Consider the dissipation of the whole system as expressed by the quality factor Q. Q accounts for internal dissipation in the suspensions, for losses due to imperfect clamping and for any other possible losses in the system. The total energy of the system will decrease in time, from its initial value E(0), as follows:
where 
, defined by 
, is the damping time of the system due to
the dissipation. If this damping time is much longer than the natural period 
of
the oscillator (i.e. if 
), then, in one natural period 
, the
energy decrease is:
which is the usual definition for the quality factor Q. The amplitude of the circular oscillation will decrease accordingly:
and the relative variation of the amplitude of oscillation, in one natural period, is:
By monitoring the amplitude of oscillation the Q of the system can be measured,
thus measuring its total dissipation. This decrease in the amplitude of the circular
oscillation can be interpreted as due to a decrease of the along track velocity, which in
turn can be considered as caused by an average damping acceleration 
, also along
track, such that:
Finally, we can consider as produced by an average damping force 
on
each mass:
where 
is the centrifugal force, equal and opposite to the elastic force of the
spring 
. The damping force is at about 
with respect to 
. We
have:
Thus, if 
the damping force is much smaller than the elastic force of the spring.
Note that Q is the experimentally measured value, thus accounting for all losses in
the system. Equation (10) also gives the ``destabilizing'' force, i.e. the force that
would be necessary, in absence of damping, to produce an exponential increase, with
quality factor -Q, of the amplitude of the oscillation. If such ``destabilizing''
force is applied to the damped oscillator it prevents the exponential decrease of the
amplitude of oscillation by pumping into the oscillator exactly the same amount of energy
it dissipates; the energy provided at any time is the same as in (4), but with the plus
sign in the exponential, so as to produce an undamped oscillator. Also the converse is
true: if an external damping force , of magnitude given by (10), is applied to
an unstable oscillator having a quality factor -Q, it will prevent in the same way
the exponential increase of the amplitude of the oscillations, that is the oscillator will
be stabilized.
(Anna Nobili- nobili@dm.unipi.it)
