Page 307 - Introduction to Naval Architecture

292 VIBRATION, NOISE AND SHOCK

In the calculation the mass per unit length must allow for the mass of
the entrained water using one of the methods described for dealing
with added virtual mass. The bending theory used ignores shear
deflection and rotary inertia effects. Corrections for these are made at
the end by applying factors, based on r s and r r, to the calculated
frequency as discussed earlier,
The energy method
This method uses the principle that, in the absence of damping, the
total energy of a vibrating system is constant. Damping exists in any real
system but for ships it is acceptable to ignore it for the present purpose.
Hence the sum of the kinetic and potential energies is constant.
In a vibrating beam the kinetic energy is that of the moving masses
and initially this is assumed to be due to linear motion only. Assuming
simple harmonic motion and a mass distribution, the kinetic energy is
obtained from the accelerations deduced from an assumed deflection
profile and frequency. The potential energy is the strain energy of
bending,
When the beam is passing through its equilibrium position the
velocity will be a maximum and there will be no bending moment at
that instant. All the energy is kinetic. Similarly when at its maximum
deflection the energy is entirely potential. Since the total energy is
constant the kinetic energy in the one case can be equated to the
potential energy in the other.
As in the deflection method the initial deflection profile is taken as
that of a uniform bar. As before allowance is made for shear deflection
and for rotary inertia. Applying this energy method to the case of the
simply supported, uniform section, beam with a concentrated mass M
at mid-span and assuming a sinusoidal deflection curve, yields a
frequency of:
4
1 fn EI\°- 5 I /48£/\°- 5
___ - _ compared with — - for the exact solution.
F
4
Since 7T /2 is 48.7 the two results are in good agreementThis simple
example suggests that as long as the correct end conditions are satisfied
there is considerable latitude in the choice of the form of the deflection
profile.

Calculation of higher modes
It might be expected that the frequencies of higher modes could be
obtained by the above methods by assuming the appropriate deflection
profile to match the mode needed. Unfortunately, instead of the
assumed deflection curve converging to the correct one it tends to