On the edge of complexity: The simplest not simple coupled mechanical system

TL;DR

This paper analyzes a coupled mechanical system of a hoop and a cylinder with an ideal spring, revealing how static friction influences the system's normal modes and the conditions for the central mass to remain at rest.

Contribution

It derives the equations of motion for the system using multiple methods and explores the role of static friction and moments of inertia in the system's behavior.

Findings

Central mass does not remain at rest during normal modes due to static friction.

Equal moments of inertia lead to zero static friction forces.

Proper positioning of spring endpoints can keep the central mass at rest.

Abstract

We show that during normal modes of an oscillatory system consisting of a hoop and a cylinder joining their centers by an ideal spring, its central mass does not remain at rest. This effect is due to the resultant external static friction forces acting on the system which disappears when the coupled rigid bodies have the same moment of inertia. However, in case of different moment of inertia by proper positioning the two fixed end points of the spring vertically, it is shown that the central mass of the system remains at rest. The equation of motion of the coupled system is derived using dynamic equations, Lagrange's equations, Hamilton's equations and even by applying the conservation laws of energy and angular momentum. The relationship between the static friction forces acting on the rigid bodies is also examined.