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A second look at experimental data suggesting gravity speed can be derived from laboratory observations


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This paper reviews experimental results achieved in the early 1970s from observation of a freely spinning sphere (2.5 mm ball bearing) at high rotational speeds in a non-contacting magnetic suspension under ultra high vacuum conditions. A significant drag of about 0.02 nHz (0.02×10-9 cycles per second) has been identified which clearly stands out from the drag related to well known sources including experimental background data scatter. That extra drag appears largely consistent in both its magnitude and frequency characteristics with rotational drag proposed by James C. Keith due to relativistic gravitational force retardation within a rotating system of discrete mass points.
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A second look at experimental data suggesting gravity speed
can be derived from laboratory observations
Johan K. Fremerey
Karl-Friedrich-Schinkel-Straße 14, D-53127 Bonn, Germany
This paper reviews experimental results achieved in the early 1970s from
observation of a freely spinning sphere (2.5 mm ball bearing) at high rotational
speeds in a non-contacting magnetic suspension under ultra high vacuum
conditions. A significant drag of about 0.02 nHz (0.02×10-9 cycles per second)
has been identified which clearly stands out from the drag related to well
known sources including experimental background data scatter. That extra
drag appears largely consistent in both its magnitude and frequency
characteristics with rotational drag proposed by James C. Keith due to
relativistic gravitational force retardation within a rotating system of discrete
mass points.
According to Einstein's general theory of relativity, gravity also is expected to
travel at the speed of light. The deceleration of the double pulsar system PSR
B1913+16 as observed in the early 1970s by R.A. Hulse and J.H. Taylor [1]
provided the first significant observational evidence supporting this prediction.
Beyond this, all efforts to find additional observational evidence, in particular
the detection of gravitational waves from remote astronomical sources, so far
have failed to materialize due to metrological limitations.
The ongoing dispute [2] on whether or not the speed of gravity can be
extracted also from the deflection of radio signals by the gravitational field of
Jupiter suggest that these investigations are at the limits of current analytical
methods. Those difficulties motivate taking a closer look at certain laboratory
results [3] achieved in the early 1970s. Those results suggest that
observational evidence in support of relativity theory can be obtained from
experiments involving small steel balls spinning at high rotational speeds. They
agree in a significant manner with predictions by J.C. Keith [4,5] of a rotational
drag due to retarded gravitational force interactions within a rotating system of
discrete mass points, or likewise between rotationally bound, accelerated mass
points and remote masses of the U niverse.
Unfortunately some experts in the field of cosmology have to date dismissed
many laboratory observations based on their theoretical interpretation of
astronomical phenomena [6,7]. In [8] it was shown that this incompatibility
may be resolved by assuming that the rotational drag as predicted by Keith
should be observable for small-sized rotors consisting of discrete point masses
that, unlike astronomical bodies, are largely transparent to gravitational force
interaction proposed by Keith.
In fact, to date, ample discussion of those experimental setups and data [3]
have been notably limited. The goal of this paper is to revisit the former results
thereby providing fresh intensive details of past experimental setup and
Experimental Setup
The spin experiments were performed using 2.5 mm steel balls (ball bearings)
suspended on axis of a cavity-type permanent-magnet support system, see
Fig.1, and vertically stabilized by an auto sensing control coil (VSC) and a
phase leading DC amplifier at the point of magnetostatic equilibrium.
Radial vibrations of the rotor are damped by lateral position control of the
magnet system which for this purpose rests on a flexible support. Radial
movements of the strongly magnetized rotor are sensed by a coil system (RP)
placed on top of the rotor. The inductive signal generated by these coils is
amplified and fed to deflection coils acting on the flexible support of the
magnet system.
Control of the rotor frequency is accomplished by a two-phase RF sine wave
generator and two corresponding pairs of coils (DM) placed at right angles
sideways around the rotor. The rotor is spun asynchroneously similar to the
armature of an induction motor. The drive engages only to get the rotor to a
new frequency test mark from where it is allowd to coast freely.
The freely spinning ball is then observed via an inductive voltage measured at
the drive coils due to the rotating component of the rotor magnetization. The
measured AC signal of the drive coils is synchronus with the rotation frequency
and serves for precisely recording actual values and drift of rotor frequency.
At the same time AC voltage amplitude is recorded by a precision voltmeter in
order to compensate measured drift rates associated with external induction
losses. An analog meter of type HP-400E [9] was used which due to a special
calibration procedure resulted in a readout error within 0.25 % full scale
For the detection of extremely low rotational drags expected in view of the
predictions by Keith, we have to minimize as far as possible each kind of
rotational friction on the rotor. Besides the non-contacting magnetic suspension
we have provided a hermetic glass tube (see Fig.1) where the rotor is enclosed
and maintained under ultra high vacuum conditions, i.e., below 10-8 Pa, by an
ion getter pump.
We further must provide extremely stable thermal conditions at the rotor in
order to ensure reliable readouts of the rotor frequency. As a rotor heats up
due to inductive drive fields, every time it reaches a new test frequency, it has
to cool down over a period of at least 12 hours to accomodate within 1 mK the
temperature of an electronically stabilized double thermostat. The heating
plates of the inner thermostat circuit are seen in Fig.1 below. In addition, a
flowing-water outer thermostat encloses the suspension system during the
period of data acquisition. Only a single measurement of drag and associated
AC voltage taken every day in the morning hours was selected for the data
evaluation. This became the preferred procedure for providing well equilibrated
temperature conditions at the rotor. While after any drive action, drive coils
adjust rather quickly to the thermostat level, the rotor itself due to its shiny
surface with low emissivity and high vacuum isolation takes many hours to
reach thermal equilibration [10]. A time period of 24 hours proved adequate to
prevent erroneous readings due to thermal expansion and related frequency
drift of the rotor.
Finally, when searching for drags within the sub Nanohertz region, we had to
provide a reference time base with adequate stability. In the experiments
described here we used an HP-105A quartz oscillator [11] as our reference
time base with specified aging rate < 5 x 10-10 per 24 hours. For each
measurement of relative frequency decay (df/dt)/f, earlier denoted as the
frequency “decay ratio“ [12], a total of 2 x 108 consecutive revolution periods
were measured at a time resolution of 1 µs for each observation described. For
easier reading we now introduce the notation “D“ for the the decay ratio
(df/dt)/f. As described earlier [13], D is determined by the simple relation,
D = (t1t2) / (t1 * t2) (1)
where t1 and t2 denote two consecutive time intervals extending over 108 rotor
revolutions. When the rotor spins at 60 kHz, each measurement covers a time
interval of about an hour. In that time we observed D with a level of
uncertainty well below 10-12/sec. [14].
It is expected that any short term drift of the time base or of the thermal
stabilization system (thermostat) as well as other accidental disturbances, e.g.,
vibration of the suspension apparatus or uncertainty in voltmeter readout,
though perhaps contributing to scatter, should produce no long term trends in
the experimental data.
Experimental Results and discussion
Under the conditions described in the previous section there still remains some
residual drag to be observed at a magnetically suspended, freely spinning rotor
which is well known, in particular, from the spinning rotor vacuum gauge
(SRG) [15]. The SRG after having been introduced as a secondary standard in
vacuum metrology [16] was commercialized in the early 1980s as a first
technological spin-off to the experimental setup and procedure described in the
present paper.
We can consider, too, residual drags due to magnetic induction losses. Those
include, (i) external losses induced in the spinning rotor environment mainly
from residual rotating radial components of the axial rotor magnetization and,
(ii) internal losses induced at the rotor itself due to asymmetries and
perturbations of the magnetic support field.
We can take account of each external induction loss and compensate for it by
taking two measurements of the decay ratio D at every test frequency under
distinct rotor magnetizations. We obtain the latter by running at frequencies
beyond the rotor material's yield stress. Applying an overspeed treatment
causes rotor material to flow plastically at the rotor's center where it
encounters maximum centrifugal stress that modifies its structure so that the
rotating component of magnetization gradually relaxes to lower amplitude.
The relative amplitudes of the rotating component's magnetization are readily
determined by measuring the corresponding pickup voltages U generated in
the drive coils. Since we know that external induction loss depends strongly on
the square of pickup voltage, we can explain each associated decay ratio and
compensate it by processing observed primary data through the relation,
(DhighD0) / (Uhigh)2 = (DlowD0) / (Ulow)2, (2a)
whence we derive the compensated decay ratio:
D0 = [ (Uhigh / Ulow)2 * Dlow Dhigh ] / [ (Uhigh / Ulow)2 – 1 ]. (2b)
Subscripts “high“ and “low“ refer to respective levels of induced voltage U and
associated drag D before and after each intermediate overspeed treatment.
The method of extrapolation according to (2b) not only clears out external
induction effects from the data, but at the same time converts upward
deviation of primary data due to relaxation effects, in particular those of plastic
flow, into corresponding downward deviation of the compensated data. Note
that the Dhigh value in (2b) comes with a negative sign. Thus, when Dhigh is
augmented by some extra drag this will result in a downward deviation of the
compensated data D0. The elastic limit of the rotor material thus becomes
evident in form of a clear maximum of drag data around 75 kHz as represented
by shaded entries in Fig.2b. At that point drag begins to fall with further
increase in rotor-frequency.
Rotational drag due to internal induction loss occurs in the rotor itself. It is
mainly caused by Earth's rotation (Coriolis drag) as described by,
DCoriolis = kela * θ0
2 / τ(3)
where θ0 denotes the equilibrium hang-off angle, i.e. the angle by which the
rotor spin axis lags behind the axis of the suspension field due to the Earth's
rotation, and τ is the time constant for establishing that lag. A first-order
correction term kela accounts for elastic deformation of the rotor due to
centrifugal stress within the elastic limits of rotor frequency [17]. Strain
hardening effects play into extending those limits. More details on elasticity
induced corrections will be discussed below.
The Coriolis drag was first analysed theoretically by J.C. Keith [18] in the early
1960s as a neccessary precondition in view of his own laboratory investigations
[19,20,21,22] on magnetically suspended steel spheres. The Coriolis drag as
predicted by Keith was confirmed subsequently by experimental results carried
out in the late 1960s at the University of Bonn [23] and also complies with
some earlier results originating from similar investigations at the University of
Virginia [12].
Apart from drag caused by induction loss, additional drags occur in high-speed
rotor experiments due to relaxation phenomena. In particular, imagine a steel
ball spun up for the first time after insertion into the magnetic suspension field.
As rotation speed increases, the magnetic axis will tend to align with the axis
of maximum inertia [13]. That in general will not be the axis of spontaneous
magnetization in case of a ball bearing which is typically made from a
precipitation hardened chrome steel.
Another significant relaxation phenomenon, as mentioned above, is associated
with plastic deformation, i.e. irreversible strain of the rotor material which
starts to develop at high rotor speed right at the center of the spinning ball,
i.e. the location of maximum centrifugal stress. By this process some central
rotor material gets plasticized in a way that the rotor experiences an
irreversible deformation with a tendency to develop the shape of a rotational
ellipsoid. The region of elastic deformation which is characterized by reversible
strain of the rotor material can be extended to higher rotational frequencies by
conditioning the rotor for some time under plastic flow conditions. After this
treatment, when the rotor comes to rest the central material will experience
pressure from outer parts of the rotor since these have not undergone plastic
deformation and tend to assume their original spherical shape. As a
consequence, the rotor will be stress-free at the center only at some
intermediate frequency of rotation, and the elastic limit will be extended to
some higher frequency of rotation as compared to the untreated rotor.
Plastic flow of the rotor material links with corresponding drop of rotational
energy which in turn dumps into the plastic deformation process. We must
beware of plastic flow phenomena that may mimic at least qualitatively the
existence of a small extra drag due to retarded gravitational force interaction,
the gravitational Keith effect that we are looking for. But fortunately, the extra
drag due to plastic flow and other relaxation phenomena is converted into a
negative deviation of the D0 data as seen by processing the primary data
according to (2b).
During the measurement period reported here, we have recorded three sets of
data as seen from respective branches labeled 1, 2 and 3 in Fig.2a below.
Branch #1 shows the first series of data recorded after the rotor is introduced
into magnetic suspension. This run is mainly characterized by relaxation
phenomena and associated drag due to initial alignment of spin and magnetic
axes with the axis of maximum inertia. The maximum rotor frequency reached
at the end of this run was 91.4 kHz, i.e., well beyond the elastic limit of the
central rotor material. Here the rotor experienced its first overstress treatment.
The second series of measurements (branch #2) was recorded at descending
frequencies between the same points where the data of the first series were
recorded at increasing frequencies. Finally, branch #3 was run from top to
lower frequency again after an intermediate re-acceleration up to 99.6 kHz, i.e.
well above the elastic limit, too.
This procedure ensures that internal stress in the rotor, as well as the rotating
component of rotor magnetization differ at the beginning of each series.
The quite satisfactory outcome of the experiments described in the present
paper showed that the observed data, after correcting for external induction
losses according to (2b), fit rather closely, i.e. within a few per cent, to the
Coriolis drag as expected in view of prior investigations [23]. The expected
drag is visualized in corresponding graph Fig.2a below.
Fig.2b displays just the deviations of the experimental data from the expected
Coriolis drag. We turned the vertical scale of the diagram in Fig.2b from
logarithmic to linear thus zooming in on the region of interest in Fig.2a
allowing negative values to be displayed while also achieving more detailed
visualization of the data characteristics.
Fig.2b also displays a set of graphs representing the extra drag expected due
to residual deviations from rotational symmetry of the magnetic suspension
field. The drags caused by magnetic field asymmetries were identified
previously [23] by their typical frequency dependent characteristics with
maximum value at lower and distinctive drop at higher rotational frequency.
The entries in Fig.2b represented by open circles were attained by processing
raw data taken from branches #1 and #2 of Fig.2a according to Eq.(2b), while
shaded entries relate to raw data taken from branches #2 and #3. Data on
branch #1 are affected severely by the initial relaxation process described
earlier. The values derived from branches #1 an #2 tend to drop off
dramatically with increasing rotor frequency. In contrast, the data derived from
processing raw data from branches #2 and #3 suggest occurence of relaxation
processes only at rotor frequencies above 75 kHz. It is evident that this
frequency marks the elastic limit of the rotor material, at which point the 2.5
mm ball bearing reaches a peripheral speed close to 600 m/s. This is probably
due to the rotor's overspeed treatment, and though of comparable magnitude
is substantially beyond cited literature data of only 480 m/s previously found
for spherical steel rotors [24].
Wheras we clearly could attribute downward slope of the experimental data
beyond the elastic limit to plastic flow phenomena, we still lack a conclusive
explanation of the even more significant upward slope of data at rotational
frequencies below the elastic limit. Data observed within respective frequency
ranges (i) stand out from other effects by way of their strong frequency
dependence, and (ii) stay in reasonable agreement in both their frequency
dependence and absolute magnitude, with the effect of retarded gravitational
force interaction as proposed by Keith [4,5] on the basis of relativity theory.
Coriolis drag as a base reference
There is something remarkable about our ability to use a tiny steel ball to search
for evidence of the speed of gravity, so disproportionate to efforts to install
kilometer-sized gravitational wave antennae. One hears a similar ring from the
past as we hearken back to times when flat spacetime gravity field theory
sought to challenge curved spacetime general relativity [25,26]. Having
juxtaposed experiments involving g-radiation reaction with purely g-radiation,
we feel obliged to add further detail to treatment of our Coriolis data, cf.,
Fig.2b, insofar as it sets an ubiquitous reference for all our drag tracking data.
We are quite aware that calculated Coriolis drag is still about 30 times larger
than the extra drag we discuss here, see Table 1. In principle, we cannot rule
out that at least part of that extra drag is due to missing first-order corrections
to calculated Coriolis drag. We also might expect first-order deviations, in
particular, in view of the effect of elastic deformation and associated stress on
the geometrical dimensions and physical parameters of the rotor material
within elastic limits of rotor frequency.
We have alreay taken account of the effect of elastic centrifugal deformation on
the Coriolis drag [17] by introducing the kela correction in Eq.(3). The blue line
labeled “ela“ in Fig.2b indicates where Coriolis drag would have fallen in
absence of the elastic correction. In that case the deviation of experimental
data is reduced to some extent but with no notable alteration of the
characteristic frequency dependence.
What has not been investigated so far is the influence of elastic centrifugal
stress on the magnetic permeability µ and electric resistivity 1/σ of the rotor
material. Those data were determined only on stress-free probes, see Fig.3,
and have been regarded as independent of rotor frequency.
From literature data [27] we learn that stress dependence of σ is negligible
compared to that of µ. When assuming µ to degrade on increasing tensile
stress, correction of µ by kela will result in a negative correction of DCoriolis. The
effect of µ-correction is entered in red in Table 2 which also gives an overview
of the parameters and algorithms involved with calculation of DCoriolis. We see
that tensile stress influence is an order of magnitude smaller than the extra
drag discussed, tends in a negative direction, and in no way can lend itself to
conditioning that drag.
We thus conclude that the stress dependence of material parameters will have
no significant impact on the calculated Coriolis drag.
Fig.3 also discloses experimental µr data derived for chrome steel 100Cr6, the
material our test rotors are made from. In view of probable errors in
determination of µr data we have graphically visualized in Fig.2b the effect of
µr data within a broad range of 40 < µr < 50 on the Coriolis drag. Note that
respective reference lines exhibit rather flat frequency dependence as
compared to the one associated with the gravitational Keith effect.
We have isolated a significant drag on a magnetically suspended freely
spinning steel sphere with a prominent maximum of about 0.02 nHz which
stands out from the background scatter of the recorded data by a full order of
magnitude. The maximum occurs at peripheral speeds close to 600 m/s. This
can be attributed to the material's elastic limit. Beyond that limit, the recorded
drag is characterized by plastic flow strain phenomena and a consistent drop of
corresponding data. Below that limit, i.e. under elastic conditions of the rotor
material, the extra drag is characterized by a steep increase with rotor
frequency in contrast to various magnetic induction effects that exhibit a
comparably weak dependence on spin frequency.
Finally, we state that at rotational frequencies within the elastic limit the
outstanding drag data fit in a reasonable way to predictions by J.C. Keith on
the effect of retarded force interaction within a rotating system of mass points
as related to the limited propagation speed of gravity. In view of these findings
it appears worthwhile to take a second look also at the theoretical background
and results derived by J.C. Keith.
The author gratefully acknowledges the brilliant ideas conceived by James C.
Keith and his powerful theories, including the one on Coriolis drag, that
encouraged the experimental search for retarded gravitational force effects at
a laboratory scale, and is grateful, too, for many valuable discussions. The
author feels indebted, in particular, to Wilhelm Groth for having initiated our
experimental investigations at University of Bonn in 1963 and for continuous
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Fig.1 – (top) Outer view of cavity-type suspension system on flexible support with deflection
coils for lateral stabilization and heating disks for thermal stabilization, (left) permanent
magnets with rotor at equilibrium position, (right) vacuum enclosure and coils: HP – pickup
system for lateral stabilization, DM – coil system for rotational drive and pickup,
VSC – pickup-and-control coil for vertical stabilization
Fig.2 as from Ref.[3] with supplementary entries: Blue line labeled “ela“ indicates Coriolis drag
as calculated without elastic correction. Fine lines labeled µr=40 and µr=50 indicate Coriolis
drag variation as due to respective variation of reversible permeability.
Fig.3 – Original sketch (1972) of setup for determination of reversible permeability µr with
material probes inserted at the secondary coil. Probes 2 mm diam. by 21.7 mm revealed
µr = 139 for yoke iron, µr = 17.4 for chrome steel 100Cr6 (needle bearing) as from factory,
and µr = 44.5 for similar probe after 6h vacuum bake at 400 °C. The latter value is used for
calculation of Coriolis drag, see Table 2.
89.566 1.1384 7.01 89.564 0.8530 4.59 .6388 .6131 .0257
87.684 1.1320 7.05 87.683 0.8468 4.59 .6370 .6082 .0288
85.738 1.1315 7.09 85.738 0.8421 4.60 .6317 .6031 .0286
83.741 1.1326 7.11 83.740 0.8372 4.60 .6245 .5978 .0267
81.677 1.1363 7.12 81.674 0.8350 4.58 .6223 .5922 .0301
79.558 1.1453 7.10 79.557 0.8350 4.56 .6171 .5864 .0307
77.379 1.1537 7.06 77.379 0.8332 4.53 .6089 .5803 .0286
75.151 1.1645 6.99 75.149 0.8356 4.47 .6080 .5740 .0340
72.863 1.1788 6.88 72.862 0.8380 4.40 .6021 .5674 .0347
70.513 1.1931 6.75 70.513 0.8394 4.31 .5959 .5605 .0354
68.112 1.2136 6.57 68.112 0.8419 4.20 .5850 .5533 .0317
65.661 1.2378 6.37 65.663 0.8478 4.07 .5788 .5458 .0330
63.163 1.2659 6.14 63.165 0.8502 3.92 .5642 .5380 .0262
60.618 1.2966 5.89 60.618 0.8575 3.76 .5555 .5299 .0256
58.023 1.3311 5.61 58.024 0.8670 3.59 .5451 .5215 .0236
55.379 1.3730 5.31 55.379 0.8792 3.40 .5361 .5126 .0235
52.690 1.4264 5.01 52.692 0.8998 3.22 .5292 .5034 .0258
49.952 1.4893 4.70 49.950 0.9198 3.03 .5148 .4938 .0210
47.213 1.5660 4.39 47.212 0.9481 2.84 .5034 .4839 .0195
44.501 1.6587 4.09 44.501 0.9874 2.65 .5017 .4738 .0279
41.558 1.7788 3.77 41.559 1.0347 2.45 .4907 .4625 .0282
38.571 1.9263 3.46 38.570 1.0932 2.25 .4828 .4506 .0322
Table 1 – Primary data of the decay ratio D2, D3 as represented along branches #2 and #3 of
Fig.2a and respective pick-up voltages U2, U3 for calculation of D0 according to Eq.(2b). In the
last column are listed the deviations of D0 from the calculated Coriolis drag DCor as represented
by shaded entries in Fig.2b. The shaded fields at the upper part of the table indicate the data
are affected by plastic flow phenomena at the rotor interior.
Table 2 – Spreadsheet for calculation of Coriolis drag including elastic corrections
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This work reviews the origin, development, completion, and outcome of a trans-elastic ultracentrifuge project of Mexico's Nuclear Center through 1971 to 1986. The project had its origin in the search for an effect that supposedly would validate Birkhoff's gravity theory over Einstein's General Relativity. For this purpose an extraordinary ultracentrifuge was built which deserved the 1973 National Award for Instruments (Mexico). The ultracentrifuge was also used to investigate the feasibility of uranium enrichment by solid state centrifugation. Highly enriched uranium was obtained, but in small quantities.
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New drag-versus-speed measurements with a freely spinning steel sphere in a cavity-type permanent-magnet suspension confirm the up-to-date theories of residual drag (Coriolis, eddy-current, and relaxation effects). A small excessive drag is of particular interest for the discussion of Keith's gravitational radiation effect.
Full-text available
A general first-order approximation correction is presented which has to be applied to the theoretical frequency decay ratio -ω&dot;∕ω of a rigid rotor in order to find the corresponding ratio for a real elastic rotor. The procedure is exemplified for the particular drags associated with gas friction and with the Coriolis effect acting on steel spheres. In these cases the decay ratio is at most about 1% smaller than that of the corresponding rigid sphere.
The a.c. field measurement technique makes use of the skin effect in which on a.c. field applied to a metal is confined to a thin layer at the surface. The surface electric field is then investigated using a voltage-difference probe. A change in the voltage gradient along the surface is interpreted as being due to a change in the path length along a field line between the probe contacts, thus indicating the presence of a crack. The technique has been described previously [1,2], and the development of this work to cover line contacts and sub-surface flaws is described elsewhere in these proceedings [3].
Compiling the expertise of nine pioneers of the field, Magnetic Bearings - Theory, Design, and Application to Rotating Machinery offers an encyclopedic study of this rapidly emerging field with a balanced blend of commercial and academic perspectives. Every element of the technology is examined in detail, beginning at the component level and proceeding through a thorough exposition of the design and performance of these systems. The roster of authors boasts an average of twenty-five years of work developing magnetic bearing technology - a truly exceptional pool of experience. The book is organized in a logical fashion, starting with an overview of the technology and a survey of the range of applications. A background chapter then explains the central concepts of active magnetic bearings while avoiding a morass of technical details. From here, the reader continues to a meticulous, state-of-the-art exposition of the component technologies and the manner in which they are assembled to form the AMB/rotor system. These system models and performance objectives are then tied together through extensive discussions of control methods for both rigid and flexible rotors, including consideration of the problem of system dynamics identification. Supporting this, the issues of system reliability and fault management are discussed from several useful and complementary perspectives. At the end of the book, numerous special concepts and systems, including micro-scale bearings, self-bearing motors, and self-sensing bearings, are put forth as promising directions for new research and development. Newcomers to the field will find the material highly accessible while veteran practitioners will be impressed by the level of technical detail that emerges from a combination of sophisticated analysis and insights gleaned from many collective years of practical experience. An exhaustive, self-contained text on active magnetic bearing technology, this book should be a core reference for anyone seeking to understand or develop systems using magnetic bearings.
The time rate of change of a point particle's kinetic energy due to gravitational interactions is calculated, together with the energy loss of a rotating mass ring due to interaction with its own gravitational field. The complete g-reaction between two spinning mass points is then obtained, and some examples of retarded gravitational reaction between two mass points are considered. Energy dissipation equations are also derived taking into account the balance of gravitational forces in a rotating system of gravitationally interacting particles. Their applicability to astronomical binaries is considered. (D.C.W.)
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Scitation is the online home of leading journals and conference proceedings from AIP Publishing and AIP Member Societies