56
2 Free Vibration of Single Degree of Freedom System
Prob. 2.30
2.31 Determine the natural frequency of the encastered beam shown in the figure.
Prob. 2.31
2.32 A platform weighing 7 × 10
2 N is supported on four columns. The columns
are identical and are clamped at both ends. It has been determined experimentally that a force of 1.75 × 10
5 N horizontally applied to the platform produces
a displacement of = 2.54 mm. Damping is 5% of critical damping. Determine from this structure the following: (a) undamped natural frequency (b)
absolute damping coefficient, (c) logarithmic decrement and (d) the number
of cycles and time required for the amplitude of motion to be reduced from
an initial value from 2.54 to 0.254 mm.
2.33 Show that for an undamped system in free vibration the logarithmic decrement
is expressed as
δ =
1
k 0
ln
x i
x i+k
where k 0 is the number of cycles separating two measured peak amplitudes
x i and x i + k .
2.34 Determine the effective torsional stiffness of the shaft and determine its
natural period.