382 Bearing Retainer Vibration  Harmonic Ratio  Calculations  for a Forced Rotation Situation  

Based on classic bearing theory, with  an analog  to planetary gear systems with  a static  outer  ring gear and  driving -torqueing sun gear, the two fundamental relationships between vibrations caused by inner/outer  ring raceway ball -contact imperfections are related  via the cage/raceway rotational speed. The ratio of  bearing cage- retainer speed/frequency  to inner rotor shaft speed/frequency  is given by  the equation:   𝜋𝜋 𝑐𝑐 𝜋𝜋 𝑟𝑟=1 2 (1− 𝐷𝐷𝑏𝑏 𝐷𝐷𝑝𝑝𝑐𝑐∗ 𝑐𝑐𝑐𝑐𝑐𝑐 𝛼𝛼𝐶𝐶𝐶𝐶 )  Where 𝜋𝜋 𝑐𝑐 is cage/ retainer rotation speed/frequency, 𝜋𝜋 𝑟𝑟 is inner ring/shaft rotation speed/frequency, 𝐷𝐷𝑏𝑏 is  nominal ball diameter, 𝐷𝐷 𝑝𝑝𝑐𝑐  is the pitch circle diameter, and  𝛼𝛼𝐶𝐶𝐶𝐶 is the nominal contact angle. This leads to   a key ratio value (for 304H) of 0.35. Similarly, the ratio relative to outer ring speed/frequency is given by:   𝐹𝐹 𝑐𝑐 𝐹𝐹 𝑟𝑟=1 2 (1+ 𝐷𝐷𝑏𝑏 𝐷𝐷𝑝𝑝𝑐𝑐∗ 𝑐𝑐𝑐𝑐𝑐𝑐 𝛼𝛼0 )  Where 𝐹𝐹 𝑐𝑐 is cage/ retainer rotation speed/frequency, 𝐹𝐹 𝑟𝑟 is effective outer ring passage speed/frequency,  and other parameters are the same as above. This leads to a second key ratio value ( for 304H ) of ~0.65.  

Non-spin- axis ratios for a 304H duplex bearing pair would all have a theoretical  basis origin due to one of  the following ratios:  0.35, 0.65, 1.0, 8.0 (same- ball inner return ratio), or 0.0588 (1/17; same- ball-retainer  geometric alignment return rat io). Other ratios are known to exist ( i.e., Fig. 5),  however, were of significantly   lower intensity  or are difficult to identify at  RWA dwell speeds prescribed in test procedures. For HST RWAs ,  IV tested/ tracked/ reported ratios  (set of 8 neglecting 0.65 detection issue) are  in Table 9 .  Table 9.  HST RWA Bearing Set Basic IV Tested/Tracked/Reported Harmonic Ratios   Theory - Based  Ratio    Empiricall y-  Established  Ratio  Description of Root Source/Cause Sig.  Figures  Known   0.3521  0.35 RATIO OF BEARING CAGE/RETAINER SPEED TO ROTOR SPEED  2  0.6479  Detection  Difficult  RATIO OF BEARING CAGE/RETAINER SPEED TO OUTER RING PASSAGE  2  1.0000  1.00 MISALIGNMENT OF ROTOR CENTER -OF-MASS & SHAFT ROTATION  POINT  AS PRECISE  AS RWA  

2.0000  2.00 SECOND HARMONIC OF 1.0 (RACE-TO-RACE MISALIGNMENT /WHIRL) AS PRECISE  AS RWA    2.8168  2.82   3  5.1832  5.18 N (# OF BALLS  - 8) TIMES 0.65  (BALL PASSAGE ON INNER RACE) 3  5.6336  5.60 SECOND HARMONIC OF 2.8 (BALL PASSAGE ON OUTER RACE) 3  -- 7.50 CAUSE NOT KNOWN WITH CERTAINTY ; ONLY EMPIRICALLY DERIVED  ~2  -- 8.50 CAUSE NOT KNOWN WITH CERTAINTY ; ONLY EMPIRICALLY DERIVED  ~2 

At first RWA IV data assessments , there was no analytical -empirica l basis correlating a harmonic’s  amplitude to another harmonic’s amplitude  thus all harmonics ( and their amplitude variations)  were  perceived  as individual  and independent uncorrelated entities . See discussion below concerning the validity  of this viewpoint . For 101-type bearing RWA assemblies , key ratios are somewhat different, as delineated   in Table 10.  

Prompted by pure- analysis deficiencies, empirical IV data have dominated quantifying RWA disturbance  macro characteristics, harmonic ratios, IV levels, variations, and relative trends. Extractions from refined IV  data studies led to a need for a comprehensive common basis to compare resultant IV macro- effects. Per  classic intrinsic physical principles, some element of bearing Geometric Runout (expressed via a geometry - related value that generally ri ses and falls in league with overall IV amplitudes) is presumed fundamentally  responsible for non- spin- axis harmonic vibrations. Since dominant effects (e.g., primary unbalance) are