Introduction

There are two general classes of
vibrations - free and forced. **Free vibration** takes place when a system
oscillates under the action of forces inherent in the system itself, and when external
impressed forces are absent. The system under free vibration will vibrate at one or more
of its **natural frequencies**, which are properties of the dynamic system
established by its mass and stiffness distribution.

Vibration that takes place under the
excitation of external forces is called **forced vibration**.* *When the
excitation is oscillatory, the system is forced to vibrate at the excitation frequency. If
the frequency of excitation coincides with one of the natural frequencies of the system, a
condition of **resonance** is encountered, and dangerously large oscillations
may result. The failure of major structures such as bridges, buildings, or airplane wings
is an awesome possibility under resonance. Thus, the calculation of the natural
frequencies of major importance in the study of vibrations.

Vibrating systems are all subject to **damping**
to some degree because energy is dissipated by friction and other resistances. If the
damping is small, it has very little influence on the natural frequencies of the system,
and hence the calculation for the natural frequencies are generally made on the basis of
no damping. On the other hand, damping is of great importance in limiting the amplitude of
oscillation at resonance.

The number of independent coordinates
required to describe the motion of a system is called **degrees of freedom** of
the system. Thus, a free particle undergoing general motion in space will have three
degrees of freedom, and a rigid body will have six degrees of freedom, i.e., three
components of position and three angles defining its orientation. Furthermore, a
continuous elastic body will require an infinite number of coordinates (three for each
point on the body) to describe its motion; hence, its degrees of freedom must be infinite.
However, in many cases, parts of such bodies may be assumed to be rigid, and the system
may be considered to be dynamically equivalent to one having finite degrees of freedom. In
fact, a surprisingly large number of vibration problems can be treated with sufficient
accuracy by reducing the system to one having a few degrees of freedom.