The result (37), which so far has been obtained on the basis of general physical
principles, is the same used in engeneering textbooks and literature on rotating machines.
On the basis of direct experience with many rotating machines (whose suspensions are
certainly not the tiny GG springs) it is concluded that friction inside rotating parts
(the suspensions) is essentially of structural nature, thus always obtaining a frequency
of whirl very close to the natural frequency; see Eqs. (11) to (14). As a consequence, the
coefficient of rotating damping (see e.g. G. Genta, Vibration of Structures and
Machines, Springer 1993, Section 4.5.5) when is given as:
where 
is the internal loss of the material at the frequency at which the
material goes through the elastic hysteresis cycle, which is 
, and not 
as stated in the
ESTEC Report. Thus:
and:
In order to stabilize the whirling motion which is known to develop because of the
rotating damping expressed by the coefficient (53) it is necessary to provide an amount of
non rotating damping, expressed by a coefficient 
which satisfies the
stability condition of the rotor (known as ``Jeffcott rotor''):
hence
From this, the required stabilizing (damping) force can be computed, since the velocity
to be damped (in the inertial reference frame) is --at any given time-- the linear
velocity of the centre of mass along the whirling circle of radius 
:
If 
then 
so that Eqs. (37),(38),(56),(57) remain
almost exactly valid also in the general case of an elliptical whirling motion (see Eqs.
(1),(2),(3)).
Next: Comparison with the ESTEC Up:
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(Anna Nobili- nobili@dm.unipi.it)
