Dynamics of a string subjected to a dual-point contact moving mass

Abstract

This study investigates the planar transient dynamics of a linearized string interacting with a moving mass. Unlike conventional models that idealize the moving system as a point mass or particle, the mass is modeled more realistically as a finite body that maintains dual contact with the string at closely spaced points. This modeling choice naturally incorporates the rotational inertia of the moving mass into the formulation. For a prescribed motion of the moving mass along the string, the coupled governing equations are derived in terms of the Dirac delta function. In the limiting case where the two contact points approach at the mass center, the formulation reduces to an equivalent monopole–dipole representation. The governing equations are discretized using the Galerkin method and integrated in time via a Runge–Kutta scheme. The formulation is first applied to a stretched string subjected to a uniformly moving mass. Particular attention is given to the trajectory paradox reported in earlier studies for moving point masses near a boundary. Using the present two-point contact model, this paradox is examined in terms of trajectory at the front contact, the separation between the actual and virtual centers of mass, and the variation in contact-point spacing induced by abrupt responses near the boundary. Subsequently, a new system, in which a string is vertically hung and carries a uniformly moving mass, is analyzed. Trajectory convergence is demonstrated for both ascending and descending motions. Similar to the trajectory paradox, a jump-like response appears near the upper fixed boundary when the mass ascends at higher velocities. At these velocities, the two-point contact model yields divergent results across the entire domain, indicating limitations in its applicability. This behavior highlights the need for future studies that incorporate nonlinear effects to more accurately capture the observed dynamics.