Falling Coupled Oscillators & Trigonometric Sums

TL;DR

This paper introduces a method to evaluate trigonometric sums applied to a system of coupled oscillators, revealing discrete velocity steps and initial motion behaviors under acceleration.

Contribution

It presents a novel application of trigonometric sum evaluation to analyze the dynamics of coupled oscillators under acceleration.

Findings

Initial motion scales as T^{2n+2} for small T

End particles remain stationary until wavefront arrival

Average velocities exhibit discrete step-like values

Abstract

A method for evaluating finite trigonometric summations is applied to a system of N coupled oscillators under acceleration. Initial motion of the nth particle is shown to be of the order ${{T}^{2n+2}}$ for small time T and the end particle in the continuum limit is shown to initially remain stationary for the time it takes a wavefront to reach it. The average velocities of particles at the ends of the system are shown to take discrete values in a step-like manner.