Page 37 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

24 Free Vibration Chap. 2

Substituting into Eq. (2.3-2)
3 w
2 —(R - r) 6 w(R - r) sin d 6 = 0
and letting sin 0 = 0 for small angles, we obtain the familiar equation for harmonic
motion
6 + -6 = 0
3 { R - r )
By inspection, the circular frequency of oscillation is
2g
3 ( R ~ r )

2.4 RAYLEIGH METHOD: EFFECTIVE MASS
The energy method can be used for multimass systems or for distributed mass
systems, provided the motion of every point in the system is known. In systems in
which masses are joined by rigid links, levers, or gears, the motion of the various
masses can be expressed in terms of the motion x of some specific point and the
system is simply one of a single DOF, because only one coordinate is necessary.
The kinetic energy can then be written as
T = \m eff^ (2.4-1)

where is the ejfective mass or an equivalent lumped mass at the specified
point. If the stiffness at that point is also known, the natural frequency can be
calculated from the simple equation
k
(2.4-2)

In distributed mass systems such as springs and beams, a knowledge of the
distribution of the vibration amplitude becomes necessary before the kinetic
energy can be calculated. Rayleigh^ showed that with a reasonable assumption for
the shape of the vibration amplitude, it is possible to take into account previously
ignored masses and arrive at a better estimate for the fundamental frequency. The
following examples illustrate the use of both of these methods.
Example 2.4-1
Determine the effect of the mass of the spring on the natural frequency of the system
shown in Fig. 2.4-1.
Solution: With x equal to the velocity of the lumped mass m, we will assume the velocity
of a spring element located a distance y from the fixed end to vary linearly with y as

^John W. Strutt, Lord Rayleigh, The Theory of Sound, Vol. 1, 2nd rev. ed. (New York; Dover,
1937), pp. 109-110.