Page 19 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

6 Oscillatory Motion Chap. 1

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.

1.1 HARMONIC MOTION
Oscillatory motion may repeat itself regularly, as in the balance wheel of a watch,
or display considerable irregularity, as in earthquakes. When the motion is re­
peated in equal intervals of time r, it is called periodic motion. The repetition time
T is called the period of the oscillation, and its reciprocal, / = 1 /r , is called the
frequency. If the motion is designated by the time function xU), then any periodic
motion must satisfy the relationship x(t) = x(t + r).
The simplest form of periodic motion is harmonic motion. It can be demon­
strated by a mass suspended from a light spring, as shown in Fig. 1.1-1. If the mass
is displaced from its rest position and released, it will oscillate up and down. By
placing a light source on the oscillating mass, its motion can be recorded on a
light-sensitive filmstrip, which is made to move past it at a constant speed.
The motion recorded on the film strip can be expressed by the equation

X = A sin 277- ( 1.1-1)
T
where A is the amplitude of oscillation, measured from the equilibrium position of
the mass, and r is the period. The motion is repeated when t = r.

Figure 1.1-1. Recording harmonic
motion.