Page 49 - Singiresu S. Rao-Mechanical Vibrations in SI Units, Global Edition-Pearson (2017)

46 Chapter 1 Fundamentals oF Vibration

x 1
x 2
x 3
etc.

FiGure 1.14 A cantilever beam (an
infinite-number-of-degrees-of-freedom system).

The coordinates necessary to describe the motion of a system constitute a set of gen-
eralized coordinates. These are usually denoted as q , q , c and may represent Cartesian
2
1
and/or non-Cartesian coordinates.
1.4.4 A large number of practical systems can be described using a finite number of degrees of
discrete and freedom, such as the simple systems shown in Figs. 1.10–1.13. Some systems, especially
Continuous those involving continuous elastic members, have an infinite number of degrees of free-
dom. As a simple example, consider the cantilever beam shown in Fig. 1.14. Since the
systems beam has an infinite number of mass points, we need an infinite number of coordinates to
specify its deflected configuration. The infinite number of coordinates defines its elastic
deflection curve. Thus the cantilever beam has an infinite number of degrees of freedom.
Most structural and machine systems have deformable (elastic) members and therefore
have an infinite number of degrees of freedom.
Systems with a finite number of degrees of freedom are called discrete or lumped
parameter systems, and those with an infinite number of degrees of freedom are called
continuous or distributed systems.
Most of the time, continuous systems are approximated as discrete systems, and solu-
tions are obtained in a simpler manner. Although treatment of a system as continuous gives
exact results, the analytical methods available for dealing with continuous systems are
limited to a narrow selection of problems, such as uniform beams, slender rods, and thin
plates. Hence most of the practical systems are studied by treating them as finite lumped
masses, springs, and dampers. In general, more accurate results are obtained by increasing
the number of masses, springs, and dampers—that is, by increasing the number of degrees
of freedom.
1.5 Classification of Vibration

Vibration can be classified in several ways. Some of the important classifications are as
follows.

1.5.1 Free Vibration. If a system, after an initial disturbance, is left to vibrate on its own, the
Free and Forced ensuing vibration is known as free vibration. No external force acts on the system. The
Vibration oscillation of a simple pendulum is an example of free vibration.

Forced Vibration. If a system is subjected to an external force (often, a repeating type
of force), the resulting vibration is known as forced vibration. The oscillation that arises in
machines such as diesel engines is an example of forced vibration.