Patent Application
20200340853
This invention relates to the field of data analysis. More particularly, this invention relates to creating estimated waveforms that can be used to fill in gaps of missing data in vibration data sets.
Machine vibration information, such as might be gathered on rotating machines like a turbine, can be very useful when monitoring the health of the machine. For example, many machine problems can be diagnosed from an analysis of the vibration information. This information is typically represented as a sinusoidal waveform, with a frequency and amplitude that increase as the rotational speed of the machine increases.
Because such machines generally rotate at a relatively high speed, a large amount of data must typically be gathered to adequately represent the vibrational characteristics of the machine, and to capture the information that would be required to diagnose any issues with the machine. Because of the large amount of data, it is problematic to both transmit the data and to store the data. Transmission problems arise because of the bandwidth that would be required by the large amount of data, and storage problems arise because of the amount of storage space that would be required by the large amount of data.
In order to reduce such problems, some vibration instruments only pass along a subset of the vibration data. This can be accomplished in one or more of a variety of different ways, for example, by either not sensing a portion of the data or by not passing along a portion of the data that is sensed. This tends to reduce the bandwidth and storage requirements.
However, it can be disconcerting to see such gaps in the data when it comes time to analyze or otherwise review the data.
What is needed therefore, is a system for reducing issues such as those described above, at least in part.
The above and other needs are met by a method for filling a gap of missing vibration data in a set of vibration data. At least one reference waveform on a first side of the gap is selected and at least one adjacent waveform on an opposing second side of the gap is selected. It is determined whether the gap is in a section of the vibration data where a frequency of the vibration data is one of increasing, decreasing, and steady state. Where the gap is in a section of the vibration data where the frequency of the vibration data is changing substantially linearly, an analytical method is applied to at least one of the at least one reference waveform and the at least one adjacent waveform to approximate the vibration data that is missing in the gap. Where the gap is in a section of the vibration data where the frequency of the vibration data is changing substantially exponentially, a numerical method is applied to at least one of the at least one reference waveform and the at least one adjacent waveform to approximate the vibration data that is missing in the gap. Where the gap is in a section of the vibration data where the frequency of the vibration data is substantially steady state, at least one of the at least one reference waveform and the at least one adjacent waveform is copied to approximate the vibration data that is missing in the gap. The approximated vibration data is presented to a user.
In some embodiments according to this aspect of the invention, when the gap is in a section of the vibration data where the frequency of the vibration data is substantially changing, the at least one reference waveform has a frequency that is slower than the at least one adjacent waveform. In some embodiments, the step of presenting the approximated vibration data includes displaying a plot of the set of vibration data to the user with the gap filled.
In some embodiments, the set of vibration data is sensed from a rotating machine. In some embodiments, the set of vibration data is sensed from a turbine. In some embodiments, the frequency of the vibration data is changing substantially linearly as a rotating speed of the turbine is increased to an operating speed. In some embodiments, the frequency of the vibration data is changing substantially exponentially as a rotating speed of the turbine is decreased from an operating speed. In some embodiments, the frequency of the vibration data is substantially steady state as the turbine is running at an operating speed.
In some embodiments, the step of approximating the data when using either the numerical method or the analytical method includes the steps of approximating a frequency rate of change of the missing vibration data in the gap, approximating tach signal locations of the missing vibration data in the gap, approximating an amplitude rate of change of the missing vibration data in the gap, and approximating a phase rate of change of the missing vibration data in the gap. Then, the gap is filled with ideal waveforms between the tach signal locations, and the ideal waveforms are adjusted using the approximated frequency, amplitude, and phase rates of change.
Further advantages of the invention are apparent by reference to the detailed description when considered in conjunction with the figures, which are not to scale so as to more clearly show the details, wherein like reference numbers indicate like elements throughout the several views, and wherein:
With reference now to
Although not indicated on the graph, the relative lengths of time for these three sections 104, 106, and 108 might not be as depicted. For example, it might take several hours for the machine to be ramped up to its operating speed, as represented by section 104, and then it might remain at the operating speed for weeks or months, as represented by section 106. Finally, it might take only a matter of minutes or a couple of hours for the machine to stop rotating once it is turned off, as represented by section 108.
As depicted in
In the second section 106 of the graph 100, re-creation of waveforms 102 in the gaps 1702 is a relatively straight-forward procedure. Because the machine is operating at steady state, each waveform 102 in section 106 looks substantially like any other waveform 102 in section 106, and so the waveform gaps 1702 in the second section 106 can be filled in with waveforms 102 that are either identical to those waveforms 102 that precede or follow the gap 1702, or are simple linear interpolations of such.
It is appreciated that any anomaly that might have occurred during such a gap 1702 cannot be recreated, regardless of the section 104, 106, or 108 in which it may have occurred, as the data to do so is probably not present in the waveform 102 data preceding or following the gap 1702 in which the anomaly occurred. Thus, the waveforms 102 produced according to the present embodiments would tend to not be useful for investigation of any such. Instead, they tend to provide a continuous representation of data that might be useful in certain circumstances.
Estimations of the waveforms 102 to fill the gaps 1702 in the first section 104 and the third section 108, however, are not as straightforward as the estimation of the waveforms 102 to fill the gaps 1702 in the second section 106, because of the change in turning speed in those sections 104 and 108.
While it might seem that a simple interpolation between preceding and following waveforms 102 could be performed to produce waveforms 102 for the gaps 1702 in the first section 104 that represents a linear increase in speed, such a method is inadequate because the speed does not increase just between successive rotations in a stepped manner. Instead, the speed increases continually throughout a given rotation, and therefore the waveform 102 produced during a given rotation changes throughout its period.
This problem is compounded further for the waveforms 102 that are produced during the third section 108, because the speed of the machine is decreasing substantially exponentially instead of substantially linearly.
Thus, according to the embodiments described below, a quadratic analytical solution is described in regard to estimating missing waveforms 102 in the gaps 1702 in the first section 104, and an iterative numerical solution is described in regard to estimating missing waveforms 102 in the gaps 1702 in the third section 108.
In each case, both the analytical solution for the linear first section 104 and the numerical solution for the exponential third section 108, a common basic approach 200 is used, as depicted in
It is then determined what section of the graph 100 the two selected waveforms belong to—be it the first section 104, the second section 106, or the third section 108, as given in block 208 of the flowchart 200.
In some embodiments, the waveform 102 next to the gap 1702 that has the slower turning speed is designated the reference waveform 1704, as depicted in
The reason for doing this is that there are more data points per unit of time in a waveform 102 that has a slower turning speed, and thus the accuracy of the computations described herein can be improved by making the designation as described in the paragraph above. However, in other embodiments, some other method besides just turning speed is used to determine which waveform 102 is designated as the reference waveform 1704 and the adjacent waveform 1706.
It is appreciated that the waveforms 1704 and 1706 and the gap 1702 are not meant to imply any kind of exact relative time scale. In general, the reference waveform 1704 will have a longer period Tr than the period Tw of the adjacent waveform 1706. The gap 1702 with a duration of Tg might have one or many missing waveforms 102 that are to be estimated.
If both the reference waveform 1704 and the adjacent waveform 1706 are from the second section 106, then the method falls to block 210, where a copy or simple linear interpolation is performed to produce the missing waveforms 102 and fill in the gap 1702. If both the reference 1704 and adjacent 1706 waveforms are from the first section 104, then the method falls to block 212, where an analytical quadratic estimation is performed to produce the missing waveforms 102 and fill in the gap 1702. Finally, If the reference waveform 1704 and adjacent waveform 1706 are from the third section 108, then the method falls to block 214, where an iterative numerical estimation is performed to produce the missing waveforms 102 and fill in the gap 1702.
The goals of the individual steps for both the analytical solution 212 and the iterative numerical solution 214 are generally the same, and so are depicted in
The general steps for both the analytical solution 212 and the iterative numerical solution 214 proceed as follows. The frequency and speed rate of change of the missing waveforms 102 in the gap 1702 are estimated, as given in blocks 216, and the rate of speed change in the gap 1702 is adjusted, as given in blocks 218. Tachometer pulses are created and placed in proper positions in the gap 1702, as given in blocks 220. The change in rate of the amplitude is estimated, as given in blocks 222, and the rate of change of the phase is estimated, as given in blocks 224. The gap 1702 is filled with an estimated waveform 102 that has been adjusted for rotational speed, amplitude, and phase changes, as given in blocks 226.
It is again mentioned that, even though these steps have similar names in each of the analytical solution 212 and the iterative numerical solution 214, they are performed in different ways, as described in more detail below.
With reference now to
The various embodiments of the present invention thus improve the basic operation of an apparatus 300 as depicted. If bandwidths of the I/O 310, for example, and the storage capacity of the medium 314 could handle the extreme amounts of data that are produced by a sensor 308, then there might be no need to only transmit and store a portion of the waveform data, and thus there might also be no gaps 1702 in the waveform 102 data that need to be filled.
The first three steps 216-220 of each method 212 and 214 have a goal of placing estimated tachometer signal pulses within the gap 1702, where two consecutive tach pulses t1 and t2 of the reference waveform 1704, as depicted in
Once the tach pulses have been placed in the gap 1702, the rate of change of both amplitude and phase are calculated, and then estimated waveforms 102 are produced in the gap 1702.
The waveform 404 within the reference block 1704 is defined by two consecutive tach signals t1 424 and t2 426. The waveform 404 includes the data gathered for one revolution of the machine. It is appreciated that the shape of the waveform 404 is representational, and in actual use might have a different and more complex shape.
Section 1702 is the gap 1702 to be filled. It is defined by the tach signal t2 426 at the end of the reference waveform block 1704 preceding the gap 1702, and the estimated (or actual) tach signal 428 at the start of the adjacent waveform 1706.
A pure sine wave 412 is placed within both sections 1704 and 1702. The sine wave 412 is given the same amplitude in both sections 1704 and 1702, but the frequency of the sine wave 412 is different in each section 1704 and 1702. In section 1704 the frequency of the sine wave 412 is set to exactly match one revolution between the two tach signals t1 424 and t2 426. In section 1702 the frequency of the sine wave 412 is set to exactly match one revolution between the two tach signals 426 and 428.
It is appreciated that if
At this point, we have two sections 1704 and 1702 that are each filled with parts of a sine wave 412, where the frequency of the sine wave 412 in section 1704 is different than the frequency of the sine wave 412 in section 1702.
A position is identified at time 414 in section 1702. In some embodiments, time 414 represents the first of a predetermined number of time divisions that will be used in each waveform 408 section (between successive tach signals) to be filled in the gap 1702. In such an embodiment, a given number of time divisions will be used for each waveform 408 in the gap 1702, no matter how short the time duration between two successive tach signals. In other embodiments, time 414 represents a specific length of time. In this embodiment, shorter waveforms in the gap 1702 (smaller distance between tach signals) are estimated using fewer time data points. In other embodiments, a combination of these two embodiments or some other method is used to determine the times 414.
The amplitude 416 of the sine wave 412 is determined at time 414. This amplitude is mapped over to the corresponding time position 418 that intersects the sine wave 412 within section 1704. Then the amplitude 420 of the actual waveform 404 is determined at that time position 418. Finally, the amplitude 420 is mapped over to position 422, which position 422 has the amplitude 420 of the actual waveform from section 1704, but at the time 414 in section 1702. By repeating this four-step process for each of the desired number of time positions 414, a complete approximated waveform 408 is constructed within the gap 1702. Using this same process, additional waveforms 408 are constructed between successive tach signals within the gap 1702, until the gap 1702 is completely filled.
These waveforms 408 are corrected for amplitude, as given in step 222 of
In addition, the phase associated with the first order turning speed changes from one waveform 102 to the next as the rotational speed of the machine changes. Therefore, the waveforms 102 that fill the gap 1702 are adjusted for this change in phase. The rate of change in phase is assumed to be either linear or exponential, according to the section 104 or 108 in which the gap 1702 is identified.
The steps as generally described in this section can be accomplished using analytical methods for the case of the linear change in speed, as depicted in section 104 of
The linear rate of change in rotational speed is initially calculated from the last two waveform cycles in the reference block 1704, according to the method 216a as depicted in
The calculations are based on the times of the last three tach pulses t0, t1, and t2, in reference block 1704, as depicted in
A linear increase in rotational speed or frequency uses the following equation, where the amplitude (y) is given by:
y=sin[(mt+w)t]=sin(mt2+wt)
where
For each complete cycle of the sine waveform 412 in
mt
1
2
+wt
1=2π and
mt
2
2
—wt
2=4π,
the amplitude is zero, which from above is expressed in an equation as:
sin(mt2+wt)=0.
Solving the two equations yields:
w=2π(1−2t12/t22)/t1(1−t1/t2), and
m=(4π−wt2)/t22.
The two equations above are used to calculate frequency and linear rate of speed change, as given in block 508 of
The rate of change m as initially calculated in this manner may not be the same as that for the waveform in the adjacent block 1706 on the opposing side of the gap 1702, and therefore needs to be adjusted according to the method 218a as depicted in
Thus, the starting rotational speed rate of change m as initially estimated is either increased or decreased as necessary, depending on a comparison of the tachometer pulse widths tw and tr, where tr is the tach pulse width between the two tach pulses that are closest to the gap 1702 in the reference block. The gap 1702 is then filled with tachometer pulses, where the final pulse within the gap 1702 Δt is compared to tw. This process is repeated iteratively until they match.
Using the equation for a linear change in rotational speed
y=sin{(mt+w)t}=sin(mt2+wt).
The equation for the pulse times in the gap 1702 is given by
sin(mt2+wt)=0, when
mt
2
+wt=2πI,
where (i) is the ith tachometer pulse.
Solving this quadratic equation gives
t
i
={−w+√(w2+8πmi)}/2m.
In regard to
The time width between the two tach pulses that are closest to the gap 1702 in the reference block 1704 and the two tach pulses that are closest to the gap 1702 in the adjacent block 1706, tr and tw respectively, are compared to see which is greater, as given in block 606. If tr is less than tw, then the right-hand side of the method 218a as depicted in
In blocks 608 and 616, tw is compared to Δt. In block 608, if Δt is not greater than tw, then the method 218a exits with the new speed m as previously determined. Similarly, in block 616, if Δt is greater than or equal to tw, then the method 218a exits with the new speed m as previously determined. However, if in either case the answer to the test of 608 or 616 is no, then the method falls respectively to block 612 or 620, where variables are initialized for a loop where m is adjusted until the criteria as described herein have been met. In block 612, m is too small, and so m is adjusted upwards by the value of Δm. In block 620, m is too large, and so m is adjusted downwards by the value of Δm.
The time t is then tested against the gap time tg, as given in block 614. If the test fails, then a new t is calculated as indicated in block 622, and as described above. The new t is again tested against the gap time tg, and when the new t eventually satisfies this condition in block 614, the method 218a returns blocks 608 and 616, where the iterations continue until the speed m is found which allows the tach signals to align.
Once the rate of change in rotational speed m has been adjusted, the next step is to create the tachometer pulses in the gap 1702. This step enables the gap 1702 to be filled with a representative wave 412.
Using the equation for a linear change in rotational speed shown above, the time for each tachometer pulse ti is given by
mt
i
2
+wt
i=2π,
Solving this quadratic equation for ti gives
t
i
={−w+√(w2+8πmi)}/(2×m)
This is depicted as method 220a in
Method 222a depicted in
y=(1+at)×sin{(mt+w)t}
where a is the linear rate of change in waveform amplitude and t is the time in seconds.
In
a=(An−Ar)*√2/Tg
where Tg is the time between the waveforms, or in other words, the gap 1702 time.
The linear change in phase Δϕ is given by
y=sin{(mt+w)t+Δϕt}
where Δϕ is the linear rate of change in waveform phase, and t is the time in seconds. Method 224a to accomplish this is depicted in
The rate of change in phase is then estimated using
Δϕr=ϕr/tr
as given in block 904, where tr is the time between the two tachometer pulses. The change in phase ϕw of the first cycle of the adjacent waveform 1706 is calculated using a SFDFT, as given in block 906. As before, the waveform samples are taken between the first two tachometer pulses in the adjacent waveform 1706. The rate of change in phase is then estimated as given in block 908, using
Δϕw=ϕw/tw
where tw is the time between the two tachometer pulses. The average rate of change in phase for the waveform in the gap 1702 is then calculated as given in block 910, using
Δϕg=(Δϕr+Δϕw)/2.
The basic concept is to compress the last tachometer cycle of the reference waveform 1704 into the tachometer cycles in the gap 1702 waveform, as given in methods 226a and 226c of
The waveform in the gap 1702 is then interpolated as described above in regard to
The amplitude of the wave is then adjusted for changes in amplitude and phase as given in flowcharts 222a and 224a. This process is repeated until the entire gap 1702 is filled.
This process is described in more detail with reference to methods 226a and 226b, as depicted in
The method then falls to a loop through method 226c, as depicted in
After returning to the method 226a, tgap is incremented by the sample time, as given in block 1012, and it is determined whether the cycle of the currently reconstructed waveform is completed, as given in block 1014. If the cycle of the current waveform is not completed, then the method returns to method 226c. If the waveform cycle has been completed (one complete period), then the method falls to block 1016, where the value of i is incremented to the next tach pulse in the gap 1702 waveform, for the possible creation of another waveform cycle by returning to block 1008. However, if the last waveform was the final waveform to be placed into the gap 1702, as tested in block 1018, then the waveform reconstruction process is finished, as given in block 1020.
In some embodiments, the change in rotational speed across the gap 1702 should not be more than about a factor of ten, as greater changes in rotational speed in some embodiments tends to reduce the number of waveform cycle samples to such an extent that detailed changes in the waveform are lost.
In some embodiments, the change in rotational speed between the two waveform cycles on either side of the gap 1702 is assumed to be linear. This is not normally an issue, as typically the change in rotational speed across the gap 1702 involves a large number of waveform cycles.
The process in this case is similar to the linear case, however, unlike the linear case that had an analytical solution to find the frequency and rate of change in rotational speed, the exponential change in rotational speed case requires a numerical iteration process to find the frequency and change in rotational speed, as depicted in method 216b of
As before, the last three tach pulses are taken from the reference block 1704, as given in block 1104. However, in this case, the reference block 1704 is, in some embodiments, the first waveform block following the gap 1702, as it contains a waveform with a lower turning speed than the block 1706 preceding the gap 1702, because the rotating machine is slowing down.
The tach pulse times are derived from the identified tach pulses, as given in block 1106, and an initial frequency is calculated, as given in block 1108, based on a linear rate of change. However, the frequency does not change linearly in this section, as previously discussed, so this calculated frequency is only a starting point for the following calculations. A small incremental value for adjusting this initial frequency is selected, as given in block 1110, and the iterative process of bocks 1112 and 1114 are commenced.
The initial estimation of frequency uses the previously defined linear equation
w=2π(1−2t12/t22)/t1(1−/t2)
With an exponential increase in frequency, the amplitude (y) is given by
y=sin(wtemt)
sin(wtemt)=0 when
wte
mt=2π,4π, etc.
Taking the natural logarithm of each side of the equation gives
ln(wt1)+mt1=ln(2π) and
ln(wt2)+mt2=ln(4π)
Solving these two equations gives
ln(wt2)−t2/t1 ln(wt1)=ln(4π)−t2/t1 ln(2π), and
m={ln(2π)−ln(wt1)}/t1
There is no analytical solution to the equation for w. Instead, w must be derived by a numerical iteration method of slowly increasing w from its initial value as set using the linear equation above, until the equation is satisfied as depicted in block 1112 of flow chart 216b of
As before, the rate of change of m may not be the same as for the adjacent waveform 1706 across the gap 1702, and therefore may need to be adjusted as depicted in flowchart 218b of
Using the equation for a linear change in rotational speed
y=sin(wtemt)
The equation for the pulse times in the gap 1702 is given by
sin(wtemt)=0 when
wt
emt=2πi
where (i) is the ith tachometer pulse.
This equation is solved iteratively by slowly increasing t until
wt
emt>=2πi
In regard to
The time width between the two tach pulses that are closest to the gap 1702 in the reference block 1704 and the two tach pulses that are closest to the gap 1702 in the adjacent block 1706, tr and tw respectively, are compared to see which is greater, as given in block 1206. If tr is less than tw, then the right-hand side of the method 218b as depicted in
In blocks 1208 and 1216, tw is compared to Δt. In block 1208, if Δt is not greater than tw, then the method 218b exits with the new speed m as previously determined. Similarly, in block 1216, if Δt is greater than or equal to tw, then the method 218b exits with the new speed m as previously determined. However, if in either case the answer to the test of 1208 or 1216 is no, then the method falls respectively to block 1212 or 1220, where variables are initialized for a loop where m is adjusted until the criteria as described herein have been met. In block 1212, m is too small, and so m is adjusted upwards by the value of Δm. In block 1220, m is too large, and so m is adjusted downwards by the value of Δm.
The time t is then tested against the gap time tg, as given in block 1214. If the test fails, then t is tested in a different manner, as indicated in block 1222, and as described above. If that value of t does not meet the secondary criteria, which in some embodiments it would not on the first pass, then t is adjusted as given in block 1224, and the criteria of block 1222 are again tested. When t eventually meets the criteria of block 1222, then the values of the variables are set as given in block 1226, and the new t is again tested against the gap time tg. When the newt eventually satisfies the condition in block 1214, the method 218b returns to the blocks 1208 and 1216, where the iterations continue until the speed m is found that allows the tach signals to align.
Once the rate of change in rotational speed m has been adjusted, the next step is to create the tachometer pulses in the waveform gap 1702. Using the equation for an exponential change in rotational speed, shown above, the time for each tachometer pulse ti is given by
wt
i
emti=2πi
There is no analytical solution to this equation for ti. In this case, ti is derived by a numerical iteration method of slowly increasing ti from a small initial value until the equation is satisfied as shown in flow chart 220b of
This is depicted as method 220b in
In this case, the exponential increase in amplitude (y) is given by
y=e
at×sin(wtemt)
where
a is the exponential rate of change in waveform amplitude, and
t is time in seconds
This is graphically illustrated in flowchart 222b of
a=(ln(An*√2)−ln(Ar*√2)/Tg
where Tg is the time between the waveforms (i.e. the waveform gap time).
The linear change in phase Δϕ is given by:
y=sin(wtemt+eΔϕt),
where
Δϕ is the linear rate of change in waveform phase, and
t is the time in seconds.
With reference now to the method 224b as depicted in
Δϕr=ϕr/tr
where tr is the time between the two tachometer pulses.
As given in block 1506, the next step is to calculate the change in phase ϕw of the first cycle of the adjacent waveform 1706 using a SFDFT. The waveform samples are taken between the two tachometer pulses in the adjacent waveform 1706 that are closest to the gap 1702. The rate of change in phase is then estimated as given in block 1508 by using:
Δϕw=ϕw/tw
where tw is the time between the two tachometer pulses.
For an exponential change in phase, the average rate of change in phase for the waveform in the gap 1702 is then calculated as given in block 1510, by using:
Δϕg=(Δφr+Δϕw)/2
A small incremental phase adjustment δϕ is set, as given in block 1512. This is the amount by which the phase will be iteratively adjusted in subsequent steps, according to the numerical method. Then, as given in block 1514, the phase Φg is calculated at tg and tw, from which ΔΦ is calculated. ΔΦ is then tested against ϕw, as given in block 1516. If ΔΦ does not equal ϕw, then Δϕg is adjusted by δϕ, as given in block 1518, and the iteration returns to block 1514. However, if ΔΦ does equal ϕw, then that current value of Δϕg is taken as the exponential phase change rate for the gap 1702, as given in block 1520.
The basic concept is to compress the nearest tachometer cycle of the reference waveform 1704 into the tachometer cycles in the gap 1702 waveform, as given in methods 226b and 226d of
The waveform in the gap 1702 is then interpolated as described above in regard to
The amplitude of the wave is then adjusted for changes in amplitude and phase as given in flowcharts 222b and 224b. This process is repeated until the entire gap 1702 is filled.
This process is described in more detail with reference to methods 226b and 226d, as depicted in
The method then falls to a loop through method 226d, as depicted in
After returning to the method 226b, tgap is incremented by the sample time, as given in block 1612, and it is determined whether the cycle of the currently reconstructed waveform is completed, as given in block 1614. If the cycle of the current waveform is not completed, then the method returns to method 226d. If the waveform cycle has been completed (one complete period), then the method falls to block 1616, where the value of i is incremented to the next tach pulse in the gap 1702 waveform, for the possible creation of another waveform cycle by returning to block 1608. However, if the last waveform was the final waveform to be placed into the gap 1702, as tested in block 1618, then the waveform reconstruction process is finished, as given in block 1620.
In some embodiments, the change in rotational speed across the gap 1702 should not be more than about a factor of ten, as greater changes in rotational speed in some embodiments tends to reduce the number of waveform cycle samples to such an extent that detailed changes in the waveform are lost.
In some embodiments, the change in rotational speed between the two waveform cycles on either side of the gap 1702 is assumed to be exponential. This is not normally an issue, as typically the change in rotational speed across the gap 1702 involves a large number of waveform cycles.
The foregoing description of embodiments for this invention has been presented for purposes of illustration and description. It is not intended to be exhaustive or to limit the invention to the precise form disclosed. Obvious modifications or variations are possible in light of the above teachings. The embodiments are chosen and described in an effort to provide illustrations of the principles of the invention and its practical application, and to thereby enable one of ordinary skill in the art to utilize the invention in various embodiments and with various modifications as are suited to the particular use contemplated. All such modifications and variations are within the scope of the invention as determined by the appended claims when interpreted in accordance with the breadth to which they are fairly, legally, and equitably entitled.
