ABSTRACT
A question one may ask is “Why do bodies vibrate?” The explanation is rooted in the
energy conservation principle.
Consider as an example a mass
m
suspended on a spring having stiffness
k
(Figure 6.1) and assume that this mass is pushed up (or down) from its static equilibrium position by the amount
y
and then released. By deforming the spring some energy is stored in it, which will be denoted by
V
. This energy is called the
potential energy
, and
V
represents the maximum energy transferred to the spring by deforming it by
y
. After releasing the mass, it will start moving back to its original position. The motion of the mass means that it acquires some
kinetic energy
, which is equal to
(6.1)
where is the time derivative of mass displacement, i.e., its velocity. At any intermediate position the potential energy of the deformed spring is equal to
(6.2)
If one assumes that the spring is ideal, i.e., its deformation does not lead to any energy losses, then according to the energy conservation principle the sum of the kinetic energy of the mass and the potential energy in the spring must be equal to the original energy introduced into this system,
V
. Thus,
. (6.3)
The above equation shows that in a spring-mass system an energy transformation takes place, from potential to kinetic and back. More than that, one can see that this process is periodic. Indeed, when then and it follows that at these extreme positions . Thus, there are two extreme positions of the mass and they are equal in magnitude, but at the opposite sides of the static equilibrium position. On the other hand, when
y =
(or, more correctly, a static displacement), . This process of a mass moving between two extreme positions in a periodic
fashion is called
oscillation
. It is characterized by two parameters: the
amplitude
of oscillation and the
period
of oscillation. The former is the maximum mass displace-
ment from the position of static equilibrium,
y
, whereas the latter is the time between two consecutive maximum displacements.
