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

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.