Update Content - 2026-01-09

content/book/wardle15_ultra_precis_bearin.md

@@ -18,12 +18,128 @@ Year
 : 2015
 
 
-## Types of error motion {#types-of-error-motion}
+## Bearing motion error {#bearing-motion-error}
 
-A distinction is made between:
+Causes:
 
--   motion errors that are harmonic of the basic rotor speed: synchronous motion error
--   those that are node: asynchronous motion errors
+-   Manufacturing Quality
+-   Bearing design
+-   External influences
+
+**Types of error motion**
+
+A distinction is made between (see Figure <fig:wardle15_synchronous_asynchronous_schematic>):
+
+-   motion errors that are harmonic of the basic rotor speed: _synchronous_ motion error
+-   those that are node: _asynchronous_ motion errors
+
+<a id="figure--fig:wardle15-synchronous-asynchronous-schematic"></a>
+
+{{< figure src="/ox-hugo/wardle15_synchronous_asynchronous_schematic.png" caption="<span class=\"figure-number\">Figure 1: </span>Effect of (a) synchronous, and (b) asynchronous motion error on surface form" >}}
+
+
+### Measurement of motion error {#measurement-of-motion-error}
+
+A capacitive sensor is typically used, and the measurement is performed over a time period corresponding to several (typically five) revolutions of the bearing.
+
+It is displayed as a polar plot of motion error amplitude versus angle of rotation (Figure <fig:wardle15_typical_error_plot>).
+
+<a id="figure--fig:wardle15-typical-error-plot"></a>
+
+{{< figure src="/ox-hugo/wardle15_typical_error_plot.png" caption="<span class=\"figure-number\">Figure 2: </span>Total error motion" >}}
+
+The Synchronous error motion (i.e. error that are harmonics of the rotational speed) can be extracted (Figure <fig:wardle15_synchronous_error_example>).
+
+<a id="figure--fig:wardle15-synchronous-error-example"></a>
+
+{{< figure src="/ox-hugo/wardle15_synchronous_error_example.png" caption="<span class=\"figure-number\">Figure 3: </span>Synchronous motion error" >}}
+
+It can then be separated into a "_fundamental error motion_" and a "_residual error motion_" (Figure <fig:wardle15_fundamental_and_residual_errors>).
+The fundamental error motion contains only one frequency corresponding to the speed of the rotation of the bearing.
+For radial measurements, it corresponds to the eccentricity, and is not always significant.
+
+<a id="figure--fig:wardle15-fundamental-and-residual-errors"></a>
+
+{{< figure src="/ox-hugo/wardle15_fundamental_and_residual_errors.png" caption="<span class=\"figure-number\">Figure 4: </span>(a) Fundamental error motion; and (b) residual synchronous error motion" >}}
+
+The Asynchronous error motion (Figure <fig:wardle15_asynchronous_error_motion_example>) contains all other motion error frequencies.
+
+<a id="figure--fig:wardle15-asynchronous-error-motion-example"></a>
+
+{{< figure src="/ox-hugo/wardle15_asynchronous_error_motion_example.png" caption="<span class=\"figure-number\">Figure 5: </span>Asynchronous motion error" >}}
+
+The measurements shown in previous figures may be quantified by a number of different parameters but it is commonplace to find the "Least Squares" best fit centre and then to place Maximum Inscribed and Minimum Circumscribed circles on the measurement.
+
+The radial separation of the centres of the circles then represents a "Peak to Peak" value of the error motion.
+
+In many cases, the displacement sensor is mounted over a rotating target surface attached to the shaft supported by the bearings.
+However, the displacement sensor now measures not only the motion error of the shaft but also any **geometrical errors present in the target surface**.
+For a radial error motion measurement, out of roundness of the target surface is recorded along with the shaft’s motion error.
+As the motion error of ultra precision bearings may be comparable in magnitude to the geometrical errors in the most accurately manufactured target surfaces then a correction must be made.
+A measurement procedure was proposed that involved two measurements, one with the target surface fixed at some angular position relative to the shaft and the second with it moved through 180 degrees.
+By adding or subtracting the two measurements, geometrical errors on the target surface can be separated from shaft motion errors.
+
+
+### Frequency Analysis {#frequency-analysis}
+
+In general, rotating systems will exhibit motion errors containing several series of harmonics, each of which relate to different aspects or components of the system.
+The main benefit of frequency analysis is therefore to obtain diagnostic information with which to identify the likely sources of motion error and to help reduce their amplitude should they be unacceptable.
+
+
+## Ball Bearings {#ball-bearings}
+
+Criterion used in this book to define ultra precision bearings: motion error of less than 100 nm peak to peak.
+
+Generally only the precision grades or low noise grades of ball bearing are
+likely to produce low motion errors. These types of ball bearing are widely
+used in high precision machine tools, quiet running electric motors,
+computer disc drives and instrumentation, where they provide good but
+not exceptional running accuracy at a competitive price.
+
+Single-row radial ball bearings are favoured in precision engineering
+applications such as computer disc drives and precision electric motors,
+where low motion errors or low noise are a primary requirement. Angular
+contact bearings, on the other hand, are widely used in precision applica-
+tions such as machine tool spindles and rotary tables where static stiffness
+is also important.
+
+
+### Motion Error {#motion-error}
+
+During the 1980s and 1990s, the computer disc drive industry emerged
+as a major application for ball bearings and motion error was recognised as
+a critical bearing performance parameter directly influencing disc capacity.
+Unlike the electric motor application, where bearings may operate under a
+diverse range of conditions, this application was focused on low cost,
+miniature bearings operating under specific conditions of light axial load
+and medium speed at near ambient temperatures. Early research work,
+performed mainly in Japan, developed an understanding of the factors that
+determine the radial motion error of disc drive ball bearings [34–38] and
+later focused specifically on reducing the ‘Non-Repeatable Run Out’
+(NRRO) [39–43]. Because in this application bearing speeds are moderate,
+the NRRO was found to be largely influenced by ball size variation.
+
+In terms of peak–peak motion error amplitudes, ball bearings can achieve
+a creditable performance. Amplitudes as low as 48 nm have been reported
+in scientific papers [41], for ball bearings used in computer hard disc drives.
+This is comparable to the motion error of some types of fluid film, but the
+**disadvantage of ball bearings is that the motion error is predominantly
+asynchronous whereas for fluid film bearings it is mostly synchronous**.
+
+The main reason is that for ball bearings, motion error frequencies relate to
+the orbital and spinning speeds of the balls and these can never be harmonic
+of shaft speed in a practical bearing design.
+
+Ball bearing motion error is
+influenced by a large number of parameters, some a function of the bearing
+design and manufacturing processes, others being dependent on application
+conditions. However, there are relatively few basic mechanisms by which
+motion error can be generated and by understanding these, the influence of
+different parameters can be more clearly defined and in many cases, even
+quantified.
+
+
+#### Dynamics model for estimating call bearing motion error {#dynamics-model-for-estimating-call-bearing-motion-error}
 
 <style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
   <div class="csl-entry"><a id="citeproc_bib_item_1"></a>Wardle, Frank. 2015. <i>Ultra Precision Bearings</i>. Elsevier.</div>