Method and apparatus for precisely controlling the periodic motion of an object

A method and apparatus precisely controlling the periodic motion of an object is specifically applicable to the control of the rotation of a nuclear magnetic resonance (NMR) spectrometer rotor, and uses a fluid source having a fluid output with a time-varying magnitude. The output of the fluid source is coupled to a stator housing within which the rotor is rotatably disposed such that the time varying fluid pressure from the fluid source impinges upon a plurality of vanes located about a circumference of the rotor. The coupling of the fluid flow to the rotor results in a periodicity of the rotor rotation being proportional to a periodicity of the time varying magnitude of the fluid flow. This, in turn, creates a plurality of stable equilibrium rotation rates at which a frequency locking effect is achieved that tends maintain the rotor periodicity at a predetermined rate, thereby achieving particularly precise and stable rotor rotation.

FIELD OF THE INVENTION 
This invention relates generally to controlling the periodic mechanical 
motion of an object and, more particularly, to a method and apparatus for 
rotating sample materials during nuclear magnetic resonance (NMR) 
experiments. 
BACKGROUND OF THE INVENTION 
As is known in the art, a nuclear magnetic resonance (NMR) spectrometer is 
a device for measuring a spectral response of a sample material under test 
to applied static magnetic and radio frequency (RF) fields. As is also 
known, nuclear magnetic resonance (NMR) is a phenomenon exhibited by a 
select group of atomic nuclei generally referred to as "gyromagnetic" 
nuclei, and is based upon the existence in these nuclei of nuclear 
magnetic moments. 
When a gyromagnetic nucleus is exposed to a relatively strong, uniform 
static magnetic field (a so-called "Zeeman Field") and perturbed by the 
magnetic field of a relatively weak radio-frequency (RF) signal, the 
gyromagnetic nucleus precesses at a resonant frequency generally referred 
to as the Larmor frequency. The Larmor frequency is characteristic of each 
nuclear type and is dependent upon the applied field strength in the 
location of the nucleus. The resonant frequencies of the nuclei can be 
observed by exposing the nuclei to a radio frequency (RF) signal pulse and 
measuring a time domain waveform corresponding to an amplitude of a 
component of a resultant magnetization vector which is transverse to the 
magnetic field over a predetermined period of time. The time domain 
waveform is typically converted to a frequency spectrum via Fourier 
transform techniques. 
Although identical nuclei have the same frequency dependence upon the 
magnetic field, differences in the chemical environment of each nucleus 
can modify the applied magnetic field in the local vicinity of the 
nucleus, so that nuclei in the same sample do not experience the same net 
magnetic field. The differences in the local magnetic field between two 
such chemically non-equivalent nuclei result in spectral shifts in the 
Larmor frequencies generally referred to as "chemical shifts." Such 
chemical shifts reveal information regarding the number and placement of 
the atoms in a molecule as well as the positioning of adjacent molecules 
with respect to each other in a compound. 
Unfortunately, it is not always possible to interpret the frequency spectra 
produced by the chemical shifts because of interfering and dominant 
molecular interactions. This is particularly true in NMR spectroscopy of 
solid material samples. In NMR testing of solid material samples, 
molecular interactions tend to obscure a desired output signal. For 
example, magnetic moments in neighboring nuclei may perturb each other, 
resulting in interactions called dipole-dipole couplings. These couplings 
tend to broaden the characteristic resonance peaks and obscure the sharply 
peaked spectral features typically produced by the chemical shifts. 
An additional problem found in NMR testing of solid material samples is 
that the orientation of molecules which make up the solid material sample 
is relatively fixed with respect to the applied static magnetic field. 
This results in anisotropic chemical shifts since the resonant frequency 
depends, at least in part, on the physical orientation of the molecules 
with respect to the applied static magnetic field. To obtain a meaningful 
output signal, it is necessary, therefore, to suppress some molecular 
interactions over others. 
In NMR testing of solid material samples for example, the aforementioned 
anisotropic chemical shift may typically be reduced by placing the solid 
material sample in a sample container having a central longitudinal axis 
aligned with respect to the applied static magnetic field at the so-called 
"magic angle" corresponding to an angle of 54.degree. 44'. The sample 
container is provided with a plurality of vanes and flutes on a portion 
thereof, and is rotatably held at the magic angle in a housing, or 
"stator." 
In conventional solid sample magic angle spinning (MAS) NMR, a steady flow 
of gas (i.e. a gas having a velocity which remains relatively constant at 
a particular point in space over time) is applied to the fluted portion of 
the sample container. The force exerted upon the fluted portion of the 
sample container by the steady gas flow causes the sample container and 
thus the solid material sample contained therein to rotate. For this 
reason, the sample container is commonly referred to as a "rotor." The 
steady flow of gas is applied to the rotor at a velocity which causes the 
rotor to rotate at a relatively constant predetermined rate of speed 
typically in the range of 2,000 to 15,000 rotations per second (rps). 
During NMR measurements made while the rotor is rotating at such 
frequencies, anisotropic field components tend to average to zero. To 
select the particular rate at which the rotor spins (e.g. 6,000 rps) the 
velocity of the steady flow gas is adjusted to thus expose the fluted 
portion of the rotor to a predetermined relatively constant torque. At any 
instant in time, the resultant pressure on the fluted portion of the rotor 
corresponds to the force produced by the steady flow gas multiplied by the 
area of the region of the rotor which the steady flow gas impacts. As the 
rotor rotates, the steady flow gas will impact different regions of the 
fluted portion of the rotor which may result in minor variations in the 
pressure applied to the rotor by the steady flow gas. 
As is also known, there exist certain two-dimensional nuclear magnetic 
resonance (NMR) techniques which utilize a rotor having disposed therein a 
solid material sample. One example is the so-called "magic angle turning" 
(MAT) experiment. In experiments of this type, it is required that the 
sample rotate at a precisely determined speed, typically less than 200 Hz, 
with the rotation axis aligned at the magic angle. These two-dimensional 
NMR techniques may be used to obtain frequency spectra of solid samples 
where the effects of anisotropic interactions are suppressed in one of the 
two dimensions. Thus, such two-dimensional NMR techniques provide the 
resolution required to separate different signal components in a first 
dimension while preserving anisotropic interactions in a second dimension. 
Typically, to rotate the rotor at such relatively low frequencies, a rotor 
manufactured for the purpose of spinning at a relatively high rate of 
speed (e.g. at rotation frequencies typically in the range of about 
2,000-15,000 Hz), is driven by a steady flow of gas which has a relatively 
low, constant pressure. The gas impacts vanes of the rotor, thereby 
causing the rotor to spin at a relatively low rate of speed (e.g. at 
rotation frequencies typically in the range of about 25-100 Hz). 
One problem with this approach, however, is that it is relatively difficult 
to control precisely the rotation frequency of the rotor. The rotation 
frequency varies primarily due to an inability to adequately control the 
relatively low pressure steady flow gas. Furthermore, relatively large 
changes in the rotation frequency come about due to relatively small 
changes in the pressure of the gas. In addition, changes in the 
temperature of the gas cause changes in its relative density, 
correspondingly changing the force it provides. This, too, may cause a 
significant change in the rotation rate of the rotor. 
Relatively elaborate schemes, generally referred to as rotor 
synchronization techniques, have been developed to adjust the timing of 
the NMR experiment to follow variations in rotor speed. A tachometer is 
used to provide a timing synchronization signal generally referred to as a 
"synchronization pulse" at equally spaced intervals during the rotation of 
the sample holder. For example, in the MAT experiment synchronization 
pulses are usually provided once for every 120 mechanical degrees of 
rotation. Thus, in this case, the tachometer provides three equally spaced 
pulses per complete revolution of the sample. The synchronization pulses 
are provided to a pulse generator or a timing controller where they are 
used to correctly time the generation of RF pulses. 
It would be desirable to provide a technique for driving a rotor at a 
constant speed over a relatively wide range of rotor frequencies. It would 
also be desirable to provide a technique for spinning a rotor at a 
constant relatively low rotation frequency. It would also be desirable to 
provide a technique for performing a slow sample spinning NMR experiment 
which does not require rotor synchronization. 
SUMMARY OF THE INVENTION 
In accordance with the present invention, an apparatus for moving an object 
with a periodic motion includes a fluid source for providing a fluid flow 
having a time-varying magnitude, and means for coupling the object to the 
fluid flow such that the object undergoes the desired periodic motion. The 
time-varying magnitude of the source is such that a periodicity of the 
object is proportional to a periodicity of the fluid flow. While the 
preferred embodiment of the invention is directed to periodic rotational 
motion, the use of a time-varying fluid flow for generating periodic 
translational motion is also envisioned. 
In one embodiment of the invention, the object undergoes periodic 
rotational motion and the fluid flow is a gas flow. The coupling between 
the rotational object and the gas flow comprises a direction of the gas 
flow toward a series of vanes located around a circumference of the 
object. Thus, the coupling strength is dependent upon the position of the 
object relative to the direction of the gas flow. In a particular 
embodiment, the rotating object is a sample container (or "rotor"), and 
holds a sample material for use with a magic angle turning probe. 
In the preferred embodiment, the fluid source comprises an audio 
loudspeaker and a fluid conduit leading to the vanes of the rotor. The 
loudspeaker generates a time-varying acoustic signal which is coupled into 
the conduit as a confined time-varying fluid pressure. A timing signal 
circuit is used to generate a time varying electrical signal which is, in 
turn, used to drive the loudspeaker. The conduit through which the 
periodic fluid flow is directed may comprise one or more fluid paths. The 
periodicity of the fluid pressure is controlled by the periodicity of the 
electrical timing signal. In turn, the periodicity of the rotational 
motion of the rotor is proportional to the periodicity of the fluid flow. 
The present invention is particularly well-suited for use with the slow 
spinning of a rotor for a magic angle turning (MAT) experiment. The 
coupling between the time-varying fluid flow and the rotor is such that a 
frequency locking effect results. This effect is such that deviations from 
an equilibrium rotation speed are opposed by a restorative torque, which 
keeps the rotor turning at a constant speed. This high degree of stability 
in the rotor speed, even at slow angular velocities, allows a higher 
degree of precision in MAT experiments.

DESCRIPTION OF THE PREFERRED EMBODIMENTS 
Referring to FIGS. 1-1B in which like elements are provided with like 
reference designations throughout the several views, a nuclear magnetic 
resonance (NMR) spectrometer 10 for generating frequency spectra of a test 
sample includes a chamber 12 having disposed therein in a magnet 13. 
Chamber 12 and magnet 13 are each cylindrical in shape and a central 
portion of chamber 12 corresponds to an open region 12a. 
A portion of a sample probe 14 is locatable within the open central region 
12a of chamber 12. Probe 14 is described in further detail below, and is 
the type into which is disposed a sample material to be tested. Probe 14 
includes a spinner assembly 16 which comprises a stator 18 in which may be 
located a sample container (or "rotor") 20. The sample material to be 
tested is deposited within rotor 20. 
An RF coil 22 is disposed within the stator assembly 18, surrounding the 
sample container 20. An RF drive circuit 23 is coupled to first and second 
ends of the coil 22. A rotor drive circuit 40 provides at output port 40a 
a gas with a time-varying fluid pressure. The gas is applied to rotor 20 
through a conduit 42. The gas generated by drive circuit 40 causes 
rotational motion of sample container 20 within the stator assembly 18. 
In a preferred embodiment, drive circuit 40 generates at output port 40a a 
stream of gas which corresponds to a sequence of gas pulses. The gas 
stream is coupled from output port 40a to a drive gas inlet of the stator 
assembly 18 via a plurality of gas pathways including a gas tube 42. From 
tube 42, the gas is delivered through channels in the stator to rotor 20. 
Thus, rotor 20 is driven at a predetermined rotation frequency by a stream 
of varying pressure gas generated by drive circuit 40. 
FIGS. 1A and 1B are partial cross-sectional views of stator assembly 18. 
Stator assembly 18 comprises a stator housing 24 having a cavity region 
there within. The cavity region includes a central portion 25a and first 
and second end portions 25b, 25c. Rotor 20, containing the sample material 
to be tested, is disposed in the cavity region 25a of stator housing 24. 
Also provided within housing 24 are a plurality of channels 26, 28. 
Disposed in the first end portion 25b of stator housing cavity 25 is a 
first or lower journal bearing 29 and disposed in the second end portion 
25c of stator housing cavity 25 is a second or upper journal bearing 30. 
Each of the lower and upper journal bearings 29 and 30 have channels 31 
formed therein such that when the bearings 29, 30, are mated to stator 
housing 24, the bearing channels 31 align with stator housing channel 28 
and gas introduced into stator housing channel 28 travels through the 
mating channels 31 and is forcefully expelled through apertures into the 
stator housing cavity. Thus, in this particular embodiment, the apertures 
form a plurality of gas jets 34. 
The gas jets 34 are positioned such that when gas is introduced to channel 
28, a gas stream emerges from each of the gas jets 34 and impacts rotor 
20. The gas expelled through jets 34 provides an air bearing which 
supports the rotor 20 with a cushion of gas within the stator housing 
cavity. 
As shown in FIG. 1B, rotor 20 has an end cap 35 coupled to a first end 
thereof. At least a portion of end cap 35 extends above upper journal 
bearing 30. Disposed over upper journal bearing 30 is a drive plate 36 
having a plurality of channels formed therein. Adjacent to drive plate 36 
is a drive plate cover 38 which terminates portions of the drive plate 
channels exposed through a top surface of drive plate 36. Drive plate 36 
is located such that drive plate inlet channel 37 aligns with stator 
housing channel 26. With the channels 26, 37 thus aligned, gas introduced 
into channel 26 travels through the mating channels 26, 37 and is 
forcefully expelled through a plurality of apertures 39 as streams or jets 
of gas into the vicinity of rotor end cap 35. Thus, in this embodiment, 
each of the plurality of apertures 39 is referred to as a gas jet 39. 
Gas jets 39 are positioned such that when gas is introduced through channel 
26, the gas stream emitted through each of the plurality of gas jets 39 
impacts end cap 35 of the rotor 20 in a manner discussed below. In 
response to the gas stream impacting the end cap 35, the rotor spins 
within cavity region 25 of stator housing 24. Since gas introduced to the 
stator cavity region 25 through gas jets 34 provides rotor 20 with a 
cushion of gas, the rotor spins within the stator cavity region 25 with 
relatively little resistance due to frictional forces which would 
otherwise occur from contact between the rotor 20 and the stator assembly 
18. 
Since a bearing gas is introduced to the rotor 20 via channel 28, channel 
28 is referred to as bearing gas channel 28. Similarly, gas introduced 
into channel 26 provides a drive force to the rotor 20 and, thus, gas 
channel 26 is referred to as drive gas channel 26. 
Referring again to FIG. 1 a bearing gas controller 44 provides a bearing 
gas at a regulated, relatively constant pressure to probe 14 via a gas 
conduit 45. The bearing gas is coupled through gas line 45 to the bearing 
gas channel 28. A tachometer 46 is coupled to spinner assembly 16 to 
detect the frequency of rotation of the rotor 20. Tachometer 46 may 
include, for example, a transducer such as an optical sensor which senses 
rotary motion of the rotor 20 and produces an electrical signal in 
response thereto. Tachometer 46 may also include an output display which 
displays the rotor rotation frequency. 
In a preferred embodiment, a solid sample is disposed in the rotor 20 and 
the rotor 20 is placed in stator assembly 18. The stator is supported by a 
pair of pivots (not shown) within the probe 14 and secured at an angle 
which positions rotor 20 at 54.degree. 44' with respect to an applied 
static magnetic field of the magnet 13. It is noted, however, that the 
present invention is not limited to solid samples or spinning at the 
"magic angle." Rather, the present invention may be used in spinning about 
any desired rotation axis. 
An NMR test is conducted by spinning a sample material in the rotor 20, and 
exposing the spinning sample to the static magnetic field generated by 
magnet 13 and to a pulsed radio frequency (RF) field generated by exciting 
RF coil 22 via a series of RF pulses generated by RF drive circuit 23. The 
NMR response signal is detected in a known way, typically by induction in 
RF coil 22. 
It should be noted that in some applications, it may be desirable to apply 
a force to rotor 20 by means other than by providing a gas flow to the 
rotor. In such applications, driver circuit 40 would provide a varying 
force of an appropriate type having an amplitude which varies as a 
function of time. 
Referring now to FIGS. 2-2B, in which like elements are provided having 
like reference designations throughout the several views, a rotor 50 
includes a cylindrically-shaped container 52 which is closed at one end 
and open at the other, and which has a cavity region 53 (FIG. 2B). In some 
applications the material samples may be solid while in others they may be 
liquid. Alternatively, the rotor itself may be a solid piece of sample 
material. 
Disposed in the open end of container 52 is an end cap 54. End cap 54 
includes a base region 55. End cap 54 has formed therein a plurality of 
flutes 57 and a corresponding plurality of vanes 56 projecting from base 
region 55. The flutes 57 may be formed in end cap 54 via machining or any 
other techniques. Alternatively, end cap 54 may be formed via injection 
molding techniques or any other desired construction technique. 
Projecting from base region 55 of the end cap is an engagement member 58 
which extends into the cavity region of the container 52. Engagement 
member 58 is cylindrical, and has a diameter which allows a press fit with 
an inner surface of container 52. 
As described above in conjunction with FIG. 1, when rotor 50 is disposed in 
stator assembly 18 the vanes 56 of end cap 54 may be engaged by a gas 
stream expelled through drive gas jets 39. The drive gas causes the rotor 
50 to rotate at a predetermined frequency. 
In an alternative embodiment, a rotor may be provided having two open ends 
and corresponding end caps. Each of the end caps may or may not have vanes 
formed therein. In the case where neither end cap is provided with vanes, 
some portion of the rotor tube may be provided with vanes formed therein. 
Regardless of the manner in which the vanes are provided in the rotor, the 
rotor spins and responds to a stream of gas or other force impinging upon 
the vanes. In still another embodiment, the rotor can have a shape which 
is not cylindrical. For example, the rotor may have a conical or truncated 
conical shape. Alternatively still, a first portion of the rotor may have 
a conical shape while a second portion of the rotor has a cylindrical 
shape. Thus, it can be seen that there are a variety of different rotors 
which all have in common a portion with vanes to which a fluid drive force 
may be applied. 
FIG. 3 is a cross-sectional top view of a stator and rotor assembly 
according to the present invention. Drive plate 36 has an inner ring 62 
and an outer ring 63, and a central aperture region 64. Inner ring 62 is 
spaced from outer ring 63 by a predetermined distance to provide a drive 
gas distribution channel 65. The gas distribution channel 65 is in fluid 
communication with drive plate inlet channel 37. In this embodiment drive 
plate 36 has a single inlet 37. However, it may also be desirable to use a 
plurality of inlets 37. 
Formed within inner ring 62 are gas jets 39. As gas is fed into the drive 
gas channel 26, it is forced through distribution ring 65 into gas jets 
39. The gas jets conduct gas from the distribution ring 65 to central 
aperture region 64 of drive plate 36 in predetermined directions and at 
predetermined locations about the rotor end cap 52. There the gas impinges 
upon the vanes 56 of rotor end cap 52. 
Different embodiments of the invention may include rotors with different 
diameters and different numbers of vanes on their end caps. Similarly, the 
number of gas jets may be varied. The particular number of drive gas jets 
provided in the drive plate and the particular number of vanes provided in 
the end cap may be selected in accordance with a variety of factors 
including, but not limited to, the desired range of rotation frequencies, 
the diameter of the rotor, the diameter of the drive plate, the diameter 
of each gas jet and the area and shape of the vanes. 
There are also applications where unequally spaced drive gas jets and/or 
vanes may be advantageous. In such applications, the output of each drive 
jet may be timed relative to the anticipated angular position of the end 
cap. In such an embodiment, each drive gas jet would fire when one or 
another of the vanes was appropriately positioned relative to that 
particular jet. It should also be noted that the angle of the drive jets 
relative to the radius of the drive plate should be selected to provide a 
controllable force or gas pressure against the vanes of the rotor. 
Furthermore, the shapes of the vanes may be selected to optimize rotation 
at particular rotation frequencies. 
The embodiment of FIGS. 1-3 provides a periodic gas flow to a plurality of 
drive jets via a common distribution channel 65. This results in the 
pulsing of gas from each of the jets 39 essentially simultaneously. 
However, it may be desirable to stagger the times at which gas is expelled 
from each of the jets. Referring now to FIGS. 4 and 5 in which like 
elements are provided having like reference designations, a valve 70 for 
expelling gas through gas jets at staggered times includes an outer sleeve 
70a having disposed therein a rotatable inner sleeve 70b. Such a valve may 
be provided as part of a stator assembly, for example. 
Inner sleeve 70b has a fitting 71 to which is coupled a first end of a tube 
72. Fitting 71 leads to an aperture 73 in the inner sleeve 70b. A second 
end of tube 72 is coupled to a gas source (not shown) with a constant gas 
pressure. Outer sleeve 71a has provided therein a plurality of apertures 
74, each located in a different angular position about outer sleeve 71a. 
Each of the apertures 74 leads to a tube 75 which connects to one of the 
drive gas inlets of a drive plate like that of FIGS. 1A, 1B and 3. 
As inner sleeve 70b rotates within outer sleeve 70a, aperture 73 is 
progressively aligned with each of outer sleeve apertures 74. This allows 
a sequential distribution of a finite amount of gas to each of tubes 75. 
For example, with apertures 73 aligned as shown, gas provided by the gas 
source travels through tube 72 and fitting 71, through aperture 73, and 
into the tube 75 with which it is shown as being aligned. The gas stream 
then flows to a respective one of the plurality of gas inlets provided in 
the drive plate. The drive plate gas inlet leads to at least one drive gas 
jet in the drive plate, and the gas stream is forcefully expelled through 
that jet. 
When inner sleeve 70b rotates further within outer sleeve 70a such that 
aperture 73 aligns with a different aperture 74, gas provided by the gas 
source travels through tube 72 and fitting 71, through aligned apertures 
73 and into the tube 75 now aligned with the aperture 73. This gas stream 
then flows to a different one (or more) of the plurality of gas jets. Thus 
valve 70, acts as a firing control means, coupled between a gas flow 
generator and the drive gas jets, for distributing gas sequentially to a 
plurality of drive gas jets. This may be seen by the partial 
cross-sectional top view of FIG. 5, in which the tubes 75 are distributed 
about the outer sleeve 70a. (For clarity, only several of the tubes are 
shown in the drawing). 
If the inner sleeve 70b is rotated at a constant rate, and the tubes 75 are 
equally distributed at angular positions about the outer sleeve, the 
firing of sequential gas jets is staggered, but equally spaced, in time. 
However, those skilled in the art will recognize that the relative angular 
locations of the gas jets, the arrangement of vanes on the rotor, the 
relative angular locations of the tubes 75 relative to the inner sleeve 
70b, and the rate of rotation of the inner sleeve may all be varied to 
achieve almost any desired timing for the firing of the jets, and the 
corresponding impact on the rotor. Furthermore, other mechanisms for 
expelling gas through gas jets at staggered or differing times may also be 
used. For example, multiple gas stream sources may each be individually 
coupled to separate tubes each of which leads to a separate drive gas 
inlet of a drive plate. Such variations are considered to be well within 
the scope of the invention. 
In one embodiment of the invention, a spinning system having two different 
drive components is provided. In such a system, the first drive component 
provides a steady gas stream, and spins the rotor at any desired frequency 
within some relatively wide range of rotation frequencies, but with 
limited frequency stability. The second drive component provides a 
pulsating gas stream which achieves a frequency lock at a desired rotation 
frequency, providing excellent frequency stability, but at a limited 
number of discrete frequencies. Each drive component in such a system 
could be optimized for its respective purpose and the combination of the 
two may provide a system which is able to accurately spin rotors at 
particular rotation frequencies within a wide range of desired rotor 
rotation frequencies. 
Referring back to FIG. 3, the rotor end cap 52, having six equally spaced 
vanes 56, is driven by a gas flow emitted from six equally spaced drive 
gas jets 39. The drive gas jets apply a torque to the rotor, causing it to 
rotate. The drive torque on the rotor depends on the gas pressure provided 
to the drive gas jets 39 (the "drive pressure"). The drive torque may also 
depend on other factors such as the rotational velocity of the rotor (the 
"rotor speed"), and may be influenced by aerodynamic or acoustical 
effects. Thus, the instantaneous drive torque is difficult to predict. 
However, the time-varying pressure applied to the rotor provides a rotor 
speed regulation effect, which may be understood with the help of a simple 
mathematical model. 
For the purposes of this model, the drive torque T.sub.D is defined as the 
product of the drive pressure P.sub.D and a "transfer coefficient"q that 
describes the dependence of the drive torque on the rotor angle. That is, 
EQU T.sub.D =P.sub.D .multidot.q 
The transfer coefficient essentially defines the responsiveness of the 
rotor to the drive pressure. Because the drive gas jets strike respective 
vanes of the rotor at a distance and angle which varies according to the 
rotor angle, the transfer coefficient will have some angular dependence. 
For the purpose of this simple model, the angular dependence may be 
assumed to be in the form of a cosine. Thus, the transfer coefficient may 
be written as a periodic function of rotor angle: 
EQU q =q.sub.0 +q.sub.1 cos(n.theta.) 
Similarly, the drive pressure, which is a periodic function of time, may be 
written as: 
EQU P.sub.D =P.sub.0 +P.sub.1 cos(.omega.t) 
where P.sub.0 is a steady state pressure component, and P.sub.1 
cos(.omega.t) is the time-varying pressure component. Therefore, the above 
representation of drive torque may be described as: 
EQU T.sub.D =P.sub.0 +P.sub.1 cos(.omega.t)!.multidot.q.sub.0 +q.sub.1 
cos(n.theta.)! 
The periodicity of the drive pressure is deliberately imposed by the 
time-varying gas source 40. The periodicity of the transfer coefficient is 
dependent on the periodic spacing of the vanes and the jets. For example, 
in the case of FIG. 3, the factor n=6. 
The relative periodicities of the drive pressure and the transfer 
coefficient may be different under different conditions. However, one 
available periodicity relationship for the embodiment of FIG. 3 is that in 
which the periodicity of the drive pressure equals that of the transfer 
coefficient. The rotor speed required to accomplish this is referred to as 
the "nominal" speed. At this speed, the periodicities are coincident, such 
that the vanes 56 of the rotor pass successive jets 39 with the same 
frequency that the jets are firing. The nominal speed is equal to 
.omega./n. In mathematical terms, the coincidence of periodicities may be 
expressed as: 
EQU n.theta.=.omega.t+n.phi. 
where .phi. is referred to as the "rotor phase." If the rotor spins at 
exactly the nominal speed of .omega./n, then .phi. is a constant. If the 
rotor spins slightly faster or slower than the nominal speed, then .phi. 
gradually increases or decreases. By substitution, the previous drive 
torque equation becomes: 
EQU T.sub.D =P.sub.0 +P.sub.1 cos(.omega.t)!.multidot.q.sub.0 +q.sub.1 
cos(.omega.t+n.phi.)! 
which may be evaluated to yield: 
EQU T.sub.D =P.sub.0 q.sub.0 +P.sub.0 .multidot.q.sub.1 
cos(.omega.+n.phi.)+P.sub.1 .multidot.q.sub.0 cos(.omega.t)+1/2P.sub.1 
q.sub.1 cos(n.phi.)+1/2 P.sub.1 q.sub.1 cos(2.omega.t+n.phi.) 
If the expression above is averaged over time, the time-averaged drive 
torque becomes: 
EQU T.sub.DAVG =P.sub.0 q.sub.0 +1/2P.sub.1 q.sub.1 cos(n.phi.) 
Finally, it may be assumed that there is a constant drag torque T.sub.CD 
which opposes the rotation of the rotor. The drag torque includes 
"mechanical load" in applications (such as motors) which derive power from 
the rotation. The rotor accelerates in response to the net torque exerted 
on the rotor, which is the difference between the drive torque and the 
drag torque. Averaged over time, this total torque is: 
EQU T.sub.NET =P.sub.0 q.sub.0 +1/2P.sub.1 q.sub.1 cos(n.phi.)-T.sub.CD 
FIG. 6 is a plot of this torque as a function of rotor phase .phi.. It is a 
periodic function, with a range from P.sub.0 q.sub.0 -1/2P.sub.1 q.sub.1 
-T.sub.CD to P.sub.0 q.sub.0 +1/2P.sub.1 q.sub.1 -T.sub.CD. For any given 
rotor phase, there is a corresponding average net torque. When the torque 
is positive, the rotor accelerates, which tends to cause .phi. to 
increase. As .phi. increases, there is a corresponding change in the 
torque. Similarly, when the torque is negative, the rotor decelerates, 
which tends to cause .phi. to decrease and, again, results in a 
corresponding change in the torque. 
As shown in FIG. 6, there may be points at which the torque is equal to 
zero. With zero torque, the rotor does not accelerate or decelerate. At 
these points, the rotor speed is constant, and they are therefore referred 
to as equilibrium points, and any corresponding rotor phase as an 
equilibrium phase. When a rotor spins at the nominal speed, and at an 
equilibrium phase, it will continue spinning at that same speed. The 
equilibrium points which are situated on a positive slope are referred to 
as unstable, since small deviations of the rotor phase result in a torque 
which causes the phase to deviate even further. The equilibrium points 
which are situated on the negative slope, however, are referred to as 
stable, since small deviations of the rotor phase result in a torque which 
causes the phase deviation to decrease. Therefore, in the vicinity of a 
stable equilibrium point, the rotor tends to remain near the equilibrium 
phase. A further consequence of operating at the stable equilibrium is 
that the rotor speed tends to remain near the nominal speed. 
Referring to FIG. 6, when a rotor rotates at the nominal frequency, the net 
torque generated at a particular stable equilibrium phase angle 
.phi..sub.E is equal to zero. In the vicinity of a stable equilibrium, the 
net torque T.sub.NET maintains the rotor spinning at the nominal 
frequency. If the rotor spins at a higher or lower frequency than the 
nominal frequency, then the rotor would be provided with either a negative 
or positive torque. For example, if the rotor began spinning slower than 
the nominal rotation frequency, then the rotor phase would fall back to 
.phi..sub.1. With the rotor at phase angle .phi..sub.1, the net torque on 
the rotor would be a positive value T.sub.1. This positive torque would 
cause a corresponding rotational speed increase, the rotor would 
eventually return to the nominal frequency, and the rotor phase would 
eventually return to the equilibrium phase .phi..sub.E. 
Similarly, if the rotor began spinning faster than the nominal frequency, 
then the rotor phase would advance to .phi..sub.2. With the rotor advanced 
to phase angle .phi..sub.2, the net torque applied to the rotor would be a 
negative value T.sub.2. Thus, the rotor would decelerate, thereby 
returning to the nominal rotation frequency, and returning to the 
equilibrium phase .phi..sub.E. 
The preceding analysis is based on a simple situation, but can be easily 
extended to more complex circumstances. For example, Fourier series 
methods can be used to extend the analysis to situations where: 1) the 
drive pressure waveform is non-sinusoidal; 2) the transfer coefficient 
waveform is non-sinusoidal; 3) the number of vanes does not equal the 
number of jets; 4) the vanes and/or jets are unequally spaced; and 5) the 
rotor speed is equal to a multiple (i.e. a harmonic or sub-harmonic) of 
the fundamental nominal speed. Furthermore, known numerical simulation 
methods may be used to give an exact treatment of: 1) aerodynamic flow 
effects; 2) turbulent flow; 3) effects of varying drag or mechanical load; 
4) acoustical wave effects; and 5) non-periodic drive pressure waveforms. 
While variations in the specific design of the invention may lead to 
variations in behavior, the fundamental principle remains the same: the 
use of a particular time-varying fluid source to drive the periodic motion 
of an object results in stable equilibrium points at which deviations from 
a given rate of motion are met by a restorative force which maintains the 
rate of motion of the object at the stable equilibrium point. 
Referring now to FIG. 7, a plot of rotor frequency vs. drive amplitude 
illustrates that within a range of drive amplitudes (here designated by 
reference character 92) the rotor frequency is held at the nominal 
rotation frequency. The vertical axis of the plot corresponds to rotor 
frequency. The horizontal axis of the plot, labeled "drive amplitude", 
corresponds to an amplitude of a drive force applied to the rotor to cause 
rotational motion of the rotor. 
The plot of FIG. 7 shows that a rotor being driven by drive gas having 
increasing steady state pressure and a time-varying pressure component 
with an increasing amplitude (as shown in FIG. 8) has a characteristic 
range of pressure within which the rotor frequency is maintained or 
"locked" at a desired rotor frequency. Thus, the locking range 92 
corresponds to the range of drive pressures over which the rotor will 
remain at the nominal frequency. The width of locking range 92 may vary in 
accordance with a variety of factors including, but not limited to, the 
angle of the drive jets, the shape of the rotor vanes and the equilibrium 
torque (i.e. the amount of drag on the rotor due to frictional forces or 
mechanical load on the rotor). Thus, as discussed above, the combination 
of the rotor torque response as a function of rotor rotation angle and a 
drive force having a time-varying amplitude component which may, for 
example, be provided as a sequence of drive gas pulses, produces a 
frequency locking effect which holds the rotor at a predetermined rotation 
frequency. 
In the present invention, a frequency must be selected at which gas pulses 
are applied to the rotor end cap. One method of determining this frequency 
is to calculate it as a product of the desired rotor rotation frequency 
and a value referred to as a "least common multiple" (LCM). The LCM is the 
smallest number which may be evenly divided by both the number of gas jets 
and the number of end cap vanes. For example, in the case of an end cap 
having six vanes and a drive plate having six gas jets, the least common 
multiple (LCM) value is six. Thus, to spin the rotor at a frequency of 30 
Hz, pulses of fluid are expelled through the gas jets at a frequency of 
6.times.30 Hz or 180 Hz. 
The smallest LCM value between two numbers results when the numbers are 
selected to be equal to each other. Consequently, selecting the number of 
gas jets equal to the number of end cap vanes results in the lowest fluid 
pulse repetition frequency for a predetermined end cap rotation frequency. 
The next lowest LCM value results from the number of vanes and the number 
of jets being related as one being the multiple of the other. 
Referring now to FIGS. 9 and 9A, one form of pulse generator is 
demonstrated. An apparatus for performing a magic angle turning experiment 
150 includes a signal source 152 having an output port coupled to an input 
port of a drive signal circuit 154. Signal source 152 may be, for example, 
a function generator which provides output signals at selectable 
frequencies and amplitudes. The signal source may be capable of providing 
a variety of waveforms including, but not limited to, sinusoidal 
waveforms, square waveforms, triangular waveforms, and the like. Signal 
source 152 feeds a signal having a time-varying amplitude to an input port 
of amplifier 154. For example, signal source 152 may provide a signal 
having a sinusoidal wave shape with a given amplitude at a predetermined 
frequency. The amplifier 154 receives the signal and provides a drive 
signal at an output terminal thereof. A control terminal of the amplifier 
is coupled to a gain control circuit 180, which provides an input signal 
that adjusts the gain provided by the amplifier 154. The gain control 
therefore allows the amplification to be easily controlled. 
The output of amplifier 154 is coupled to an input port of a transducer 
assembly 155 which, in this particular embodiment, includes an audio 
loudspeaker 156. The amplified output signal is fed to an input port of 
the speaker 156, which converts the electrical signal to an acoustical 
wave. The acoustical wave is coupled into tube 42, and thereby takes the 
form of a controllable fluid pressure. The fluid pressure propagates 
through tube 42 to magic angle turning probe 14 (FIG. 1), where the tube 
42 is coupled to a fluid input fitting of the probe 14. 
Referring to FIG. 9A, transducer assembly 155 includes a speaker 156 having 
a cone shaped membrane 156a and a gasket 157. Speaker 156 is mounted to a 
plate 190 via screws or other fastening means, such that gasket 157 forms 
a gas tight seal to a first surface of plate 190. Plate 190 may be, for 
example, a material such as plastic having a thickness of about 6 mm. 
Speaker membrane 156a vibrates in response to drive signals fed to the 
speaker, and a gas flow produced by movement of membrane 156a is forced 
through tube 42. In this particular embodiment, tube 42 is coupled to 
plate 190 via a tubing adapter 192, which is hermetically sealed about the 
circumference of the tube 42. External screw threads on tubing adapter 192 
mate with a threaded aperture 191 in plate 190. Thus, in response to drive 
signals, the vibration of the membrane 156a causes pulses of gas, such as 
air, to be coupled into tube 42. 
Although speaker 156 is shown mounted to a flat surface of plate 190, those 
skilled in the art will recognize that plate 190 could also have a horn 
shaped surface to provide a better acoustical coupling between speaker 156 
and tube 42. Alternatively, or in addition to the horn shaped plate 
surface, a proximal end of tube 42 could have a tapered shape. Certain 
advantages might be gained by providing such tapering, since the pressure 
increases as an acoustical wave travels along a tube which is tapering 
down to a smaller cross-section. Furthermore, once the pressure wave 
enters probe 14, it may travel through different paths of varying 
diameter, such as tubes internal to the probe, and channels through the 
stator. 
Referring again to FIG. 9, bearing gas module 44 provides a bearing gas to 
the probe 14. A controller 162 is also provided which includes a 
tachometer 170 that is coupled to probe 14 via signal line 166. The 
tachometer 170 measures the speed of rotation of the rotor, and provides a 
signal indicative thereof. The output signal of the tachometer 170 is 
received by a frequency detector 172 and a phase detector 174 which 
detect, respectively, the frequency and phase of the tachometer signal. 
This information is then provided to feedback controller 176, which 
signals the gain control circuit 180 in response to the tachometer 
feedback information to increase or decrease the gain as necessary. As 
shown in FIG. 9, a phase control circuit 181 is also provided, and may be 
used to control the relative phase of the signal output by the signal 
source 152 via feedback controller 176. 
The rotor disposed in the probe 14 is driven by the airflow provided from 
the acoustic wave produced by the speaker 156. In a preferred embodiment, 
the air flow is a pulsating airflow at a constant pulse rate with each of 
the pulses having a constant pulse width. The pressure amplitude of the 
pulsating air flow is adjusted as needed to provide a stable equilibrium 
condition. When the system locks into the equilibrium condition, the rotor 
spins at a frequency proportional to the frequency of the pulsating 
airflow to provide a system having exceptional rotational stability. 
Referring now to FIGS. 10-10B, it can be seen that a plurality of different 
drive pressure waveforms may be used to produce the desired frequency 
locking effect. FIG. 10 shows a pressure waveform which may be used after 
a frequency lock has been achieved. FIGS. 10A-10B show pressure signal 
waveforms which may be used to locate pressure amplitudes at which a 
frequency lock may be achieved. For example, FIG. 10A illustrates that a 
steady state component of the drive pressure signal may be increased until 
a magnitude at which a frequency lock occurs is reached. It should be 
noted that, in FIG. 10A, peak-to-peak amplitude of the time-varying 
component of the drive pressure signal does not change. FIG. 10B, on the 
other hand, illustrates that the steady state component of the drive 
pressure signal may remain constant while the the peak-to-peak amplitude 
of the time-varying component of the drive pressure signal may be adjusted 
to locate an amplitude at which a frequency locking condition occurs. Once 
a frequency locking condition is found, however, (i.e., amplitude 
parameters are found which produce a frequency locking effect), the shape 
of the drive pressure signal may be similar to that as shown in FIG. 10. 
Having described preferred embodiments of the invention, it will now become 
apparent to one of ordinary skill in the art that other embodiments 
incorporating their concepts may be used. It is felt therefore that these 
embodiments should not be limited to disclosed embodiments, but rather 
should be limited only by the spirit and scope of the appended claims. For 
example, the number of jets and the number of vanes used with a system 
according to the invention may be varied. However, many magic angle 
turning experiments conducted at relatively low sample spinning 
frequencies require three-fold symmetry. Thus, in some applications, it 
may be desirable to use a system which includes a drive plate having three 
jets and an end cap rotor having three vanes. With relatively few vanes on 
the end cap, there is a relatively large spacing between the vanes. Such 
relatively large vane spacing may improve the frequency locking behavior 
of the system. 
Another variation of the present invention is its application to a 
non-rotary system. The use of a time-varying fluid pressure to drive a 
periodic system having a linear or otherwise non-rotary direction 
component could provide similar desired frequency locking effects.