Page 293 - Introduction to Naval Architecture

278 VIBRATION, NOISE AND SHOCK

5
2
The period becomes T d = &r/ [(A/M) - (^/2JW) ]° , frequency being
l/T d , where/i is a damping coefficient such that damping force equals
ju (velocity).
Successive amplitudes decay according to the equation

As the damping increases the number of oscillations about the mean
position will reduce until finally the body does not overshoot the
equilibrium position at all. The system is then said to be dead beat.

Regular forced vibrations
Free vibrations can occur when for instance, a structural member is
struck an instantaneous blow. More generally the disturbing force will
continue to be applied to the system for a longish period and will itself
fluctuate in amplitude. The simplest type of disturbing force to assume
for analysis purposes is one with constant amplitude varying sinusoi-
dally with time. This would be the case where the ship is in a regular
wave system. The differential equation of motion, taking x as the
displacement at time t, becomes:

The solution of this equation for x is the sum of two parts. The first
part is the solution of the equation with no forcing function. That is,
it is the solution of the damped oscillation previously considered. The
second part is an oscillation at the frequency of the applied force. It
is x = B sm(a)t - y).
After a time the first part will die away leaving the oscillation in the
frequency of the forcing function. This is called a. forced oscillation. It is
important to know its amplitude, B, and the phase angle, y. These can
be shown to be:

In these expressions A is called the tuning factor and is equal to
0-5
a)/ (k/M) . That is the tuning factor is the ratio of the frequency of the
applied force to the natural frequency of the system. Since k represents
the stiffness of the system, F 0/k is the displacement which would be
caused by a static force F 0. The ratio of the amplitude of the dynamic