Dynamical Properties of the Multiple Elastic Pendulum

This article studies the dynamical properties of the multiple elastic pendulum formed by an arbitrary number of point masses connected in a chain by elastic springs. First, using the Lagrangian formulation, the equations of motion governing the system's behavior, accounting for both large-angle displacements and elastic deformations, are derived. It is shown that well-established, low-dimensional systems, such as the classical double elastic pendulum, can be systematically recovered as particular cases directly from the general equations of motion after applying the appropriates particular conditions. The behavior of the system under small oscillations near its stable equilibrium is also investigated. For this case, the linearized equations allow the solution of the eigenvalue problem, demonstrating that in this case uncoupled behavior between the angular displacements and the spring elongations.