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A simple method of approximating the natural frequency of cantilever beams is shown
below. The method also estimates equivalent stiffness and equivalent mass of the beam.
Recall the generic expression of natural frequency in rad/sec is
. To find the
natural frequency of a cantilever beam, the equivalent stiffness and equivalent mass are needed.
As given in section 2.1.2, the deflection w at the tip of a cantilever beam (x=L) is
Using Hook’s law, the deflection at the end of the cantilever can be expressed as
where k is the stiffness of the cantilever beam. Combining eq. 17 and eq. 18, k can be
given as
Therefore, the frequency of a cantilever with a point load m at length x can be given as
The same frequency can be provided by a load
at the end of beam
Consider a cantilever beam with constant cross section and uniformly distributed mass of
value m per meter along the length. At any time t during vibration, the relationship between
generic deflection (measured at an abscissa y from free end), denoted by
and the
deflection at the free end, denoted by
can be expressed as:
The kinetic energy of the distributed parameter cantilever is expressed as:
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