Aspects Concerning Validation of Theoretical Solution of Generalised Ladder Problem

One of the most well-known problems of dynamics is the “ladder problem”. In this paper, a theoretical model is proposed followed by the experimental validation of the predicted solution. The model refers to a rod of negligible thickness with the ends leaning frictionless on two walls. By approximating the rod as a segment, the problem is simplified, and the Lagrange equations can be applied. The experimental validation of the model had to address several challenges: the actual rod–wall contacts are singular points, friction cannot be neglected, and the rod’s motion must remain confined to the vertical plane. The physical “ladder” was designed as a cylindrical rod with two identical balls of well-controlled geometry, fixed at the ends. These spheres make contact with two half-cylinder grooves—one vertical and one horizontal—ensuring that the motion remains parallel to the vertical plane. The presence of dry friction in the sphere–groove contacts leads to a complex, strongly nonlinear differential equation of motion, requiring numerical methods of integration. A test-rig was designed and constructed for the experimental study of motion, and an aspect overlooked by the theoretical model was emphasised: the interruption of contact with the vertical wall. An excellent agreement was found between the experimental data and the theoretical results.