  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.
