trajectory moving on a string

Abstract

This paper deals with the paradoxical properties of the solution of string vibration under a moving mass. The solutions published to date are not simple enough and cannot be applied to investigations in the entire range of mass speeds, including the overcritical range. We propose a formulation of the problem that allows us to reduce the problem to a second-order matrix differential equation. Its solution is characteristic of all features of the critical, subcritical, and overcritical motion. Results exhibit discontinuity of the mass trajectory at the end support point, which has not been previously reported in the literature. The closed solution in the case of a massless string is analyzed and the discontinuity is proved. Numerical results obtained for an inertial string demonstrate similar features. Small vibrations are analyzed, which is why the effect discussed in the paper is of purely mathematical interest. However, the phenomenon results in complexity in discrete solutions.

Numerical modelling of structure vibrations under inertial moving load

“…The analysis of results exhibits a jump of the mass in the neighbourhood of the end support. In the papers [13,14] the discontinuity of the mass trajectory near the end support was mathematically proved. The proof was given in the case of massless string.…”

“…We consider first the integral term of (14). We use the same linear interpolation of the velocity (12).…”

“…This important property is visible both in semi-analytical and numerical results and influences the response of more complex systems. Moreover, it is mathematically proved in [13,14]. The final solution is obtained as a matrix differential equation of the second order.…”

Space–time finite element approach to general description of a moving inertial load

“…8 and 9 we observe a discontinuity in the trajectory of the moving mass particle near the end support. This phenomenon was explained in detail in [4,29]. Results of displacements in time in the middle of a string are presented in Figs.…”

“…The correct solution of the differential equation of motion of the analysed system exhibits unexpected phenomena, for example a sharp gradient of the mass displacement near the end support. In the paper [4] the discontinuity of the mass trajectory was mathematically proved. The proof was given in the case of a massless string.…”