Page 38 - Thomson, William Tyrrell-Theory of Vibration with Applications-Taylor _ Francis (2010)

Sec. 2.4 Rayleigh Method: Effective Mass 25

follows:
. y

The kinetic energy of the spring can then be integrated to
1 fU
1 m,
T - ^ f‘i y\ . . _ 1 •;
Tadd 2 2 3 ^
and the effective mass is found to be one-third the mass of the spring. Adding this to
the lumped mass, the revised natural frequency is

m +

7 ^ -i/2— H
dy

Figure 2.4-1. Figure 2.4-2. Effective mass of beam.
Effective mass
of spring.
Example 2.4-2
A simply supported beam of total mass has a concentrated mass M at midspan.
Determine the effective mass of the system at midspan and find its fundamental
frequency. The deflection under the load due to a concentrated force P applied at
midspan is PP/A^EL (See Fig. 2.4-2 and table of stiffness at the end of the chapter.)
Solution: We will assume the deflection of the beam to be that due to a concentrated load
at midspan or
3jc
^HT)1 (f<^)
The maximum kinetic energy of the beam itself is then
le beam itself is then
2
2
] | ^ ^ ^ (0.4857 Wb)y,max
The effective mass at midspan is then equal to
m.ff = M + 0.4857 rriu
and its natural frequency becomes
48 £ /
/7 M + 0.4857