User`s guide
99
The lumped load
at the end of beam has the kinetic energy:
The two kinetic energies of Eq. 29 and Eq.30 need to be equal. The equivalent mass is:
Therefore, the natural frequency in rad/sec is expressed as:
The error of the estimation is within 2%.
2.2.2 Mode Shapes of a Cantilever Beam under Free Vibration
The mode shapes of a vibrating beam can be determined through solving the relevant
equations. The video below shows the vibration mode shapes of a simply supported beam and a
cantilever beam.
http://www.youtube.com/watch?v=kun62B7VUg8
2.2.3 Damping Factor of a Cantilever Beam under Free Vibration
The vibrating object dissipates energy through damping, and the oscillation amplitude
decays with time as a result. The damping ratio is a dimensionless measure describing how
rapidly the oscillations decay during each cycle.
Where the system is completely lossless, the mass would oscillate indefinitely, with
constant amplitude. This hypothetical case is called undamped.
If the system contained high losses, for example if the system vibrates in a viscous fluid,
the mass could slowly return to its rest position without ever overshooting. This case is called
overdamped. Commonly, the mass tends to overshoot its starting position, and then return,
overshooting again. With each overshoot, some energy in the system is dissipated, and the
oscillations die towards zero. This case is called underdamped. Between the overdamped and
Eq.30
Eq.31
Eq.32