Page 47 - Singiresu S. Rao-Mechanical Vibrations in SI Units, Global Edition-Pearson (2017)

44 Chapter 1 Fundamentals oF Vibration
O

u
l y

3 m 1
2 l (1 cos u)
Datum
x

mg

FiGure 1.10 A simple pendulum.

1.4.3 The minimum number of independent coordinates required to determine completely the
number of positions of all parts of a system at any instant of time defines the number of degrees of
degrees freedom of the system. The simple pendulum shown in Fig. 1.10, as well as each of the
systems shown in Fig. 1.11, represents a single-degree-of-freedom system. For example,
of Freedom the motion of the simple pendulum (Fig. 1.10) can be stated either in terms of the angle
u or in terms of the Cartesian coordinates x and y. If the coordinates x and y are used to
describe the motion, it must be recognized that these coordinates are not independent.
2
2
2
They are related to each other through the relation x + y = l , where l is the constant
length of the pendulum. Thus any one coordinate can describe the motion of the pendulum.
In this example, we find that the choice of u as the independent coordinate will be more
convenient than the choice of x or y. For the slider shown in Fig. 1.11(a), either the angular
coordinate u or the coordinate x can be used to describe the motion. In Fig. 1.11(b), the
linear coordinate x can be used to specify the motion. For the torsional system (long bar
with a heavy disk at the end) shown in Fig. 1.11(c), the angular coordinate u can be used
to describe the motion.

x
k
u u
m
x
(a) Slider-crank- (b) Spring-mass system (c) Torsional system
spring mechanism

FiGure 1.11 Single-degree-of-freedom systems.