A possibility of an effective damping of rotor nutations by modulating the field of the moment transducers in synchronism with the nutation frequency is considered. The algorithms for forming the control moments are proposed and their application is discussed

Komarov, Valentine N.

English

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DAMPING ROTOR NUTATION OSCILLATIONS
IN A GYROSCOPE WITH MAGNETIC SUSPENSION 1
Valentine N. Komarov
Research Institute for Applied Mathematics and Cybernetics
of Nizhny Novgorod State University, Nizhny Novgorod, Russia
22 -27
e
SUMMARY
A possibility of an effective damping of rotor nutations by modulating the field of the
moment transducers in synchronism with the nutation frequency is considered. The algorithms
for forming the control moments are proposed and their application is discussed.
INTRODUCTION
Practical use ofnoncontact suspensions in gyroscopy is possible if the following
requirements, at least, are fulfilled: the rotor must be stably suspended and stably rotate with
given parameters; information about the angular motion of the body must be available.
The second problem of making the rotor rotate is, in turn, divided into three other problems:
providing a necessary angular momentum for the rotor, which, most frequently, is equivalent to
providing a certain rotation speed f_ for the rotor, providing a small, zero in the limit, nutation
angle O, and maintaining the achieved motion parameters within the given limits in the further
operation of the unit. A way for solving these problems with respect to a gyro with a
magnetoresonance suspension can be found in (ref. 1).
The small dissipative moments in a noncontact gyro leads to the fact that after acceleration
the rotor rotates with a nutation angle defined mainly by the initial conditions. The damping of
the nutation oscillations over a finite time can be achieved if, for the formation of the control
moments (not only for nutation damping), we use modulation or self-modulation methods, when
parameters of the suspension field or of the additional coil field (magnitude, direction, rotation
speed, etc.) vary depending on an angular position of the aspheric rotor (ref. 2, 3, 4), due to
which a nonconservative component appears in the moment effect on the rotor, which also
provides an active damping of nutation. Let us consider one of the possible methods for an
effective damping of the rotor nutation relative to a gyro with magnetic suspension.
_This research has been done under the support of the Russian Foundation for Fundamental
Research (Project 95-01-00002) 327
https://ntrs.nasa.gov/search.jsp?R=19960052929 2020-06-16T03:45:19+00:00Z
TIlE EQUATIONS OF MOTION
Assuming the gyro rotor to be stably suspended in the suspension fiehl, we represent the
equations of its angular motions in the form of six first order equations with respect to the
following phase variables (ref. 5): the value of angular momentum K , two angles p and er ,
which determine its orientation (trihedron O}_) relative to the suspension (trihedron OZk), and
The Euler angles q0, _b and 0 , which prescribe the position of the rotor (trihedron (OXi))
relative to the angular momentum I( ; qo is the angle of the proper rotation of the rotor about
the axis of dynamic symmetry (OX3) ; _h is the angle of precession of this axis about the
angular momentum I_ ; t9 is the nutation angle or the angle between the angular momentum and
the axis OXa.
Using the main equation of gyroscopy
d -+fix =M
written in the coordinate system OYj anti projecting the vector I_ on the rotor axes OXi
(I_. _i) = K. b_a = & " _i,
where Ii is the rotor's moment of inertia along OXi, and I_i is the projection of the rotor's
al)solute angular velocity on the same axis, for an axisymmetric rotor (12 = I2 = A _ la = C)
and the arbitrary moments we obtain
K dp+[l,_ =All, K sinP-d-/- [_'a =M2, d--t-= '
dO
K
dt - (312 cos t/, + M_ sin _/,) = -M_o,, ,
(01)
(02)
(0:_)
d--_ - K ( ldtC A) c°s o = M_ c°s _l' - M_ sin _Ksin 0
d4' K M_ cos 4' - h12 sin _/'cotan0
dt A K
where Mj = (l_l-ffj), £/a = Q.,._+ (]tt_/l()cotanp , a,,d Q.,j are the projections onto the
axes 0}'[/ of the absolute angular velocity _, of the coordinate system OZk. This velocity can be
determined by the earth's rotation, by the rotation of an object on which the gyro is installed,
by a forced rotation of the device casing performed to make its interaction with t.he rotor
symmetric, etc.
The presence of different time-scale motions in real gyros, such as a "slow" precession m<>tion
of the angular momentum and a "fast" rotor rotation allows us to use, fi)r studying the gyro
dynamics, the averaging method in which even the first al>proximation reveals the basic laws of
gyro behavior (ref. 5).
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Omitting the standard procedure for normalizing the equations anti finding in the latter a
small parameter e,. which characterizes the ratio of the work of external forces per one rotation
of tile rotor to the kinetic energy accumulated by tile gyro rotor, we write their standard form,
which allows the asymptotic methods to be applied:
('k' ) tr'sinp = e (-_ +wo,), k" =¢,n3, (01)p* = E -- Ods2
k0" = -e (m, cos ¢ + 7.,2Sill¢) ,
m2 cos ¢ - ml sin 4,)qo*+TkcosO=e ksiuO '
(m2 c°s _b - mt sin ¢cotanO _ w3) .tl," - k = -_ K
It is evident from the equations that free motion of tile rotor (generating solution) rel)resents
a regular precession or the Euler-Poinsot motion with the constant angular velocities of the
proper rotation _,g and precession tb_ for a constant nutation angle 0, such that
( C- A) (05)
_,; = k0, _; = - _kocos00, 7 - _ ,
where k0 is the constant value of the angular momentum if: which is immobile in the inertial
space. For the periodic dependence of the right-hand sides of the equations of motion on the fast
variables ¢ and _, , to obtain the first approximation over the small parameter e , the right-hand
sides of the equations for the slow variables are averaged over the variables _/, and _, , whose
dependences on r are assumed to be the same as those in the unperturbed motion.
Let us now specify the requirements to be met by the moment controlling the rotor motion
to perform the damping of the rotor's nutation oscillations over a finite period of time.
After the averaging of the equations of motion over the fast variables ¢ and q0 , the nutation
angle varies according to
kO ° = -e,n_o, (p, a, O, k) . (06)
Since the initial values of the nutation angle 0 are closer to 0 rather than 7r/2 , and this
angle must still decrease with time, we introduce variable x = sin 0 and linearize Eq. (06) with
respect to this variable
= +
wh,-re ,,,0(p,_, 0, _-)= ,._o.(p,_,, o, k), ,,,' = O,._o./O.rl.=o.
Therefore, a monotonic decrease of the nutation angle 0 to zero over a finite period of time is
possible only if the condition m ° > Im'l is fidfilled. In this case, there is no equilibrium state for
the angle 0 or x , and their variation rates are always negative, which allows the required effect
to be achieved.
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TIlE FORMATION OF THE CONTROl. MOMENT
Let us discussa possibility for dampingthe rotor nutationsthro,gh its interactionwith tile
field of a transducer for the angular position of the rotor. II1 a gyro with a magnetoresonance
suspension, this transducer is embodied as a system of two four-leg magnetic circ,tits located on
the same axis but on the different sides of the rotor with a coil (ref. 1) at each leg. An axial
opening in the rotor leads to the fact that from the moment standpoint, the rotor will behave as
anaspheric body, whose surface is described by a set of even harmonics (even Legendre
polynomials), the first of which is the second harmonic responsible for the ellipsoidality or the
uniaxial anisotropy. The moment defined by this harmonic will be considered below.
This moment is described by
= - 2a (E. 171)[ff × hi, (07)
where a > 0 is the interaction amplitude determined, in particular, by the field amplitude: _"and
la are the unit vectors of the symmetry axes of the ellipsoid and the field. (Since we allow for the
anisotropic properties of the rotor determined by its opening, from the magnetic viewpoint it
should be considered as an oblate ellipsoid tending to position itself with its axis perpendicular
to the field's axis, which is reflected by the minus sign in Eq. (07).)
It can be assumed that the unit vector ff coincides with the unit vector of the dynamic
symmetry axis OX3 of the rotor, and, thus,
ff = K3 = 371sin O cos tl, + _2 sin 0 sin _, + 373cos O .
Using the spherical angles or, and 3. , we prescribe the position of the unit vector 1_ . which
deternfines the symmetry axis of the magnetic field generated by the n-th pair of the coils
belonging to different magnetic circuits but located on the same axis that passes through the
center of the rotor
la = fit sin ¢_. cos/3,, + if2 sin ,_. sin/t_ + _'n cos o. ,
where c_,_= c_ is the same for all n and 13. =/3o + rn/2.
Let us substitute these unit vectors into the expression for the moment (07) and find its
component m_o, = rnt cos _/, + m2 sin r/, responsible for the variation of the uutation angle ,)
where
M_o. = 2a,, (hi sin _, - h2 cos _,) h3 cos ,)-
,1
(_1. 2_j) , ht = cospsinctcos(cr -/3,_)- sin pcos cr .hj
h2 = - sin osin (cy -3,), h3 = sinpsin c_cos (cy -,3,) + cos pcos c_ .
(0_)
From Eq. (0,q) it follows that 31,._,_ is a periodic function of the precession an_le ,." without
,Othe constant component M_o,_ (p,_r, 0, K) , which can appear only in the case of a pulsed
330
switching-on of the field over the time periods, during which tile moment does not change its
sign. As it is evident from Eq. (08), the moment has a harmonic component with frequency _/,"
proportional to cos 2 O and a component with double frequency 2_b° proportional to sin 20. For
O = 0 this results in
Alco,_ = 2a, (hi sin _b - h2 cos _b) h3 .
Apparently, for the effective damping of nutation, i.e., dropping the angle 0 down to zero
over a finite period of time, the coil field with frequency r/, ° must be changed. (The speed of the
rotor rotation about the symmetry axis OX3 is equal to _0 = K/C , the nutation frequency
4, ° = K]A , and, consequently, _h" = _oC/A ). It should be noted that nutation oscillations of a
cryogenic gyro can be damped in a similar manner (ref. 3).
Let us write hi sin _, - h2 cos _, in the form:
hi sin _ - h2 cos _/, = - sin pcos a sin _/, +
1
+ z[(1 + cos p) sin (_ + a-/3,)- (1 - cos p) sin (_]; - cr + fl,)] sin _.
Z
From the above expression it follows that the optimal switching-on of the field should follow
the algorithm
{°,_(_,) = (09)O,
i.e., the field is switched on over half the period of the nutation oscillations with frequency _,°.
2kr < _/,+ o -/L < (2k + 1)Tr
(2k + l)rr < _/, + or- fl,, _< 2(k + 1)rr,
For this operation mode, two pairs of the control coils are switched on at any instant, which
greatly improves the efficiency but leading, however, to a higher heat release that can be reduced
due to a decrease in either the field amplitude, or the switching-on duration.
(_--- _'<q,+cr--gn<_ _ 2k+ r+_
outside the intervals
(1o)
Ilere r < rr is the switching-on duration. Apparently, the algorithm (09) is a particular case of
(10) for r = re.
If all the coil pairs and their switching-on algorithms are identical, they create a total control
lll()Fllellt,
7"
(M_o,,) = 2a-(lr + cos p) (2 cos p - 1) sin 2o sin _ cos 2 O, (11)
which leads to equation
7"
K9"=-2-a(lr + cos p) (2 cos p - l)sin2osin_cos29. (12)
331
Apart from the variable 0 , this equation also incorporates other quantities K(t) and p(t)
that do not remain constant. If we write tile rotor-affecting moments via the force function IV
(ref. 5), then, instead of the corresponding equations in (03), we can obtain
l(,W ,..)K sin p- - --0--_-c°sp -_a '
dK OW
dt O_b '
,lo (ow ow
a--/= It" sin6 \-_- cosO O_ )
Since tile rotor opening is symmetric about its dynamic axis, then, consequently, O|,l"/O_ = 0
The summation of the moments from all tile pairs of tile symmetric coils also results in
OIV/Oa = 0 . These conditions yield the equations
_t 1 OW dK OW dO 1 OIVK
sill p Orb dt 0_[, dt K sin 1_ 0_,
from which we obtain two integrals
K_3 = K cos p = Const , K_.3 = K cos 0 = Const , (13)
the physical meaning of which is evident, i.e., on the strength of the control system and rotor
symmetries, the projections of the angular momentum onto the symmetry axes of the suspension
(K.,._) and tile rotor (It'_3) remain constant. Representing K and cos p from Eq.(13) and
sul)stituting them into Eq. (12), we can find the law of the nulalion angle _ variation with time.
To simplify the solution, let us use the smallness of the misalignment angle p , which, along with
the use of the second integral, yields
( 4a sin 2a sin 2) (l,1)
_e = _ vcos 3 _ , v -- 7lit'z3
from which it is evident that the time of the nutation angle decrease from the initial value 00 to
0 is given by
,( ,,,,,0 [ (: ;_0)]/T(Oo)= _ 2cos_O o+_ln tan + . (15)
At least up to 0o _ 200 , the dependence T (0o) is insignificantly different from the linear
dependence T(,_o) = Oo/v , which simplifies the calculations.
Since on the right side of Eq. (14) there are no terms that are linear in sin0 , the interaction
amplitude a (or the coil field value) can be any value other than zero, which affects only the time
of the nutation damping rather than its character.
332
POSSIBLE REALIZATIONS
To conchlde the study of the nutation damping, let us make some final remarks relative to
the application of the method proposed:
a) As it is evident, the working part of the control moment of the TT-th coil pair
sin (d' + a - _/,). At the same time, the projection of the rotor symmetry axis onto the
direction if,, = _'1 cos B, + z_ sin fin , that corresponds to this pair is
(if- F,) = l_ cos (q, + cl - ft,) + p cos (a -/7,) for small 17and p . Since the moment transducer in
the gyro incorporates four coil pairs, it is apparent that. the control moment of any pair can be
formed based on the signal taken from an adjacent pair.
lh>wever, this mode of operation, in which the signal is taken from one pair and there are
switchings-over in the other pair, can lead to an electrical engagement of both channels
disrupting thereby their normal operation. In this respect, we need to obtain a signal with
frequency _/," by another method, for example, using a transducer to ot)tain a. signal on one side
of the rotor and a transducer to ensure control on the other side of the rotor. In this case,
however, along with the moment effect, there will also be a force effect on the rotor or tension.
b) Another method is the switching-over mode of operation, in which, for a certain period
of time, the transducers are used for obtaining the information and then for control with
periodic repetitions of the process. There is no input signal in the control mode, and it results in
a decrease of K. Consequently, the control signal can be formed by a device of an underexcited
oscillator type to the input of which an external signal with frequency _/," is initially applied. In
this case, control will be efficient only over the time period until the phase difference between
the real signal from the rotor with frequency _," and the signal from the oscillator achieves the
critical value (A_0 = z/2) .
If at the initial time it was the frequency _v0 , anti the rotor deceleration could l>e simulated
l>y the viscous friction when _v (t) = w0 exp (-At) , the phase difference between the signals from
the rotor and from the oscillator can be calculated as
If 7_ = 2_'/_o , T:_ = A-I , then
,_/_o. = T_ (16)
Let us consider the following example: if the rotor speed is ll = 1000 r.p.s.
(g.,° = 2_" • 1200 sec -t) , and the damping is such that Tx = 1 hr , the nutation can be damped
during T_o, _ 1.22 sec until the phase difference becomes A_ = r/2 , after which we again need
to switch over to the regime of signal reception and control moment formation.
c) The proposed mechanism of nutation damping is active for both constant and variable
magnetic fiehts generated by the coils, but the constant field efficiency is greater. The maximum
333
gain in efficiency of the effect on the rotor can be obtained, provided the transducer coils are
used for control alone and the input signal with frequency _/,° for the control moment formation
is taken from other devices.
REFERENCES
1. Odintsov, A. A.: Magnetic Gyroscope and Inertial System on it.s Basis. Sudoslroitcl'naya
promyshlennost', set. Navigatsiya i giroscopiya, no. 1, 1991, pp. 51-58 (in Russian).
2. Komarov, V.N.: On the Influence of the Self-Modulation on Nutation of a Noncontact
Gyroscope. Prikladnaya mekhanika, vol. 20, no. 5, 1984, pp.104-109 (ill Russian).
3. Komarov, V.N.; Urman, Yu. M.: Active Damping of Noncontact Gyroscope Rotor
Nutation. lzv. vuzov. PriboTwstroyeniye, vol. 25, no.12, 1982, pp.- 44-47 (in Russian).
,t. Komarov, V.N.: Regulating the Magnetic Gyroscope's Motion. The Fourth International
Symposium on Magnetic Bearings. Zurich. [Iochschulverlag AG an ETN,199,1, pp.19-22.
5. Beletskiy, V.V.: I)ynamics of Fast Rotations. Proc. lDs. Inst. for Mechanics. Moscow
University, no. 29, 1973, pp.97-118 (in Russian).
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Session 10 -- Bearing/Actuator Design
Chairman: James Downer
Satcon Technology Corporation
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Damping Rotor Nutation Oscillations in a Gyroscope with Magnetic Suspension

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