Page 35 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

22 Free Vibration Chap. 2

2.3 ENERGY METHOD
In a conservative system, the total energy is constant, and the differential equation
of motion can also be established by the principle of conservation of energy. For
the free vibration of an undamped system, the energy is partly kinetic and partly
potential. The kinetic energy T is stored in the mass by virtue of its velocity,
whereas the potential energy U is stored in the form of strain energy in elastic
deformation or work done in a force field such as gravity. The total energy being
constant, its rate of change is zero, as illustrated by the following equations:
T U = constant (2.3-1)

i ( T + U ) - 0 (2.3-2)
If our interest is only in the natural frequency of the system, it can be
determined by the following considerations. From the principle of conservation of
energy, we can write
+ i/i = 7^2 + U2 (2.3-3)

where ^ and 2 represent two instances of time. Let ^ be the time when the mass is
passing through its static equilibrium position and choose = 0 as reference for
the potential energy. Let 2 be the time corresponding to the maximum displace­
ment of the mass. At this position, the velocity of the mass is zero, and hence
T2 = 0. We then have
+ 0 = 0 + i/2 (2.3-4)
However, if the system is undergoing harmonic motion, then and U2 are
maximum values, and hence
T = i/ (2.3-5)
^ max
'-^max
The preceding equation leads directly to the natural frequency.
Example 2.3-1
Determine the natural frequency of the system shown in Fig. 2.3-1.

Figure 2.3-1.