Analytic solution to the motion of the mass-spring oscillator subjected to external force

TL;DR

This paper provides an analytic solution for the motion of a mass-spring oscillator with a heavy spring under external force, revealing how spring mass affects vibrational frequency and addressing cases where renormalization fails.

Contribution

It introduces an analytic method for evanescent vibration analysis of heavy-spring oscillators, including a generalized orthogonal basis and insights into frequency changes.

Findings

Vibrational frequency increases when spring mass is considered.

Analytic solutions are derived for cases where renormalization fails.

Heavy spring effects are characterized in the oscillator's motion.

Abstract

The non-evanescent vibration, evanescent vibration and forced vibration of an oscillator attached to the massless spring are always discussed in general mechanics courses. In this article, we focus on the heavy-spring conditions. We first investigate the general situation where both viscous resistance and applied force are considered under the perspective of the renormalization group theory. Then we deal with the special case where renormalization method fails to work. We use analytic method to study the evanescent vibration of an oscillator attached to the heavy spring. Generalized orthogonal and over-complete base set is established. Two independent vibration modes indicate the vibrational frequency becomes bigger than simple harmonic oscillator if the mass of spring is considered.