Page 33 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

20 Free Vibration Chap. 2

The statical deflection of the spring suspending the ^-kg mass is obtained from the
relationship mg = kA
mg 0.25 X 9.81
16.0 mm
0.1533

^N /m m
Example 2.2-2
Determine the natural frequency of the mass M on the end of a cantilever beam of
negligible mass shown in Fig. 2.2-2.
M
I n -
2
_1_ Figure 2.2-2.
Solution: The deflection of the cantilever beam under a concentrated end force P is
pp p
^ 3EI k
where El is the flexural rigidity. Thus, the stiffness of the beam is k = 3EI/P, and
the natural frequency of the system becomes
J _ / 3 ^
'
■'' 2-17 V mP

Example 2.2-3
An automobile wheel and tire are suspended by a steel rod 0.50 cm in diameter and
2 m long, as shown in Fig. 2.2-3. When the wheel is given an angular displacement
and released, it makes 10 oscillations in 30.2 s. Determine the polar moment of
inertia of the wheel and tire.
Solution: The rotational equation of motion corresponding to Newton’s equation is
Je = -K6
where J is the rotational mass moment of inertia, K is the rotational stiffness, and 6
is the angle of rotation in radians. Thus, the natural frequency of oscillation is equal

Figure 2.2-3.