Numerical solutions for point masses sliding over analytical surfaces: Part 1

Abstract

In this study, we introduce a system of differential equations describing the motion of a single point mass or of two interacting point masses on a surface, that is solved by a fourth-order explicit Runge–Kutta (RK4) scheme. The forces acting on the masses are gravity, the reaction force of the surface, friction, and, in case of two masses, their mutual interaction force. This latter is introduced by imposing that the geometrical distance between the coupled masses is constant. The solution is computed under the assumption that the point masses strictly slide on the surface, without leaping or rolling. To avoid complications stemming from numerical errors related to real topographies that are only known over discrete grids, we restrict our attention to simulations on analytical continuous surfaces. This study sets the basis for a generalization to more complex systems of masses, such as chains or matrices of blocks that are often used to model complex processes such as landslides and rockfalls. The results shown in this paper provide a background for a companion paper in which the system of equations is generalized, and different geometries are presented. Keywords: Point-mass sliding, Two point-mass systems, Analytical surfaces, Numerical methods, Runge–Kutta method