KARACHUN
V.V.
NATIONAL TECHNICAL
UNIVERSITY OF UKRAINE "IGOR SIKORSKY KYIV POLYTECHNIC INSTITUTE"
INFLUENCE OF ACOUSTIC VIBRATION OF GYROSCOPE FLOAT ENDS ON MEASURING ERROR
We will analyze the side surface of the float gimbal and and
its resilient co-operating with acoustic radiation penetrating
from outside.
A sound-wave unavoidable results in the resilient
moving of the surface in radial 
and tangential 
directions (relative motion),
that in the conditions of the
portable angular motion of the aircraft fuselage with velocity 
will be responsible Euler forces (forces of Coriolis inertia, and more precisely, moments of inertia forces ) and, naturally, inciting moments 
and 
(Fig. 1).
Constituents 
and 
of these moments will result in precession of the main axis and, accordingly, the error of measuring will appear :
; 
;
,(1)
where 
- is a moment of inertia of the float, 
- is the
angular momentum of the gyroscope.
Now will find out the
degree of influence of the forced bend moving 
of the surface of head ends of the float. The presence
of relative and portable motions, as in the previous case, will
result in inciting moment of Coriolis inertia forces of 
(Fig. 2)
.
(2)
In accordance with the Resal theorem this moment will form the instrumental errors
. (3)
Thus, the integral error of the gyroscope will equal:
. (4)
Will analyse the inciting motion of the float in flight. We will hard bind the system of co-ordinates 
with
the body of aircraft:
will be sent along the axis of the aircraft, 
and 
will be placed in the former plane. For the
supporting system of co-ordinates we will choose the axes which are related to Earth. Axis 
will be sent vertically downward, axis 
- horizontally (for example, directed on the line of the set course), axis 
constitutes the right three of axes
with
the first two axes.
Let the aircraft at the moment of start occupies a
free position. Let’s draw a plane through its mass
center, perpendicular to the longitudinal axis (the
former plane) to the crossing with the
horizontal plane . For the crossing lines of these planes 
(lines of
knots) we will direct axis 
and will draw in the
horizontal plane axis
, perpendicular to . For the Euler angles we will
choose the angle of turning round the vertical line of
the horizontal co-ordinate plane 
to
its coinciding with the axes of the
system 
(let’s
call it the angle of
yaw 
), the angle of turning round
the line of knots 
of
the co-ordinate plane to coinciding of axis with
the longitudinal axis
of the airplane. We will call - the angle of pitch (in this case axis 
will occupy the position of 
in
the former’s plane) and the angle of turning of
the plane 
about
the longitudinal axis of the fuselage (angle of roll 
). The corresponding angle
rate will be directed along the vertical line 
, the line of knots 
and along the
axis of the vehicle 
.
The angle rate of the aircraft can be
shown as hands for
the unitary vector 
of axes 
, 
and 
,
(5)
or in
projections on the axes which are connected to the body of
the vehicle -
. (6)
When the
aircraft starts from an immobile base (the
axes are immovable), the
projections of angle rate on axis 
, which are connected with the
fuselage, are calculated by the
formulas (fig. 3, fig. 4) :
; 
;
; 
; (7)
; 
,
where 
; 
; 
.
Conversely, when the start
is carried out from a mobile base (for
example, a carrier-aircraft), at
first it is necessary to resolve angle
rate into the axes. It also conserns the case, when
it is necessary to take into account the angle rate of day's
rotation of Earth.
Assume that the angles 
and , and also their
derivatives in time, small. The angle rate of yaw we will show as -
, (8)
where 
- the size
which is measured by the angle rate sensor, for
example, at the aircraft rotation, and 
<<
is rather small
indignation of this angulator.
Obviously, that constituents 
and 
do not carry out
the influence on the error of
gyroscope, because they coincide after direction from the figure.
At the same time, kinematic inciting 
and 
will result
in the additional error of measuring (Fig. 4):
. (9)
The angle rate vector 
and angular
acceleration 
are directed
along the initial axis of the unit.
The
analysis proves that the angle rate 
of
the fuselage during the acoustic vibration of the float
butt ends results in a spiral motion, that, itself, is a
necessary factor, because it reduces dry friction on
the float axis. But together with the
resilient radial motion 
of the
side surface of the float, the
angle rate will start the
appearance of the moment of Euler forces
(forces of Coriolis inertia)
and, naturally, will result in the origin of angle rate 
, directed parallel to the ingoing
axis, the axis of sensitivity, and
the device (Fig. 5), :
. (10)
The tangential resilient displacements 
of
the side surface, at the given angle
rate 
, lead to appearance
of forces of Coriolis inertia, which lines of
action will cross the center of the
gimbal and will not create the inciting moment.
Thus, the aircraft angular
motion with velocities 
and serves to negative influence of acoustic
vibration on the device as a factor 
on the
outgoing axis. In
its turn, the angle rate of
the fuselage will underline the radial resilient movements of
the side surface of the float 
, imitating the presence of "erroneous" input value of the
device 
.
Fig. 4. Origin of angular acceleration 
Fig. 5. Origin joint action to
gyroscope
kinematic
and acoustic action